Complex analysis and CR geometry

Giuseppe Zampieri

To my children Caterina, Sebastiano and Emanuele Zampieri “brevissimus quisque dilucidissimus est”

Contents

Preface vii

Chapter 1. Several Complex Variables 1 1.1. Analytic functions in complex spaces 1 1.2. Cauchy formula in polydiscs 5 1.3. Analytic functions and power series 11 1.4. Subharmonic functions 14 1.5. Separate analyticity 22 1.6. Levi forms—continuity principle (I)—Hartogs extension

theorem 24 1.7. Logarithmic supermean of Taylor radius of holomorphic

functions—continuity principle (II)—propagation of holomorphic extendibility 30

1.8. Domains of holomorphy and pseudoconvex domains 39 1.9. L^{2}-estimates for ¯∂ on q-pseudoconvex domains of ℂ^{n} 51 1.10. Subelliptic estimates for ¯∂ 68

Chapter 2. Real Structures 77 2.1. Euclidean spaces and their diffeomorphisms 77 2.2. Integration of vector fields and vector bundles—Frobenius

theorem 83 2.3. Real symplectic spaces—Frobenius-Darboux theorem 90 2.4. Subelliptic estimates and hypoellipticity of systems of

vector fields 96 2.5. Miscellanea: foliations—orbits 102

Chapter 3. Real/Complex Structures 109

3.1. Complex structures—real underlying structures—

complexifications 109 3.2. CR manifolds 115 3.3. CR functions and CR mappings 125 3.4. The Levi form of a submanifold M ⊂ ℂ^{n} and an abstract

CR structure 134 3.5. Real/complex symplectic spaces 138 3.6. Approximation of CR functions by polynomials 146 3.7. Analytic discs and the extension of CR functions: the

“edge of the wedge” theorem, the deformation of discs for manifolds of type 2 and the Levi extension 153

3.8. Iterated commutators, finite type, Bloom-Graham normal

form: deformation of discs for manifolds of higher type 161 3.9. Partial lifts of analytic discs and CR curves 171 3.10. Defect of analytic discs—deformation of non-defective

discs—wedge extension from minimal manifolds 177 3.11. Propagation of CR extendibility 180 3.12. Separate real analyticity 186

Bibliography 193

Subject Index 199

Symbols Index 201

Preface

The geometry of the domains of holomorphic and CR functions is one of the primary topics in contemporary mathematics. It is a field where various techniques converge: PDE’s, real/complex vector fields, induced systems, boundary values, analytic discs in symplectic spaces, just to mention a few. The excellent interplay of these tools serves as a model both for research and education. This book collects my lectures addressed to graduate students at the University of Padova during the last three years as well as an introduction to part of my present research. The purpose is twofold: to provide a basic foundation in the theory of several complex variables and CR functions and to give some new insight and tools in current research.

Chapter 1 introduces the theory of complex analytic functions. It starts with classical facts about power series and plurisubharmonic functions along the guidelines of Oka and Hörmander. Next, it begins the discussion of the equivalence between pseudoconvex domains and domains of holomorphy and proves by geometric evidence the coincidence of the pseudoconvexity of a domain with that of its boundary. Finally, it solves the ¯∂-Neumann problem by the method of Kohn. This implies the solution of the Levi problem and provides the complete proof of the equivalence mentioned above. It also includes some new results concerning the regularity of ¯∂ at a weakly q-pseudoconvex boundary.

Chapter 2 deals with real structures. Along with the classical Frobenius theorem, it presents its symplectic counterpart, that is, the Darboux-Frobenius theorem. Finally, it discusses subellipticity and hypoellipticity of systems of real vector fields.

The content of Chapter 3 is CR structures. Particular emphasis is devoted to the analysis of the conormal bundle to a real submanifold of ℂ^{n} under a canonical transformation. This serves to describe Hamiltonian/Levi foliations and also to interchange a higher-codimensional submanifold with a hypersurface. Next, it recalls the theory of analytic discs attached to real submanifolds and of their infinitesimal deformations. Furthermore, it introduces a new family of discs with singular boundary. They yield a new proof and a refined statement of the classical results on the extension of CR functions from manifolds of finite type. The problem of the construction of lifts and partial lifts of analytic discs is also addressed: the discussion fully discloses the symplectic-geometric character of these objects. By means of them, there is obtained in a natural way a “connection” on the conormal bundle to a real submanifold of ℂ^{n}. The coupling between lifts and infinitesimal deformations is then exploited. It first shows that CR extendibility evolves along a CR curve according to the dual-inverse connection of the above one. Also, when M is “minimal” in the sense of Tumanov, it yields the wedge extension of CR functions. The discussion ends with the theory of separate real analyticity, the real counterpart of the Hartogs theorem, which takes a geometric explanation here.

I am greatly indebted to Luca Baracco and Alexander Tumanov for having taught me much of the material developed here.

Padova, December 21, 2006

CHAPTER 1 Several Complex Variables

Summary and Notes to Sections 1.1–1.6. These sections are classical, so we comment about them all together. We recall in Sections 1.1 and 1.2 the definition and basic properties of analytic functions of several complex variables which come from the extension of Cauchy’s integral formula to product domains of ℂ^{n}. We discuss in Section 1.3 the relation between the domains of holomorphic functions and of convergent power series. This exhibits the first example of domains from which any analytic function can be extended to a larger set. We then discuss in Section 1.4 some basic properties of subharmonic functions and prove in Section 1.5 Hartogs’s theorem: analyticity is separable in each single variable. We end this discussion in Section 1.6 where we introduce the Levi form, prove the normal form of a real hypersurface, and present the Hartogs extension theorem. It states that all holomorphic functions extend from the concave side of a boundary.

We refer to the basic book by Gunning and Rossi [46] and to Hörmander [50] from which a large part of the presentation is inspired. The reader can find there a greater account of the material. We also cite the books by Gunning [45], Krantz [63], Narasimhan [71], Range [74] and Wells [98] where the reader can also test his knowledge of the subject by clever exercises.

1.1. Analytic functions in complex spaces

We study here complex functions f : ℂ^{n} → ℂ. Being identified with functions f : ℝ^{2n} → ℝ^{2}, they are also particular real vector functions in even dimension. But we are mainly interested in those which preserve the complex structure. For a choice of a root i = \sqrt{−1} we write the coordinates in ℂ^{n} as

z = x + iy for (x, y) ∈ ℝ^{2n};

here z = (z_{1}..., z_{n}), x = (x_{1}, ..., x_{n}), y = (y_{1}, ..., y_{n}). We have thus obtained an identification ℝ^{2n} →^{∼} ℂ^{n} given by

(1.1.1)  (x, y) ↦ z = x + iy.

In this correspondence, we write x = Re z and y = Im z; thus, the inverse of (1.1.1) is written as z ↦ (Re z, Im z). If we want it to look “algebraic”, we need to call into play the other root −i = −\sqrt{−1} and define by z̄ = x − iy the “conjugate” of z. Then, in the pair (z, z̄), the arbitrariness of the choice of ±\sqrt{−1} disappears, and we can describe the real structure underlying ℂ^{n} through the correspondence

(1.1.2)  (x, y) ↦ (z, z̄) = (x + iy, x − iy)

whose inverse has now become linear

(1.1.3)  (z, z̄) ↦ (x, y) = (\frac{z + z̄}{2}, \frac{z − z̄}{2i}) .

With the change (1.1.2) and its inverse (1.1.3) in hand, let us see how derivatives and differentials behave. In general, associated to a change of real coordinates t̃ = t̃(t) in ℝ^{m}, the chain rule yields the transformation for the coordinate-derivatives ∂_{t_{i}} ↦ ∑_{j=1,...,m} \frac{∂t̃_{j}}{∂t_{i}} ∂t̃_{j}. Hence, having in mind the transformations (1.1.2) and (1.1.3), we define ∂_{z} and ∂z̄ by

(1.1.4)  \begin{cases}∂_{x} = \frac{∂z}{∂x}∂_{z} + \frac{∂z̄}{∂x}∂z̄ = ∂_{z} + ∂z̄, \\ ∂_{y} = \frac{∂z}{∂y}∂_{z} + \frac{∂z̄}{∂y}∂z̄ = i(∂_{z} − ∂z̄), \end{cases}

which reads explicitly, by inversion,

(1.1.5)  \begin{cases}∂_{z} = \frac{∂x}{∂z}∂_{x} + \frac{∂y}{∂z}∂_{y} = \frac{1}{2}(∂_{x} − i∂_{y}), \\ ∂z̄ = \frac{∂x}{∂z̄}∂_{x} + \frac{∂y}{∂z̄}∂_{y} = \frac{1}{2}(∂_{x} + i∂_{y}). \end{cases}

Here we use the notation ∂_{x} = (∂_{x_{j}})_{j}, \frac{∂z}{∂x} = (\frac{∂z_{i}}{∂x_{j}} )_{i j} and similarly for

the other variables y, z and z̄. Thus, defined by (1.1.4) and (1.1.5), the derivatives (∂_{z}, ∂z̄) correspond to (∂_{x}, ∂_{y}) via the chain rule under the transformations (1.1.2) and (1.1.3). We have for the dual basis of differentials the evident correspondences

(1.1.6)  \begin{cases}dx = \frac{dz+dz̄}{2}, \\ dy = \frac{dz−dz̄}{2i}, \end{cases}

with inverse

(1.1.7)  \begin{cases}dz = dx + idy, \\ dz̄ = dx − idy. \end{cases}

Again, we write dx for (dx_{j})_{j} and similarly for the other variables. In particular, for a function f : ℂ^{n} → ℂ identified with a function f : ℝ^{2n} → ℂ, we have

(1.1.8)  ∂_{x}fdx + ∂_{y}fdy = ∂_{z}fdz + ∂z̄fdz̄

that we also write as

(1.1.9)  df = ∂f + ¯∂f.

Equation (1.1.9) serves as a definition of ∂f and ¯∂f. Note that in (1.1.8) ∂_{x}fdx + ∂_{y}fdy stands for ∑_{j} (∂_{x_{j}}fdx_{j} + ∂_{y_{j}}fdy_{j}) and ∂_{z}fdz + ∂z̄fdz̄ stands for ∑_{j} (∂_{z_{j}}fdz_{j} + ∂z̄_{j}fdz̄_{j}). Equation (1.1.8) states that the system of derivatives ∂_{z} and ∂z̄ is the dual system to the system of differentials dz and dz̄. It is often used, in alternative to (1.1.4), (1.1.5), to define ∂_{z} and ∂z̄ from dz and dz̄.

Remark 1.1.1. The length \sqrt{2} of dz_{j} and dz̄_{j}, which is imposed by the notation, forces the dual system ∂_{z_{j}} and ∂z̄_{j} to have the unnatural length \frac{1}{\sqrt{2}}; this is why the chain rule is preferable to duality in their definition.

We have to notice that in case f is real valued then (1.1.8) yields

(1.1.10)  df = 2Re ∂f.

There is no doubt that (1.1.10) is great: it tells us that the “real” differentials are the real parts of the “complex” differentials. It serves the intrinsic version of the identification ℝ^{2n} ≃ ℂ^{n} between the euclidean structure and the real part of the hermitian structure: xx̃+yỹ = Re z\overline{z̃}. As for (1.1.9), it says that the complexified differentials, that is, the set of the combinations of the dx_{j}’s and dy_{j}’s with complex coefficients, are identified with the combinations of the dz_{j}’s and dz̄_{j}’s. Also, (1.1.10) identifies real differentials with “diagonal” combinations, in which the coefficients of the dz̄_{j}’s are the conjugates of those of the dz_{j}’s. We will come back to this point in Chapter 3.

Differential forms which are combinations of the dz_{j}’s are said to be of type (1, 0) and those which are combinations of the dz̄_{j}’s of type (0, 1). Thus, ∂f and ¯∂f are the components of df of type (1, 0) and (0, 1), respectively. Let Ω be a domain in ℂ^{n}.

Definition 1.1.2. A function f ∈ C^{1}(Ω) is said to be holomorphic or complex analytic when df = ∂f, that is, when f satisfies the differential system ∂z̄_{j}f = 0, j = 1, ..., n. The set of holomorphic functions in Ω is denoted by hol(Ω).

The above system is called the Cauchy-Riemann system (CR system for short). Thus, df, which is in general ℝ-linear, is in this case ℂ-linear, that is, we have the commutative diagram

(1.1.11)  \begin{matrix}ℝ^{2n} & →^{df} & ℝ^{2} \\ || & & || \\ ℂ^{n} & →^{∂f} & ℂ. \end{matrix}

If we decompose f = Re f + iIm f, the CR system is a system of 2n real equations

(1.1.12) ∂_{x_{j}}Re f − ∂_{y_{j}}Im f = 0, ∂_{y_{j}}Re f + ∂_{x_{j}}Im f = 0, j = 1, ..., n.

The elliptic estimates for the system of the ∂z̄_{j}’s will tell us in Section 1.10 that the generalized solutions—in any possible sense of the distributions or even of the hyperfunctions—are indeed “classical” solutions. In other words, to be C^{1} is a consequence of being a solution of the system ∂z̄_{j}, j = 1, ..., n. But we can say more: any function which is “separately” holomorphic, that is, holomorphic with respect to each z_{j} when the other arguments are kept fixed, is in fact holomorphic. Also, by the elliptic regularity of each single ∂z̄_{j}, it suffices that each restriction is just a generalized solution. Note that a separate solution is not even a “weak” solution, in general. Thus, complex analyticity is “separable” in contrast to any kind of regularity of real functions.

We now treat the inversion of holomorphic functions. We start from an elementary remark. A holomorphic function of a single variable f(z), z ∈ ℂ, such that ∂_{z}f(z_{o}) ≠ 0 at some z_{o} ∈ ℂ is locally invertible because df = ∂f ≠ 0 at z_{o}. (To show off our ability in the ℝ/ℂ switch, we can invoke that det ∂_{(x,y)}(Re f, Im f)|_{z_{o}} = |∂_{z}f(z_{o})|^{2} ≠ 0.) On the other hand, differentiation in z̄ and z of the identity f^{−1} ∘ f = id yields

∂z̄f^{−1}\overline{∂_{z}f} = 0 and ∂_{z}f^{−1}∂_{z}f = 1.

Thus, f^{−1} is holomorphic and ∂_{z}f^{−1} = (∂_{z}f)^{−1}. This has an immediate generalization to ℂ^{n}.

Theorem 1.1.3. Let f(z) = (f_{j}(z))_{j=1,...,n} be analytic functions in a neighborhood of z_{o} for which we assume that

det (∂_{z_{i}}f_{j})_{ji} ≠ 0 at z_{o}.

Then, there exists g(z) = (g_{j}(z))_{j=1,...,n} analytic in a neighborhood of w_{o} = f(z_{o}) such that g ∘ f = id.

Proof. We “double” our variables from z ∈ ℂ^{n} to (z, w) ∈ ℂ^{n}×ℂ^{m} and prove a more general statement. For an analytic function h : ℂ^{n} × ℂ^{m} → ℂ^{n}, h = (h_{j}(z, w))_{j}, such that h = 0 and det (∂_{z}h) ≠ 0 at (z_{o}, w_{o}), the equations (h_{j}(z, w))_{j=1,...,n} = 0 have uniquely determined analytic solutions z = (z_{j})(w) in a neighborhood of w_{o} with z(w_{o}) = z_{o}.

This implies the theorem when applied for m = n and h_{j} = w_{j} − f_{j}(z). To prove the claim, we have to show that dh_{j} = 0 and dw_{k} = 0 for any j = 1, ..., n and k = 1, ..., m, imply dz_{j} = 0, j = 1, ..., n: but this follows from det ∂_{z}h ≠ 0. Thus, by the implicit function theorem, the equations h_{j} = 0 define uniquely the solutions z = (z_{j})(w) with z(w_{o}) = z_{o}. For these solutions z(w), the identity dh = ∂_{z}hdz +∂_{w}hdw = 0 implies that the dz_{j}’s are combinations of the dw_{k}’s; thus the z_{j}’s are holomorphic.

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1.2. Cauchy formula in polydiscs

Let z^{o} = (z_{1}^{o}, ..., z_{n}^{o}) be a point of ℂ^{n} and r = (r_{1}, ..., r_{n}) a “multiradius” in (ℝ^{+})^{n}.

Definition 1.2.1. Let Δ_{z_{j}^{o} r_{j}} be the disc {z_{j} ∈ ℂ : |z_{j} − z_{j}^{o}| < r_{j}} of center z_{j}^{o} and radius r_{j}. We define

P_{z^{o} r} = Π_{j=1,...,n}Δ_{z_{j}^{o} r_{j}}

and call it the “polydisc” of center z^{o} and multiradius r.

The boundary ∂P is stratified by strata of codimension 1, 2, ... which correspond to single equations |z_{j_{1}} − z_{j_{1}}^{o}| = r_{j_{1}}, pairs of equations |z_{j_{1}} − z_{j_{1}}^{o}| = r_{j_{1}}, |z_{j_{2}} − z_{j_{2}}^{o}| = r_{j_{2}} and so on. The highest codimensional (n-codimensional) stratum is called the “distinguished” or “Shilov” boundary of P and is denoted by

∂_{0}P = {|z_{j} − z_{j}^{o}| = r_{j}, j = 1, ..., n}.

Holomorphic functions on polydiscs admit the Cauchy integral representation on ∂_{0}P. This also enables us to start our process of relaxing the initial assumption of C^{1}-regularity for f.

Theorem 1.2.2. Let f be a continuous function on a polydisc P̄ which is, for any j, a holomorphic function of z_{j} (that is, a C^{1} solution of ∂z̄_{j}f = 0) when the other variables z_{k} for k ≠ j are fixed. Then for any z ∈ P we have

(1.2.1) f(z) = (2πi)^{−n} ∫_{∂_{0}P} \frac{f(ζ)}{(ζ_{1} − z_{1}) · ... · (ζ_{n} − z_{n})}dζ_{1} ∧ ... ∧ dζ_{n}.

Proof. We proceed by induction on the dimension n of ℂ^{n} and first treat the case n = 1. To simplify notation, we assume z^{o} = 0 and r = 1; thus P coincides with the standard disc Δ. Under this choice, we want to prove the general Cauchy integral formula, which applies to a function g ∈ C^{1} not necessarily holomorphic and that we state as a separate lemma.

Lemma 1.2.3. Let g ∈ C^{1}(Δ̄); for any z ∈ Δ we have

(1.2.2)  g(z) = \frac{1}{2πi} [∫_{∂Δ} \frac{g(ζ)}{ζ − z} dζ + ∫∫_{Δ} \frac{∂ζ̄g(ζ)}{ζ − z}dζ ∧ dζ̄] .

Proof. For z ∈ Δ, let ϵ < dist(z, ∂Δ); thus Δ^{ϵ} := Δ_{z ϵ} satisfies Δ̄^{ϵ} ⊂⊂ Δ. We note that ∂Δ and ∂Δ^{ϵ}, as parts of the boundary of Δ \ Δ^{ϵ}, carry opposite orientation: +∂(Δ \ Δ^{ϵ}) = +∂Δ − ∂Δ^{ϵ}. We use the notation ω for the 1-form \frac{g(ζ)}{ζ−z}dζ; Stokes’s formula yields

(1.2.3)  \frac{1}{2πi} ∫_{+∂Δ^{ϵ}} ω = \frac{1}{2πi} [∫_{+∂Δ} ω − ∫∫_{Δ\Δ_{ϵ}} dω] .

We note that the last integrand in (1.2.3) is −dω = \frac{∂ζ̄g(ζ)}{ζ−z}dζ ∧ dζ̄. Since dω is integrable at its singular point z, then

∫∫_{Δ\Δ^{ϵ}} dω → ∫∫_{Δ} dω.

We also note that

\frac{1}{2πi} ∫_{+∂Δ^{ϵ}} ω → g(z),

because g is continuous at z. Thus, (1.2.3) at the limit for ϵ → 0 turns into (1.2.2).

□

In particular, if instead of a general C^{1} function g we take a solution f of ∂z̄f = 0, (1.2.2) yields

f(z) = \frac{1}{2πi} ∫_{∂Δ} \frac{f(ζ)}{ζ − z} dζ,

that is, (1.2.1) for n = 1.

End of proof of Theorem 1.2.2. We carry on our inductive argument and suppose that (1.2.1) has been proved for holomorphic functions of n − 1 arguments z_{1}, ..., z_{n−1}. We denote by P′ the polydisc Π_{j≤n−1}Δ_{z_{j}^{o} r_{j}} and write (1.2.1) for z_{1}, ..., z_{n−1} with parameter z_{n}:

(1.2.4)

f(z_{1}, ..., z_{n}) = (\frac{1}{2πi})^{n−1}∫_{∂_{0}P′} \frac{f(ζ_{1}, ..., ζ_{n−1}, z_{n})}{(ζ_{1} − z_{1}) · ... · (ζ_{n−1} − z_{n−1})}dζ_{1}∧....∧dζ_{n−1}.

But in the above integral we can make the substitution which we are allowed by the 1-dimensional Cauchy formula:

(1.2.5)  f(ζ_{1}, ..., ζ_{n−1}, z_{n}) = \frac{1}{2πi} ∫_{|ζ_{n}−z_{n}|=r_{n}} \frac{f(ζ_{1}, ..., ζ_{n})}{ζ_{n} − z_{n}}dζ_{n}.

From (1.2.4), (1.2.5) we get (1.2.1) by Fubini’s theorem.

□

Remark 1.2.4. The separate C^{1} regularity in each z_{j} is needed for Stokes’s formula, and the joint C^{0} regularity is needed for Fubini’s theorem.

For a function f ∈ C^{0}(Ω) separately (C^{1} and) holomorphic in each z_{j}, we wish to apply (1.2.1) to all possible P̄ ⊂⊂ Ω. Since the functions of z produced by the integrals in ζ are C^{∞}, as one can check by bringing derivatives in the z_{j}’s inside the convergent integral, we get at once

Corollary 1.2.5. If f is C^{0} on Ω and separately holomorphic in each z_{j}, then it is C^{∞} and in particular jointly holomorphic.

We write the point ranging in ∂_{0}P (z, r) as ζ = (ζ_{j}) for ζ_{j} = z_{j} + r_{j}e^{iθ_{j}}. Then we have

(2πi)^{−n} ∫_{∂_{0}P (z,r)} \frac{f(ζ)}{(ζ_{1} − z_{1}) · ... · (ζ_{n} − z_{n})}dζ_{1} ∧ ... ∧ dζ_{n}

= ∫_{[0,1]^{n}} f(z + (r_{1}e^{i2πθ_{1}}, ..., r_{n}e^{i2πθ_{n}}))dθ_{1} ∧ ... ∧ dθ_{n}.

In particular, (1.2.1) says that a holomorphic function satisfies the mean property: the value of f at z equals its mean value on the distinguished boundary of any polydisc contained in the region where f is holomorphic. Passing from f to |f| in (1.2.1), we see that |f(z)| is estimated by the mean value of |f| over ∂_{0}P (z, r): we say that |f| enjoys the submean property, in this case. In particular, if |f| attains a maximum at a point z of the domain Ω where f is holomorphic, then |f| must be constant with the same value |f(z)| in ∂_{0}P (z, r) for any P (z, r) ⊂⊂ Ω. Thus, |f| is constant in a neighborhood of z. (In fact f itself is constant since |∂_{z}f|^{2} = ∂_{z}∂z̄|f|^{2} = 0.) It follows that f satisfies the so-called “maximum modulus principle”, a property that absolute values of holomorphic functions share with all functions satisfying the submean property:

Theorem 1.2.6. Let Ω be bounded and f be continuous on Ω̄ and holomorphic on Ω. Then |f| attains its maximum on ∂Ω.

Proof. The function f is locally constant at any point in the interior of Ω where |f| attains its maximum. So the set of these point, being also closed, cannot be non-empty unless it is a full connected component of Ω. But then the same maximum value for |f| is attained at some boundary point.

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We need to introduce now the notions of multiindices, multipowers and multiderivatives. For a multiindex α = (α_{1}, ..., α_{n}) ∈ ℕ^{n} we put α! = α_{1}! · ... · α_{n}! and |α| = α_{1} + α_{2} + ... + α_{n}. We define multipowers by r^{α} = r_{1}^{α_{1}} · ... · r_{n}^{α_{n}} and multiderivatives by ∂_{z}^{α}f = ∂_{z_{1}}^{α_{1}} … ∂_{z_{n}}^{α_{n}}f that we also denote sometimes by f^{(α)}. We write α + 1 for (α_{1} + 1, ..., α_{n} + 1) and, if β is another multiindex such that β_{j} < α_{j} for any j, we define α − β := (α_{1} − β_{1}, ..., α_{n} − β_{n}).

We say that a power series ∑_{α} a_{α}(z − z_{o})^{α} converges normally in the polydisc P (z_{o}, r) when for any r′ with r′_{j} < r_{j}, for all j we have ∑_{α} |a_{α}|r^{′α} < +∞. For any fixed β, we denote by (∑_{α} a_{α}(z − z_{o})^{α})^{(β)} the series of β-derivatives, that is, the series ∑_{α≥β} \frac{α!}{(α−β)!}a_{α}(z − z_{o})^{α−β}; it

is a routine application of Abel’s theorem to show that if a power series centered at z_{o} converges in the polydisc P (z_{o}, r), then its derivative of any index β converges in P (z_{o}, r). By this remark we can realize any power series as a holomorphic function wherever it converges.

Theorem 1.2.7. Let ∑_{α} a_{α}(z − z_{o})^{α} converge in P (z_{o}, r); then its sum f is a holomorphic function on P (z_{o}, r).

Proof. The partial sums S_{ν} := ∑_{|α|≤ν}a_{α}(z−z_{o})^{α} converge uniformly

together with their derivatives in any compact subset of P. In particular, the relation ∂z̄_{j}S_{ν} = 0 for any j passes to the limit and turns into ∂z̄_{j}f = 0.

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We want to prove now the converse statement: any holomorphic function on Ω is, in a neighborhood of any point z_{o} ∈ Ω, the sum of a power series convergent in a polydisc P (z_{o}, r), necessarily its own Taylor series at z_{o}.

Theorem 1.2.8. Let f ∈ hol(Ω) and let z_{o} ∈ Ω. Then

f(z) = ∑_{α} \frac{f^{(α)}(z_{o})}{α!}(z − z_{o})^{α},

with normal convergence in any polydisc P (z_{o}, r) ⊂ Ω.

Note that the choice of the polydisc is arbitrary provided that it is contained in Ω. If one needs to stress attention to a specific variable, say z_{j}, it is convenient to stretch the polydisc along that direction so that r_{j} approaches the distance of z_{o} to ∂Ω along the z_{j}-plane at the expenses of the other radii r_{k} for k ≠ j which possibly approach 0.

Proof. We choose z_{o} = 0. We write

\begin{aligned}\frac{1}{(ζ_{1} − z_{1}) · ... · (ζ_{n} − z_{n})} & = \frac{1}{ζ_{1} · ... · ζ_{n}} \frac{1}{(1 − \frac{z_{1}}{ζ_{1}} ) · ... · (1 − \frac{z_{n}}{ζ_{n}})} \\ & = ∑_{α} \frac{z^{α}}{ζ^{α+1}} , \end{aligned} ; (1.2.6)

with normal convergence when |\frac{z_{j}}{ζ_{j}} | < 1 for any j. In particular, the convergence is absolute and uniform for z ranging on compact subsets of P and ζ on ∂_{0}P. Thus, we substitute (1.2.6) into Cauchy’s formula and get

\begin{aligned}f(z) & = (2πi)^{−n} ∫_{∂_{0}P} \frac{f(ζ)}{(ζ_{1} − z_{1}) · ... · (ζ_{n} − z_{n})}dζ_{1} ∧ ... ∧ dζ_{n} \\ & = (2πi)^{−n} ∫_{∂_{0}P} ∑_{α} \frac{z^{α}}{ζ^{α+1}} f(ζ)dζ_{1} ∧ ... ∧ dζ_{n} \\ & = ∑_{α} ((2πi)^{−n} ∫_{∂_{0}P} \frac{f(ζ)}{ζ^{α+1}}dζ_{1} ∧ ... ∧ dζ_{n}) z^{α}, \end{aligned} ; (1.2.7)

with normal convergence for z ∈ P. Thus, f is the sum of a power series centered at z_{o} = 0. But iterated derivation of the identity of f with the sum of the series and evaluation at z_{o} yields that the coefficients in (1.2.7) are indeed the Taylor coefficients of f at z_{o} = 0, that is, \frac{f^{(α)}(0)}{α!}.

□

The proof of the theorem gives in fact the “Fourier integral” expression of the coefficients:

(1.2.8)  \frac{f^{(α)}(z_{o})}{α!} = (2πi)^{−n} ∫_{∂_{0}P} \frac{f(ζ)}{(ζ − z_{o})^{α+1}} dζ_{1} ∧ ... ∧ dζ_{n}.

Now that we have described locally an analytic function as the sum of a convergent power series, we realize that the zeroes of f “propagate” from an open part of Ω to the whole of its connected component.

Corollary 1.2.9 (Analytic continuation). Let Ω_{o} ⊂ Ω be open subsets of ℂ^{n} with Ω_{o} ≠ ∅, Ω connected and let f ∈ hol(Ω) with f|_{Ω_{o}} ≡ 0. Then f ≡ 0 all over Ω.

In other words, two holomorphic functions on Ω which coincide over an open non-empty set must coincide everywhere.

Since in a neighborhood of any point z_{o} ∈ Ω, f is the sum of its Taylor series, then in fact the corollary says that

f^{(α)}(z_{o}) = 0 for any α implies f ≡ 0 on Ω.

Proof. Let Ω_{1} be the maximal open subset of Ω where f|_{Ω_{1}} ≡ 0. (“Maximal” makes sense: just take the union of all such subsets.) It is non-empty since it contains Ω_{o}. We must have Ω_{1} = Ω. If not, let ξ be a point in Ω_{1} whose distance to ∂Ω is bigger than its distance to ∂Ω_{1}: such a ξ must exist because of the connectedness of Ω. Now, the Taylor series of f at ξ is 0 and it represents f in any polydisc contained in Ω, in particular in those stretched along the plane in which the distance of ξ to ∂Ω_{1} is attained. But these polydiscs go out of Ω_{1} and bring the zeroes of f beyond their maximal set.

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In the following statement there is special emphasis on the center z_{o} and the multiradius r of P = P (z_{o}, r): the bigger the radii of the discs that we can insert in the domain where f is holomorphic, the better the estimates of its derivatives.

Theorem 1.2.10 (Cauchy’s inequalities). Let f be holomorphic in P and continuous on P̄. Then, for any α and for suitable c = c_{α r}

(1.2.9)  |f^{(α)}(z_{o})| ≤ \frac{α!}{r^{α}} sup_{∂_{0}P (z_{o},r)}|f|,

(1.2.10)  |f^{(α)}(z_{o})| ≤ c||f||_{L^{1}(P (z_{o},r))}.

Proof. Write ζ_{j} = z_{o j} + r_{j}e^{iθ_{j}}; then (1.2.8) becomes

\frac{f^{(α)}(z_{o})}{α!} = (2πi)^{−n}i^{n} ∫_{[0,2π]^{n}} \frac{f(z_{o} + re^{iθ})}{r^{α}}e^{−iθα}dθ_{1} ∧ ... ∧ dθ_{n}.

Taking absolute values, we get

(1.2.11) |f^{(α)}(z_{o})| ≤ \frac{α!}{r^{α}} (2π)^{−n} ∫_{[0,2π]^{n}} |f(z_{o} + re^{iθ})|dθ_{1} ∧ ... ∧ dθ_{n},

which yields (1.2.9). By moving the multiradius r′ with the constraint 0 ≤ r′ ≤ r, multiplying both sides of (1.2.11) by r^{′α+1} and integrating with respect to r′, we get (1.2.10).

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In particular, in the space hol(Ω) the C^{∞} convergence coincides with the L_{loc}^{1} convergence, indeed with any “weak” (except from pointwise) convergence. But the main consequence is about the so-called “normal families” of holomorphic functions.

Theorem 1.2.11 (Stjelties-Vitali). Let {f_{ν}} be a bounded sequence in hol(Ω). Then, there is a subsequence {f_{ν_{k}}} uniformly convergent on compact subsets of Ω to a limit f ∈ hol(Ω).

Proof. By Theorem 1.2.10, “{f_{ν}} bounded” implies “{∂_{z_{j}}f_{ν}} bounded” on compact subsets. Hence {f_{ν}} is equicontinuous and by Arzela’s theorem it converges to a limit. The limit is also holomorphic, just by interchanging ∂z̄_{j} with “lim”.

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1.3. Analytic functions and power series

The domain of convergence of a power series of one complex variable is a disc. For a power series in several complex variables it shows a more interesting structure. It also serves as the first non-trivial example of a “forced” extension for holomorphic functions which is not possessed by functions of one variable. Let

(1.3.1)  ∑^{+∞} ; |α|=0a_{α}z^{α}

be a power series centered at 0.

Definition 1.3.1. We denote by D the domain of convergence of the power series (1.3.1) which consists of the set of points in whose neighborhood the power series converges normally, that is, absolutely uniformly.

Definition 1.3.2. We denote by B the set of “boundedness” of (1.3.1), that is, the set of points z such that |a_{α}z^{α}| < c for any α.

The two above sets are related by the following theorem, which is an elementary application of Abel’s theorem.

Theorem 1.3.3. We have D = B^{∘}.

Proof. D is obviously open and contained in B^{∘}. Conversely, if

z_{o} ∈ B^{∘}, then there are a point w ∈ B, positive numbers k_{j} < 1 and a neighborhood V of z_{o} such that |z_{j}| ≤ k_{j}|w_{j}| for any j and z ∈ V.

Let k = (k_{1}, ..., k_{n}); it follows that

\begin{aligned}∑_{α} |a_{α}z^{α}| & ≤ c ∑_{α} k^{α} \\ & = c Π_{j=1,...,n}(1 − k_{j})^{−1} for any z ∈ V. \end{aligned} ; (1.3.2)

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Therefore, the domain of convergence D of a power series shows its first two interesting features:

• It is invariant under rotations: if z ∈ D, then e^{iθ}z ∈ D for any θ ∈ [0, 2π]^{n}.

• If w ∈ D, then, for any z which satisfies |z_{j}| ≤ |w_{j}| for all j,

we also have z ∈ D. An open set which enjoys the first property is called a “Reinhardt domain”: it can be described through its representative in the space of moduli |D| ⊂ (ℝ^{+})^{n} defined by |D| := {(|z_{1}|, ..., |z_{n}|) : z ∈ D}. A Reinhardt domain which enjoys the second property is called “complete”: it is therefore characterized as a union of polydiscs centered at 0. Therefore, for a domain of ℂ^{n} to be the domain of convergence of a power series, we have to require it to be Reinhardt and complete. But not all complete Reinhardt domains are precisely the convergence domain of a power series. Some additional geometric requirements must be fulfilled.

Theorem 1.3.4. D is “logarithmically convex” in the sense that the set D^{∗} := {ξ ∈ ℝ^{n} : ξ_{j} = log|z_{j}|, j = 1, ..., n, for z ∈ D} is convex.

Proof. Remember that D = B^{∘}; hence, what we have to prove is that

(1.3.3)  \begin{cases}|a_{α}z^{α}| < c, \\ |a_{α}w^{α}| < c \end{cases}

implies for λ ∈ [0, 1]

(1.3.4)  |a_{α}z^{λα}w^{(1−λ)α}| < c.

But this is a straightforward calculation.

□

We compare now analytic functions and convergent power series. We have already seen that they coincide over polydiscs, hence over the union of those which are centered at the same point: this follows easily from analytic continuation. In particular, over a Reinhardt complete (non-empty) domain, a holomorphic function is always represented as a sum of a power series centered at 0. (Of course for a general domain Ω ⊂ ℂ^{n}, a holomorphic function is a family of convergent power series but not a single one, in general.) We see now that a Reinhardt domain need not be complete for this conclusion to be true.

Theorem 1.3.5. Let Ω be a connected Reinhardt domain which contains 0 and let f ∈ hol(Ω); then we have

f(z) = ∑_{α} \frac{f^{(α)}(0)}{α!}z^{α},

with normal convergence in Ω.

Proof. We denote by Ω_{ϵ} the connected component of 0 in the set {z ∈ Ω : d(z, ℂ^{n} \ Ω) > ϵ|z|}. The sets Ω_{ϵ} are increasing for ϵ ↘ 0 and their union over ϵ covers the whole of Ω since this is connected. We denote by P_{1+ϵ} the polydisc P_{1+ϵ} = {t ∈ ℂ^{n} : |t_{j}| < 1 + ϵ for any j = 1, ..., n} and define

(1.3.5)  g(z) = (2πi)^{−n} ∫_{∂_{0}P_{1+ϵ}} \frac{f(t_{1}z_{1}, ..., t_{n}z_{n})}{(t_{1} − 1)...(t_{n} − 1)}dt_{1} ∧ ... ∧ dt_{n}.

Since z ∈ Ω_{ϵ}, then (1 + ϵ)z ∈ Ω; since t ∈ ∂_{0}P_{1+ϵ} and Ω is Reinhardt, then tz ∈ Ω. Thus, the integral is defined and produces a smooth function of z. By interchanging ∂z̄_{j} with integration, we see that g is holomorphic. When t leaves ∂_{0}P_{1+ϵ} and enters into P_{1+ϵ}, then tz is no longer contained in Ω, in general. But this is true for small z since Ω contains a neighborhood of 0. For these values of z, the substitutions ζ := tz and P_{1+ϵ}^{z} = {tz : t ∈ P_{1+ϵ}} inside (1.3.5) yield g(z) = (2πi)^{−n} ∫_{∂_{0}P_{1+ϵ}^{z}} \frac{f(ζ)}{(ζ_{1}−z_{1})...(ζ_{n}−z_{n})}dζ_{1} ∧ ... ∧ dζ_{n}. But the term in the right side is the Cauchy integral of f at z for the polydisc P_{1+ϵ}^{z} which is completely contained in the region where f is holomorphic. Hence f(z) = g(z) for those values of z, but then in fact f ≡ g on Ω_{ϵ} by analytic continuation. We turn our attention now to the power series

\frac{1}{(t_{1} − 1)...(t_{n} − 1)} = ∑_{α} t^{−α−1},

with normal convergence for t ∈ ∂_{0}P_{1−ϵ}. If we take ∑_{α} out of the integral on the right side of (1.3.5), we get f = ∑_{α} f_{α} where the f_{α} are the analytic functions defined by

f_{α}(z) = (2πi)^{−n} ∫_{∂_{0}P_{1+ϵ}} \frac{f(tz)}{t^{α+1}}dt_{1} ∧ ... ∧ dt_{n}.

In the same way as for the function g previously discussed, we can see that f_{α}(z) = \frac{f^{(α)}(0)}{α!}z^{α} on Ω_{ϵ}. But then f is a sum of a power series.

□

Since a holomorphic function on a Reinhardt domain containing 0 is represented by a power series, then the function extends analytically wherever the power series converges. Now, the first points to be added to the initial domain are those of its “completion”, the points z such that for some w ∈ Ω we have |z_{j}| ≤ |w_{j}| for any j and then also the points of its “logarithmically convex hull”, the points z for which there are w^{1} and w^{2} in Ω such that log |z_{j}| = λ log |w_{j}^{1}| + (1 − λ) log |w_{j}^{2}| for any j ≤ n and for λ ∈ [0, 1]. All these points were unnaturally “missing” from the initial domain of f. These remarks are collected in the following statement.

Theorem 1.3.6. Let Ω be a connected Reinhardt domain containing 0 and let Ω̃ be the smallest complete logarithmically convex Reinhardt domain containing Ω. Then we have an extension mapping

hol(Ω) −−→^{ext} hol(Ω̃).

Remark 1.3.7. We have purposely included the pleonastic word “complete” in the above statement: but a Reinhardt domain logarithmically convex and containing 0 is always complete. In fact, let w ∈ Ω: we know that ϵ^{λ}|w|^{1−λ} := (ϵ_{1}^{λ}|w_{1}|^{1−λ}, ..., ϵ_{n}^{λ}|w_{n}|^{1−λ}) must belong to |Ω| for any real small ϵ and for any λ ∈ [0, 1]. But the set of these points whose image in the space of moduli is ϵ^{λ}|w|^{1−λ} covers the whole polydisc {z : |z_{j}| ≤ |w_{j}| for any j}.

We conclude this discussion by remarking that, conversely, Ω̃ is the maximal set of analytic extension or, in other words, there is a holomorphic function which does not extend beyond Ω̃. This is a consequence of Theorem 1.8.8 whose proof can be found in detail for instance in [50, Corollary 2.5.8]. On the other hand this function is indeed represented by a unique power series. Thus, a Reinhardt, complete, logarithmically convex domain is the precise domain of convergence of a power series.

1.4. Subharmonic functions

It is clear from the end of the last section that our most attractive task is now to describe the natural “domains” of holomorphic functions. For this purpose we need to develop some preliminary technology; it will also be the main tool in solving the problem of separate analyticity. Let us declare from the very beginning that this is strictly a one-variable theory. We recall that a C^{2} function h on a domain Ω of ℂ with coordinate z = x + iy is said to be harmonic when ∂_{z}∂z̄h = 0.

Definition 1.4.1. A real function φ on Ω ⊂ ℂ with values in [−∞, +∞) is subharmonic when (i) φ is upper semicontinuous, i.e., for any z_{o}, φ(z_{o})≥lim sup_{z→z_{o}} φ(z);

(ii) for any K ⊂⊂ Ω and for any h continuous on K and har-

monic on K^{∘},

(1.4.1)  φ|_{∂K} ≤ h|_{∂K} implies φ|_{K} ≤ h|_{K}.

In particular, by taking as h the constant function on K with value max_{∂K} φ, which exists and is finite by the upper semicontinuity of φ, we have that subharmonic functions satisfy the maximum principle. We denote by SH(Ω) the set of subharmonic functions on Ω. A function φ is said to be superharmonic when −φ is subharmonic. The harmonic functions are precisely those which are at the same time sub/superharmonic. This is always true once we know that any continuous function on ∂Δ is endowed with a unique harmonic extension to Δ (given by the Poisson integral as explained in Section 3.7). An easier argument for C^{2} functions is based on Theorem 1.4.9.

Subharmonicity does not appear to be a local property at first sight but it turns out to be. Let Δ be the standard disc in ℂ and Δ_{z_{o} r} the disc of center z_{o} and radius r. Here is the main characterization.

Theorem 1.4.2. Let φ : Ω → [−∞, +∞) be an upper semicontinuous function. The following are equivalent,

(i) φ ∈ SH(Ω).

(ii) For any disc Δ_{z_{o} r} ⊂⊂ Ω and for any polynomial P = P (z),

φ|_{∂Δ_{zo r}} ≤ Re P |_{∂Δ_{zo r}} implies φ|Δ̄_{zo r} ≤ Re P |Δ̄_{zo r}.

(iii) (spherical submean) For any Δ̄_{z_{o} r} ⊂⊂ Ω, we have

φ(z_{o}) ≤ (2πr)^{−1} ∫_{∂Δ_{zo r}} φds,

where ds is the element of the arc. (iv) (solid submean) For any Δ̄_{z_{o} r} ⊂⊂ Ω, we have

φ(z_{o}) ≤ (−2iπr^{2})^{−1} ∫∫_{Δ_{zo r}} φdτ ∧ dτ̄.

(iv)_{loc} (Local solid submean) For any z_{o} there is r_{o} < dist(z_{o}, ∂Ω) such that for any r ≤ r_{o}, (iv) holds.

Proof. (i) ⇒ (ii) and (iv) ⇒ (iv)_{loc} are obvious, whereas (iii) ⇒ (iv) is immediate by integration on dr.

(ii) ⇒ (iii): We take a continuous function f(θ) ≥ φ(z_{o} + re^{iθ}) and

approximate f by a real trigonometric polynomial g(θ) = ∑^{+N} ; k=−Na_{k}e^{ikθ} with a_{−k} = ā_{k}:

(1.4.2)  φ(z_{o} + re^{iθ}) ≤ f(θ) ≤ g(θ) ≤ f(θ) + ϵ.

We write P (z) = a_{0} + 2∑^{N} ; k=1a_{k}(\frac{z−z_{o}}{r})^{k}; thus g = Re P |_{∂Δ_{zo r}} and therefore

(1.4.2) can be rewritten as

φ|_{∂Δ_{zo r}} ≤ Re P |_{∂Δ_{zo r}}.

By (ii) the inequality must hold in the whole of Δ̄_{z_{o} r}, in particular at the center z_{o} which yields

\begin{aligned}φ(z_{o}) & ≤ Re P (z_{o}) \\ & = a_{0} = \frac{1}{2π} ∫_{0}^{2π} g(θ)dθ, \end{aligned} ; (1.4.3)

where the last equality follows from the fact that all non-constant harmonics vanish when integrated over complete cycles. Finally, by the third inequality of (1.4.2), we have

(1.4.4)  \frac{1}{2π} ∫_{0}^{2π} g(θ)dθ ≤ \frac{1}{2π} ∫_{0}^{2π} f(z_{o} + re^{iθ})dθ + ϵ.

Combining (1.4.3) and (1.4.4) and then taking infimum in f and ϵ we get (iii).

(iv)_{loc} ⇒ (i): Let K ⊂⊂ Ω and let h be harmonic on K^{∘}, continuous on K and such that

(1.4.5)  φ|_{∂K} ≤ h|_{∂K}.

Let M := max_{K} (φ − h); we reason by contradiction and assume M > 0.

We define the function v := φ − h and the set F := v^{−1}(M): v is upper

semicontinuous and F is a compact subset of K^{∘} because of (1.4.5). Let z_{o} be a point of ∂F. Then, there is some disc Δ(z_{o}, r) such that Δ(z_{o}, r) ∩ (K \ F) ≠ ∅. Also, we can suppose that r is taken so small that (iv)_{loc} holds for such a z_{o} and r. It follows that

\begin{aligned}M & = v(z_{o}) \\ & ≤ (πr^{2})^{−1} ∫_{0}^{2π} ∫_{0}^{r} tvdt dθ. \end{aligned}

But the mean value on the right hand side is taken over a disc which has an open portion outside F, the set of maximum value M for v. Hence this mean value must be < M, a contradiction.

□

Example 1.4.3. If φ is not upper semicontinuous, submean is not equivalent to local submean. For example, the function φ on the disc Δ which is 0 for Re τ ≤ 0 and 1 for Re τ > 0 satisfies (iii)_{loc} and (iv)_{loc}, that is (iii) and (iv) in local version, but neither (iii) nor (iv). On the other hand, if we force φ to be upper semicontinuous by giving it value 1 at Re τ = 0, it looses the local submean property. In general, apart from subharmonic functions, upper semicontinuity and the submean property are in contrast to one another.

When φ is upper semicontinuous, Theorem 1.4.2 shows that any characterization of subharmonicity can be expressed in local terms: subharmonicity reveals therefore its local character. Also, on account of the submean property, subharmonicity is stable under summation, multiplication by positive constants and “sup” of arbitrary families provided that the “sup” is already known to be < +∞ and upper semicontinuous. We also have a similar statement for the “lim sup” though its proof is a little less evident.

Proposition 1.4.4. Let {φ_{ν}} be subharmonic and uniformly bounded. Then lim sup_{ν} φ_{ν} is subharmonic if it is upper semicontinuous.

Proof. For any z we have

(1.4.6)  lim sup_{ν} φ_{ν}(z) ≤ \frac{1}{2π} ∫_{0}^{2π} lim sup_{ν} φ_{ν}(z + re^{iθ})dθ,

because of the subharmonicity of the φ_{ν}’s and Fatou’s lemma.

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Remark 1.4.5. If lim sup_{ν} φ_{ν} is not known to be upper semicon-

tinuous, nevertheless it is integrable and still satisfies (1.4.6); thus, it always enjoys the submean property.

Hence, apart from a minor adjustment, subharmonicity is stable under “sup” and “lim sup”. It is also stable under monotonic “inf”:

Lemma 1.4.6. If {φ_{ν}}_{ν} is a decreasing sequence of subharmonic functions, then φ = inf_{ν} φ_{ν} is also subharmonic.

Proof. We note that for any c we have {z : φ(z) < c} = ⋃_{ν}{z : φ_{ν}(z) < c} which is an open set: hence φ is upper semicontinuous. If now h|_{∂K} ≥ φ|_{∂K} and ϵ > 0, then K_{ν} := {z ∈ ∂K : φ_{ν}(z) ≥ h(z) + ϵ} is compact and decreasing. Since the limit for ν → +∞ is empty, then K_{ν} must be empty for large ν. Hence φ_{ν}|_{K} ≤ h|_{K} + ϵ for these values of ν.

□

Subharmonicity is preserved under composition with convex increasing functions.

Theorem 1.4.7. Let χ be a function from ℝ to ℝ such that χ̇ ≥ 0 and χ^{··} ≥ 0 (extended at −∞ by χ(−∞) := lim_{t→−∞} χ(t)). Let φ be

subharmonic; then χ(φ) is also subharmonic.

Proof. Since χ is convex, then for any t_{o} and for suitable c, we have

(1.4.7)  χ(t_{o}) ≤ χ(t) − c(t − t_{o}).

Replacing t with φ(z + re^{iθ}) in (1.4.7) and integrating over θ, we get (1.4.8)

χ(t_{o}) ≤ (2π)^{−1} [∫_{0}^{2π} χ(φ)(z + re^{iθ})dθ − c(∫_{0}^{2π} φ(z + re^{iθ})dθ − 2πt_{o})] .

We choose now t_{o} = (2π)^{−1} ∫_{0}^{2π} φ(z + re^{iθ})dθ and get

\begin{aligned}χ(φ(z)) & ≤ χ ((2π)^{−1} ∫_{0}^{2π} φ(z + re^{iθ})dθ) \\ & ≤ (2π)^{−1} ∫_{0}^{2π} χ(φ)(z + re^{iθ})dθ, \end{aligned}

where the first inequality follows from the subharmonicity of φ and the fact that χ is increasing and the second from (1.4.8) under the already-mentioned choice of t_{o}.

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We have said that subharmonicity is preserved under composition with a convex increasing χ but it is indeed strengthened: χ(φ) is “more” subharmonic than φ is and how much more depends on how convex χ is. According to [43], φ is said to be “strongly subharmonic” when there exists χ positive, increasing, strictly convex, such that χ^{−1}(φ) is still subharmonic. Thus, for instance, if ψ is subharmonic and positive, then any power ψ^{a} for a > 1 is strongly subharmonic. It is another useful exercise to prove that if φ is C^{2}, strongly subharmonic and satisfies ∂φ ≠ 0, then ¯∂∂φ > 0.

We recall now that it is proved in [43] that if φ is strongly subharmonic and has a finite harmonic majorant in the unit ball, then its boundary measure is absolutely continuous: just plain subharmonicity would not suffice for this conclusion. A very intense case, corresponding to χ = exp, occurs when φ is logarithmically subharmonic, that is, log φ is subharmonic. Let us further discuss logarithmic subharmonicity. We begin by the obvious remark that for f ∈ hol(Ω), the mean property of f, and hence the submean of |f|, implies that |f| ∈ SH(Ω). We can say more.

Lemma 1.4.8. If f ∈ hol(Ω), then log|f| ∈ SH(Ω).

Proof. Let K be a compact of Ω and P a polynomial such that

(1.4.9)  log |f||_{∂K} ≤ Re P |_{∂K}.

We apply “exp” and observe that e^{Re P} = |e^{P} |; hence (1.4.9) yields

(1.4.10)  |\frac{f}{e^{P}} ||_{∂K} ≤ 1.

By the maximum principle, (1.4.10) remains true when passing from ∂K to K and next, the reverse of the implication from (1.4.9) to (1.4.10) yields

log |f||_{K} ≤ Re P |_{K}.

Thus, log |f| is subharmonic by (ii) of Theorem 1.4.2.

□

In particular, since |f|^{\frac{1}{ν}} = exp \frac{1}{ν}log(|f|), we have that |f|^{\frac{1}{ν}} is subharmonic for any ν. This will be the main tool in the problem of separate analyticity that we will solve in Section 1.5. However, the subharmonicity of all the |f|^{\frac{1}{ν}} ’s, that is, the strong subharmonicity of f in any degree, does not suffice in proving such a delicate property as propagation of CR extendibility. We will see that logarithmic subharmonicity of |f| is needed.

We want to give a differential description of subharmonicity. Though the characterization applies to a more general situation, we want to confine ourselves to the case of φ ∈ C^{2}.

Theorem 1.4.9. Let φ ∈ C^{2}(Ω); then φ is subharmonic if and only if

(1.4.11)  ∂_{z}∂z̄φ ≥ 0.

Note that ∂_{z}∂z̄ coincides, up to a constant factor, with the celebrated Laplace operator

∂_{x}^{2} + ∂_{y}^{2} = 4∂_{z}∂z̄.

Proof. We first prove that the submean property implies (1.4.11). We rewrite the submean contained in (iii) of Theorem 1.4.2 as

(1.4.12)  ∫_{0}^{2π} [φ(z_{o} + re^{iθ}) − φ(z_{o})] dθ ≥ 0.

If we express the difference between brackets in (1.4.12) by Taylor expansion up to order 2, (1.4.12) becomes

(1.4.13) ∫_{0}^{2π} ( [∂_{z}φ(z_{o})re^{iθ} + ∂z̄φ(z_{o})re^{−iθ}] + \frac{1}{2} [∂_{z}^{2}φ(z_{o})r^{2}e^{2iθ}

+∂2 ; z̄φ(z_{o})r^{2}e^{−2iθ} + 2∂_{z}∂z̄φ(z_{o})r^{2}] )dθ + o(r^{2}) ≥ 0.

Now, the only term which survives after integration is the non-harmonic one; thus (1.4.13) implies for r sufficiently small, ∂_{z}∂z̄φ(z_{o}) ≥ 0.

Conversely, assume ∂_{z}∂z̄φ ≥ 0. Write x = r cos θ, y = r sin θ, with inverse correspondence r = \sqrt{x^{2} + y^{2}}, θ = arctg \frac{y}{x}. The chain rule yields

∂_{x} = \frac{∂r}{∂x}∂_{r} + \frac{∂θ}{∂x}∂_{θ} = cos θ∂_{r} − \frac{sin θ}{r}∂_{θ},

∂_{y} = \frac{∂r}{∂y} ∂_{r} + \frac{∂θ}{∂y} ∂_{θ} = sin θ∂_{r} + \frac{cos θ}{r}∂_{θ}.

Substitution into ∂_{z}∂z̄ = \frac{1}{4}(∂_{x}^{2} + ∂_{y}^{2}) yields the expression of ∂_{z}∂z̄ in (r, θ) coordinates

∂_{z}∂z̄ = \frac{1}{4}(∂_{r}^{2} + \frac{1}{r}∂_{r} + \frac{1}{r^{2}} ∂_{θ}^{2}).

Our assumption implies that

(1.4.14)  ∫_{0}^{2π} (∂_{r}^{2} + r^{−1}∂_{r} + r^{−2}∂_{θ}^{2}) φ(z_{o} + re^{iθ})dθ ≥ 0.

Note that under integration the term which contains the derivation in θ vanishes. We denote by M(r) the mean value of φ over the r-circle with center z_{o}:

M(r) := (2π)^{−1} ∫_{0}^{2π} φ(z_{o} + re^{iθ})dθ.

Thus, (1.4.14) reads

rM^{··}(r) + Ṁ(r) ≥ 0.

But the left hand side term above is the derivative of rṀ(r). Thus, rṀ(r) is increasing and since it is 0 at r = 0, it follows that Ṁ(r) ≥ 0 and hence M(r) itself is increasing. But M(0) = φ(z_{o}) and therefore the inequality M(0) ≤ M(r) is precisely the submean property.

□

Note that through its differential characterization provided by the theorem above, subharmonicity reveals in full its local character.

Remark 1.4.10. The statement of Theorem 1.4.7 is immediate if φ is C^{2} in addition to subharmonic. In fact, for χ̇ ≥ 0 and χ^{··} ≥ 0, we have

\begin{aligned}∂_{z}∂z̄χ(φ) & = χ^{··}∂_{z}φ∂z̄φ + χ̇∂_{z}∂z̄φ \\ & ≥ 0. \end{aligned}

However, in Section 1.7, we will have a crucial application of composition in which φ is not C^{2}.

We end this section with an application of Fatou’s lemma to the integrals which enter into the submean property. It reflects a general principle. For sequences of functions which admit integral representation, or estimates by integrals like submeans, in the presence of a uniform bound, the pointwise “lim sup” enters into the integral and becomes “uniform”.

Theorem 1.4.11. Let {φ_{ν}}_{ν} be a sequence of subharmonic functions which are uniformly bounded on any compact subset of Ω. Let lim sup_{ν} φ_{ν}(z) ≤ l for any z ∈ Ω; then for any ϵ and for any K ⊂⊂

Ω there is ν_{ϵ K} such that

(1.4.15)  sup_{z∈K} φ_{ν}(z) ≤ l + ϵ for any ν ≥ ν_{ϵ K}.

Proof. Fix z_{o} ∈ K; for |z − z_{o}| < δ and r < d(z_{o}, ∂Ω) − δ with δ << r, we have by the submean property

\begin{aligned}φ_{ν}(z) & ≤ (πr^{2})^{−1} ∫_{0}^{2π} ∫_{0}^{r} tφ_{ν}(z + te^{iθ})dt dθ \\ & ≤ (π(r + δ)^{2})^{−1} ∫_{0}^{2π} ∫_{0}^{r+δ} tφ_{ν}(z_{o} + te^{iθ})dt dθ + O(δ), \end{aligned} ; (1.4.16)

where the control O(δ) for the error in passing from integration over B(z, r) to B(z_{o}, r + δ) follows from the uniform boundedness of the φ_{ν}’s. By Fatou’s lemma we can see that (1.4.16) implies

lim sup_{ν} sup_{z∈B(z_{o},δ)}φ_{ν}(z) ≤ (π(r + δ)^{2})^{−1}

× ∫_{0}^{2π}∫_{0}^{r+δ} lim sup_{ν} tφ_{ν}(z_{o} + te^{iθ})dt dθ + O(δ)

≤ l + O(δ).

By a finite covering argument of K and by the arbitrariness of δ, we conclude that

lim sup_{ν} sup_{z∈K} φ_{ν}(z) ≤ l.

□

Theorem 1.4.11 is a variant of the famous Hartogs lemma.

Remark 1.4.12. The use of Fatou’s lemma inside the spherical submean estimate would not suffice for the conclusion. In that case we should apply Fatou’s lemma to the function θ ↦ sup_{z} φ_{ν}(z + re^{iθ}) but

we do not know if its lim sup is still ≤ l. However, a proof which uses 1-dimensional integrals is also available: notice that it uses the Poisson kernel and not just the submean (cf. [71]) as is hereby done.

1.5. Separate analyticity

If f is continuous on Ω ⊂ ℂ^{n} and separately holomorphic in each variable when the others are kept fixed, then we have already seen in Section 1.2, as a consequence of Cauchy’s integral formula, that it is holomorphic. We wish now to get rid of the hypothesis of continuity. We start by proving that in this context continuity is in fact a consequence of boundedness:

Lemma 1.5.1. Let f be separately holomorphic and bounded on compact subsets of Ω. Then f is continuous and hence holomorphic on Ω.

Proof. By using iteration, we may assume that Ω is a subset of ℂ^{2}. Let z^{o} ∈ Ω, let z_{1}^{ν} → z_{1}^{o}, let z_{2} move near z_{2}^{o} so that (z_{1}^{ν}, z_{2}) stays at distance > r from ∂Ω, and let c be a uniform bound for |f| in the r-neighborhood of these points. Define F_{ν}(z_{2}) := f(z_{1}^{ν}, z_{2}); Cauchy’s inequalities yield

|∂_{z_{2}}F_{ν}(z_{2})| ≤ \frac{c}{r}.

Hence {F_{ν}}_{ν} is equicontinuous and in particular, if we also take z_{2}^{ν} → z^{o}, we conclude that

f(z_{1}^{ν}, z_{2}^{ν}) → f(z_{1}^{o}, z_{2}^{o}).
□

We are ready for the main result of the section.

Theorem 1.5.2 (Hartogs theorem). Let f be a function on a domain Ω ⊂ ℂ^{n} which is separately holomorphic; then it is holomorphic.

Proof. The statement is local and can be obtained by recurrence: so we assume from the beginning that Ω ⊃⊃ Δ̄ × Δ̄ in ℂ^{2} and prove that f is holomorphic on Δ × Δ. For large positive l, let us define E_{l} := {z_{2} ∈ Δ : sup_{z_{1}∈Δ}|f(z_{1}, z_{2})| ≤ l}. The continuity of f(z_{1}, ·) for fixed

z_{1} implies that if z_{2}^{ν} → z_{2}^{o} with z_{2}^{ν} ∈ E_{l}, then z_{2}^{o} ∈ E_{l}. Thus E_{l} is closed.

The continuity on Δ̄ of f(·, z_{2}) for fixed z_{2} implies that ⋃_{l}E_{l} = Δ. Thus,

Baire’s theorem applies and implies that E_{l} has non-empty interior for large l. At this point we apply the conclusion of Lemma 1.5.1: f

is holomorphic in Δ × E^{∘}_{l}. We can repeat the same construction as above by defining different sets E_{l} on any open subset of Δ; hence f is holomorphic on Δ × B for an open dense subset B ⊂ Δ. This set contains a horizontal strip arbitrarily close to z_{2} = 0. At this stage we can even forget that f is separately holomorphic in z_{1} (outside the strip) while keeping the separate analyticity with respect to z_{2} ∈ Δ and prove that f is holomorphic in Δ×Δ. Also, for this statement, it is not restrictive to assume that the strip is centered at 0, that is, of the form Δ×Δ_{ϵ}. (Otherwise, we have just to center expansion (1.5.1) at z_{2} = z_{2}^{o}, for |z_{2}^{o}| arbitrarily small, instead of z_{2} = 0.) Thus Theorem 1.5.2 is a consequence of the following.

Theorem 1.5.3. Let f be holomorphic in Δ × Δ_{ϵ} and separately holomorphic in z_{2} ∈ Δ when z_{1} is fixed in Δ. Then f ∈ hol(Δ × Δ).

Proof. Since we have to prove that f is holomorphic on every relatively compact polydisc, we can assume that f is in fact holomorphic over a strip slightly bigger than Δ × Δ_{ϵ}. We then consider the Taylor series of f with respect to z_{2} at z_{2} = 0:

(1.5.1)  ∑_{ν} \frac{∂_{z_{2}}^{ν}f(z_{1}, 0)}{ν!}z_{2}^{ν}.

It converges normally for (z_{1}, z_{2}) ∈ Δ×Δ_{ϵ}, that is, absolutely uniformly on compact subsets of Δ × Δ_{ϵ}. Write

φ_{ν}(z_{1}) := (\frac{|∂_{z_{2}}^{ν}f(z_{1}, 0)|}{ν!})^{\frac{1}{ν}} .

The family of functions {φ_{ν}}_{ν∈ℕ} are subharmonic and satisfy, by Cauchy’s inequalities,

lim sup_{ν} sup_{z_{1}∈Δ} φ_{ν} ≤ ϵ^{−1}.

By our assumption of separate analyticity in z_{2}, they also satisfy, for any fixed z_{1},

lim sup_{ν} φ_{ν}(z_{1}) ≤ 1.

But then, by Theorem 1.4.11 they satisfy

lim sup_{ν} sup_{Δ} φ_{ν} ≤ 1.

Thus by Hadamard’s theorem, the power series (1.5.1) converges normally for (z_{1}, z_{2}) ∈ Δ × Δ and thus its sum produces a holomorphic function on Δ × Δ. This completes the proof of Theorem 1.5.3 and therefore of Theorem 1.5.2 also.

□

Remark 1.5.4. We have used here the subharmonicity of all

(\frac{|∂_{z_{2}}^{ν}f(z_{1},0)|}{ν!})^{\frac{1}{ν}}. This was obtained as a consequence of the subharmonic-

ity of \frac{1}{ν} log \frac{|∂_{z_{2}}^{ν}f(z_{1},0)|}{ν!} to which we have applied the convex increasing function exp. However the subharmonicity of \frac{1}{ν} log \frac{|∂_{z_{2}}^{ν}f(z_{1},0)|}{ν!} itself is not needed here.

1.6. Levi forms—continuity principle (I)—Hartogs extension theorem

We want to describe an impressive phenomenon of the forced extension of holomorphic functions of which we have taken a small taste in Section 1.3. In contrast to the case of real functions, some domains of ℂ^{n} are not “domains” of holomorphic functions. We start from the local case and develop some preliminary discussion.

Definition 1.6.1. For a real C^{2} function r in a domain of ℂ^{n}, we define the Levi form of r at z_{o} as the hermitian form

(1.6.1)  L_{r}(z_{o}) = ∑_{ij} ∂_{z_{i}}∂z̄_{j}r(z_{o})dz_{i} ⊗ dz̄_{j}.

Theorem 1.6.2. The Levi form (1.6.1) is invariant under holomorphic changes of coordinates: if z ↦ z̃ = F (z) is a holomorphic diffeomorphism and if r̃ is r in the new system (that is, F^{∗}r̃ = r), then

(1.6.2)  F^{∗}Lr̃ = L_{r}.

(Here F^{∗} denotes the pull-back by F.)

Proof. We have to prove that

(1.6.3)  Lr̃(F′u, \overline{F′v}) = L_{r}(u, v̄) for any (u, v̄) ∈ ℂ^{n} × ℂ̄^{n}.

Now, the coefficients of L_{r} = L_{r̃∘F} are calculated by the chain rule. First, writing z̃(z) for F (z), we have

∂_{z_{h}}∂z̄_{k}(r̃ ∘ F) = ∂_{z_{h}} [∑_{i} ∂z̃_{i} r̃∂z̄_{k} z̃_{i} + ∑_{i} ∂_{\overline{z̄}_{i}} r̃\overline{∂_{z_{k}} z̃_{i}}] .

But the first sum vanishes because ∂z̄_{k} z̃_{i} = 0. We have likewise

\begin{matrix}∂_{z_{h}} ∑_{i} & (∂_{\overline{z̃}_{i}} r̃)\overline{∂_{z_{k}} z̃_{i}} = ∑_{ji} ∂z̃_{j}∂_{\overline{z̃}_{i}} r̃∂_{z_{h}} z̃_{j}\overline{∂_{z_{k}} z̃_{i}} \\ + ∑_{ji} ∂_{\overline{z̃}_{j}}∂_{\overline{z̃}_{i}} r̃∂_{z_{h}} ¯z̃_{j}\overline{∂_{z_{k}} z̃_{i}} + ∑_{ji} ∂z̄_{i} r̃\overline{∂z̄_{h}∂_{z_{k}} z̃_{i}}. \end{matrix}

Again, the sums in the second line cancel. Writing F (z) for z̃(z), we get the identity for matrices

L_{r} = ^{t}\overline{J(F)} · Lr̃ · J(F),

which is another way of stating (1.6.3).

□

The Levi form is called the “complex hessian” of r. The full hessian, the quadratic term of the expansion of r, written in coordinates (z, z̄) ∈ ℂ^{n} × \overline{ℂ}^{n} is

hess r = \frac{1}{2} ∑_{ij}∂_{z_{i}}∂_{z_{j}}rdz_{i}⊗dz_{j}+\frac{1}{2} ∑_{ij}∂z̄_{i}∂z̄_{j}rdz̄_{i}⊗dz̄_{j}+∑_{ij}∂_{z_{i}}∂z̄_{j}rdz_{i}⊗dz̄_{j}.

The sum of the first two terms on the right side is a real harmonic polynomial called the “Levi polynomial” that we denote by LP_{r} := Re ∑_{ij} ∂_{z_{i}}∂_{z_{j}}rdz_{i}⊗dz_{j}; thus hess r = LP_{r}+L_{r}. By a coordinate change we can see that the Levi polynomial can be canceled from a graphing function: the Levi form is therefore the holomorphic invariant of the hessian. A first rough way of getting rid of the Levi polynomial consists in taking the mean of the value of the hessian over a vector u and iu.

Proposition 1.6.3. We have

L_{r}(u, ū) = \frac{1}{2} [hess r(u, u) + hess r(iu, iu)] .

Proof. We have

LP_{r}(iu, iu) = −LP_{r}(u, u), (1.6.4)

L_{r}(iu, \overline{iu}) = L_{r}(u, ū). (1.6.5)

It follows that hess r(u, u) + hess r(iu, iu)

= (LP_{r}(iu, iu) + LP_{r}(u, u)) + (L_{r}(iu, \overline{iu}) + L_{r}(u, ū))
= 2L_{r}(u, ū).
□

We go on now to a geometric notion, that of the Levi form of a real hypersurface M ⊂ ℂ^{n} and refer to Section 3.4 for the case of higher codimension. We assume that M is locally defined by an equation r = 0 with ∂r ≠ 0. We denote by T^{ℂ}M the complex tangent bundle to M defined by T^{ℂ}M = T M ∩ iT M. For any z ∈ M its fiber T_{z}^{ℂ}M is the space of vectors orthogonal to ∂r(z) under the hermitian product. Sometimes, we also denote it by ∂r(z)^{⊥}.

Definition 1.6.4. The Levi form of the hypersurface M at a point z ∈ M is defined by

L_{M}(z) := L_{r}(z)|_{T_{z}^{ℂ}M}.

We have to note at the very beginning that the conformal class of L_{M} does not depend of the choice of the equation r = 0 of M; thus the rank of L_{M} and, under the choice of the orientation, the signature are well-defined. In fact, let r · g = 0 be another equation with g ≠ 0, and consider

L_{r·g} = gL_{r} + rL_{g} + ∂r ⊗ ¯∂g + ¯∂r ⊗ ∂g.

First, we have rL_{g} = 0 on r = 0, and second, both ∂r ⊗ ¯∂g and ¯∂r ⊗ ∂g vanish when restricted to ∂r^{⊥}. Thus

L_{r·g}|_{T^{ℂ}M} = g · L_{r}|_{T^{ℂ}M}.

We denote by s_{r}^{+}, s_{r}^{−}, s_{r}^{0} the numbers of the eigenvalues of L_{r} which are > 0, < 0, = 0, respectively. We denote by s_{M}^{+}, s_{M}^{−}, s_{M}^{0} the corresponding numbers for L_{M}. Here there is an ambiguity on the sign since M is defined both by ±r = 0. We always agree to think of M as the boundary of Ω : {z : r(z) < 0} with the orientation induced by the outward conormal ∂r.

Theorem 1.6.5 (Normal coordinates for a hypersurface). Let M be a hypersurface of ℂ^{n} and z_{o} a point of M. Then, by a holomorphic, indeed quadratic, change of coordinates and with the notation z = (z_{1}, z′), we can put locally at z_{o} = 0 the equation of M in the form

(1.6.6)  y_{1} = (− ∑^{s^{+}+1} ; j=2|z_{j}|^{2} + ∑^{s^{+}+s^{−}+1} ; j=s^{+}+2|z_{j}|^{2}) + o^{2}(x_{1}, z′).

Proof. By a linear change of coordinates we can assume that z_{o} = 0 and that the equation r = 0 of M satisfies 2Re ∂r(0) = dy_{1}. Thus, M is locally a graph

y_{1} = f(z′, z̄′, x_{1}) with ∂f(0) = 0.

Hence f = O^{2} and we can therefore decompose

f = (Q(z′, z̄′) + L(z′, z̄′)x_{1} + Cx_{1}^{2}) + o^{2}(x_{1}, z′),

where Q is quadratic, L linear, C constant with all of them real. We write explicitly

Q = ∑_{ij} a_{ij}z′_{i}z′_{j} + ∑_{ij} ā_{ij}z̄′_{i}z̄′_{j} + ∑_{ij} 2b_{ij}z′_{i}z̄′_{j}

(with b_{ji} = b̄_{ij}) and

L = ∑_{i} c_{i}z′_{i} + c̄_{i}z̄′_{i}.

We make now a quadratic coordinate change

\begin{cases}z_{1} ↦ z_{1} − i (∑_{i} 2c_{i}z_{i}z_{1} + Cz_{1}^{2}) , \\ z′ ↦ z′. \end{cases}

By the implicit function theorem the equation of M becomes

y_{1} = Q(z′, z̄′) + o^{2}(x_{1}, z′).

We now want to eliminate the Levi polynomial inside Q. We perform the quadratic transform

{z_{1} ↦ z_{1} − i ∑_{ij} 2a_{ij}z′_{i}z′_{j},

and our equation becomes

y_{1} = 2 ∑_{ij} b_{ij}z′_{i}z̄′_{j} + o^{2}(x_{1}, z′).

We are ready to conclude. For this we perform a last, orthonormal, change of coordinates inside the plane ℂ^{n−1} ≃ T_{0}^{ℂ}M of the z′ variables which diagonalizes the Levi matrix (b_{ij})_{ij} = L_{M} and next a dilation which makes the eigenvalues unitary. This provides the normal form (1.6.6) in the statement of the theorem.

□

Example 1.6.6. Let Ω be defined by y_{1} < −2x_{2}^{2} +y_{2}^{2} +... where the dots denote terms of degree ≥ 3 in z_{2}, z̄_{2}; we also write the inequality which defines Ω as r < 0. Hence, for the real structure of the x_{2}, y_{2} plane, the set M = ∂Ω is neither convex nor concave. But it is “more” convex than concave and thus it is convex in the complex sense. In fact, if e_{2} = (0, 1, 0...), then

L_{M}(e_{2}, ē_{2}) = \frac{1}{2} [hess r(e_{2}, e_{2}) + hess r(ie_{2}, ie_{2})]

= \frac{1}{2} > 0.

Hence e_{2} is a positive eigenvector for the Levi form. According to the normal form proved in Theorem 1.6.5 there must be a coordinate change z̃ = z̃(z) under which the inequality which defines the domain changes into

(1.6.7)  ỹ_{1} < −|z̃_{2}|^{2} + ....

For this purpose, it is enough to consider the transformation

\begin{cases}z̃_{1} = z_{1} + i\frac{3}{2}z_{2}^{2}, \\ z̃_{2} = \frac{z_{2}}{\sqrt{2}}, \end{cases}

and the initial inequality changes into (1.6.7).

Thus, the convexity of the Levi form represents the “complex” convexity. It is invariant under complex holomorphic change of coordinates since the Levi form is, whereas the real convexity can be affected by these transformations as we have just seen.

Remark 1.6.7. The C^{2} regularity of M = ∂Ω at some point z_{o} implies the property of Ω to contain a ball which is tangent to M at z_{o} from the interior of Ω. Its radius is the minimum curvature radius in different directions. On the other hand, strict positivity of the Levi form L_{M}(z_{o}) > 0 is equivalent, under a holomorphic change of coordinates, to the fact that Ω is contained in a neighborhood of z_{o} = 0 in the set defined by

\begin{aligned}y_{1} & < −ϵ|z′|^{2} − x_{1}^{2} + o^{2} \\ & < −\frac{ϵ}{2}(|z′|^{2} + x_{1}^{2}). \end{aligned}

Thus, instead, Ω is now contained in a ball (of center −i\frac{2}{ϵ} and radius \frac{2}{ϵ}).

We describe now the (deformed) Hartogs figures. This technique is far weaker than the “continuity principle” presented in the next section, but we think it is worth talking about it because of its simplicity and its historical relevance. A holomorphic disc is the holomorphic bounded image into ℂ^{n} of the standard disc Δ of ℂ. We consider a holomorphic foliation of a domain Ω ⊂ ℂ^{n} by means of holomorphic discs. This can be described by taking a domain U ⊂ ℂ^{n−1} and a holomorphic one-to-one mapping Φ:

Δ × U →^{Φ} Ω, w ↦ z = Φ(w),

with each disc being defined by A_{w′} := Φ_{w′}(Δ) for Φ_{w′}(·) := Φ(·, w′). We can also assume that Φ extends as a holomorphic diffeomorphism to a neighborhood of ∂Δ and set ∂A_{w′} := Φ_{w′}(∂Δ). For an open subset Ω_{o} ⊂ Ω we assume

(1.6.8)  \begin{cases}∂A_{w′} ⊂⊂ Ω̄_{o} & for any w′ ∈ U, \\ Ā_{w′} ⊂⊂ Ω̄_{o} & for w′ in an open subset U′ ⊂ U. \end{cases}

Under these circumstances, we encounter one of the most elementary phenomenon of holomorphic extension which is essentially a consequence of Cauchy’s formula:

hol(Ω_{o}) −−→^{ext} hol(Ω).

In fact, we begin by remarking that we have the freedom of shrinking our discs a little. This is the same as replacing Ω̄_{o} by Ω_{o} in (1.6.8). Next, we define Φ̃^{∗}f(w) := \frac{1}{2πi} ∫_{∂Δ} \frac{Φ^{∗}f(ζ_{1},w′)}{ζ_{1}−w_{1}}dζ_{1}. The integral makes

sense because Φ_{w′}(∂Δ) ⊂⊂ Ω_{o} where f is defined. The function Φ̃^{∗}f is holomorphic for w ∈ Δ × U, coincides with Φ^{∗}f on Δ × U′, and hence it coincides on the whole of Δ × U by analytic continuation. Thus f̃:= Φ̃^{∗}f ∘ Φ^{−1} is the holomorphic extension of f from Ω_{o} to Ω. This elementary principle suffices for the following celebrated theorem.

Theorem 1.6.8 (Hartogs extension theorem). Let Ω be a domain with C^{2} boundary, let z_{o} ∈ M = ∂Ω and assume that L_{M} has at least one negative eigenvalue. Then for any neighborhood B of z_{o} there is another neighborhood B′ ⊂ B such that

hol(Ω ∩ B) −−→^{ext} hol((Ω ∩ B) ∪ B′).

Proof. By our assumption about the Levi form, we can assume that Ω contains in a neighborhood of z_{o} (that we suppose, by rescaling, to be the unit ball) the set y_{n} < |z_{1}|^{2} − Q(z_{2}, ..., z_{n−1}) − o^{2}(x_{n}, z_{2}, ..., z_{n−1}) where Q denotes a quadratic term in z_{2}, ..., z_{n−1}. We consider this quadratic term and o^{2} as an error term E = E(x_{n}, z_{2}, ...) and define U := {z′ ∈ ℂ^{n−1} : y_{n} < 1 − E} and U′ := {z′ : y_{n} < −E}. Now, the family of straight discs in which z_{1} ranges through Δ and z′ is fixed in U satisfies

∂A_{z′} ⊂⊂ Ω for z′ ∈ U, A ⊂⊂ Ω for z′ ∈ U′.

On the other hand it is clear that this family {A_{z′}} covers a full neighborhood B′ of z_{o} = 0. The conclusion is then a consequence of the extension criterion herewith proved.

□

1.7. Logarithmic supermean of Taylor radius of holomorphic

functions—continuity principle (II)—propagation of holomorphic extendibility

Summary. Cauchy’s inequalities and the Cauchy-Hadamard theorem relate the holomorphic extension of a function with the estimates of the moduli of its derivatives. Since these are logarithmically subharmonic, their inverses, which represent the radii of convergence of the Taylor expansion, are logarithmically superharmonic. By this, we get the proof of the theorem of propagation of holomorphic extendibility: a complex curve in a hypersurface M is a propagator of holomorphic extendibility from either side of M.

Logarithmic supermean of Taylor radius of holomorphic functions.

Definition 1.7.1. We say that a real upper semicontinuous function φ in Ω ⊂ ℂ^{n} is plurisubharmonic when its restriction to any disc A = Φ(Δ) ⊂ Ω is subharmonic.

We say that φ is plurisuperharmonic when −φ is plurisubharmonic. When φ is C^{2}, plurisubharmonicity is characterized by ∂_{z_{i}}∂z̄_{j}φ ≥ 0 for any i, j. Thus, subharmonicity along straight discs τ ↦ w_{o} + τw, indeed just coordinate rays in which w = (...0, 1, 0...), suffices. But this is in fact always true, not just for C^{2} functions.

Proposition 1.7.2. Let φ ≠ −∞ be upper semicontinuous and subharmonic along each cartesian ray. Then, there is a sequence {φ_{ν}}_{ν} of C^{∞} plurisubharmonic functions on Ω_{ν} := \{z ∈ Ω : d(z, ∂Ω) > \frac{1}{ν}\} such that φ_{ν} ↘ φ. In particular, φ is plurisubharmonic.

Proof. We take a function χ ∈ C_{c}^{∞} of |z_{1}|, ..., |z_{n}| with support contained in |z| < 1, satisfying χ ≥ 0 and ∫ χ\frac{dζ∧dζ̄}{(−2i)^{n}} = 1, and set

φ_{ν}(z) = ∫_{Δ×...×Δ} φ(z − \frac{ζ}{ν})χ(ζ)\frac{dζ ∧ dζ̄}{(−2i)^{n}}.

It is obvious that every φ_{ν} is C^{∞} in Ω_{ν}. We prove now that it is plurisubharmonic. Since it is smooth, it suffices to check the property along each cartesian complex plane. By iterated integration, it suffices to prove it in dimension 1 for the integration in a single variable ζ ∈ Δ. But in fact,

\frac{1}{2π} ∫_{0}^{2π} φ_{ν}(w_{o} + re^{iθ})dθ

= ∫∫_{Δ} (\frac{1}{2π} ∫_{0}^{2π} φ(w_{o} + re^{iθ} − \frac{ζ}{ν})dθ) χ(ζ)\frac{dζ ∧ dζ̄}{−2i}

≥ ∫∫_{Δ} φ(w_{o} − \frac{ζ}{ν})χ(ζ)\frac{dζ ∧ dζ̄}{−2i} = φ_{ν}(w_{o}).

We come back to general dimension n. Since the φ_{ν}’s are plurisubharmonic, then the (φ_{ν})_{\frac{1}{ϵ}} ’s are decreasing with ϵ ↘ 0 (by iterated 2dimensional submean over ϵ-discs). By letting ν → ∞, the φ_{\frac{1}{ϵ}} ’s themselves are decreasing: writing ν in place of \frac{1}{ϵ}, we conclude φ_{ν} ↘ φ. Last, we show that φ is plurisubharmonic. It is upper semicontinuous for the same reason as in one variable (cf. the beginning of the proof of Lemma 1.4.6). Moreover, by taking composition with the parametrization of a disc A = Φ(Δ), we have

φ_{ν} ∘ Φ ↘ φ ∘ Φ.

Since each φ_{ν} ∘ Φ is subharmonic, then also the monotonic limit φ ∘ Φ is subharmonic on account of Lemma 1.4.6.

□

Let Ω be an open domain, v a unit vector in ℂ^{n}, f a holomorphic function on Ω. We consider the function

(1.7.1)  r_{f}^{v}(z) = ( lim sup_{ν} |\frac{∂_{v}^{ν}f(z)}{ν!}|^{\frac{1}{ν}} )^{−1} , z ∈ Ω.

The function r_{f}^{v}(z) represents the radius of the maximal disc in the v direction in which the Taylor series of f at z is separately convergent and hence f is separately holomorphic. When we uniformize over z′ ranging in a neighborhood B of z in the (n − 1)-plane orthogonal to v passing through z, we get

Proposition 1.7.3. We have

(1.7.2)  inf_{z′∈B}r_{f}^{v}(z′) = sup{r : f is holomorphic in vΔ_{r} × B}.

Proof. The inequality “≥” follows from Cauchy’s inequalities. As for “≤”, we suppose that the direction of v is that of the z_{1} axis and denote by z′ the complementary variables. In any polydisc centered at ξ_{o} and contained in Ω, f is a sum of a double power series in z_{1} − ξ_{1} and z′ − ξ′ that we may rearrange as

(1.7.3)  ∑_{ν} \frac{∂_{v}^{ν}f(z′)}{ν!}(z_{1} − ξ_{1})^{ν}.

Let r < inf_{z′∈B}r_{f}^{v}(z′), that is, r < ( lim sup_{ν} |\frac{∂_{v}^{ν}f(z′)}{ν!} |^{\frac{1}{ν}} )^{−1} for any

z′ ∈ B. It follows that

(1.7.4)  sup_{z′∈B} lim sup_{ν} |\frac{∂_{v}^{ν}f(z′)}{ν!}|^{\frac{1}{ν}} ≤ r^{−1}.

Hence, if we apply Theorem 1.4.11 we may interchange “sup” and “lim sup” in (1.7.4); in particular the power series (1.7.3) converges normally for z_{1} − ξ_{1} ∈ Δ_{r}, uniformly for z′ ∈ B and thus defines a holomorphic function on vΔ_{r} × B which matches f; hence r stays in the set on the right side of (1.7.2) which completes the proof of Proposition 1.7.3.

□

The property expressed by (1.7.2) is essential in the present section. As for the purpose of the subsequent Section 1.8, we need not to uniformize on z′ ∈ B but, instead, on v ∈ S^{2n−1}, the unit (2n − 1)-sphere in ℂ^{n}.

Proposition 1.7.4. Let f be holomorphic in a neighborhood of z. Then

(1.7.5)  inf_{v∈S^{2n−1}}r_{f}^{v}(z) = sup{r : f ∈ hol(𝔹^{n}(z, r))},

where 𝔹^{n}(z, r) is the n-ball with center z and radius r.

Proof. “≥” is clear by Cauchy’s inequalities. For the opposite, let r < r_{f}^{v}(z) for any v. Then f is separately holomorphic along each ray z + vΔ_{r} and, in addition, holomorphic in a neighborhood of z. By an obvious variant of Theorem 1.5.3 it is holomorphic in 𝔹^{n}(z, r). In fact, let us prove that f is holomorphic in a conical neighborhood in B(z, r) of z + vΔ_{r} for each v. We can suppose, without loss of generality, that v = (1, ..., 0). Under the transformation, (z_{1}, z′) ↦ (z_{1}, \frac{z′}{z_{1}} ), the ϵ-conical neighborhood of z + vΔ_{r} is transformed into the polydisc Δ_{r} × (Δ_{ϵ} × Δ_{ϵ} × ...). By applying Theorem 1.5.3 we get the proof of our claim. By a finite covering we conclude that f is holomorphic in B(z, r).

□

We will make essential use of the following elementary consequence of Proposition 1.7.3. Let z ∈ Ω and z_{o} ∈ ∂Ω be a pair of points such that z_{o} − z is normal to ∂Ω at z_{o}. We write v := \frac{z_{o}−z}{|z_{o}−z|} and d(z) := |z_{o} − z|; thus, d(z) is the distance of z to ∂Ω. In this situation, for a holomorphic function on Ω we have

if inf_{z′∈B}r_{f}^{v}(z′) > d(z) for some B, then f extends holomorphically across ∂Ω at z_{o}.

In fact, take r with inf_{z′∈B}r_{f}^{v}(z′) > r > d(z); then, the Taylor series of f at z converges in vΔ_{r} × B and provides the extension of f across the boundary ∂Ω at z_{o}. This will be especially useful in combination with the following statement which is also a consquence of the results of Section 1.4

Theorem 1.7.5. The function log r_{f}^{v} enjoys the supermean property along any disc.

Proof. Set φ_{ν} := \frac{1}{ν} log |\frac{∂_{v}^{ν}f}{ν!} |. Since

− log( lim sup_{ν} ·) = − lim sup_{ν} (log ·),

then

log r_{f}^{v} = log ( lim sup_{ν} |\frac{∂_{v}^{ν}f}{ν!} |^{\frac{1}{ν}} )^{−1}

= − lim sup_{ν} φ_{ν}.

Here

\begin{cases}φ_{ν} are plurisubharmonic (by Lemma 1.4.8), \\ φ_{ν} are uniformly bounded (by Cauchy’s inequalities). \end{cases}

Restrict the φ_{ν}’s to a disc A, that is, take the composition φ_{ν} ∘ Φ with its parametrization Φ. By Remark 1.4.5, log r_{f}^{v} ∘ Φ = − lim sup_{ν} φ_{ν} ∘ Φ

enjoys the supermean property.

□

Note that in the above proof we cannot apply Proposition 1.4.4 as it stands because we do not know if log r_{f}^{v} ∘ Φ is lower semicontinuous. On the other hand, we can force it to be semicontinuous by replacing r_{f}^{v} by

r^{∨}_{f}^{v} := sup_{B} inf_{z′∈B}r_{f}^{v}(z′).

This is the maximal lower semicontinuous minorant of r_{f}^{v}. In this way, we gain semicontinuity but, as a price, we may loose supermean property (cf. Example 1.4.3). We call r^{∨}_{f}^{v}(z) the Taylor radius of f in the v direction in a neighborhood of z.

Continuity principle. We now wish to take advantage of the powerful tool that we have just developed and generalize the procedure of extension by means of families of analytic discs. This has already been exploited, in a particular case, in the course of the proof of the Hartogs extension theorem. We consider a family of discs {A_{t}}_{t} parametrized over a connected space of parameters T. We suppose that the parametrization is a foliation of a domain Ω ⊂ ℂ^{n} that is a C^{1}-diffeomorphism Φ : T × Δ →^{∼} Ω which is holomorphic with respect to τ ∈ Δ and which extends as a diffeomorphism to a neighborhood of ∂Δ. Differently from the case of the Hartogs figure, Φ is not necessarily a holomorphic diffeomorphism. We set Φ_{t} = Φ(t, ·), A_{t} = Φ_{t}(Δ) and ∂A_{t} = Φ_{t}(∂Δ).

Theorem 1.7.6. In the above situation, let Ω_{o} be a subdomain of Ω which satisfies

(i) ∂A_{t} ⊂⊂ Ω̄_{o}, (ii) Ā_{t} ⊂⊂ Ω̄_{o} for t ranging in an open set of parameters T_{o} ⊂ T.

Then, there is an extension mapping

(1.7.6)  hol(Ω_{o}) −−→^{ext} hol(Ω).

Proof. It is not restrictive to reduce Ω_{o} to

Φ ((T × (Δ \ Δ̄_{1−ϵ})) ∪ (T_{o} × Δ))

for a suitable annulus Δ \ Δ̄_{1−ϵ} = {τ : 1 − ϵ < |τ| < 1}. Also, by exhausting T × Δ by relatively compact subsets T′ × Δ′, it suffices to prove extension to the set Φ(T′ × Δ′). In this way we may assume that our discs are shrunk so that ∂A_{t} ⊂⊂ Ω_{o}. Finally, though it is not essential, we suppose T connected by arcs.

Fix t_{1} and let γ be a connected curve connecting t_{1} to a point t_{o} ∈ T_{o}. Fix a holomorphic function f ∈ hol(Ω_{o}) and let γ′ = {t ∈ γ : f is holomorphic in a neighborhood of A_{t}}. We have

(a) γ′ ≠ ∅ (by (ii)),

(b) γ′ is open.

But γ′ is also closed. In fact, let t_{ν} → ξ for t_{ν} ∈ γ′ and let z_{o} be a generic point in A_{ξ}. By reparametrizing Δ, we may assume that z_{o} is the “center”, that is, corresponds to the value τ = 0. We also denote by z_{ν} the centers of the discs A_{t_{ν}}’s. Set v := T_{z_{o}}(Φ(γ′ × {0})); by (i) we have for a neighborhood B of 0 and a constant c > 0 independent of ν

r_{f}^{v}|_{∂A_{tν} +B} ≥ c.

Since (r_{f}^{v})^{−1} satisfies the maximum principle, this implies

r_{f}^{v}|_{z_{ν}+B} ≥ c.

It follows that the Taylor expansion of f at z_{ν} converges in polydiscs of a fixed radius in the v direction; in particular these polydiscs contain z_{o} for z_{ν} → z_{o}. Since the intersection of these polydiscs with Ω_{o} ∪ ⋃_{t<ξ}A_{t}

is connected, the sum of this series coincides with the extension of f for t < ξ. It follows that f extends holomorphically to a neighborhood of z_{o}. In conclusion, γ′ is closed and hence it must coincide with the whole of γ by connectedness.

□

It is quite obvious that the assumption that Φ is a C^{1}-diffeomor-phism is not essential. We can give a similar proof for any family of discs which is topologically equivalent to T × Δ.

Remark 1.7.7. The statement can be adapted in many ways. The key point in the proof is that the Taylor series of f in the polydiscs which contain z_{o} coincides with f previously extended. For this reason there do not arise “monodromy” problems in the above construction. This shows that what is crucial are the geometric properties of the region Ω swept by the discs rather than those of their parametrization. Thus, for instance, if Ω is pseudoconvex in the sense of Section 1.8, a very general extension principle should hold. In fact what is true for a general family of discs, regardless of their parametrization, is an extension to a domain sheeted over ℂ^{n}, the envelop of holomorphy, whose projection contains Ω. Moreover, the assumption of pseudoconvexity is exactly that the envelop is single-sheeted. The book [83] should be of help in understanding this matter.

As an application, we give a “global” version of Theorem 1.6.8.

Corollary 1.7.8. Let Ω be a convex bounded domain of ℂ^{n}. Then holomorphic functions extend from a neighborhood of ∂Ω to the whole of Ω.

Proof. We write the variable as z = (z_{1}, z′) and denote by Ω′ the projection of Ω in the z′-plane and by Ω_{o} a neighborhood of ∂Ω inside Ω. We “slice” Ω by the sets

{A_{z′}}_{z′∈Ω′} = {(ℂ × {z′}) ∩ Ω}_{z′∈Ω′}.

These are convex bounded subsets of ℂ × {z′}, hence straight discs by the Riemann mapping theorem. They satisfy ∂A_{z′} ⊂⊂ Ω̄_{o} and also Ā_{z′} ⊂⊂ Ω̄_{o} when (z_{1}, z′) approaches a point of strong convexity. Finally, they are parametrized by a C^{1}-diffeomorphism over Ω′ × Δ. The conclusion follows from Theorem 1.7.6.

□

Remark 1.7.9. Corollary 1.7.8 holds in wider generality and applies to any bounded domain Ω ⊂⊂ ℂ^{n} with connected boundary ∂Ω (Bochner-Hartogs theorem). There are several proofs. One uses the Bochner-Martinelli kernel, another the solution of ¯∂ with compact support. Our proof has an immediate generalization to domains which are pseudoconvex in the sense of Section 1.8, but, according to [38], it does not cover all cases.

Remark 1.7.10. If we are solely interested to the case of a convex set as in Corollary 1.7.8, we can follow an alternative proof. It is not restrictive to assume ∂Ω ∈ C^{ω}; in fact, we can approximate ∂Ω by a C^{ω}- hypersurface inside the “shell” where f is holomorphic. We then define f̃(z_{1}, z′) := (2πi)^{−1} ∫_{∂A_{z′}} \frac{f(ζ,z′)}{ζ−z_{1}}dζ; this is a C^{ω}-function, holomorphic in z_{1} for any z′ and coinciding with f when z belongs to a neighborhood of a point of strong convexity. In particular, ∂z̄_{j} f̃ = 0 for these values of z. But then, the principle of analytic continuation applied to the ∂_{z_{j}} f̃’s for j = 1, ..., n implies that f̃ is holomorphic in Ω and coincides with f wherever defined.

Propagation of holomorphic extendibility. We start from a general principle of propagation of analyticity. We consider in the beginning a domain Ω = Δ×Ω′ and a function f of z = (z_{1}, z′) ∈ ℂ^{n} defined over Δ × Ω′ and such that

• f is holomorphic with respect to z_{1},

• f is holomorphic in a neighborhood of {0} × Ω′. We want to show that f is in fact holomorphic on Δ × Ω′. If f ∈ C^{1}, then the conclusion is elementary. In fact,

\begin{cases}df ∧ dz_{1} ∧ ... ∧ dz_{n} has holomorphic coefficients, \\ df ∧ dz_{1} ∧ ... ∧ dz_{n} ≡ 0 for |z_{1}| < ϵ. \end{cases}

By analytic continuation the above form is ≡ 0 for any z_{1}; thus, f is holomorphic in Δ × Ω′. This argument can be extended in two directions. First, we can give the discussion in the “weak sense”. If f is a distribution endowed with distributional restriction to each plane z_{1} ≡ const, which is certainly the case if f is e.g. continuous, then, in the above situation, f is still holomorphic. (Note here that when f is bounded, then it is easily verified to be in fact continuous because of a “normal family” argument.) In fact, we have to define in this case

g(z_{1}) := ⟨f(z_{1}, ·), dψ(·)⟩,

where ψ is any (n − 1, n − 2)-form with C_{c}^{∞}-coefficients in z′ and ⟨·, ·⟩ is the pairing between (0, 0)- and (n − 1, n − 1)-forms. Again, since g is holomorphic and is 0 for |z_{1}| < ϵ, then it is ≡ 0; thus, f is holomorphic. Another generalization consists in deforming the complex lines z′ ≡ const into a general foliation by holomorphic curves, the graphs z_{1} ↦ φ(z_{1}, z′) of a function φ which is holomorphic in z_{1}, smooth in (z_{1}, z′) and normalized, say, by φ(0, z′) = z′. Assume f is holomorphic along each curve {(z_{1}, φ(z_{1}, z′)) : z_{1} ∈ Δ} and holomorphic in (z_{1}, z′) for |z_{1}| < ϵ; it then follows that f is holomorphic. In this case the proof consists in using the C^{∞} change of coordinates (z_{1}, z′) →^{F} (z_{1}, z̃′) = (z_{1}, φ(z_{1}, z′)) whose inverse F^{−1} is the straightening of the curves. We then define

g(z_{1}) = ⟨f ∘ F, F^{∗}dψ⟩_{{z_{1}}×Ω′},

for any (n − 1, n − 2)-form ψ with C_{c}^{∞}-coefficients in z′. As before, g(z_{1}) ≡ 0. Thus f annihilates ∂\overline{z̃}′_{j} for j ≥ 2 (and F_{∗}∂_{\overline{z}_{1}} from the beginning) and is C^{0} and hence it is holomorphic. In what follows, we see a much more refined principle of propagation in which in particular the presence of a single complex curve, not of a full foliation, suffices.

What we aim to describe is a phenomenon of propagation of holomorphic extendibility across a hypersurface M for functions which are holomorphic on one side, say Ω, of M. Complex curves γ ⊂ M are “propagators” of extendibility: if f ∈ hol(Ω) extends across M at a point z_{o} ∈ γ, then it extends across M at any other point z ∈ γ. We could rephrase the propagation of extendibility in terms of propagation of “microlocal” regularity in the inward conormal for the generalized boundary value of f on M. It then looks like a particular case of a general phenomenon of propagation of regularity for the induced system of ¯∂ on M. But it is not. First, according for instance to [78] or [53], the propagators of the regularity of the solutions of the P.D.E.’s should be the complex curves of the characteristic set, in this case T_{M}^{∗}ℂ^{n}, the conormal bundle to M in ℂ^{n} (cf. for instance Definition 3.2.5 and the remark which follows). But the correspondence induced by the projection π : T^{∗}ℂ^{n} → ℂ^{n} between complex curves of T_{M}^{∗}ℂ^{n} and M is one-to-one only if M is pseudoconvex (cf. Sections 1.8 and 3.5). For instance, the hypersurface defined by y_{1} = z_{2}z̄_{3} + \overline{z_{2}z̄_{3}} contains the complex curve z_{2} = 0 but its conormal bundle is “totally real” lacking therefore any complex tangency; this follows from the fact that the Levi form is non-degenerate (cf. Section 3.5). Also, the general theory of the propagation, such as in [78] and [53], concerns the case of foliations by complex leaves, bicharacteristic leaves of involutive submanifolds of the characteristic set, whereas, as we have already said, we are here in the presence of a single leaf.

Theorem 1.7.11. Let M be a hypersurface of ℂ^{n}, Ω one side of M, γ a complex curve of M, z_{o} a point of γ, f a holomorphic function on Ω. Then, if f extends across M at z_{o}, it also extends at any other point z^{1} ∈ γ.

(Cf. Hanges and Treves [47] for the case in which f has slow growth and hence a distributional boundary value at M.)

Proof. We first assume that γ is a disc A and suppose that the point z^{1} to which we wish to propagate is its “center”. By “reducing” A, that is, taking the restriction of Φ to a piece of the parametric disc Δ, we can suppose that the point of initial extension z_{o} is a boundary point. We denote by v the unit normal to M at z^{1} pointing outside Ω and set for small η > 0

A_{η} = −ηv + A, z_{o η} = −ηv + z_{o}, z_{η}^{1} = −ηv + z^{1}.

We assume that the diameter of A is small, say ϵ, and that the extension at z_{o} is to the δ-ball with η << δ << ϵ. Let η′ be the maximal radius in the v-plane for a disc centered at points of ∂A_{η} and contained in Ω; this is a small perturbation of η. In fact, if φ ∼ ϵ is the angle between v and the conormal to M along ∂A, we have \frac{η′}{η} = cos φ for cos φ = 1 − O(ϵ^{2}). We have therefore for a neighborhood B of 0

\begin{aligned}r_{f}^{v}|_{∂A_{η}+B} & ≥ η′ \\ & = η(1 − O(ϵ^{2})). \end{aligned} ; (1.7.7)

But we also have in an α-neighborhood Γ_{η} of z_{o η} in ∂A_{η}

(1.7.8)  r_{f}^{v}|_{Γ_{η}+B} ≥ δ.

Taking the “log” of (1.7.7), (1.7.8) and using the supermean property of log r_{f}^{v}, we get

(1.7.9)  log r_{f}^{v}|_{z_{η}^{1}+B} ≥ \frac{α log δ + (2πϵ − α) log η′}{2πϵ} .

Applying “exp” back, we get

\begin{aligned}r_{f}^{v}|_{z_{η}^{1}+B} & ≥ η′ (\frac{δ}{η′} )^{\frac{α}{2πϵ}} \\ & > η, \end{aligned} ; (1.7.10)

1.8. DOMAINS OF HOLOMORPHY AND PSEUDOCONVEX DOMAINS 39

for small η and η′. This shows that f is holomorphic in a neighborhood of z^{1}. If we pass from a disc A to a general curve γ, we take a real curve Γ ⊂ γ connecting z_{o} to z^{1} and denote by ξ the “extremal” point along Γ of the holomorphic extension for f. We have that ξ must coincide with z^{1} for otherwise by means of discs centered at ξ, the argument of propagation by discs would violate the “extremality” of ξ.

□

Remark 1.7.12. We are using here the full power of the submean property of log((r_{f}^{v})^{−1}). Note that the submean property of (r_{f}^{v})^{−1} would not suffice. In fact, this would imply, on account of (1.7.7), (1.7.8)

\begin{aligned}r_{f}^{v}|_{z_{η}^{1}+B} & ≥ \frac{2πϵ}{(η′)^{−1}(2πϵ − α) + δ^{−1}α} \\ & = η′ \frac{1}{1 − \frac{α}{2πϵ} + \frac{η′α}{δ2πϵ}} = η′(1 + O(α)). \end{aligned} ; (1.7.11)

Thus, the loss from η to η′ = η(1 − O(ϵ^{2})) is not balanced by the gain from η′ to η′(1 + O(α)) and the last term of (1.7.11) is possibly smaller than η regardless of how small η is chosen.

Suggested research. Develop the discussion contained in Remark 1.7.7 about families of discs which sweep out pseudoconvex domains (after having a look at Section 1.8).

Notes. The continuity principle belongs to mathematical folklore and can be found in many basic books such as the one of Vladimirov [96]. It underlies an enormous variety of technical passages in SCV (several complex variables). However, for a clever example of how “not everything is doable with it” see the counterexample by Fornaess [38]. Notice also that it becomes dangerous when “monodromy” is missing. The proof that we propose here is a byproduct of the preparation of the propagation theorem. The latter is due to Hanges and Treves [47] where it is obtained, instead, by means of microlocal techniques. Our proof, fully geometric, is new.

1.8. Domains of holomorphy and pseudoconvex domains

Summary. We have already encountered in Section 1.3 domains where all holomorphic functions extend to a larger set. We call a domain for which this does not happen a domain of holomorphy. It turns out that this is, in fact, the domain of a single holomorphic function, a function which does not have analytic continuation anywhere across the boundary. Next, we define holomorphically convex domains and prove their equivalence with the domains of holomorphy. A more explicit convexity property, pseudoconvexity, is introduced and its equivalence with Levi convexity of the boundary is proved. Also, holomorphic convexity implies pseudoconvexity, essentially by the use of the analytic tools prepared in Section 1.7. But the inverse implication, the solution of the so-called Levi problem, is only stated here whereas its proof is left to Section 1.9.

Our main task is to characterize the “domains of holomorphic functions”. To introduce this notion properly, we need new terminology. We say that a function f ∈ hol(Ω) does not extend beyond ∂Ω at z_{o} ∈ ∂Ω (or is critical at z_{o}) when for any neighborhood B_{z_{o}} of z_{o} there is no function f̃ ∈ hol(B_{z_{o}}) which induces f in some open component of B_{z_{o}} ∩ Ω. By abuse of terminology, we say in this situation that f has no “extension” f̃ at z_{o}.

Definition 1.8.1. A domain Ω ⊂ ℂ^{n} is said to be a domain of holomorphy if for any boundary point z_{o} ∈ ∂Ω there exists f_{z_{o}} ∈ hol(Ω) which does not extend beyond ∂Ω at z_{o}.

Any domain Ω of ℂ is a domain of holomorphy (having f_{z_{o}} = \frac{1}{z−z_{o}} as critical functions at the various z_{o} ∈ ∂Ω). In ℂ^{n}, n > 1, according to the conclusions of Sections 1.3 and 1.6, not all domains enjoy the property of Definition 1.8.1. A striking phenomenon is that in a domain of holomorphy there is indeed a single function f ∈ hol(Ω) which is critical at any boundary point; thus, Ω is “the domain” of f. Along with the discussion on various types of non-extendibility, there naturally arises the question if Ω always admits a “holomorphy hull” that is a domain of holomorphy Ω̃ ⊃ Ω such that hol(Ω) −−→^{ext} hol(Ω̃). Thus, holomorphic functions extend from Ω to Ω̃ but some of them not beyond ∂Ω̃. This is true if we allow Ω̃ to “go out” of ℂ^{n} on manifolds spread over ℂ^{n}; we refer to [50] and [82] for this point.

The property of being a domain of holomorphy concerns the local non-extendibility at the boundary points z_{o} ∈ ∂Ω but it involves global sections f_{z_{o}} ∈ hol(Ω). In particular, it is not clear whether it is local: it holds equivalently for Ω or for the various Ω ∩ B where {B} is a covering of ∂Ω. The point is that non-extendibility for sections on Ω∩B is easier than for Ω. But they eventually turn out to be equivalent (cf. Proposition 1.8.14 and the comments which follow).

We now want to get better insight into the property expressed by Definition 1.8.1. This requires some preparation.

Definition 1.8.2. If K ⊂⊂ Ω, we define the “hol(Ω)-hull” of K by

K̂_{Ω} = {z ∈ Ω : |f(z)| ≤ sup_{K} |f| for any f ∈ hol(Ω)}.

If Ω_{1} ⊂ Ω_{2}, then K̂_{Ω_{1}} ⊂ K̂_{Ω_{2}}; also, it is easy to check that regardless of how Ω is chosen, the set K̂_{Ω} is in any case contained in the convex hull that we denote K̂. In fact, let K be contained in H_{z_{o},ζ}, the halfspace through z_{o} with conormal ζ:

K ⊂ {z : Re ⟨z − z_{o}, ζ⟩ ≤ 0}.

Here ⟨·, ·⟩ is the hermitian product in ℂ^{n} defined by ⟨z, ζ⟩ = ∑_{j} z_{j}ζ̄_{j}. Since we can consider among all possible holomorphic functions f the entire functions f(z) = e^{⟨z−z_{o},ζ⟩}, we then have just by our definition of hull

sup_{K̂_{Ω}} |e^{⟨z−z_{o},ζ⟩}| = sup_{K} e^{Re ⟨z−z_{o},ζ⟩}
≤ 1.

Thus also K̂_{Ω} ⊂ H_{z_{o},ζ}. Since K̂ is an intersection of half-spaces H_{z_{o},ζ}, the inclusion K̂_{Ω} ⊂ K̂ follows. Fix a function f ∈ hol(Ω) and a direction v ∈ ℂ^{n}; recall the function r_{f}^{v} defined in (1.7.1). Despite the fact that K̂_{Ω} may be bigger than K, we have the following anyhow.

Lemma 1.8.3.

(1.8.1)  inf_{K} r_{f}^{v} = inf_{K̂_{Ω}} r_{f}^{v}.

Proof. “≥” is obvious. As for “≤”, let r < inf_{K} r_{f}^{v}. We then have

(1.8.2)  |∂_{v}^{ν}f(w)| < ν!r^{−ν} for any w ∈ K and ν ≥ ν_{o}.

But by definition of K̂_{Ω} and on account of the fact that any ∂_{v}^{ν}f ∈ hol(Ω), we have that (1.8.2) holds indeed for any w ∈ K̂_{Ω}. This implies

r < inf_{K̂_{Ω}} r_{f}^{v}.

□

Lemma 1.8.4. Let Ω be a domain of holomorphy. Then

d(z, ∂Ω) = inf_{v∈S^{2n−1}, f∈hol(Ω)}r_{f}^{v}(z).

Proof. “≤” follows from Cauchy’s inequalities. Conversely, Ω being a domain of holomorphy, if r > d(z, ∂Ω), there must exist f ∈ hol(Ω) such that f ∉ hol(𝔹^{n}(z, r)). By Proposition 1.7.4 there must then exist v such that r_{f}^{v}(z) < r.

□

The set K̂_{Ω} is always closed in Ω but need not be compact in Ω. However, this is the case when Ω is a domain of holomorphy. More precisely, if d denotes the euclidean distance, then

Theorem 1.8.5. Let Ω be a domain of holomorphy and K ⊂⊂ Ω be a compact subset. Then

d(K, ∂Ω) = d(K̂_{Ω}, ∂Ω).

Proof. We apply Lemmas 1.8.3 and 1.8.4 and get

d(K, ∂Ω) = inf_{z∈K} d(z, ∂Ω)

= inf_{z∈K} inf_{v, f} r_{f}^{v}(z)

= inf_{z∈K̂_{Ω}} inf_{v, f} r_{f}^{v}(z)

= d(K̂_{Ω}, ∂Ω).

□

Definition 1.8.6. A domain Ω is said to be holomorphically convex when for any K compact in Ω, the set K̂_{Ω} is still compact in Ω.

In particular, Theorem 1.8.5 says that a domain of holomorphy is holomorphically convex. Also, since K̂_{Ω} is controlled by the convex hull K̂, it follows that a convex domain is holomorphically convex. Holomorphic convexity is stable under intersection.

Proposition 1.8.7. If Ω_{j} is a domain of holomorphy for any j, then the interior Ω of ⋂_{j} Ω_{j} is holomorphically convex.

Proof. We have

\begin{aligned}d(K̂_{Ω}, ∂Ω_{j}) & ≥ d(K̂_{Ω_{j}}, ∂Ω_{j}) \\ & = d(K, ∂Ω_{j}), \end{aligned} ; (1.8.3)

where the first inequality follows from the inclusion K̂_{Ω_{j}} ⊃ K̂_{Ω} and the second equality is a consequence of Theorem 1.8.5. If we take “inf_{j}” in

(1.8.3) we conclude

(1.8.4)  d(K̂_{Ω}, ∂Ω) = d(K, ∂Ω).
□

Theorem 1.8.8. The following are equivalent:

(i) Ω is holomorphically convex. (ii) Ω is a domain of holomorphy. (iii) There is f ∈ hol(Ω) which does not extend holomorphically to

any larger set Ω̃ ⊃ Ω.

Proof. (iii) ⇒ (ii) is obvious and (ii) ⇒ (i) has already been proved in Theorem 1.8.5. We prove now that (i) ⇒ (iii). In fact, let P be a polydisc with center at 0 and denote by P_{z} for z ∈ Ω the largest “P -shaped” polydisc centered at z and contained in Ω. Thus, P_{z} = z + rP for a suitable r > 0 and P̄_{z} hits ∂Ω at some point. Let T ⊂ Ω be countable dense: we want to construct f ∈ hol(Ω) which cannot be continued to a neighborhood of P̄_{z} for any z ∈ T. Let z_{1}, z_{2}... be a sequence in T repeating each term in T infinitely many times and K^{1} ⊂⊂ K^{2}... a sequence of compact sets of Ω such that any compact subset of Ω belongs to some of the K^{j}’s. Since K̂^{j}_{Ω} ⊂⊂ Ω, then there is ζ_{j} ∈ P_{z_{j}} such that ζ_{j} ∉ K̂^{j}_{Ω}. In particular, there is f_{j} ∈ hol(Ω) such that sup_{K^{j}} |f_{j}| < |f_{j}(ζ_{j})|. By replacing f_{j} by its powers, we may assume

f_{j}(ζ_{j}) = 1, sup_{K^{j}} |f_{j}| < 2^{−j}.

We can also suppose that the f_{j}’s are not ≡ 1 in any component of

Ω. Define f := Π^{+∞}_{j=1}(1 − f_{j})^{j}; the product converges uniformly on each

K^{l} since ∑ j2^{−j} < +∞ and defines a holomorphic function f on Ω. This function has a zero of order j at ζ_{j}. It cannot therefore have continuation to a neighborhood of any P̄_{z}’s for z ∈ T because otherwise it would have there a zero of infinite order and hence be identically 0 in a component of Ω. The zero of infinite order would be taken at any subsequential limit of ζ_{j_{k}} if z_{j_{k}} is the sequence which repeats z infinitely many times. These limits are dense in ∂Ω since T is dense.

□

Remark 1.8.9. Proposition 1.8.7 in combination with Theorem 1.8.8 ensures that the interior of the intersection of domains of holomorphy is a domain of holomorphy. In particular, if Ω is a domain of holomorphy, then for any boundary point z_{o} there is a system of neighborhoods {B} such that B ∩ Ω is a domain of holomorphy. (For this we just have to take a system of balls B.) This answers the easy half of the question about localizability of the property.

Example 1.8.10. Let Ω be a “holomorphic polyhedron” in the form Ω = {z : |f_{j}| < 1} for a family of holomorphic functions f_{j} ∈ hol(ℂ^{n}), j = 1, ..., n; then Ω is a domain of holomorphy. In fact, let K ⊂⊂ Ω; we must have |f_{j}||_{K} ≤ 1 − ϵ for any j and for suitable ϵ. This inequality remains true in K̂_{ℂ^{n}} which is therefore compact in Ω. Hence K̂_{Ω} is also compact because K̂_{Ω} ⊂ K̂_{ℂ^{n}}.

The example has the following generalization.

Proposition 1.8.11. Let Ω and Ω′ be domains of holomorphy and F : Ω → ℂ^{n} a holomorphic mapping. Then F^{−1}(Ω′) is a domain of holomorphy.

Proof. For K ⊂⊂ F^{−1}(Ω′) we have

(1.8.5)  K̂_{F^{−1}(Ω′)} ⊂ F^{−1}(F̂ K_{Ω′}),

because F^{∗}(hol(Ω′)) ⊂ hol(F^{−1}(Ω′)) (where F^{∗} denotes the pull-back by F). Now, F̂ K_{Ω′} is compact in Ω′, hence closed in ℂ^{n} and thus F^{−1}(F̂ K_{Ω′}) is closed in Ω. In combination with (1.8.5) this implies that K̂_{F^{−1}(Ω′)}, which is closed in F^{−1}(Ω′), is in fact closed in Ω. But then, since

K̂_{F^{−1}(Ω′)} ⊂ K̂_{Ω} ⊂⊂ Ω,

we conclude that K̂_{F^{−1}(Ω′)} ⊂⊂ F^{−1}(Ω′).

□

Notice that Example 1.8.10 is obtained from Proposition 1.8.11 under the choice Ω′ = Δ × ... × Δ, F = (f_{1}, ..., f_{n}). Notice also that when F is proper, Ω is automatically a domain of holomorphy.

We remember that an upper semicontinuous function is plurisubharmonic when the restriction to any disc is subharmonic; we denote by d(z) the distance to ∂Ω. We go on to define the most crucial notion in several complex variables.

Definition 1.8.12. A domain Ω is said to be pseudoconvex when − log d is plurisubharmonic.

Every domain of ℂ is pseudoconvex. Also, since plurisubharmonicity is stable under “sup”, then pseudoconvexity is stable under inter-

section: for a family {Ω_{j}}_{j} of pseudoconvex domains, we have that ⋂_{j}Ω_{j}^{⌢}^{∘}

is pseudoconvex. Since balls, and more generally convex sets, are pseudoconvex (cf. Theorem 1.8.13), any pseudoconvex domain is “locally pseudoconvex”. The opposite is also true and will be proved in Proposition 1.8.14. Before the statement we need to define the notion of an “exhaustion” function, that is, a function ψ : Ω → ℝ such that the sets {z : ψ(z) ≤ c} for c ↗ +∞ are a fundamental system of compact subsets of Ω. We also need to define the cohomology of the ¯∂ system on Ω. Let

(1.8.6)  ...C^{∞}(Ω)^{k−1} →^{¯∂^{k−1}} C^{∞}(Ω)^{k} →^{¯∂^{k}} C^{∞}(Ω)^{k+1}...

be the ¯∂-complex acting on antiholomorphic forms f = \begin{matrix}∑ & ′ \\ |J|=k \end{matrix}f_{J}dz̄_{J}

where the suffix ′ denotes ordered multiindices J = j_{1} < ... < j_{k} and dz̄_{J} stands for dz̄_{j_{1}} ∧ ... ∧ dz̄_{j_{n}}. Here the coefficients f_{J} are in C^{∞}(Ω)

and the action of ¯∂ is defined via ¯∂f = ∑_{j=1,...,n} \begin{matrix}∑ & ′ \\ |J|=k \end{matrix}∂z̄_{j}f_{J}dz̄_{j} ∧ dz̄_{J}; in

particular, ¯∂^{k} ∘ ¯∂^{k−1} = 0. We denote the cohomology of (1.8.6) by

K^{k}(Ω) := ker ¯∂^{k} ; range ¯∂^{k−1} .

The complete characterization of pseudoconvexity is contained in the following theorem, which is referred to as the “solution of the Levi problem”. Note that in this section we prove the full characterization apart from the fact that pseudoconvexity implies K^{k}(Ω) = 0, k ≥ 1 ((ii) ⇒ (iii) in the next statement) which will be the content of Section 1.9.

Theorem 1.8.13. The following are equivalent:

(i) Ω is pseudoconvex.

(ii) There is an exhaustion function ψ of Ω which is continuous

and plurisubharmonic. (iii) K^{k}(Ω) = 0, k ≥ 1. (iv) Ω is a domain of holomorphy.

Proof. (i) ⇒ (ii): If Ω is bounded, we just have to set ψ := − log d and otherwise, ψ := − log d + λ|z|^{2}, which is continuous plurisubharmonic and exhaustive.

(ii) ⇒ (iii): This is the most difficult part that will be proved in the next section.

(iii) ⇒ (iv): We proceed by induction over the dimension n and use the hypothesis in three occurrences. We fix a boundary point z_{o} ∈ ∂Ω, say z_{o} = 0, and suppose that Ω has non-empty intersection ω := Ω∩{z : z_{n} = 0}. Let j : ω ↪ Ω and π : ℂ^{n} ↠ {z : z_{n} = 0} be the natural injection and projection, respectively. We first prove that any k-form f in ω which is ¯∂-closed, that is, satisfies ¯∂f = 0, can be “lifted” to a ¯∂-closed form f̃ on Ω: this means ¯∂f̃ = 0 and j^{∗}f̃ = f. To prove it, we note that ω and Ω \ π^{−1}(ω) are relatively closed in Ω and take χ ∈ C^{∞}(Ω) which is 1 at ω and 0 at Ω \ π^{−1}(ω); thus, χπ^{∗}f is well defined on Ω and induces f on ω. It is not necessarily ¯∂-closed; to obtain it, we have to use a “correction” term of type z_{n}v for a solution v of ¯∂v = \frac{¯∂χ∧π^{∗}f}{z_{n}} and set

f̃:= χπ^{∗}f − z_{n}v.

The existence of the solution v to the above equation is assured by the vanishing hypothesis of the cohomology groups ( for degree k + 1 if f has degree k). Thus, f can be “lifted” to f̃ ¯∂-closed. Also, ω inherits from Ω the vanishing of the cohomology:

(1.8.7)  K^{k}(ω) = 0, k ≥ 1.

In fact, if f satisfies ¯∂f = 0 on ω, we first lift it to Ω, then solve ¯∂ũ = f̃ on Ω and finally define u := j^{∗}ũ. This is the solution of ¯∂u = f on ω.

By the inductive assumption, (1.8.7) implies that ω is a domain of holomorphy (which is trivially true at the first inductive step ω ⊂ ℂ^{1}) and therefore there is a function f ∈ hol(ω) which does not extend at z_{o}. Next, using again that K^{1}(Ω) = 0, we can find f̃ ∈ hol(Ω) such that f̃|_{ω} = f; hence f̃ cannot extend holomorphically and Ω is a domain of holomorphy.

(iv) ⇒ (i): In a domain of holomorphy Ω we have (Lemma 1.8.4)

d(z) = inf_{v, f} r_{f}^{v}(z), z ∈ Ω.

Thus − log d = sup_{v, f}(− log r_{f}^{v}); now, the functions − log r_{f}^{v}’s enjoy the

submean property along discs according to Theorem 1.7.5 and the same is true for their “sup”. Since, in addition, − log d is continuous, then it is plurisubharmonic. The proof of the theorem is complete.

□

Proposition 1.8.14. Let Ω be a domain of ℂ^{n} and suppose that for every point in ∂Ω there is a basis of neighborhoods {B} such that B ∩ Ω satisfies (ii) of Theorem 1.8.13; then Ω also satisfies (ii).

Proof. We denote by ψ_{B∩Ω} the plurisubharmonic exhaustion function on each B ∩ Ω. By taking a locally finite covering of ∂Ω by the B’s and gluing the various ψ_{B∩Ω} by taking their sup over overlappings, we find ψ^{1} plurisubharmonic on a neighborhood Ω_{1} of ∂Ω in Ω, such that ψ^{1} → +∞ as z → ∂Ω, and ψ^{1} is locally bounded at ∂(Ω\Ω_{1}). Let ψ^{2} be a plurisubharmonic function on ℂ^{n} with ψ^{2} → +∞ if |z| → +∞ such that ψ^{2} > ψ^{1} over a neighborhood V of ∂(Ω\Ω_{1}). Define ψ to be ψ^{2} on Ω_{2} := (Ω \ Ω_{1}) ∪ V and sup(ψ^{1}, ψ^{2}) on Ω_{1}; ψ is an exhaustion function for Ω and it is plurisubharmonic because this property is stable under “sup” over finite indices.

□

Proposition 1.8.14 reveals that pseudoconvexity is a local property. In combination with Theorem 1.8.13, it shows that the property of being a domain of holomorphy is also local. Moreover, it implies a version of this property in terms of germs.

Remark 1.8.15. If Ω is a domain of holomorphy, then for any z_{o} ∈ ∂Ω there does not exist a forced extension map for germs of holomorphic functions

(1.8.8)  hol(Ω)_{z_{o}} −−→^{ext} hol(ℂ^{n})_{z_{o}}.

This means that there is a neighborhood B of z_{o} and a holomorphic function f ∈ hol(Ω ∩ B) (indeed f ∈ hol(Ω)), such that f ∉ hol(B′) for any B′ ⊂ B. It is an open question whether the opposite implication holds, that is, whether non-extendibility passes from germs to global sections. Of course the question makes sense only if it is formulated for any boundary point z_{o} ∈ ∂Ω. In fact, for a single point z_{o} we can well have extension for global holomorphic functions but not for germs. For instance, if Ω is defined in ℂ^{2} by y_{2} > φ(y_{1}) where φ ≡ 0 for |y_{1}| < 1 and φ = (|y_{1}| − 1)^{2} for |y_{1}| ≥ 1, then hol(Ω) −−→^{ext} hol(ℂ^{2}) holds but hol(Ω)_{0} −−→^{ext} hol(ℂ^{2})_{0} does not.

We come back to Theorem 1.8.13. We have proved, in conjunction with the conclusions of Theorem 1.8.13, that if there exists an exhaustion plurisubharmonic function ψ for Ω, then − log d is also plurisubharmonic. This is a deep point in which we use the full strength of the characterization. A direct but not simple proof could be given along the lines of [50, Theorem 2.3.7, p. 46]. Note that in the next geometric characterization and for ∂Ω of class C^{2} we deal in fact with − log(−r) where r is a defining function for ∂Ω with r < 0 on Ω; but it is clear that this function could replace − log d in Theorem 1.8.13. We also need an extra amount of Levi positivity which is provided by an additional term λ|z|^{2} for large λ.

Let Ω be a domain with C^{2} boundary defined by r < 0 with ∂r ≠ 0 at points where r = 0, set M = ∂Ω, let L_{M} = L_{r}|_{T^{ℂ}M} be the Levi form of M and let s_{M}^{+}, s_{M}^{−}, s_{M}^{0} be the number of its eigenvalues which are > 0, < 0, = 0, respectively. We describe M, in a neighborhood of a boundary point z_{o}, as a graph y_{1} = f(x_{1}, z′, z̄′), set r = −y_{1} + f, and denote by z ↦ z^{∗} the projection on M along the y_{1}-axis. The following relates, in a neighborhood of z_{o}, the signature of the Levi form of the exhaustion function ψ = − log(−r) + λ|z|^{2} for suitable λ to the signature of the Levi form of the boundary L_{M}.

Theorem 1.8.16. We have

(1.8.9)  \begin{cases}s_{ψ}^{−}(z) = s_{M}^{−}(z^{∗}), \\ s_{ψ}^{+}(z) = s_{M}^{+}(z^{∗}) + s_{M}^{0}(z^{∗}) + 1. \end{cases}

Hence, in particular, s_{ψ}^{0} ≡ 0.

All degeneracy of L_{M} has turned into positivity of L_{ψ} with an extra gain of 1 because of the jump of dimension from M to ℂ^{n}.

Proof. To relate L_{ψ}(z) to L_{M}(z^{∗}), we first note that for r = −y_{1}+ f we have

(1.8.10)  \begin{cases}(∂_{z_{i}}∂z̄_{j}r(z))_{ij} = (∂_{z_{i}}∂z̄_{j}r(z^{∗}))_{ij}, \\ ∂r^{⊥}(z) = ∂r^{⊥}(z^{∗}). \end{cases}

We then remark that

(1.8.11) ∑_{ij}∂_{z_{i}}∂z̄_{j}ψdz_{i} ⊗ dz̄_{j} = |r|^{−1}∑_{ij}∂_{z_{i}}∂z̄_{j}rdz_{i} ⊗ dz̄_{j}

+ |r|^{−2}∂r ⊗ ¯∂r + λ∑_{j}dz_{j} ⊗ dz̄_{j}.

For a vector w ∈ ℂ^{n} and for a point z^{∗} ∈ M we decompose w = (w^{τ}, w^{υ}) where w^{τ} and w^{υ} are the tangential and normal components to M at z^{∗}, respectively. We have

L_{ψ}(w, w̄) ≥ |r|^{−1}L_{r}(w^{τ}, w̄^{τ}) (1.8.12)
+ [(|r|^{−2} − c|r|^{−1})|w^{υ}|^{2} − c|r|^{−1}|w^{τ}||w^{υ}| + λ|w|^{2}] ,

where c is a bound for the second derivatives of r. Then for suitable λ = λ_{c} we have

(1.8.13)  L_{ψ}(w, w̄) ≥ |r|^{−1}L_{r}(w^{τ}, w̄^{τ}) + \frac{λ}{2}|w|^{2}.

This implies s_{ψ}^{−}(z) ≤ s_{M}^{−}(z^{∗}) whereas the opposite is obvious: thus, the first of (1.8.9) is proved. The second also follows immediately from (1.8.13).

□

In particular, we can relate the pseudoconvexity of a domain Ω, that is, the positivity of the Levi form of an exhaustion function of Ω, to the pseudoconvexity of its boundary, that is, the positivity of L_{M}. Since the exhaustion function is now meant to be global, we have to modify a little the argument of Theorem 1.8.16 which has only a local character: for this, we replace −r by d, the Eucledian distance to ∂Ω.

If z ↦ z^{∗}, Ω → ∂Ω is now the orthogonal projection (the projection along the vector field ∂d), the equality of ∂ ¯∂r at z and z^{∗} is converted into

(1.8.14)  ∂ ¯∂d(z)|_{∂d^{⊥}(z)} ≤ ∂ ¯∂d(z^{∗})|_{∂d^{⊥}(z^{∗})}.

In fact, the inclusion {z ∈ Ω : d(z) > ϵ} ⊂ {z ∈ Ω : r < −ϵ}, forces inequality for real hessians in real tangential directions hess d(z) ≤ hess(−r)(z^{∗}). Use of the inequality for u and iu with u ∈ ∂d^{⊥} and application of Proposition 1.6.3 yields (1.8.14). Thus, L_{M} ≥ 0 implies that − log d+λ|z|^{2} is plurisubharmonic on a neighborhood of ∂Ω inside Ω; notice that, in contrast to − log(−r) + λ|z|^{2}, this is defined globally and not only on a local patch. We can say better

Corollary 1.8.17. We have L_{M} ≥ 0 if and only if L_{− log d} ≥ 0. In other words, pseudoconvexity of M and Ω are equivalent.

Proof. The argument above shows that − log d + λ|z|^{2} is plurisubharmonic over an annulus in the interior side of the boundary. This can be continued to a plurisubharmonic exhaustion function to the whole Ω (cf. Proposition 1.8.14). Thus Ω is pseudoconvex (cf. Theorem 1.8.13). Conversely, if Ω is pseudoconvex and hence a domain of holomorphy, then L_{M} cannot take any negative eigenvalue (cf. 1.6.8).

□

In the terms of the Corollary, pseudoconvexity reveals, at least for C^{2} boundaries, its fully local character.

Remark 1.8.18. The relation between s_{ψ}^{+}, s_{ψ}^{−}, s_{ψ}^{0} and s_{M}^{+}, s_{M}^{−}, s_{M}^{0} in Theorem 1.8.16 does not depend on the choice of r but could depend on that of λ (and of the specific structure of the exhaustion function as ψ = − log(−r) + λ|z|^{2}). However, if L_{ψ} ≥ 0 for ψ = − log(d) + λ|z|^{2}, then it is also ≥ 0 for ψ = − log d. Therefore, the term λ|z|^{2} may eventually be discarded; but it did a great job in the course of the proof of Theorem 1.8.16 and Corollary 1.8.17.

Corollary 1.8.19. Let Ω be a domain of ℂ^{n} with C^{2} boundary; the following two conditions are equivalent:

(i) Ω is pseudoconvex.

(ii) There is no analytic disc A such that

(1.8.15)  ∂A ⊂⊂ Ω, A ∩ ∂Ω ≠ ∅.

Proof. (ii) ⇒ (i): If Ω has a boundary point, say z_{o} = 0, where s^{−} > 0, then by putting the equation of ∂Ω in canonical form (1.6.6), that is, y_{1} = −|z_{2}|^{2} +O^{2}(z_{3}, ...)+o^{2}(x_{1}, z_{2}, z_{3}, ...), we have that the disc {0} × Δ_{ϵ} × {0}... contradicts (1.8.15).

(i) ⇒ (ii): Let A be a disc which satisfies (1.8.15). Assume without loss of generality that the point where it touches ∂Ω is its center A(0) and that it coincides with the origin 0. Then the family of translated discs {A_{z}} for A_{z} = z + A, |z| < ϵ, satisfies (1.6.8) and provides an extension of holomorphic functions from Ω to a neighborhood of 0. Thus Ω is not a domain of holomorphy, or equivalently, it is not pseudoconvex.

□

In ℝ^{n}, a domain Ω is convex if and only if there is no line segment whose boundary is in Ω but not all points in between. Hence, the characterization of Corollary 1.8.19 is the perfect complex-invariant analogous to this condition with the line segments replaced by discs.

Exercise 1.8.20. Prove that a domain Ω ⊂ ℂ^{n} with C^{2} boundary is pseudoconvex if and only if the slice with every 2-dimensional complex plane is pseudoconvex. Hint: use the characterization of Corollary 1.8.17.

Try a different argument which sustains the case of general boundary (not known).

Suggested research. (1) For a pseudoconvex domain Ω ⊂ ℂ^{n}, is the function − log(−r) plurisubharmonic even without the additional term λ|z|^{2}? We know that this is true for − log d.

(2) (Generalization of (1)) For a general domain Ω ⊂ ℂ^{n}, is the conclusion of Theorem 1.8.16 true for ψ = − log(−r) + λ|z|^{2} replaced by − log(−r) or − log d?

(3) When ∂Ω ∈ C^{2}, the absence of an extension map (1.8.8) for germs of holomorphic functions at any z_{o} ∈ ∂Ω assures that Ω is pseudoconvex (cf. the proof of Corollary 1.8.19). Is the same conclusion true if ∂Ω is not C^{2}?

1.9. L^{2}-ESTIMATES FOR ¯∂ 51

Notes. The characterization of the domains of holomorphy as holomorphically convex, stated in Theorem 1.8.8, is due to Cartan and Thullen [31]. That domains of holomorphy are pseudoconvex was essentially found by Levi and Hartogs. We also refer to the fundamental work by Oka contained in the papers [73]. The presentation that we follow here is the one of the book by Hörmander [50] revisited by the method of Section 1.7. The link between the pseudoconvexity of a domain and its boundary, Theorem 1.8.16, follows the approach of Zampieri [101] (cf. also Hörmander [50, Theorem 2.6.12] for another proof).

1.9. L^{2}-estimates for ¯∂ on q-pseudoconvex domains of ℂ^{n}

Summary. We study the existence on a bounded domain Ω ⊂ ℂ^{n} of solutions to the Cauchy-Riemann equations

(1.9.1)  ¯∂u = f for f satisfying ¯∂f = 0,

where the datum f and the solution u are (0, k)- and (0, k − 1)-forms, respectively. We prove solvability of (1.9.1) in C^{∞}(Ω̄) for all k ≥ q + 1 when Ω is bounded, has smooth boundary, and is “q-pseudoconvex”. If Ω is unbounded or does not have C^{∞} boundary, but can be exhausted by bounded, C^{∞}, q-pseudoconvex domains Ω_{ν} ↗ Ω, then (1.9.1) can be solved in C^{∞}(Ω) in corresponding degrees k. In particular, all pseudoconvex domains have this exhaustion and this implies the vanishing of all cohomology groups of ¯∂ in degree k ≥ 1 (the implication (ii) ⇒ (iii) of Theorem 1.8.13, that is, the solution of the Levi problem).

The key point in solving (1.9.1) is to also call into play the adjoint operator ¯∂^{∗}: in particular, we take full advantage of the ellipticity in the interior of the system (¯∂, ¯∂^{∗}).

To study equation (1.9.1), the most effective tool is the Hilbert space techniques in the context of the ¯∂-Neumann problem. To exploit them, we first establish the a priori estimates of Morrey, Kohn and Hörmander in the weighted L^{2} spaces. By a suitable choice of the weight, these ensure the existence of the ¯∂-Neumann operator and its continuity in the Sobolev spaces H^{s}. The C^{∞} regularity up to the boundary of ¯∂ follows by an approximation argument.

Weighted H^{0} estimates for the (¯∂, ¯∂^{∗})-system. Let T^{∗}ℂ^{n} be the cotangent bundle to ℂ^{n}; we decompose forms of the fiberwise complexified bundle ℂ ⊗_{ℝ} T^{∗}ℂ^{n} into (1, 0)- and (0, 1)-forms and denote by ℂ⊗_{ℝ}T^{∗}ℂ^{n} = (T^{1,0}ℂ^{n})^{∗} ⊕(T^{0,1}ℂ^{n})^{∗} this decomposition. We take an orthonormal basis ω_{1}, ..., ω_{n} of (1, 0)-forms and the dual basis ∂_{ω_{1}}, ..., ∂_{ω_{n}} of (1, 0) vector fields. We write (0, k)-forms (or k-forms for short) as

u = \begin{matrix}∑ & ′ \\ |J|=k \end{matrix}u_{J} ω̄_{J} where ω̄_{J} := ω̄_{j_{1}} ∧ ... ∧ ω̄_{j_{k}}; here J = j_{1} < ... < j_{k}

is a multiindex with ordered entries and ∑′ denotes summation over ordered indices. When the multiindices are no more ordered, then they are assumed to be alternant: if J decomposes into jK, then u_{jK} := sign \binom{J}{jK}u_{J}. We define |u|^{2} := \begin{matrix}∑ & ′ \\ |J|=k \end{matrix}|u_{J}|^{2}; this norm is inde-

pendent of the choice of the orthonormal basis. We consider a bounded domain Ω ⊂⊂ ℂ^{n} with boundary ∂Ω of class C^{∞} defined by r(z, z̄) < 0 with ∂r ≠ 0 over the points where r = 0. We choose our orthonormal frame of (1, 0)-forms such that ∂_{ω_{n}}r = 1 (and for this we need possibly to renormalize r) and ∂_{ω_{j}}r = 0 for any j ≤ n − 1; hence ∂_{ω_{j}}, j ≤ n − 1, are a basis for T^{1,0}∂Ω. The coefficients of our forms are taken in various spaces Λ such as C^{∞}(Ω̄), C^{∞}(Ω), C_{c}^{∞}(Ω), L^{2}(Ω), H^{s}(Ω) and the corresponding spaces of k-forms are denoted by Λ^{k}. Though our a priori estimates are proved over smooth forms, they are stated in Hilbert norms. Thus, let ||u||_{H^{0}} be the H^{0} = L^{2}-norm and, for a real function φ, let the weighted L^{2}-norm be defined by

(1.9.2)  ||u||_{H_{φ}^{0}}^{2} := \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∫_{Ω} e^{−φ}|u_{J}|^{2}dV,

where dV is the element of volume in ℂ^{n}. We begin by noticing that ¯∂ is a closed densely defined operator on (H_{φ}^{0})^{k}. Moreover, smooth forms certainly belong to the domain D ¯∂ of ¯∂ and

(1.9.3) ¯∂u = \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix} ∑_{i<j^{ij=1,...,n}}(∂ω̄_{i}u_{jK} − ∂ω̄_{j}u_{iK})ω̄_{i} ∧ ω̄_{j} ∧ ω̄_{K} + ...,

where the dots denote terms in which no differentiation of u occurs. These error terms just come from differentiation of the coefficients of the vector fields ∂_{ω_{j}}.

We denote by δ_{ω_{j}} the formal H_{φ}^{0}-adjoint of −∂ω̄_{j}. (“Formal adjoint” means adjoint in the H_{φ}^{0} scalar product restricted to compactly supported test functions.) We have

δ_{ω_{j}} = ∂_{ω_{j}} − φ_{j} + ...,

where φ_{j} stands for ∂_{ω_{j}}φ and the dots denote a 0-order operator which does not involve φ. We denote by ¯∂_{φ}^{∗} the H_{φ}^{0}-adjoint of ¯∂. The operator ¯∂_{φ}^{∗} is still closed, densely defined but it is no longer true that smooth forms belong to D ¯∂_{φ}^{∗}. For this, they must satisfy certain boundary conditions. Namely, there must exist g ∈ (H_{φ}^{0})^{k−1} such that

(1.9.4)  ⟨u, ¯∂ψ⟩ = ⟨g, ψ⟩ for any ψ ∈ (H_{φ}^{0})^{k−1}.

If this is the case, one sets ¯∂_{φ}^{∗}u := g. But integration by parts shows that (1.9.4) cannot hold unless

∑_{j=1,...,n} ∫_{∂Ω} e^{−φ}∂_{ω_{j}}(r)u_{jK} ψ̄_{K}dS = 0 for any K and ψ_{K},

that is, ∑_{j=1,...,n}∂_{ω_{j}}(r)u_{jK}|_{∂Ω} ≡ 0 for any K. (Here dS is the element of

hypersurface in ∂Ω.) Since we have chosen our basis with the property ∂_{ω_{j}}r|_{∂Ω} = κ_{jn}, we then conclude

(1.9.5)  u belongs to D ¯∂_{φ}^{∗} iff u_{J}|_{∂Ω} = 0 whenever n ∈ J.

As a set, H_{φ}^{0} coincides with H^{0}; from (1.9.5) it also follows that D ¯∂_{φ}^{∗} = D ¯∂^{∗}. We call a form which belongs to D ¯∂^{∗} “tangential”. Over such a form the action of the Hilbert adjoint of ¯∂ coincides with that of its “formal adjoint” and is therefore expressed by a “divergence operator”:

(1.9.6)  ¯∂_{φ}^{∗}u = −\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{j}δ_{ω_{j}}(u_{jK})ω̄_{K} + ... for any u ∈ D ¯∂^{∗},

where the dots denote an error term in which u is not differentiated and φ does not occur. When taking the restriction of (1.9.6) to ∂Ω, the index j does not take value n because, in that case, the coefficient u_{nK} is missing. We leave unspecified in what follows the indices such as i, j, meaning that they run from 1 to n when we are in the interior of Ω and from 1 to n − 1 over the boundary ∂Ω. We get from (1.9.6) and (1.9.3)

(1.9.7)  \begin{cases}||¯∂_{φ}^{∗}u||_{H_{φ}^{0}}^{2} = \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij} ∫_{Ω}e^{−φ}δ_{ω_{i}}u_{iK} \overline{δ_{ω_{j}}u_{jK}}dV + R, \\ ||¯∂u||_{H_{φ}^{0}}^{2} = −\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij} ∫_{Ω}e^{−φ}∂ω̄_{j}u_{iK} \overline{∂ω̄_{i}u_{jK}}dV \\ + \begin{matrix}∑ & ′ \\ |J|=k \end{matrix}∑_{j} ∫_{Ω}e^{−φ}|∂ω̄_{j}u_{J}|^{2}dV + R, \end{cases}

where we denote by R different error terms which involve integration of products of type ∂ω̄_{j}(u_{iK})ū or δ_{ω_{j}}(u_{jK})ū with coefficients independent of φ and only depending on the derivatives of the coefficients of the ω_{j}’s. (We omit indices in the u’s when unnecessary.) The key technical result is contained in the following proposition.

Proposition 1.9.1 (Hörmander, Kohn and Morrey). Let u ∈ C^{∞}(Ω̄)^{k} ∩ D ¯∂^{∗} and fix arbitrarily an index q_{o} with 0 ≤ q_{o} ≤ n − 1; then for a suitable C > 0, independent of u and φ,

\begin{matrix}||¯∂u||_{H_{φ}^{0}}^{2} + ||¯∂_{φ}^{∗}u||_{H_{φ}^{0}}^{2} + C||u||_{H_{φ}^{0}}^{2} \\ ≥ +\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij} ∫_{Ω} e^{−φ}φ_{ij}u_{iK} ū_{jK}dV − \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}} ∫_{Ω} e^{−φ}φ_{jj}|u_{J}|^{2}dV \\ + \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij} ∫_{∂Ω} e^{−φ}r_{ij}u_{iK} ū_{jK}dS − \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}} ∫_{∂Ω} e^{−φ}r_{jj}|u_{J}|^{2}dS \\ + (1 − ϵ) (\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}}||δ_{ω_{j}}u_{J}||_{H_{φ}^{0}}^{2} + \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≥q_{o}+1}||∂ω̄_{j}u_{J}||_{H_{φ}^{0}}^{2}) . \end{matrix} (1.9.8)

(Note that there is no relation between k and q_{o} in this statement.)

Proof. We replace the suffix H_{φ}^{0} by φ in all our norms. We recall (1.9.7); we wish to rewrite the integrals δ_{ω_{i}}u_{iK} \overline{δ_{ω_{j}}u_{jK}} and ∂ω̄_{j}u_{iK} \overline{∂ω̄_{i}u_{jK}} in the right sides of the two lines of (1.9.7). By integration by parts we get (1.9.9)

\begin{cases}∫_{Ω} e^{−φ}δ_{ω_{i}}u_{iK} \overline{δ_{ω_{j}}u_{jK}}dV = ∫_{∂Ω} e^{−φ}r̄_{j}δ_{ω_{i}}(u_{iK})ū_{jK}dS \\ − ∫_{Ω} e^{−φ}∂ω̄_{j}δ_{ω_{i}}(u_{iK})ū_{jK}dV + R, \\ − ∫_{Ω} e^{−φ}∂ω̄_{j}u_{iK} \overline{∂ω̄_{i}u_{jK}}dV = − ∫_{∂Ω} e^{−φ}r_{i}∂ω̄_{j}(u_{iK})ū_{jK}dS \\ + ∫_{Ω} e^{−φ}δ_{ω_{i}}∂ω̄_{j}(u_{iK})ū_{jK}dV + R. \end{cases}

Here r_{j} stands for ∂_{ω_{j}}r. Recall that ∂_{ω_{j}}r = ⟨∂_{ω_{j}}, ω_{n}⟩ and this is 0 or 1 according to j < n or j = n. On the other hand, for j = n, it is the coefficient u_{nK} which vanishes when restricted to ∂Ω. We thus conclude that the boundary integrals vanish in both equalities of (1.9.9). Note also that by taking the sum of the two terms in the right side of (1.9.9), after discarding the boundary integrals, we put in evidence the commutators [δ_{ω_{i}}, ∂ω̄_{j}]. In the last sum of the second equality of (1.9.7), we now want to interchange ∂ω̄_{j} with δ_{ω_{j}} for indices j ≤ q_{o}. A double integration by parts yields

(1.9.10) ||∂ω̄_{j}u_{J}||_{φ}^{2} = ||δ_{ω_{j}}u_{J}||_{φ}^{2} − ∫_{Ω} e^{−φ} [δ_{ω_{j}}, ∂ω̄_{j}] u_{J} ū_{J}dV + R,

since, again, the boundary integrals vanish because r_{j}|_{∂Ω} ≡ 0 for j ≤ q_{o}. Thus, the sum of the two right hand sides of (1.9.9) is ∫_{Ω} e^{−φ}[δ_{ω_{i}}, ∂ω̄_{j}](u_{iK})ū_{jK}dV + R. What we have obtained so far is

\begin{aligned}||¯∂u||_{φ}^{2} + ||¯∂_{φ}^{∗}u||_{φ}^{2} & = \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij} ∫_{Ω} e^{−φ}[δ_{ω_{i}}, ∂ω̄_{j}](u_{iK})ū_{jK}dV \\ & − \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}} ∫_{Ω} e^{−φ}[δ_{ω_{j}}, ∂ω̄_{j}](u_{J})ū_{J}dV \\ & + \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} (∑_{j≤q_{o}}||δ_{ω_{j}}u_{J}||_{φ}^{2} + ∑_{j≥q_{o}+1}||∂ω̄_{j}u_{J}||_{φ}^{2}) + R. \end{aligned} (1.9.11)

We have to describe now the commutators which appear in (1.9.11). Recall that ω_{n} = ∂r and hence ∂ω̄_{n} = ∂ ¯∂r. Observe that

(1.9.12)  ∂ ¯∂r ∼ L_{r}

in the sense that

(1.9.13)  ⟨∂ ¯∂r, u ∧ v̄⟩ = L_{r}(u, v̄) for any u and v in T ℂ^{n}.

(Incidentally, we remark that the identification contained in (1.9.13) looks strange because L_{r} is hermitian whereas ∂ ¯∂r is alternant. They coincide just because their action is restricted to (T^{1,0}ℂ^{n} ⊕ {0}) × ({0} ⊕ T^{0,1}ℂ^{n}).) Denote by (r_{ij})_{ij} the matrix of ∂ ¯∂r or L_{r} in the basis ω_{1}, ..., ω_{n}; we wish to relate r_{ij} for j ≤ n − 1 to the component of [∂_{ω_{i}}, ∂ω̄_{j}] along ∂_{ω_{n}}. For this, we recall Cartan’s formula

\begin{aligned}⟨ω_{n}, [∂_{ω_{i}}, ∂ω̄_{j}]⟩ & = −⟨¯∂ω_{n}, ∂_{ω_{i}} ∧ ∂ω̄_{j}⟩ \\ & + ∂_{ω_{i}}(⟨ω_{n}, ∂ω̄_{j}⟩) − ∂ω̄_{j}(⟨ω_{n}, ∂_{ω_{i}}⟩). \end{aligned}

By the orthogonality ⟨ω_{n}, ∂ω̄_{j}⟩ = 0 and ⟨ω_{n}, ∂_{ω_{i}}⟩ = 0, and by the identity ∂ ¯∂ = −¯∂∂, we have

\begin{aligned}⟨ω_{n}, [∂_{ω_{i}}, ∂ω̄_{j}]⟩ & = −⟨¯∂ω_{n}, ∂_{ω_{i}} ∧ ∂ω̄_{j}⟩ \\ & = r_{ij}. \end{aligned}

Similarly

\begin{aligned}⟨ω̄_{n}, [∂_{ω_{i}}, ∂ω̄_{j}]⟩ & = −⟨∂ω̄_{n}, ∂_{ω_{i}} ∧ ∂ω̄_{j}⟩ \\ & = −r_{ij}. \end{aligned}

It follows that

(1.9.14)  [∂_{ω_{i}}, ∂ω̄_{j}] = r_{ij}∂_{ω_{n}} − r_{ij}∂ω̄_{n} + ...,

where the dots denote combinations of ∂_{ω_{h}}’s and ∂ω̄_{h}’s for h ≤ n − 1. We denote by c_{ij}^{h} = ⟨ω_{h}, [∂_{ω_{i}}, ∂ω̄_{j}]⟩(= −⟨¯∂ω_{h}, ∂_{ω_{i}} ∧ ∂ω̄_{j}⟩), resp. d_{ij}^{h} = ⟨ω̄_{h}, [∂_{ω_{i}}, ∂ω̄_{j}](= −⟨∂ω̄_{h}, ∂_{ω_{i}} ∧∂ω̄_{j}⟩), the coefficients with which the ∂_{ω_{h}}’s and ∂ω̄_{h}’s, respectively, occur in the dotted terms of (1.9.14). (Incidentally, they are related by the relation

d_{ij}^{h} = ⟨ω̄_{h}, [∂_{ω_{i}}, ∂ω̄_{j}]⟩
= \overline{⟨ω_{h}, [∂ω̄_{i}, ∂_{ω_{j}}]⟩}
= −\overline{⟨ω_{h}, [∂_{ω_{j}}, ∂ω̄_{i}]⟩} = −\overline{c_{ji}^{h}}.)

Let (φ_{ij})_{ij} be the matrix of L_{φ} in the basis ω_{1}, ..., ω_{n}. We have

\begin{aligned}φ_{ij} & = ∂_{ω_{i}}∂ω̄_{j}(φ) + (∂ω̄_{n})_{ij}φω̄_{n} + ∑^{n−1} ; h=1(∂ω̄_{h})_{ij}φω̄_{h} \\ & = ∂_{ω_{i}}∂ω̄_{j}(φ) + r_{ij}φω̄_{n} − ∑^{n−1} ; h=1d_{ij}^{h}φω̄_{h}. \end{aligned} ; (1.9.15)

It follows that

\begin{aligned}[−φ_{i}, ∂ω̄_{j}] & = ∂ω̄_{j}∂_{ω_{i}}(φ) \\ & = −[∂_{ω_{i}}, ∂ω̄_{j}](φ) + ∂_{ω_{i}}∂ω̄_{j}(φ) \\ & = φ_{ij} − r_{ij}φ_{ω_{n}} − ∑^{n−1} ; h=1c_{ij}^{h}φ_{ω_{h}}. \end{aligned} ; (1.9.16)

Using (1.9.14) together with (1.9.16), we get

\begin{aligned}[δ_{ω_{i}}, ∂ω̄_{j}] & = [∂_{ω_{i}} − φ_{i}, ∂ω̄_{j}] \\ & = φ_{ij} + r_{ij}δ_{ω_{n}} − r_{ij}∂ω̄_{n} + ∑^{n−1} ; h=1c_{ij}^{h}δ_{ω_{h}} + ∑^{n−1} ; h=1d_{ij}^{h}∂ω̄_{h}. \end{aligned} ; (1.9.17)

With (1.9.17) in hand, we wish to rewrite the third line of (1.9.11). What we want is to interchange δ_{ω_{n}} with ∂ω̄_{n} in the expressions for [δ_{ω_{i}}, ∂ω̄_{j}] since the other derivatives, of either type δ or ¯∂, are all error terms, as we will see soon. Now,

\begin{aligned}∫_{Ω} e^{−φ}r_{ij}δ_{ω_{n}}(u_{iK})ū_{jK}dV & = ∫_{∂Ω} e^{−φ}r_{ij}r_{n}u_{iK} ū_{jK}dS \\ & − ∫_{Ω} e^{−φ}r_{ij}u_{iK} \overline{∂ω̄_{n}(u_{jK})}dV, \end{aligned} ; (1.9.18)

where r_{n} = ⟨∂_{ω_{n}}, ∂r⟩ = 1 under our normalization. All terms involving squares of u or products of u by δ_{ω_{j}}u for j ≤ n − 1 or ∂ω̄_{j}u for j ≤ n are denoted by R. Also, to keep track of former errors described after (1.9.7) coming from ||¯∂^{∗}u||_{φ}^{2}, we include into R integrals of products δ_{ω_{n}}(u_{n K})ū. We also denote by S the sum in the last line in (1.9.11). Thus, we can rewrite the right side of (1.9.11) as

\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix} (∑_{ij} ∫_{Ω} e^{−φ}φ_{ij} · −∑_{j≤q_{o}} ∫_{Ω} e^{−φ}φ_{jj} ·)

+ \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} (∑_{ij} ∫_{∂Ω} e^{−φ}r_{ij} · −∑_{j≤q_{o}} ∫_{∂Ω} e^{−φ}r_{jj} ·) + S + R,

where · denotes products of the u’s. To conclude our proof, we only need to prove that

(1.9.19)  R ≤ ϵS + C_{ϵ}||u||_{φ}^{2}.

In fact, if we point our attention at terms which involve ∂ω̄_{j}(u)ū for q_{o} + 1 ≤ j ≤ n or δ_{ω_{j}}(u)ū for j ≤ q_{o}, then (1.9.19) is clear since S carries the corresponding square ||∂ω̄_{j}u||_{φ}^{2} or ||δ_{ω_{j}}u||_{φ}^{2}. Otherwise, we note that for j ≤ n − 1 we may interchange ∂ω̄_{j} with δ_{ω_{j}} by means of integration by parts: boundary integrals do not occur because r_{j}|_{∂Ω} = 0 for j ≤ n−1. As for δ_{ω_{n}}, notice that it only hits coefficients whose index contains n and hence u_{nK}|_{∂Ω} = 0. So δ_{ω_{n}}(u_{nK})ū is also interchangeable with u_{nK}∂ω̄_{n}(ū) by integration by parts. This concludes the proof of Proposition 1.9.1. □

Remark 1.9.2. The term (1 − ϵ)S in the last line of (1.9.8) is not relevant in our line of discussion but it is by itself very significant. In fact, let us point our attention at forms u ∈ (C_{c}^{∞}(Ω))^{k}. Then, each ||∂ω̄_{j}u||_{φ}^{2} can be interchanged with ||δ_{ω_{j}}u||_{φ}^{2} even for j = n by the vanishing of the boundary integral which occurs because u has compact support. Thus,

S + C||u||_{φ}^{2} ≥ \frac{1}{2}\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} (∑_{j≤n} ||∂ω̄_{j}u_{J}||_{φ}^{2} + ∑_{j≤n} ||δ_{ω_{j}}u_{J}||_{φ}^{2}) .

For the same reason, in the right side of (1.9.8), the boundary integrals vanish. If we then apply the variant of (1.9.8) with, say, φ = 0, we get an estimate which fully expresses the elliptic regularity in the interior of the system (¯∂, ¯∂^{∗}):

(1.9.20) ||¯∂^{∗}u||^{2} + ||¯∂u||^{2} + ||u||^{2} >_{∼} ||u||_{H^{1}}^{2} for any u ∈ (C_{c}^{∞}(Ω))^{k},

where “>_{∼}” means “≥” up to multiplicative constant and where H^{1} is the Sobolev space of index 1.

Remark 1.9.3. In the proof of (1.9.8) we have interchanged all ||∂_{ω_{j}}u_{J}||_{φ}^{2} with ||δ_{ω_{j}}u_{J}||_{φ}^{2} for any j ≤ q_{o}. If we do it only when j belongs to J, that is, for J = jK, we get an obvious variant of (1.9.8), in which, in particular, −\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}}· is replaced by −\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix} ∑_{j≤q_{o}}· in the second and

third lines.

As we have already said, (1.9.8) is true for any q_{o} and any k. We clarify now how they must be specified in order to exploit the full strength of (1.9.8).

Weighted H^{s} estimates on q-pseudoconvex domains. Let M = ∂Ω and let (r_{ij})_{ij=1,...,n−1} be the matrix of L_{M} under a choice of a C^{2}, orthonormal basis of (1, 0)-forms ω_{1}, ..., ω_{n} with ω_{n} = ∂r. We order the eigenvalues of L_{M}(z) as λ_{1}(z) ≤ λ_{2}(z) ≤ ... ≤ λ_{n−1}(z) and denote by s_{M}^{+}(z), s_{M}^{−}(z), s_{M}^{0}(z) the numbers of them which are positive, negative and null, respectively. We go on to describe our geometric hypotheses on M. Let q ≤ n − 2; following [101] and [1], we introduce

Definition 1.9.4. We say that M is q-pseudoconvex (for the orientation induced from Ω) if there exists a covering of the boundary and, on each patch, a C^{2} smooth bundle V ⊂ T^{1,0}M of rank q_{o} ≤ q, say V = Span {∂_{ω_{1}}, ..., ∂_{ω_{qo}}}, such that

(1.9.21)  ∑_{j=1}^{q+1} λ_{j}(z) − ∑_{j=1}^{q_{o}} r_{jj}(z) ≥ 0.

The index q_{o} may vary on different patches. It is evident that (1.9.21) implies λ_{q+1} ≥ 0; hence (1.9.21) is still true if we replace the first sum ∑_{j=1}^{q+1} · by ∑_{j=1}^{k} · for any k such that q + 1 ≤ k ≤ n − 1.

Example 1.9.5. A rough estimate of the index q = q_{o} for which (1.9.21) holds in a neighborhood B of a boundary point z_{o} is

(1.9.22)  q = s^{−}(z_{o}) + s^{0}(z_{o}).

In some cases we can do better. For instance, assume that s^{−}(z) is constant; then λ_{s^{−}} < 0 ≤ λ_{s^{−}+1} and hence the negative eigenvectors span a bundle V^{s^{−}} that, identified with the span of the first s^{−} coordinate vectors, yields ∑_{j≤s^{−}+1} λ_{j}(z) − ∑_{j≤s^{−}} r_{jj}(z) ≥ 0. Note that a pseudoconvex domain is characterized by s^{−}(z) ≡ 0; thus, it is 0-pseudoconvex in our sense.

Now, (1.9.21) is equivalent to

\begin{matrix}∑ & ′ \\ |K|=q \end{matrix}∑_{ij}r_{ij}u_{iK} ū_{jK} − \begin{matrix}∑ & ′ \\ |J|=q+1 \end{matrix} ∑_{j≤q_{o}}r_{jj}|u_{J}|^{2} ≥ 0

for any u tangential of degree q + 1.

(1.9.23)

By what was remarked after (1.9.21), it follows that (1.9.23) is in fact true for any u tangential of degree k ≥ q + 1. Sometimes, we also use a variant of (1.9.23) which is still sufficient for our existence problem:

(1.9.24) \begin{matrix}∑ & ′ \\ |K|=q \end{matrix}∑_{ij}r_{ij}u_{iK} ū_{jK} − \begin{matrix}∑ & ′ \\ |K|=q \end{matrix} ∑_{j≤q_{o}}r_{jj}|u_{jK}|^{2} ≥ 0

for any u tangential of degree q + 1.

Again, (1.9.24) is in fact true for any tangential form of degree k with q + 1 ≤ k ≤ n − 1.

Theorem 1.9.6. Let Ω be q-pseudoconvex or let it satisfy (1.9.24); then, for φ_{t} := (t + C)|z|^{2} and for any u ∈ C^{∞}(Ω̄)^{k} ∩ D ¯∂^{∗}, we have

(1.9.25)  t||u||_{H_{φ}^{0}}^{2} ≤ ||¯∂u||_{H_{φ}^{0}}^{2} + ||¯∂_{φ}^{∗}u||_{H_{φ}^{0}}^{2} if k ≥ q + 1.

Proof. We choose q_{o} as in (1.9.21); we have (1.9.26)

\begin{cases}\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij}r_{ij}u_{iK} ū_{jK} − \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}}r_{jj}|u_{J}|^{2} ≥ 0, \\ \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij}φ_{ij}u_{iK} ū_{jK} − \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}}φ_{jj}|u_{J}|^{2} ≥ (k − q_{o})(t + C)|u|^{2}. \end{cases}

Using (1.9.8) and noticing that k − q_{o} ≥ 1, we get (1.9.25).

If we are assuming (1.9.24) instead of (1.9.21), then we replace u_{J} by u_{jK} in the second sums of the two lines of (1.9.26). Taking into account Remark 1.9.3, we conclude that (1.9.25) is still true.

□

For s integer, let H_{φ_{t}}^{s} be the weighted s-Sobolev space: recall that u ∈ H_{φ_{t}}^{s} when D^{α}u ∈ H_{φ_{t}}^{0} for any |α| ≤ s. As a set, H_{φ_{t}}^{s} coincides with H^{s} and the two norms are equivalent. If we wish to bring the estimate (1.9.25) into H^{s}-norm, we have to control the commutators [¯∂, D^{α}] and [¯∂_{t}^{∗}, D^{α}] where we are writing ¯∂_{t}^{∗} for ¯∂_{φ_{t}}^{∗}. To this end, we use the full power of the weight φ_{t}.

Theorem 1.9.7. Let Ω be q-pseudoconvex or let it satisfy (1.9.24). Then for any s, for suitable t = t_{s} and for any u ∈ C^{∞}(Ω̄)^{k} ∩ D ¯∂^{∗}, we have

(1.9.27)  ||u||_{H^{s}}^{2} <_{∼} ||¯∂u||_{H^{s}}^{2} + ||¯∂_{t}^{∗}u||_{H^{s}}^{2} if k ≥ q + 1.

Proof. We show how to derive (1.9.27) from (1.9.25). We denote by T a general tangential vector field, that is, a combination of ∂_{ω_{j}}, ∂ω̄_{j}, j ≤ n − 1, and ∂_{ω_{n}} − ∂ω̄_{n} and by T^{s} a tangential operator of order s. We denote by N the normal vector field ∂_{ω_{n}} − ∂ω̄_{n}. We first remark that for any u with H^{s} cefficients and for any i ≥ 1, we have for any choice of t (1.9.28) ||N^{i}T^{s−i}u||_{H_{φ_{t}}^{0}}^{2} ≤ c_{s}(||¯∂u||_{H_{φ_{t}}^{s−1}}^{2} + ||¯∂_{t}^{∗}u||_{H_{φ_{t}}^{s−1}}^{2} + ||T^{s}u||_{H_{φ_{t}}^{0}}^{2}) + c_{st}||u||_{H_{φ_{t}}^{s−1}}^{2}.

(It is meant here that the inequality is true for any T^{s−i} and for suitable T^{s}.) Classically, this formula is referred to as a consequence of the “noncharacteristicity” of (¯∂, ¯∂^{∗}). If i = 1 and s = 1 it immediately follows from the splitting

(1.9.29)  N = ∂ω̄_{n} + T ,

and the fact that the “energy” ||¯∂u||^{2}+||¯∂^{∗}u||^{2}+||u||^{2} contains ||∂ω̄_{n}u||^{2}. For general i and s, one replaces ∂ω̄_{n} inside (1.9.29) (or ∂_{ω_{n}} inside the twin decomposition N = ∂_{ω_{n}} + T for a new T ) according to the (noncharacteristicity) relations

(1.9.30)  \{ \begin{aligned}N u_{j K} & = ∂ω̄_{n}u_{j K} + T u_{j K} \\ & = ∂ω̄_{j}u_{n K} − (¯∂u)_{j n K} + T u_{j K} + P^{0}u, \end{aligned} ; \begin{aligned}N u_{n K} & = ∂_{ω_{n}}u_{n K} + T u_{n K} \\ & = (¯∂_{t}^{∗}u)_{K} − ∑_{j=1}^{n−1} ∂_{ω_{j}}u_{j K} + T u_{n K} + P_{t}^{0}u, \end{aligned}

where P^{0} and P_{t}^{0} are 0-order operators. If we then apply N^{i−1}T^{s−i} to the two terms of (1.9.30), remark that ∂ω̄_{j} and ∂_{ω_{j}} j ≤ n − 1 are derivatives of type T , and put into an s − 1 order term the error from commutators for bringing N in head position, we get (1.9.28). Now, (1.9.28), together with induction, shows that in order to estimate the various ||N^{i}T^{s−i}u||’s, that is, the full norm||u||_{s}, it is sufficient to estimate the ||T^{s}u||’s. This is the central point of the proof. To see it, we begin by remarking that T^{s}u still belongs to D ¯∂^{∗}. Next, we observe that we can describe the commutator [¯∂_{t}^{∗}, T^{s}] as A_{s} + A_{s−1}^{t} where A_{s} and A_{s−1}^{t} are operators of order s and s−1, respectively, the first being independent of t and involving at most one non-tangential derivative. Also, [¯∂, T^{s}] has order ≤ s and is independent of t; hence it can be “absorbed” by A_{s}. It follows that

\begin{aligned}t||T^{s}u||_{H_{φ_{t}}^{0}}^{2} & ≤ ||¯∂T^{s}u||_{H_{φ_{t}}^{0}}^{2} + ||¯∂_{t}^{∗}T^{s}u||_{H_{φ_{t}}^{0}}^{2} \\ & ≤ ||T^{s} ¯∂u||_{H_{φ_{t}}^{0}}^{2} + ||T^{s} ¯∂_{t}^{∗}u||_{H_{φ_{t}}^{0}}^{2} + ||A_{s}u||_{H_{φ_{t}}^{0}}^{2} + ||A_{s−1}^{t}u||_{H_{φ_{t}}^{0}}^{2}. \end{aligned} (1.9.31)

Now, if A_{s} is fully tangential, it can be absorbed in the left side of (1.9.31) by a suitable choice of t = t_{s}. Otherwise, if A_{s} carries some normal derivative N^{i}, we use (1.9.28) and get the estimate (1.9.32)

||A_{s}u||_{H_{φ_{t}}^{0}}^{2} ≤ c_{s}(||¯∂u||_{H^{s−1}}^{2} + ||¯∂_{t}^{∗}u||_{H^{s−1}}^{2} + ||T^{s}u||_{H^{0}}^{2}) + c_{st}||u||_{H^{s−1}}^{2}.

Again, the term involving T^{s}u can be absorbed in the left side of (1.9.31). This proves

(1.9.33)  t||T^{s}u||_{H^{0}}^{2} <_{∼} ||¯∂u||_{H^{s}}^{2} + ||¯∂_{t}^{∗}u||_{H^{s}}^{2} + c_{s t}||u||_{H^{s−1}}^{2}.

As it has already been said, (1.9.33), inserted inside (1.9.28) implies (1.9.27) apart from an error of order s − 1 which can be removed by induction.

This completes the proof of (1.9.27).

□

In contrast to the statement of Theorem 1.9.6, when passing from H^{0}- to H^{s}-norm, we need a multiplicative constant in estimate (1.9.27).

Remark 1.9.8. The constant t has disappeared from (1.9.27) except from the adjoint operator ¯∂_{t}^{∗}. However, the weight φ_{t} did a great job: by means of its part C|z|^{2}, it served in Proposition 1.9.1, in combination with the sum of derivatives that we denoted by S, to control the error terms R. Also, through its part t|z|^{2}, it took care of the errors given by the commutators [¯∂, T^{s}] and [¯∂_{t}^{∗}, T^{s}].

The ¯∂-Neumann operator. We give the outline of the construction by Kohn. We begin by recalling from [50, Lemma 4.1.3] and, more closely, [36, Lemma 4.3.2] that in a domain Ω with C^{∞} boundary, C^{∞}(Ω̄)^{k} ∩ D ¯∂^{∗} is dense in H^{0}(Ω)^{k} ∩ D ¯∂^{∗} ∩ D ¯∂ for the triplet norm

||u||_{H^{0}}^{2} + ||¯∂u||_{H^{0}}^{2} + ||¯∂_{t}^{∗}u||_{H^{0}}^{2}.

For this reason, (1.9.25) is in fact true for any u ∈ H^{0}(Ω)^{k} ∩ D ¯∂_{t}^{∗} ∩ D ¯∂ instead of C^{∞}(Ω̄)^{k} ∩ D ¯∂_{t}^{∗}. We define now the “complex Laplacian” by

□_{t} := ¯∂¯∂_{t}^{∗} + ¯∂_{t}^{∗} ¯∂.

This is a linear closed densely defined operator on forms with coefficients in H^{0} which leaves unchanged their degree. A form u belongs to D_{□_{t}} iff u ∈ D ¯∂_{t}^{∗} and ¯∂u ∈ D ¯∂_{t}^{∗}, that is, u satisfies the celebrated “¯∂-Neumann conditions”. In our normalization ω_{n} = ∂r, they are described over a smooth form u by

(1.9.34)  \{u_{n K}|_{∂Ω} ≡ 0, ; \begin{aligned}∂ω̄_{n}u_{j K}|_{∂Ω} & = (∂ω̄_{n}u_{j K} − ∂ω̄_{j}u_{n K})|_{∂Ω} \\ & = (¯∂u)_{n K}|_{∂Ω} ≡ 0. \end{aligned}

It is readily checked that

⟨□_{t}u, u⟩_{H^{0}} = ||¯∂u||_{H^{0}}^{2} + ||¯∂_{t}^{∗}u||_{H^{0}}^{2}.

Thus, (1.9.27) for the choice s = 0 (extended for all u ∈ H^{0} ∩ D_{□_{t}}) can be rewritten as

\begin{aligned}||u||_{H^{0}}^{2} & <_{∼} ⟨□_{t}u, u⟩_{H^{0}} \\ & ≤ ||□_{t}u||_{H^{0}}||u||_{H^{0}}, \end{aligned}

which yields

||u||_{H^{0}} <_{∼} ||□_{t}u||_{H^{0}} for any u ∈ H^{0}(Ω)^{k} ∩ D_{□_{t}}.

The range of □_{t} is therefore closed and □_{t} is injective. There is then a well-defined “¯∂-Neumann” inverse operator N_{t} := □_{t}^{−1} which acts as

N_{t} : H^{0}(Ω)^{k} → D_{□_{t}}

and which satisfies

t||N_{t}u||_{H^{0}} ≤ ||u||_{H^{0}} for any u ∈ H^{0}(Ω)^{k}.

It is easy to check that it satisfies, among other properties

(i) ¯∂N_{t} = N_{t} ¯∂ and

(ii) ¯∂_{t}^{∗}N_{t} = N_{t} ¯∂_{t}^{∗}.

We have used (1.9.27) for s = 0 (or rather (1.9.25)) so far. Coming to a general s, we have a statement similar as to Theorem 1.9.7 with the system (¯∂, ¯∂_{t}^{∗}) replaced by the operator □_{t}, that is,

(1.9.35) ||u||_{H^{s}}^{2} <_{∼} ||□_{t}u||_{H^{s}}^{2} when u ∈ D_{□} ∩ C^{∞}(Ω̄)^{k} for k ≥ q + 1.

The proof is the same as above and consists in getting control of the commutators [Λ^{s}, □_{t}] by a suitable choice of t = t_{s}. By the density of C^{∞}(Ω̄) into H^{s}, (1.9.35) can be rewritten as (1.9.36)

||N_{t}u||_{H^{s}} <_{∼} ||u||_{H^{s}} for any u ∈ H^{s}(Ω) such that N_{t}u ∈ H^{s}(Ω)^{k}.

Note that we only know that N_{t}u ∈ H^{0}; so, the restraint N_{t}u ∈ H^{s} is substantial. To eliminate it from (1.9.36), we have to use the method of the “elliptic regularization” (cf. [57]) and define a perturbed elliptic operator N_{t}^{ϵ} : H^{s} → D_{□_{t}} ∩ H^{s+2} which satisfies

(1.9.37)  ||N_{t}^{ϵ}u||_{H^{s}}^{2} ≤ c_{s}||u||_{H^{s}}^{2} for any u ∈ H^{s}(Ω)^{k},

where c_{s} is independent of ϵ. Let u ∈ H^{s}(Ω)^{k}; because of (1.9.37) we have a subsequential H^{s}-weak limit N_{t}^{ϵ}u → v as ϵ → 0. It is clear that N_{t}^{ϵ}u → N_{t}u in H^{0}. In particular N_{t}u = v, which proves that

(1.9.38)  N_{t}u ∈ H^{s}(Ω)^{k} if u ∈ H^{s}(Ω)^{k}.

We are ready to prove

Theorem 1.9.9. Let Ω be q-pseudoconvex or let it satisfy (1.9.24). Then, if k ≥ q+1, for any f ∈ H^{s}(Ω)^{k} with ¯∂f = 0, we can find u = u_{s} in H^{s}(Ω)^{k−1} such that

(1.9.39)  \begin{cases}¯∂u = f, \\ ||u||_{H^{s}} <_{∼} ||f||_{H^{s}}. \end{cases}

Proof. We set u := ¯∂_{t}^{∗}N_{t}f for t = t_{s}; we have

¯∂(¯∂_{t}^{∗}N_{t}f) = □_{t}N_{t}f − ¯∂_{t}^{∗} ¯∂N_{t}f
= f − ¯∂_{t}^{∗}N_{t} ¯∂f
= f.
□

Remark 1.9.10. If under the choice t = t_{s}, we have (1.9.27) for H^{s}, we also have it for any H^{j}, j ≤ s. In particular, the solution u of Theorem 1.9.9 satisfies not only the estimate in the second line of (1.9.39) but also ||u||_{H^{j}}^{2} <_{∼} ||f||_{H^{j}}^{2} for any j ≤ s.

We did not just find a solution in H_{φ_{t}}^{s}, but the one which is orthogonal to the kernel of ¯∂; in fact ¯∂_{t}^{∗}u = 0 because ¯∂_{t}^{∗} ∘ ¯∂_{t}^{∗} = 0. This serves to define the “Bergman projection”

\begin{aligned}B_{s} : & H^{s} & → & ker ¯∂, \\ & f & ↦ & f − ¯∂_{t}^{∗}N_{t} ¯∂f. \end{aligned}

In particular, the estimate ||¯∂_{t}^{∗}N_{t} ¯∂f||_{H_{φ_{t}}^{s}} <_{∼} ||f||_{H_{φ_{t}}^{s}} shows that B_{s} is continuous in H^{s}-norm. However, we do not know if the canonical solution in H^{0} (unweighted), u = ∂^{∗}N_{0}f, belongs to H^{s} nor if the projection B_{0} accordingly defined, is H^{s}-continuous. Let us denote by K^{k}(Ω̄) (resp. K^{k}(Ω)) the cohomology in degree k of the complex ¯∂ over forms with coefficients in C^{∞}(Ω̄) (resp. C^{∞}(Ω)) (cf. the lines after (1.8.6)).

Theorem 1.9.11. Let Ω be bounded, smooth and q-pseudoconvex or let it satisfy (1.9.24). Then

K^{k}(Ω̄) = 0 if k ≥ q + 1.

Proof. We follow [58]. We have to solve the equation ¯∂u = f for f ∈ C^{∞}(Ω̄)^{k} such that ¯∂f = 0. We claim that we can find a sequence {u_{ν}}_{ν} of H^{ν} solutions such that ¯∂_{t}^{∗}u = 0 for t = t_{ν} and

||u_{ν+1} − u_{ν}||_{H^{ν}} ≤ 2^{−ν}.

Suppose we have carried out our construction up to u_{ν} and now look for u_{ν+1}. Take ũ_{ν+1} ∈ H^{ν+1} with (¯∂ũ_{ν+1} = f, ¯∂_{t}^{∗}ũ_{ν+1} = 0) according to Theorem 1.9.9. Put h := −ũ_{ν+1} + u_{ν}; the problem that we have to overcome is that h is only H^{ν} and not yet H^{ν+1}. Thus, in each patch of a finite covering {B_{j}}_{j≥1} of ∂Ω we “push points of Ω̄ inward to Ω” by means of the mapping

Ω̄ ∩ B_{j} → Ω, z ↦ z + ϵv_{j},

where v_{j} is an “inner normal” to Ω in B_{j}. Let B_{0} be a neighborhood of Ω \ ⋃_{j≥1}B_{j}, take a partition of the unity {ψ_{j}}_{j≥0} subordinate to the

covering {B_{j}}_{j≥0} and put

h_{ϵ}(z) = ψ_{0}(z)h(z) + ∑_{j≥1}ψ_{j}(z)h(z + ϵv_{j}).

Note that by the ellipticity of (¯∂, ¯∂_{t_{ν}}^{∗}) in the interior of Ω, the function h_{ϵ} is C^{∞}(Ω̄). We also rewrite the above sum as ∑_{j≥0}ψ_{j}h_{ϵ}^{j}. We have

\begin{aligned}¯∂h_{ϵ} & = ∑_{j≥0}¯∂ψ_{j} ∧ h_{ϵ}^{j} \\ & = ∑_{j≥0}¯∂ψ_{j} ∧ (h_{ϵ}^{j} − h), \end{aligned}

where the last equality follows from ∑_{j≥0}¯∂(ψ_{j})h = ¯∂(∑_{j≥0}ψ_{j}h) = 0. Thus,

||¯∂h_{ϵ}||_{H^{ν}} → 0 as ϵ → 0. At this point we recall Theorem 1.9.7 and find a solution v_{ϵ} of

\begin{cases}¯∂v_{ϵ} = ¯∂h_{ϵ}, \\ ||v_{ϵ}||_{H^{ν}} <_{∼} ||¯∂h_{ϵ}||_{H^{ν}}. \end{cases}

The solution v_{ϵ} can be found in any Sobolev space, in particular in H^{ν+1}, because ¯∂h_{ϵ} ∈ C^{∞}(Ω̄); its norm is controlled by the norm of ¯∂h_{ϵ} in any H^{j} for j ≤ ν +1, in particular in H^{ν}. (Cf. Theorem 1.9.9 and the subsequent remark.) Define u_{ν+1} := ũ_{ν+1} +h_{ϵ} −v_{ϵ}; by choosing ϵ small,

we can achieve ||u_{ν+1} − u_{ν}||_{H^{ν}} ≤ 2^{−ν}. Define u = u_{ν_{o}} + ∑_{ν≥ν_{o}}(u_{ν+1} − u_{ν});

then the above power series converges and defines u ∈ H^{ν_{o}} for any ν_{o}; thus, u is in C^{∞}(Ω̄).

□

We have thus achieved our goal of solving the ¯∂-problem with smooth solutions up to the boundary over a q-pseudoconvex domain provided that it is bounded and has smooth boundary. Note that this does not mean that the canonocal solution in H^{0} is smooth; there are counterexamples in case of the “worm” domains (cf. e.g. [36]). We go on to consider the situation in which Ω no longer has a C^{∞} boundary but can be exhausted by a sequence Ω_{ν} ↗ Ω of C^{∞}, bounded, q-pseudoconvex domains. In this context we are not able to prove the regularity of ¯∂ at the boundary, that is, the existence of C^{∞}(Ω̄) solutions. However, our techniques suffice to produce solutions in C^{∞}(Ω). We start with a definition. For a domain Ω, possibly with non-smooth boundary and unbounded, and for a smooth exhaustion function ψ of Ω, we denote by λ_{1}^{ψ} ≤ λ_{2}^{ψ} ≤ ... ≤ λ_{n}^{ψ} the ordered eigenvalues of the Levi form L_{ψ}|_{∂ψ^{⊥}}, where ∂ψ^{⊥} is the complex (n − 1)-bundle orthogonal to ∂ψ. We introduce a notion related to q-pseudoconvexity which applies to a boundary which is not necessarily smooth. Let ψ be a C^{∞} exhaustion function in B ∩ Ω for a neighborhood B of ∂Ω.

Definition 1.9.12. The function ψ is said to be q-plurisubharmonic if there is a smooth bundle V^{q_{o}} on Ω̄ ∩ B orthogonal to ∂ψ, say V^{q_{o}} = Span {∂_{ω_{1}}, ..., ∂_{ω_{qo}}}, such that

(1.9.40)  ∑_{j=1}^{q+1} λ_{j}^{ψ}(z) − ∑_{j=1}^{q_{o}} ψ_{jj}(z) ≥ 0 for any z ∈ B ∩ Ω.

We have also a variant of this definition on the line of (1.9.24). In contrast to Definition 1.9.4, the latter Definition 1.9.12 is a condition on Ω instead of ∂Ω; but it is not substantially different because it is uniform up to ∂Ω.

The case q = 0 deserves particular attention. First, if we assume in addition that ∂Ω is C^{∞}, Definition 1.9.4 mathches Definition 1.9.12 for the choice ψ = − log d (Corollary 1.8.17). In general, we have already introduced in Section 1.8 another notion of pseudoconvexity which also applies to non-smooth domains. We wish to show that it is consistent with the new one.

Proposition 1.9.13. Let Ω have a continuous plurisubharmonic exhaustion function ψ. Then we can regularize it to a C^{∞}, strictly plurisubharmonic exhaustion function ψ̃.

In particular, pseudoconvexity is equivalent in the sense of Definition 1.8.12 and Definition 1.9.12.

Proof. (Cf. [50, Theorem 2.6.11].) Set Ω_{j} := {z ∈ Ω : ψ(z) < j}; let χ_{ϵ} = ϵ^{−2n}χ(\frac{z}{ϵ}) be a set of smooth functions which approximate the Dirac measure δ as defined in Section 1.7. We wish to regularize ψ by convolution with χ_{ϵ}; however, some care is needed when approaching the boundary ∂Ω. Let

ψ_{j}(z) = ∫_{Ω_{j+1}} ψ(ζ)χ_{ϵ}(z − ζ)dV (ζ) + ϵ|z|^{2}.

For sufficiently small ϵ = ϵ_{j}, we have that ψ_{j} ∈ C^{∞}(ℂ^{n}) and ψ < ψ_{j} < ψ+1 on a neighborhood of Ω̄_{j}; also, from Δψ_{j} = Δψ∗χ_{ϵ} +ϵ, we deduce that Δψ_{j}|Ω̄_{j} > 0. Next, we take the composition χ(ψ_{j} + 1 − j) where χ is a C^{∞} function such that χ(t) ≡ 0 for t ≤ 0 and χ̇ > 0, χ̈ > 0 for t > 0; this is a strictly plurisubharmonic function on Ω̄_{j} \ Ω_{j−1}. Hence we can find a sequence of positive numbers {a_{j}} such that

ψ_{μ} := ψ_{j_{o}} + ∑^{μ} ; j=j_{o}+1a_{j}χ(ψ_{j} + 1 − j),

is > ψ, and is strictly plurisubharmonic on Ω̄_{μ}. Since different ψ_{μ_{i}} and ψ_{μ_{k}} agree over Ω̄_{j} when μ_{i}, μ_{k} > j, we have that ψ̃ := lim_{μ}ψ_{μ} is well

defined, > ψ, and strictly plurisubharmonic C^{∞} on Ω.

□

We come back to general q and are ready for the conclusive statement.

Theorem 1.9.14. Let Ω have a smooth q-plurisubharmonic exhaustion function; then

K^{k}(Ω) = 0, k ≥ q + 1.

Notice that Ω is not necessarily smooth and bounded.

Proof. Let ψ be the C^{∞} exhaustion function as in Definition 1.9.12, and define Ω_{ν} := {z ∈ Ω : ψ(z) < ν}; the Ω_{ν}’s are bounded, C^{∞}, q-pseudoconvex. We have to solve the equation ¯∂u = f in the spaces of forms with C^{∞}(Ω)-coefficients, when deg(f) = k ≥ q+1 and ¯∂f = 0. We first assume that f ∈ H^{0}(Ω)^{k} where H^{0} = H_{φ_{t}}^{0} for φ_{t} = e^{−(C+t)|z|^{2}} and show how to find a solution u ∈ H^{0}(Ω)^{k−1}. In the present discussion, t can be chosen as t = 0. According to Theorem 1.9.6, for any ν, there is a solution u_{ν} := ¯∂_{t}^{∗}N_{t}f ∈ H^{0}(Ω_{ν})^{k−1}; in particular, u_{ν} satisfies ¯∂_{t}^{∗}u_{ν} = 0. Also, the u_{ν}’s have the estimate

\begin{aligned}t||u_{ν}||_{H^{0}(Ω_{ν})} & ≤ ||f||_{H^{0}(Ω_{ν})} \\ & ≤ ||f||_{H^{0}(Ω)}. \end{aligned}

Hence, there is a subsequence u_{ν_{j}} which converges in H^{0}(Ω) to a weak limit u which is a solution of (¯∂u = f, ¯∂_{t}^{∗}u = 0). If we assume that the coefficients of f are C^{∞}(Ω) (in addition to H_{φ_{t}}^{0}(Ω)), then (1.9.20) (elliptic regularity in the interior) implies u ∈ C^{∞}(Ω)^{k−1}.

We pass to the general case in which f is not necessarily in H_{φ_{t}}^{0}. We then change the weight into φ_{t} + η where η is plurisubharmonic and so rapidly increasing at ∂Ω and at ∞ that f ∈ H_{φ_{t}+η}^{0}. As before, we first solve in H_{φ_{t}+η}^{0} and next regularize in C^{∞}.

□

For the case q = 0, when Ω has a plurisubharmonic exhaustion funtion which is C^{0}, then it has also another which is C^{∞} (Proposition 1.9.13) and hence K^{k}(Ω) = 0 for any k ≥ 1, which is the implication (ii) ⇒ (iii) of Theorem 1.8.13. This completes the last detail in the solution of the “Levi problem”.

Notes. The ¯∂-Neumann problem was suggested by Garabedian and Spencer [44]. The basic a priori estimates were first proved for 1-forms by Morrey [69]. Next, Kohn [55] derived the general estimates and proved the regularity up to the boundary on strongly pseudoconvex domains. The L^{2} estimates with weights which are 0 at the boundary were used by Hörmander [49] to bypass the boundary regularity problem. The regularity at the boundary by use of the weighted ¯∂- Neumann problem, that is, the conclusion of Theorem 1.9.11 for q = 0, was proved by Kohn in [57]. A very clear and exhaustive explanation of the ¯∂-Neumann method by Kohn can be found in the book by Chen and Shaw [36]. As for the condition of q-pseudoconvexity and its use for the ¯∂-Neumann weighted estimates, we refer to Zampieri [101] and Ahn [1]. Another approach, through the integral method, is mainly due to Henkin (cf. [4]) and is very well explained in the last six chapters of the book by Boggess [27].

We now want to make some complementary comments. The solution u of ¯∂u = f that we have constructed in Theorem 1.9.11 is obtained by approximation and is not necessarily the “canonical” solution. This is the one in the form ¯∂^{∗}Nf and hence, it satisfies ¯∂^{∗}u = 0; in particular, it is orthogonal to the kernel of ¯∂. As pointed out by Kohn, the regularity of the canonical solution would have significant consequences for mapping problems. In fact, for f ∈ C^{∞}(Ω̄), it would imply that the Bergman projection B(f) := f − u where u is the solution of ¯∂u = ¯∂f which satisfies u ⊥ Ker ¯∂ would continue to belong to C^{∞}(Ω̄). But then, according to a result by Bell and Ligocka [23], any holomorphic one-to-one self-mapping of Ω would extend smoothly up to Ω̄. However, Barret discovered smooth pseudoconvex domains, the so called “worm” domains, for which the canonical H^{0} solution is not smooth. Even more delicate is the problem of the regularity of the solution u in the part of ∂Ω where f is regular. Consider for example (cf. Kohn [58]) a domain Ω ⊂ ℂ^{2} with smooth boundary defined by y_{2} < 0 for |z_{1}| < 1. Let χ be a real C^{∞} function which satisfies χ ≡ 1 for |z_{1}| < \frac{1}{4} and χ ≡ 0 for |z_{1}| > \frac{1}{2}. Then, f := ¯∂(\frac{χ}{z_{2}} ) is a ¯∂-closed 1-form which is C^{∞} in the part of ∂Ω where |z_{1}| < \frac{1}{4} and |z_{1}| > \frac{1}{2}. However, no solution u of ¯∂u = f can be regular in that part of the boundary, because, otherwise, the function u − \frac{χ}{z_{2}} would be holomorphic in Ω but would be uniformly bounded in the boundary of the disc ∂Δ × {−iη} and unbounded at their center in violation of the maximum principle.

1.10. Subelliptic estimates for ¯∂

Summary. We say a few words about the subellipticity and hypoellipticity of the system (¯∂, ¯∂^{∗}) on bounded domains Ω ⊂ ℂ^{n} with smooth boundary. This is “1-subelliptic”, that is, elliptic in the interior of Ω as we have noticed in Remark 1.9.2: so our interest is confined to the boundary ∂Ω. If Ω is strongly q-pseudoconvex, we prove \frac{1}{2}- subelliptic estimates. They are of type |||u|||2_{\frac{1}{2}} ∼^{<} ||¯∂u||_{0}^{2}+||¯∂^{∗}u||_{0}^{2}+||u||_{0}^{2}

for any form in the domain of ¯∂^{∗} of degree ≥ q + 1. These are classical in a situation which can be referred to as the celebrated “Z-condition” of Kohn [42]. We end the section by proposing a new and simple proof of subellipticity of “decoupled finite type” domains. For these domains, the finite type conditions in the sense of commutators or contact order of complex curves do coincide. In whatever case, subelliptic estimates entail local hypoellipticity at the boundary for (¯∂, ¯∂^{∗}), that is, any H^{0} solution is C^{∞}(Ω̄) precisely in the part of Ω̄ where the datum is C^{∞}(Ω̄).

1.10. SUBELLIPTIC ESTIMATES FOR ¯∂ 69

We keep the notation of Section 1.9, suppose that M is defined by r = 0 with ∂r ≠ 0 and choose an orthonormal system of (1, 0)- vector fields ∂_{ω_{j}}, j = 1, ..., n, with ∂_{ω_{j}}r = κ_{jn}; thus, ∂_{ω_{1}}, ..., ∂_{ω_{n−1}} and ∂ω̄_{1}, ..., ∂ω̄_{n−1} when restricted to M are a basis of the space of complex vector fields tangential to M of type (1, 0) and (0, 1), denoted by T^{1,0}M and T^{0,1}M respectively. Let ∂_{ω} be a vector field in T^{1,0}M; the commutator [∂_{ω}, ∂ω̄] is still a tangent vector of ℂ ⊗ T M but not necessarily of ℂ ⊗ T^{ℂ}M = T^{1,0}M ⊕ T^{0,1}M. If [∂_{ω}, ∂ω̄] ∈ T^{1,0}M ⊕ T^{0,1}M, we iterate the commuation and consider expressions of type [∂_{ω}^{ϵ_{1}}, [∂_{ω}^{ϵ_{2}}[..., ∂_{ω}^{ϵ_{m}}]]] where the suffix ϵ_{j} in ∂_{ω}^{ϵ_{j}} denotes either choice of ∂_{ω} or ∂ω̄. We call these expressions commuators of order m.

Definition 1.10.1. ∂_{ω} is said to be of finite type m if iterated commutators of ∂_{ω} and ∂ω̄ of order m produce the “totally real tangential” vector field T_{ℝ} = \frac{i}{\sqrt{2}}(∂_{ω_{n}} − ∂ω̄_{n}).

Recall the Levi form L_{M} whose matrix in the ω_{j}’s basis we have denoted by (r_{ij})_{ij}. Remember that (cf. (1.9.14)) by contraction with the (1, 0)-form ω_{n} = ∂r, Cartan formula yields

\begin{aligned}[∂_{ω}, ∂ω̄] & = (r_{ij})(u, ū)∂_{ω_{n}} − (r̄_{ji})(u, ū)∂ω̄_{n} + ... \\ & = \frac{\sqrt{2}}{i}(r_{ij})(u, ū)T_{ℝ} + ... \end{aligned} ; (1.10.1)

where the dots denote combinations of the ∂_{ω_{i}}’s and ∂ω̄_{i}’s for i ≤ n−1. Thus, [∂_{ω}, ∂ω̄] produces T_{ℝ} if and only if (r_{ij})(u, ū) ≠ 0. There is a “real” construction of iterated commutators underlying the complex one. We write ∂_{ω} = X − iY and identify

(1.10.2)  \begin{aligned}T^{1,0}ℂ^{n} & & & (a + ib)∂_{ω} \\ & ↘^{∼} & & & ↘^{∼} \\ & & T ℝ^{2n}, & & & aX + bY. \\ & ↗_{∼} & & & ↗_{∼} \\ T^{0,1}ℂ^{n} & & & (a − ib)∂ω̄ \end{aligned}

This identification induces an identification between tangent vector fields

T^{1,0}M

↘^{∼}

T^{ℂ}M.

↗_{∼}

T^{0,1}M

We call these identifications “Re”. We give the “real” counterpart of the properties of the iterated commutators of ∂_{ω} and ∂ω̄. First, the iterated commutators of the X and Y can go out of T^{ℂ}M but stay inside T M. Also, if we denote by L^{ϵ_{j}} either choice of X or Y, then [∂_{ω}^{ϵ_{1}}, [∂_{ω}^{ϵ_{2}}[..., ∂_{ω}^{ϵ_{m}}]]] produces T_{ℝ} if and only if the same is true of [L^{ϵ_{1}}, [L^{ϵ_{2}}[..., L^{ϵ_{m}}]]].

Since we need to work with fractional or negative index s, we can not hesitate any more to introduce the tangential Sobolev space with the corresponding tangential norm ||| · |||_{H_{φ}^{s}(Ω)} (or ||| · |||_{s} for short); we also denote by (·, ·)^{Tan} the associated scalar product. For this, we use the defining function r of M as the last coordinate, supplement it to a full system of coordinates (a, r) ∈ ℝ^{2n−1} × ℝ and denote by ξ = (ξ_{a}, ξ_{r}) the dual coordinates. We then consider the standard s-order elliptic symbol Λ^{s}(ξ_{a}) = (1+|ξ_{a}|^{2})^{\frac{s}{2}} , denote by F_{Tan} the partial Fourier transform with respect to a, define the tangential microdifferential operator Λ^{s} = Λ_{Tan}^{s} with symbol Λ^{s}(ξ_{a}) by Λ^{s}(u) = F_{Tan}^{−1}(Λ^{s}(ξ_{a})F_{Tan}(u)), and set

(1.10.3)  |||u|||_{s}^{2} := ∫_{∞}^{0} ∫_{ℝ^{2n−1}} Λ^{s}(ξ_{a})^{2}|F_{Tan}u(ξ_{a}, r)|^{2}dξ_{a}dr.

When ∂_{ω} is of finite type m there is a remarkable consequence. For any function u ∈ C_{c}^{∞}(Ω̄ ∩ B) we have the celebrated “subelliptic estimate”

(1.10.4)  |||T_{ℝ}u|||_{−1+ \frac{1}{2^{m−1}}}^{2} ≃ ||∂_{ω}u||_{0}^{2} + ||∂ω̄u||_{0}^{2} + ||u||_{0}^{2}.

Thus we get control of “(T_{ℝ})^{\frac{1}{2^{m−1}}} ” by means of ∂_{ω} and ∂ω̄. Estimate (1.10.4) follows from Theorem 2.4.6 below which is in turn a consequence of Lemma 2.4.7 applied to the pair Re ∂_{ω} and Im ∂_{ω}. We call the attention to this point. Finite type of systems of complex vector fields implies subellipticity but under the strict assumption that the system is closed under conjugation (cf. the counterexample by Kohn [60]). We postpone to Remark 2.4.9 below the analysis of this point.

We state now \frac{1}{2}-subelliptic estimates. In these estimates, the Sobolev spaces are unweighted and the forms u have compact support in Ω̄. They apply to a domain Ω which is strongly q-pseudoconvex in the sense that (1.9.21) holds with the inequality “≥ 0” replaced by “≥ c” for c > 0. In particular, L_{M} is not the 0-form and M has type 2: we have \frac{1}{2}-subelliptic estimates in this case:

Theorem 1.10.2. Let Ω be smooth and strongly q-pseudoconvex in a neighborhood B of z_{o}. Then, for any u ∈ C_{c}^{∞}(Ω̄ ∩ B)^{k} ∩ D ¯∂^{∗}, we have

(1.10.5) |||u|||2_{\frac{1}{2}} + |||∂_{r}u|||_{− \frac{1}{2}} <_{∼} ||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2} + ||u||_{0}^{2} if ≥ q + 1.

Proof. We go back to the proof of Proposition 1.9.1 with the choice φ = 0, and thus in particular δ_{ω_{j}} = ∂_{ω_{j}}, and modify a little the argument therein. We interchange the third sums of the right side of (1.9.8) according to the following scheme:

\begin{matrix}||∂_{ω_{j}}u||_{0}^{2} → & (1 − ϵ)||∂_{ω_{j}}u||_{0}^{2} + ϵ||∂ω̄_{j}u||_{0}^{2} \\ + ϵ ∫_{Ω}[∂_{ω_{j}}, ∂ω̄_{j}](u)ūdV, +... for j ≤ q_{o}, \end{matrix}

\begin{matrix}||∂ω̄_{j}u||_{0}^{2} → & (1 − ϵ)||∂ω̄_{j}u||_{0}^{2} + ϵ||∂_{ω_{j}}u||_{0}^{2} \\ − ϵ ∫_{Ω}[∂_{ω_{j}}, ∂ω̄_{j}](u)ūdV + ... for q_{o} + 1 ≤ j ≤ n − 1. \end{matrix}

Hence, the estimate becomes

\begin{aligned}||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2} & ≥ ϵ\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ( ∑_{j≤n−1}||∂_{ω_{j}}u_{J}||_{0}^{2} + ∑_{j≤n}||∂ω̄_{j}u_{J}||_{0}^{2}) + Ru \\ & + \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}∑_{ij} ∫_{∂Ω} e^{−φ}r_{ij}u_{iK} ū_{jK}dS \\ & − (1 − ϵ)\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}} ∫_{∂Ω} r_{jj}|u_{J}|^{2}dS \\ & − ϵ\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{q_{o}+1≤j≤n−1} ∫_{∂Ω} r_{jj}|u_{J}|^{2}dS, \end{aligned} (1.10.6)

where Ru is an error term. We denote by ϵSu the term before Ru in the right side of (1.10.6). For suitable ϵ, the sum of the boundary integrals is positive because Ω is strongly q-pseudoconvex. Also, |Ru| ≤ \frac{ϵ}{2}Su + C_{ϵ}||u||_{0}^{2}. Thus, we get in conclusion

(1.10.7)  \frac{ϵ}{2}Su − C_{ϵ}||u||_{0}^{2} ≤ ||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2}.

By the strong q-pseudoconvexity of M, its Levi form has at least one positive eigenvalue: in particular, commutators of the ∂_{ω_{j}}’s and ∂ω̄_{j}’s for j ≤ n−1 generate the vector field T_{ℝ} = \frac{i}{\sqrt{2}}(∂_{ω_{n}} −∂ω̄_{n}) which belongs to ℂ⊗T M but is transversal to ℂ⊗T^{ℂ}M when restricted to M. Along the same lines as in the proof of (1.10.4), we get

(1.10.8)  |||T_{ℝ}u|||_{− \frac{1}{2}}^{2} <_{∼} ||∂_{ω_{j}}u||_{0}^{2} + ||∂ω̄_{j}u||_{0}^{2} + ||u||_{0}^{2}.

On the other hand, ∂ω̄_{n} is not in ℂ ⊗ T M since ∂ω̄_{n}r = 1. Thus, the system (∂_{ω_{j}})_{j≤n−1}, (∂ω̄_{j})_{j≤n} supplemented by T_{ℝ} spans the whole of ℂ ⊗ T ℂ^{n} which implies

(1.10.9)  |||u|||2_{\frac{1}{2}} + |||∂_{r}u|||_{− \frac{1}{2}}^{2} <_{∼} Su + ||u||_{0}^{2}.

Combining (1.10.9) with (1.10.7), we get the conclusion.

□

Remark 1.10.3. Estimate (1.10.5) is said to be the “\frac{1}{2}-subelliptic estimate” for (¯∂, ¯∂^{∗}). It ensures the “hypoellipticity” of the system (¯∂, ¯∂^{∗}), that is, the property

\begin{matrix}¯∂u ∈ (C^{∞}(Ω̄))_{z_{o}}^{k+1}, ¯∂^{∗}u ∈ (C^{∞}(Ω̄))_{z_{o}}^{k−1}, \\ u ∈ (H^{0}(Ω))_{z_{o}}^{k} ⇒ u ∈ C^{∞}(Ω̄)_{z_{o}}^{k}. \end{matrix}

This can be checked essentially by the same argument as in Theorem 2.4.5 below but requires a little extra work (cf. [61]). As a result of hypoellipticity, not only do we have a solution u ∈ (C^{∞}(Ω̄))^{k} to ¯∂u = f when f ∈ (C^{∞}(Ω̄))^{k} satisfies ¯∂f = 0, but any H^{0} solution u with ¯∂^{∗}u = 0, such as the canonical solution, is smooth in Ω̄. We regain, in the case of strongly pseudoconvex domains, that is, for q = 0, the celebrated conclusion that the Bergman projection acts continuously from (C^{∞}(Ω̄))^{k} into itself for any k ≥ 1.

The conclusion of Theorem 1.10.2 is classical since, when Ω is strongly q-pseudoconvex, then λ_{q+1} > 0 and hence s^{+} ≥ n − 1 − q. In this situation, and for any k ≥ q + 1, Ω enjoys the celebrated “Z(k) condition” (see [42]) which yields subelliptic estimates. We present now a new result (cf. [2]) which deals with the intermediate situation between q-pseudoconvexity and strong q-pseudoconvexity. Note that, in the absence of additional assumptions, the weak q-pseudoconvexity does not suffice for (1.10.5) as the example by Kohn shows (cf. the Notes of Section 1.9). Let Ω be q-pseudoconvex and let ∂_{ω_{jo}} be a vector field in V^{⊥} (the orthogonal bundle to V in T^{1,0}M for the metric). We will deal with the property

(1.10.10) (∑_{ij=1}^{n−1} r_{ij}(z)u_{iK} ū_{jK} − ∑_{j=1}^{q_{o}} r_{jj}(z)|u|^{2}) − \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}r_{j_{o}j_{o}}(z)|u_{j_{o}K}|^{2} ≥ 0

for any z ∈ M and u ∈ D ¯∂^{∗} of degree k ≥ q + 1.

In Theorem 1.10.2 it has been discussed the finite type 2. As for the higher type, we have

Theorem 1.10.4. Let Ω be a q-pseudoconvex domain and assume that for any vector field ∂_{ω_{jo}} of a system {∂_{ω_{j}}}_{j=q+1,...,n−1} ⊂ V^{⊥} of rank n − 1 − q, we have

(i) (1.10.10) is fulfilled, (ii) ∂_{ω_{jo}} is of finite type m_{j_{o}}.

Then, for k ≥ q + 1 and for any u ∈ C_{c}^{∞}(Ω̄ ∩ B)^{k} ∩ D ¯∂^{∗}, we have (1.10.11)

|||u|||_{ϵ}^{2}+|||∂_{r}u|||_{−1+ϵ} <_{∼} ||¯∂u||_{0}^{2}+||¯∂^{∗}u||_{0}^{2} if ϵ < \frac{1}{2^{m−1}} where m = max_{j_{o}} m_{j_{o}}.

Remark 1.10.5. According to Remark 2.4.8 below, we have that (1.10.11) holds in fact for any ϵ < \frac{1}{m}; note that, according to [32], ϵ = \frac{1}{m} is optimal.

Remark 1.10.6. The theorem applies in particular to a pseudoconvex domain, that is, for q = 0: ϵ-subelliptic estimates for ϵ = \frac{1}{2^{m−1}} (m = max_{j} m_{j}) hold over forms of degree k ≥ 1; in fact, they also hold for ϵ = \frac{1}{m} according to the remark above. But in this situation we have that complex curves have order of contact ≤ m with M and the conclusion follows from [33]; also in order to apply [33], (i) is needless. However our description of ϵ is more accurate and the method more elementary.

Remark 1.10.7. Similarly to the conclusions of Remark 1.10.3, the estimate (1.10.11) implies the hypoellipticity at the boundary for the system (¯∂, ¯∂^{∗}). In particular, the canonical solution that we already know from (1.9.25) to exist in H^{0}, is in fact smooth.

Proof of Theorem 1.10.4. We assume, by reordering, that V = Span{∂_{ω_{1}}, ..., ∂_{ω_{qo}}}. We fix ∂_{ω_{jo}} for which (i) and (ii) are satisfied and assume by reordering that ∂_{ω_{jo}} = ∂_{ω_{n−1}}. We go back to the basic estimate in Proposition 1.9.1 that we apply twice. Once as it stands and another time by interchanging (1.10.12)

\begin{matrix}∑ & ′ \\ |J|=k \end{matrix}||∂ω̄_{jo}u_{J}||_{0}^{2} = _{\begin{matrix}∑ & ′ \\ J∌j_{o} \end{matrix}}||∂ω̄_{jo}u_{J}||_{0}^{2}+\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}||∂_{ω_{jo}}u_{j_{o}K}||_{0}^{2}−\begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix} ∫_{∂Ω} r_{j_{o}j_{o}}|u_{j_{o}K}|^{2}dS.

In both cases, the term which involves boundary integrals can be discarded since it is positive. What we get is (1.10.13)

\{ \begin{aligned}(\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}}||∂_{ω_{j}}u_{J}||_{0}^{2}+ & \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{q_{o}+1≤j≤n}||∂ω̄_{j}u_{J}||_{0}^{2}) \\ & <_{∼} ||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2} + ||u||_{0}^{2}, \end{aligned} ; \begin{aligned}(\begin{matrix}∑ & ′ \\ |J|=k \end{matrix} ∑_{j≤q_{o}} & ||∂_{ω_{j}}u_{J}||_{0}^{2} + \begin{matrix}∑ & ′ \\ |J|=k \end{matrix} _{∑_{q_{o}+1≤j≤n} ; j≠j_{o} or j∉J}||∂ω̄_{j}u_{J}||_{0}^{2} + \begin{matrix}∑ & ′ \\ |K|=k−1 \end{matrix}||∂_{ω_{jo}}u_{j_{o}K}||_{0}^{2}) \\ & <_{∼} ||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2} + ||u||_{0}^{2}, \end{aligned}

in the two cases respective. We denote by S_{1} and S_{2} the two terms between brackets involving derivatives in the first and third line of (1.10.13) respectively. We select a multiindex K; by combining the two inequalities in (1.10.13) we get in particular

(1.10.14) ||∂_{ω_{jo}}u_{j_{o}K}||_{0}^{2} + ||∂ω̄_{jo}u_{j_{o}K}||_{0}^{2} <_{∼} ||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2} + ||u||_{0}^{2}.

Let T_{ℝ} = \frac{i}{\sqrt{2}}(∂_{ω_{n}} −∂ω̄_{n}) be the totally real vector field tangential to M. Since ∂_{ω_{jo}} has finite type m_{j_{o}}, we know that T_{ℝ} is obtained by m_{j_{o}} − 1 iterated commutators of ∂_{ω_{jo}} and ∂ω̄_{jo}. We then have (cf. Section 2.4)

\begin{aligned}|||T_{ℝ}u_{j_{o}K}|||_{−1+_{\frac{1}{2^{m_{jo} −1}}}}^{2} & <_{∼} ||∂_{ω_{jo}}u_{j_{o}K}||_{0}^{2} + ||∂ω̄_{jo}u_{j_{o}K}||_{0}^{2} + ||u||_{0}^{2} \\ & <_{∼} ||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2} + ||u||_{0}^{2}. \end{aligned} ; (1.10.15)

Recall what has already been said: subellipticity follows from finite type for systems of real vector fields. Therefore, the same is true for systems of complex vector fields when these are closed under conjugation such as the system ∂_{ω_{jo}} and ∂ω̄_{jo} (but it is not necessarily true for general complex systems). We carry out our proof. We reason in the same way for any index j of the system {∂_{ω_{j}}} ⊂ V^{⊥} for which (i) and (ii) of the statement of Theorem 1.10.4 hold. By changing the metric in M (which does not affect the ¯∂-Neumann conditions) we can assume that this system is orthonormal. We remark now that, u having degree k ≥ q +1, any coefficient u_{J} can be written as u_{J} = u_{jK} for some index j of the system or else u_{J} = u_{nK}. We start from the first; they satisfy (1.10.16)

|||T_{ℝ}u_{J}|||_{−1+ϵ}^{2} <_{∼} ||¯∂u||_{0}^{2} + ||¯∂^{∗}u||_{0}^{2} + ||u||_{0}^{2} for ϵ = inf_{j} 2^{−(m_{j}−1)}.

In this situation, we say that the energy, that is, the quantity in the right of (1.10.16) contains ϵ derivative T_{ℝ} of these coefficients. The energy also contains a full ∂_{ω_{j}} (resp. ∂ω̄_{j}) derivative for 1 ≤ j ≤ q_{o} (resp. q_{o} + 1 ≤ j ≤ n). The missing barred and unbarred derivatives, except from ∂_{ω_{n}}, can be “ϵ-interchanged” with the complemantary ones on account of

\begin{aligned}||||∂_{ω_{j}}u_{J}|||_{−1+ϵ}^{2} − |||∂ω̄_{j}u_{J}|||_{−1+ϵ}^{2}| & ≤ |([∂_{ω_{j}}, ∂ω̄_{j}]u_{J}, u_{J})_{−1+ϵ}| + ||u_{J}||_{0}^{2} \\ & <_{∼} ||T_{ℝ}u_{J}||_{−1+ϵ}^{2} + ||u_{J}||_{0}^{2}. \end{aligned} (1.10.17)

This concludes the proof of (1.10.11) for any coefficient u_{J} such that n ∉ J. As for the coefficients u_{nK}, we have that barred and unbarred derivatives are fully interchangeable on account of the ¯∂-Neumann condition u_{nK}|_{M} ≡ 0. In other terms, the energy contains the full ||u_{nK}||_{1}- norm.

□

We have a result of subellipticity for an important class of domains, the so-called “finite type decoupled” domains. We choose q = q_{o} and write z′ = (z_{1}, ...z_{q}) and z′′ = (z_{q+1}, ..., z_{n−1}).

Theorem 1.10.8. Let Ω be defined by r < 0 for r = 2Re z_{n} −

h(z′) + ∑^{n−1} ; j=q+1|f_{j}(z′′_{j})|^{2} where h_{ij}(z′) ≥ 0 in ℂ^{q_{o}} and the f_{j}(z′′_{j})’s are

holomorphic functions with vanishing order m_{j} at z_{o} = 0. Then ϵsubelliptic estimates hold in degree k ≥ q + 1 for ϵ = \frac{1}{2^{2m−1}} for m = max_{j} m_{j}.

Proof. Let π : ℂ^{n} → ℂ^{n−1} be the projection over the first n − 1 coordinates; since h and the f_{j}’s do not depend on z_{n}, that is Ω is “rigid”, then we have an identification

L_{r}|_{T^{ℂ}M} →^{∼}^{π′} ∂ ¯∂r|_{ℂ_{z_{1},...,z_{n−1}}^{n−1}}.

We take ∂_{ω_{j}} = ∂_{z_{j}} − r_{z_{j}}∂_{z_{n}}, j = 1, ..., n − 1 and ∂_{ω_{n}} = ∂_{z_{n}} as a basis of (1, 0) vector fields and its dual system ω_{1}, ..., ω_{n} = ∂r as a basis of (1, 0) forms; we choose a Hermitian metric in which this is orthonormal. In this basis we have

(r_{ij})_{i,j≤n−1} = [\begin{matrix}−(h_{ij}) & 0 \\ 0 & m_{j}^{2}|z′′_{j}|^{2(m_{j}−1)}κ_{ij} \end{matrix}] .

It readily follows that Ω is q-pseudoconvex and that (1.10.10) is fulfilled.

□

Example 1.10.9. Let z′ ∈ ℂ^{q}, z′′ ∈ ℂ^{n−1−q} and define

Ω = {z : y_{n} > −∑^{q} ; j=1|z′_{j}|^{2} + ∑^{n−1} ; j=q+1|z′′_{j}|^{2m_{j}}}.

Then (1.10.11) holds for k ≥ q + 1 and with ϵ = \frac{1}{2^{2m−1}} .

According to Remark 2.4.8, we have in fact subelliptic estimates for any ϵ < \frac{1}{2m}. Again, this result could be obtained from [33] but for a worse index ϵ.

Suggested research. Suppose that Ω is q-pseudoconvex and satisfies the finite contact order condition (in the sense of D’Angelo) with every (k − q_{o})-dimensional complex submanifold transversal to V. In this situation, do we have subelliptic estimates for forms of degree k ≥ q+1? Are the finite contact conditions necessary for the subelliptic estimates?

Notes. “Finite type” is here understood in the sense of bracket as was originally introduced by Kohn in [56]. Subsequently, Kohn introduced “finite ideal type” and proved that it is sufficient for subelliptic estimates. D’Angelo [37] proposed another notion of finite type through the order of contact of complex varieties. The necessity of this for subelliptic estimates was proved by Catlin who gave subsequently in [33] the ultimate characterization. All this literature deals with pseudoconvex domains. As for strongly q-pseudoconvex domains, subelliptic estimates were established by Folland and Kohn in [42] under the socalled “Z-condition” which turns out to hold for any k ≥ q + 1. Our approach is inspired by the guidelines [42] assimilated into the style of Section 1.9. For weakly q-pseudoconvex domains of finite type which satisfy the conditions of Theorem 1.10.4 the estimates have been recently established by Ahn, Baracco and Zampieri [2].

CHAPTER 2 Real Structures

Summary of Chapter 2. In Section 2.1 we start with analysis on Euclidean spaces and then pass to manifolds, spaces which locally look like ℝ^{n}. We then present, in Section 2.2, systems of vector fields and deal with the problem of their integration. It turns out that the formal integrability, the so-called involutivity, ensures the effective integrability (Frobenius theorem). Next, in Section 2.3, we present real symplectic manifolds and prove that they are all symplectically diffeomorphic to T^{∗}ℝ^{n} (Darboux theorem). We also give the canonical form of the submanifolds of T^{∗}ℝ^{n} over which the canonical 2-form σ has constant rank. In particular, this describes all involutive/isotropic/lagrangian submanifolds. We then present, in Section 2.4, the Hörmander numbers associated to a system of vector fields and the related condition of finite type m. We prove that it ensures subelliptic estimates with a gain of \frac{1}{2^{m−1}} derivatives; these latter, regardless of the amount of the gain, imply in turn local hypoellipticity. We end the discussion by introducing, in Section 2.5, the notion of the orbit of a set of vector fields. In case the set is C^{ω}, the orbit is uniquely defined and coincides with the integral manifold of the Lie span (Nagano theorem). Otherwise, for a smooth set, we can find in any case a unique minimal manifold, the local orbit, stable under the set (Sussmann’s theorem).

2.1. Euclidean spaces and their diffeomorphisms

Let F be a map of class C^{k}, k ≥ 1, from Ω ⊂ ℝ^{n} to Ω̃ ⊂ ℝ^{m}. Let x_{o} be a point of Ω and F′(x_{o}) : ℝ^{n} → ℝ^{m} be the tangent map to F at x_{o}. It is defined over tangent vectors v of ℝ^{n}, say column vectors, as

(2.1.1)  F′ : v → lim_{t→0} \frac{F (x_{o} + tv) − F (x_{o})}{t}.

Let (f_{j})_{j=1,...,m} be the components of F and let

JF = [\begin{matrix}∂_{x_{1}}f_{1} & . & . & . & ∂_{x_{n}}f_{1} \\ . & & & & . \\ . & & & & . \\ ∂_{x_{1}}f_{m} & . & . & . & ∂_{x_{n}}f_{m} \end{matrix}]

be the Jacobian matrix of F; sometimes, we also denote it by J_{x}F or ∂_{x}F. The linear map F′ can be identified with the matrix JF in the sense that

(2.1.2)  F′(v) = JF · v,

where in the right side we have the usual product between matrices and vectors. Let ^{t}F′(x̃_{o}) : ℝ^{m} → ℝ^{n} be the cotangent application to F at x̃_{o} = F (x_{o}) defined over cotangent vectors w at x̃_{o}, say column vectors w ∈ ℝ^{m}, by the identity over tangent vectors v at x_{o}

⟨^{t}F′(x̃_{o})w, v⟩_{x_{o}} = ⟨w, F′(x_{o})v⟩x̃_{o}.

Here ⟨·, ·⟩ denotes the duality between tangent and cotangent vectors. These dualities at x_{o} and x̃_{o} are in fact identified with the inner product which defines the euclidean structure in ℝ^{n} and ℝ^{m}

⟨w, v⟩ = ∑_{j} w_{j}v_{j}.

It is obvious that the matrix associated to the linear map ^{t}F′ is t(J_{x}F), the transpose of the Jacobian.

Remark 2.1.1. Sometimes, it is more convenient to denote cotangent vectors as row vectors ^{t}w; in this case the inner product is written

⟨w, v⟩ = ^{t}w · v

and the map ^{t}F′ is now represented by J_{x}F.

We now assume that F is a diffeomorphism of an open subset Ω ⊂ ℝ^{n}; this is also called a change of coordinates F : x ↦ x̃ = x̃(x). In particular, we now have n = m. Along with F′(x_{o}) : ℝ^{n} → ℝ^{m} we now also have at our disposal ^{t}F^{′−1}(x_{o})(= ^{t}(F^{−1})′(x_{o})) : ℝ^{n} → ℝ^{m}. (Because of inversion, ^{t}F^{′−1}(x_{o}) “goes in the same sense” as F′(x_{o}).) By means of F′ and ^{t}F^{′−1} we can thus describe the effect of a change of coordinates over tangent and cotangent vectors. How does it act over derivations and differentials? First, the coordinate derivatives transform according to the rule

(2.1.3)  ∂_{x_{j}} → ∑_{i} ∂_{x_{j}}(x̃_{i})∂x̃_{i}.

In fact, for a function f = f(x̃), under the change x̃ = x̃(x), the chain rule yields ∂_{x_{j}} (f(x̃(x))) = (∑_{i} ∂_{x_{j}}(x̃_{i})∂x̃_{i}f) (x̃(x)). For a general derivation along the direction v = (v_{j})_{j}, that is, ∂_{v} := ∑_{j} v_{j}∂_{x_{j}}, if we apply (2.1.3), we get

(2.1.4)  ∂_{v} → ∑_{i}(∑_{j} ∂_{x_{j}}(x̃_{i})v_{j})∂x̃_{i}.

But in the basis ∂x̃_{i} the coefficients on the right side of (2.1.4) are exactly those of ṽ := J_{x}x̃ · v. We have thus obtained

∂_{v} → ∂ṽ.

Hence, under coordinate change, derivations and tangent vectors transform accordingly. This allows us to identify, as we intensively do, vectors v and derivations ∂_{v} = ∑_{j} v_{j}∂_{x_{j}}. As for the basis of differential dx_{j}, dual to ∂_{x_{j}} in the inner product ⟨·, ·⟩, the chain rule applied to the inverse transformation x_{j} = x_{j}(x̃) yields

dx_{j} → ∑_{i} ∂x̃_{i}(x_{j})dx̃_{i}.

For a general differential, that is, a combination w = ∑_{j} w_{j}dx_{j}, we get

(2.1.5)  w → ∑_{i} (∑_{j} ∂x̃_{i}(x_{j})w_{j}) dx̃_{i}.

But the coefficients on the right are those of (^{t}Jx̃x)w = ^{t}(J_{x}x̃)^{−1}w which we also denote by w̃, and therefore (2.1.5) reads

w → w̃.

We are ready to enlarge the setting of our analysis and deal with the diffeomorphic images of the euclidean space which is the local structure of an immersed manifold. In greater generality, a manifold M is the union of the homeomorphic images U_{j} →^{∼}_{Φ_{j}} M_{j} of open sets U_{j} ⊂ ℝ^{n} such

that if Φ_{jk} (resp. Φ_{kj}) is the homeomorphism induced by Φ_{j} (resp. Φ_{k}) on overlappings M_{j} ∩ M_{k} ≠ ∅, then

Φ_{jk} ∘ Φ_{kj}^{−1} is a diffeomorphism.

We are mainly interested in immersed manifolds, which, in real geometry, is not a significant restriction.

Definition 2.1.2. A real l-codimensional submanifold M of ℝ^{n} of class C^{k}, k ≥ 1, is, in a neighborhood B ⊂ ℝ^{n} of each point x_{o} ∈ M, the image U →^{∼}_{Φ} M ∩ B of an open set U ⊂ ℝ^{n−l} under a map Φ whose differential has maximal rank.

Thus, M is locally a C^{k} smooth deformation of an open set of the euclidean space ℝ^{n−l}. The mapping Φ is a diffeomorphism of U with its image and is called a local chart or a local parametrization of M and its inverse Φ^{−1} is called a local system of coordinates. There is another possible definition, so-called implicit, of a manifold.

Definition 2.1.3. A subset M of ℝ^{n} is said to be a submanifold of codimension l and class C^{k} if for any x_{o} ∈ M there is V ⊂ ℝ^{n} and a system of C^{k} functions g_{1}, ..., g_{l} with independent differentials such that M′ = {x ∈ V : g_{j}(x) = 0 for any j}.

We want to show that the parametrization can be chosen to be “cartesian”, that is, with U being an open set of a coordinate plane in a group x′ of variables or equivalently that the equations can take the form x′′ − E(x′) = 0 where the x′′ are the complementary variables. (For this reason it has been better to take U instead of the whole of ℝ^{n−l} in Definition 2.1.2.)

Theorem 2.1.4 (Implicit function theorem). Definitions 2.1.2 and 2.1.3 are equivalent.

Proof. We set G = ^{t}(g_{1}, ..., g_{l}) and denote by JΦ and JG the Jacobians of Φ and G, respectively; these are n × (n − l) and l × n matrices, respectively. Remember that according to Definitions 2.1.2 and 2.1.3 they are assumed to have rank n−l and l, respectively. Thus, we can choose a group of n − l variables x′ and the complementary variables x′′ such that J_{x′}Φ and J_{x′′}G have at x_{o} maximal rank n − l and l, respectively. In both cases it follows that the projection π : ℝ^{n} → ℝ^{n−l}, x ↦ x′ induces a local diffeomorphism M′ →^{∼}_{π} U between an open

neighborhood M′ of x_{o} on M and an open set U ⊂ ℝ^{n−l}. We denote by x′ ↦ (x′, E(x′)) the inverse of π|_{M′}: thus, M′ is now parametrized by Φ = id × E or defined by G = 0 for G = x′′ − E(x′).

□

We also deal with manifolds with boundary M^{+}. They are manifolds in the usual sense at interior points whereas, at a boundary point x ∈ ∂M^{+}, there is a diffeomorphism U^{+} →^{∼}_{Φ} M^{+} where U^{+} is a half-space in

ℝ^{n−l} and Φ extends as a diffeomorphism beyond the boundary. They are therefore manifolds truncated by a transversal inequality g_{l+1} > 0.

The tangent space to M at x, denoted by T_{x}M, is defined as the space of vectors of ℝ^{n} proportional to the limits

(2.1.6)  v = lim_{x_{ν}∈M^{xν →x}} \frac{x_{ν} − x}{|x_{ν} − x|}.

Since M is defined by G = 0, then differentiation yields

(2.1.7)  T_{x}M = {v ∈ ℝ^{n} : JG · v = 0}.

On the other hand, differentiation of the identity G ∘ Φ ≡ 0 yields range(JΦ) ⊂ ker(JG); but the two spaces have the same dimension by our assumption, which implies

(2.1.8)  range(JΦ) = ker(JG).

So far, there could be some ambiguity in our description of tangent vectors. First, if we change coordinates in ℝ^{n}, how does this affect the limit of our differences (2.1.6)? Next, if we change parametrization Φ or equation G = 0, how does this affect the description of T_{x}M? Now, we have seen that under a change x̃ = x̃(x) the limits change as ṽ = J_{x}x̃·v. As for the case of two different parametrizations

\xymatrix{U_{1} \ar[dd]_{Θ} \ar[dr]^{Φ_{1}}_{∼} \\  & M′, \\ U_{2} \ar[ur]^{∼}_{Φ_{2}}}

differentiation of the identity Φ_{2} = Φ_{1} ∘ Θ^{−1} yields

JΦ_{2} = JΦ_{1} · (JΘ)^{−1}.

Last, two different systems of equations G = 0 and G^{1} = 0 would be related by g_{j}^{1} = ∑_{i} α_{ji}g_{i} for an invertible square matrix (α_{ji})_{j,i=1,...,l}. This yields

JG^{1} = J(α_{ji})_{ji} · JG,

which shows again the independence of the tangent space from the choice of the equation of M.

The distribution of the tangent planes T_{x}M, x ∈ M, is called the tangent bundle and is denoted by T M. The distributions T_{x}^{∗}M, x ∈ M, of linear forms on T_{x}M, is called the cotangent bundle and is denoted by T^{∗}M. A C^{k} smooth section L ∈ T M, resp. θ ∈ T^{∗}M, is called a vector field, resp. a differential form. It consists of a smooth function which assigns to each x ∈ M a vector L_{x} ∈ T_{x}M, resp. θ_{x} ∈ T_{x}^{∗}M. In a local coordinate system x = (x_{1}, ..., x_{n}), L and θ can be represented by

L_{x} = ∑_{j} a_{j}(x)∂_{x_{j}} and θ_{x} = ∑_{j} α_{j}(x)dx_{j},

where each coefficient a_{j} and α_{j} is defined over a local chart U → M′, t ↦^{Φ} x(t) and depends in C^{k} fashion on t. A vector field L is described by its action over smooth functions f that we denote by L(f) or Lf, and a differential θ is described by the pairing with vector fields ⟨L, θ⟩. Sometimes, different coordinate systems are in use near the same x_{o} ∈ M. Our previous work explains how the coordinates of a vector field or a differential are described in two different systems x on M′ and x̃ on M′′ at common points of M′ ∩M′′. Let L be written as ∑_{j} a_{j}∂_{x_{j}} and ∑_{j} b_{j}∂x̃_{j}: we have already seen that the coefficients are related by (b_{j}) = (J_{x}x̃) · (a_{j}). (This could be reproved by remarking that ∂x̃_{j} x̃_{k} = κ_{jk} (Kroneker symbol), and therefore

\begin{aligned}b_{j} & = L(x̃_{j}) \\ & = ∑_{k} a_{k}∂_{x_{k}}(x̃_{j}).) \end{aligned}

In the same way, if θ is written in the two systems as θ = ∑_{j} α_{j}dx_{j} and θ = ∑_{j} β_{j}dx̃_{j}, then the coefficients are related by (β_{j}) = ^{t}(Jx̃x)(α_{j}).

Let F be a C^{k} map from a manifold M to a manifold N. Recall that tangent vectors (cotangent vectors) are the same as derivations (differentials). Our previous discussion explains that F induces a tangent map

T M → M ×_{N} T N, (x, v) ↦ (x, F (x); F′(x) · v),

whose action over a vector field L on M, denoted by F_{∗}L and called the “push forward” of L by F, is defined by means of composition with F:

F_{∗}L(g) = L(g ∘ F) for any g smooth on N.

We identify F_{∗} = F′. In the same way, F induces a cotangent map

M ×_{N} T^{∗}N → T^{∗}M, (x, F (x); w) ↦ (x,^{t}F′(x) · w),

whose action on smooth sections θ, called the “pull back” of θ and denoted by F^{∗}θ, is given by

⟨F^{∗}θ, L⟩_{x} = ⟨θ, F_{∗}L⟩_{F (x)} for any L.

Again, we identify F^{∗} = ^{t}F′. Note that under the choice F = Φ, where Φ is a parametrization of M, we can define by the aid of F_{∗} and F^{∗} two “distinguished” bases {L_{j}} and {θ_{j}} of T M and T^{∗}M, dual to each other. If t = (t_{j})_{j=1,...,n−l} are coordinates in ℝ^{n−l}, these are defined by

L_{j} = F_{∗}∂_{t_{j}}, j = 1, ..., n − l,

resp.

θ_{j} = (F^{−1})^{∗}dt_{j}, j = 1, ..., n − l.

2.2. Integration of vector fields and vector bundles—Frobenius theorem

A vector field of class C^{m}, m ≥ 1, over an open domain Ω ⊂ ℝ^{n}, is a C^{m}-section of T ℝ^{n}|_{Ω}; in a system of local coordinates x it takes the form L = ∑_{j} a_{j}(x)∂_{x_{j}} with the a_{j}’s of class C^{m}. We know from the local existence and uniqueness of the solution of the Cauchy problem that for any point x ∈ Ω there is a unique integral curve of L, that is, a curve Γ parametrized, say, by t ↦ φ(t), such that φ̇(t) is parallel to L(φ(t)) for any t. Also, if Σ is any hypersurface transversal to L, that is, satisfying T_{x}Σ ∩ L(x) = {0} for any x, then we have a smooth parametrization of this system of curves by means of the intersection point ξ of Γ with Σ. In other words, we have a local diffeomorphism

(2.2.1)  (ξ, t) ↦ x = φ_{ξ}(t).

This tells us that (ξ, t) serves as a new system of coordinates in which the integral curves Γ_{ξ} are “straightened”, that is, they take the trivial equation

ξ ≡ const.

We want to generalize this conclusion to general systems of vector fields.

Definition 2.2.1. A Frobenius distribution or bundle E of rank k on Ω is a distribution of vector spaces E_{x}, x ∈ Ω, of constant dimension k.

Let L be a vector field; if L(x), which we also denote L_{x}, belongs to E_{x} for any x, then L is said to be a “section” of E.

Definition 2.2.2. A Frobenius distribution or bundle E of rank k over Ω is said to have class C^{m} when it is spanned by a system of vector fields L_{1}, ..., L_{k} of class C^{m}, a basis of E.

A vector field L is in a natural way an operator of derivation: its action on a smooth function g is denoted by L(g).

Definition 2.2.3. The commutator or Lie bracket of two vector fields L and W is defined by

[L, W] = L ∘ W − W ∘ L.

Proposition 2.2.4. Let L = ∑_{j} a_{j}(x)∂_{x_{j}} and W = ∑_{j} b_{j}(x)∂_{x_{j}} be vector fields of class C^{m}, m ≥ 2, on a domain Ω ⊂ ℝ^{n}. We have

(2.2.2)  [L, W] = ∑^{n} ; j=1 (∑^{n} ; i=1a_{i}∂_{x_{i}}(b_{j}) − b_{i}∂_{x_{i}}(a_{j})) ∂_{x_{j}}.

Proof. Let g be a smooth function on Ω. We have

\begin{aligned}∑^{n} ; j,i=1 (a_{i}∂_{x_{i}}(b_{j}) & − b_{i}∂_{x_{i}}(a_{j})) ∂_{x_{j}}(g) \\ & = ∑^{n} ; i=1a_{i}∂_{x_{i}} (∑^{n} ; j=1b_{j}∂_{x_{j}}(g)) − ∑^{n} ; i,j=1a_{i}b_{j}∂_{x_{i}x_{j}}^{2}(g) \\ & − ∑^{n} ; i=1b_{i}∂_{x_{i}} (∑^{n} ; j=1a_{j}∂_{x_{j}}(g)) + ∑^{n} ; j,i=1a_{j}b_{i}∂_{x_{j}x_{i}}^{2}(g) \\ & = ∑^{n} ; i=1a_{i}∂_{x_{i}} (∑^{n} ; j=1b_{j}∂_{x_{j}}(g)) − ∑^{n} ; i=1b_{i}∂_{x_{i}} (∑^{n} ; j=1a_{j}∂_{x_{j}}(g)) \\ & = (L ∘ W − W ∘ L)(g) = [L, W](g), \end{aligned} (2.2.3)

where the second equality follows from the commutativity of the second derivatives ∂_{x_{i}x_{j}}^{2}(g) = ∂_{x_{j}x_{i}}^{2}(g).

□

Remark 2.2.5. The Lie bracket [L, W] between vector fields of class m ≥ 2 enjoys the following properties whose proof is straightforward:

(i) [L, W] = −[W, L] and in particular [L, L] = 0 (alternance). (ii) [L + W, Z] = [L, Z] + [W, Z] and the same for the second term

(additivity). (iii) For C^{1} functions f and g, we have [fL, gW] = fL(g)W −

gW (f)L + fg[L, W]. (iv) [L, [W, Z]] + [W, [Z, L]] + [Z, [L, W]] = 0 (Jacobi’s identity).

Definition 2.2.6. A C^{m}, m ≥ 2, bundle E is said to be “involutive” or “integrable” when for any pair of sections L, W of E, their Lie bracket [L, W] is still a section of E.

Thus, E is involutive when it is stable under Lie bracket. To check the involutivity of E, it suffices to test the property over the elements of a basis. In fact we have

Proposition 2.2.7. Let L_{1}, ..., L_{k} be a basis of E, that is, at any x ∈ Ω, the vectors L_{1}(x), ..., L_{k}(x) are a basis of E_{x}. Then, E is involutive if (and only if) for any j = 1, ..., k the Lie bracket [L_{i}, L_{j}] is a section of E.

Proof. We have to prove only the “if” part of the statement. For this, we remark that for general vector fields L = ∑_{j} a_{j}L_{j}, W = ∑_{j} b_{j}L_{j}, we have

\begin{aligned}[L, W] & = [∑_{i} a_{i}L_{i}, ∑_{j} b_{j}L_{j}] \\ & = ∑_{i,j} a_{i}b_{j}[L_{i}, L_{j}] + ∑_{i,j} a_{i}L_{i}(b_{j})L_{j} − ∑_{i,j} b_{j}L_{j}(a_{i})L_{i}. \end{aligned}

The last two terms in the second equality belong to E. Thus, we have the fact that [L, W] ∈ E is reduced to [L_{i}, L_{j}] ∈ E for any i, j.

□

Definition 2.2.8. A k-dimensional manifold Γ is said to be an integral leaf of the bundle E when T_{x}Γ = E_{x} for any x ∈ Γ.

Our problem is to state under which condition on E there is the unique integral leaf of E through each point. If E has rank k = 1, we have already seen that the problem has a positive answer. In the general situation, the answer is contained in the following

Theorem 2.2.9 (Frobenius theorem). Let E be a bundle on Ω of rank k and class C^{m}, m ≥ 2. The following are equivalent:

(i) E is involutive.

(ii) For any x_{o} ∈ Ω there is a unique local integral leaf Γ of E

passing through x_{o}. (iii) For any x_{o} ∈ Ω there are local coordinates x = (x′, x′′), x′ = (x′_{1}, ..., x′_{k}), such that E = Span{∂_{x′_{1}}, ..., ∂_{x′_{k}}} in a neighborhood of x_{o}.

Remark 2.2.10. A system of manifolds {Γ} as in (ii), that is, a local covering of Ω by disjoint manifolds, is said to be a “local foliation” of Ω. According to (iii), it is possible to reduce a foliation, under a local coordinate change, to the system of k-planes defined by x′′ = const. In fact, reducing the bundle to Span{∂_{x′_{j}}}_{j=1,...,k} is equivalent to reducing its integral leaves to x′′ = const. Thus, under a C^{1} diffeomorphism, the leaves of a foliation are always “straightenable”. We remark now that the changes of coordinates which preserve the leaves x′′ = const, or the bundle Span{∂_{x′_{j}}}_{j=1,...,k}, are those in the form

(2.2.4)  \begin{cases}x̃′ = x̃′(x′, x′′), \\ x̃′′ = x̃′′(x′′). \end{cases}

We also remark that these changes do not need to preserve each single ∂_{x′_{j}} but, as has already been said, the set of their linear combinations. This freedom is the key point in the proof of Theorem 2.2.9, which consists of two steps: linear algebra to adjust the basis {L_{j}} so that the terms commute, that is, [L_{i}, L_{j}] = 0 and a coordinate change of type (2.2.4) provided by the solution of Cauchy’s problem, which transforms L_{j} for j ≤ k into ∂_{x′_{j}}.

We start from this latter point, which we present as a separate statement.

Proposition 2.2.11. Let L_{j}, j = 1, ..., k, be a system of independent vector fields which satisfy

[L_{i}, L_{j}] = 0, i, j = 1, ..., k.

Then, in new local coordinates x, we have L_{j} = ∂_{x′_{j}}, j = 1, ..., k.

Proof. We reason by induction on k. When k = 1, on one hand we have a bundle spanned by a single vector field which is always involutive. On the other hand, this can always be reduced to a coordinate derivative ∂_{x_{1}} by Cauchy’s theorem. Suppose now that the proposition is true for k. There are coordinates x = (x′, x′′) with x′ = (x′_{1}, ..., x′_{k}), x′′ = (x′′_{k+1}, ..., x′′_{n}), such that L_{1} = ∂_{x′_{1}}, ..., L_{k} = ∂_{x′_{k}}. Let L_{k+1} = ∑_{j} a_{j}(x)∂_{x_{j}}; from the hypothesis [L_{k+1}, L_{j}] = 0 for j ≤ k, it follows that a_{j}(x) = a_{j}(x′′). We may also assume without loss of generality that a_{k+1} ≠ 0. We look for new coordinates x̃ such that L_{j} = ∂_{x_{j}} → ∂x̃_{j} for j ≤ k and L_{k+1} → ∂x̃_{k+1}. To obtain the first, the transformation must be of type (2.2.4). Moreover, since we are requiring in fact that each single ∂_{x_{j}}, j ≤ k, is preserved, we must ask that the transformation be of the type

(2.2.5)  \begin{cases}x̃′_{j} = x_{j} + b_{j}(x′′), \\ x̃′′ = x̃′′(x′′). \end{cases}

As for the second point, the reduction of L_{k+1} to ∂x̃_{k+1}, we are asking that

∑_{j} a_{j}∂_{x_{j}} = ∑_{j} ∂x̃_{k+1}(x_{j})∂_{x_{j}},

that is,

∂x̃_{k+1}(x_{j}) = a_{j}, j = 1, ..., n.

Now, to find x = x(x̃), whose inverse is the transform we are looking for, we just have to solve the ordinary system of differential equations in the variable x̃_{k+1}, with the other arguments x̃_{j} as parameters, with initial values at x̃_{k+1} = 0:

(2.2.6)  \begin{cases}∂x̃_{k+1}(x_{j}) = a_{j}, j = 1, ..., n, \\ x_{j}|_{x̃_{k+1}=0} = \{ \begin{matrix}x̃_{j}, j ≠ k + 1, \\ 0, j = k + 1. \end{matrix} \end{cases}

Since a_{j} = a_{j}(x′′), then the solutions of (2.2.6) satisfy x′′ = x′′(x̃′′) which is the crucial point of the proof. These solutions yield the explicit description of the change of coordinates

(2.2.7) \begin{cases}x_{j} = x̃_{j} + a_{j}(x′′(ξ̃^{j}, x̃′′_{k+2}, ..., x̃′′_{n}))x̃_{k+1} for j ≠ k + 1, \\ x_{k+1} = a_{k+1}(x′′(ξ̃^{k+1}, x̃′′_{k+2}, ..., x̃′′_{n}))x̃_{k+1}, \end{cases}

where ξ̃^{j} is the point in the interval (0, x̃′′_{k+1}) of mean value for ∂x̃_{k+1}x_{j}(·, x̃′′_{k+1}, ..., x̃′′_{n}); thus (2.2.7) is certainly of type (2.2.5).

□

Proof of Theorem 2.2.9. (iii) ⇒ (ii) is obvious. (ii) ⇒ (i) is easy: if r_{j} = 0 is a system of equations for Γ with T_{x}Γ = E_{x}, then from Lr_{j}|_{Γ} ≡ 0 and W r|_{Γ} ≡ 0, we get (L ∘ W − W ∘ L)r_{j}|_{Γ} ≡ 0.

(i) ⇒ (iii): let L_{1}, ..., L_{k} be a basis for E and write L_{i} = ∑_{j} a_{ij}(x)∂_{x_{j}}. We want to find a new basis which commutes. The matrix (a_{ij}(x))_{ij} has rank k at every point. By a linear change of coordinates, we can assume that the k × k submatrix of the first k columns is non-degenerate. In particular,

Span{L_{1}, ..., L_{k}} ∩ Span{∂_{x_{k+1}}, ..., ∂_{x_{n}}} = {0}.

We can then solve in the unknown (z_{i}^{h})_{i=1,...,k} the linear systems

∑_{i=1}^{k} a_{ij}z_{i}^{h} = κ_{jh}, j = 1, ..., k,

where κ_{jh} is the Kronecker symbol. These are a set of k systems with

index h = 1, ..., k. We then put L̃_{h} = ∑^{k} ; i=1z_{i}^{h}L_{i}. We have

\begin{aligned}L̃_{h} & = ∑^{k} ; i=1z_{i}^{h}∑^{n} ; j=1a_{ij}(x)∂_{x_{j}} \\ & = ∑^{k} ; i=1z_{i}^{h} (∑^{k} ; j=1a_{ij}(x)∂_{x_{j}} + ∑^{n} ; j=k+1a_{ij}(x)∂_{x_{j}}) \\ & = ∑^{k} ; j=1 (∑^{k} ; i=1a_{ij}(x)z_{i}^{h}) ∂_{x_{j}} + ∑^{n} ; j=k+1 (∑^{k} ; i=1a_{ij}(x)z_{i}^{h}) ∂_{x_{j}} \\ & = ∂_{x_{h}} + ∑^{n} ; j=k+1c_{hj}∂_{x_{j}}, \end{aligned} ; (2.2.8)

where we have used the notation c_{hj} := ∑^{k} ; i=1a_{ij}(x)z_{i}^{h} for any j ≥ k +

1. Hence, [L̃_{i}, L̃_{j}] belongs to Span{∂_{x_{k+1}}, ..., ∂_{x_{n}}} but it also belongs to Span{L̃_{1}, ..., L̃_{k}} which is transversal to the former. It follows that [L̃_{i}, L̃_{j}] = 0; we can thus apply Proposition 2.2.11, which yields the conclusion of the proof of Theorem 2.2.9.

□

For further application to symplectic geometry we need the following variant.

Corollary 2.2.12. Let E = Span{L_{1}, ..., L_{n}} be an involutive bundle and let the constants c_{ijr} be defined by the relations [L_{i}, L_{j}] = ∑_{r} c_{ijr}L_{r}. Let h_{1}, ..., h_{k} ∈ C^{∞} satisfy

(2.2.9)  L_{i}h_{j} − L_{j}h_{i} = ∑_{r} c_{ijr}h_{r}.

Let S be a manifold with tangent plane supplementary to E and pick a function u_{o} ∈ C^{∞}(S). Then, there is a unique solution u ∈ C^{∞} of the initial value problem

(2.2.10)  \begin{cases}L_{j}u = h_{j}, j = 1, ..., k, \\ u|_{S} = u_{o}. \end{cases}

Note that (2.2.9) in clearly necessary for (2.2.10) (regardless of the boundary condition on S).

Proof. The hypothesis (2.2.9) as well as the conclusion (2.2.10) is obviously invariant under change of coordinates. According to Theorem 2.2.9, we can find coordinates in which E = Span{∂_{x_{n−k+1}}, ...∂_{x_{n}}}. It then suffices to prove the theorem for the system ∂_{x_{n−k+1}}, ..., ∂_{x_{n}}. Under this choice, (2.2.9) becomes

∂_{x_{i}}h_{j} − ∂_{x_{j}}h_{i} = 0.

By this, for fixed x_{1}, ..., x_{n−k}, the form ∑_{j≥n−k+1}h_{j}dx_{j} is closed. Let S be

graphed over x′ = x_{1}, ..., x_{n−k} by x′′ = s(x′) and consider the function

u(x) = u_{o}(x′, s(x′)) + ∫_{s(x′)}^{x′′} ∑_{j≥n−k+1}h_{j}dx_{j},

where integration is performed along a curve of ℝ^{k} connecting s(x′) to x′′. This is well defined and provides the unique solution to (2.2.10).

□

Example 2.2.13. Let M ⊂ ℝ^{n} be a manifold of class C^{2}. We can see that in the neighborhood of any point of M there is a C^{1} “orthogonal projection”:

p : ℝ^{n} → M, x ↦ p(x),

which associates to x the point of minimal distance p(x) on M. In fact, let T_{x}M, x ∈ M, be the tangent space at x and T_{x}M^{⊥} the orthogonal space in the euclidean product. We set

(T M^{⊥})_{ϵ} = {(x, ξ) : x ∈ M, ξ ∈ (T_{x}M)^{⊥}, |ξ| = ϵ}, (2.2.11) (ℝ^{n} \ M)_{ϵ} = {x ∈ ℝ^{n} \ M : d(x, M) = ϵ}, (2.2.12)

where d is the euclidean distance. We then have

(ℝ^{n} \ M)_{ϵ} →^{∼} (T M^{⊥})_{ϵ} ↠ M, (2.2.13)

y ↦ (x, ξ) ↦ x, (2.2.14)

where the first correspondence identifies y to the unique (x, ξ) such that y = x + ξ and the second is the projection on the first component. Note that we have a natural foliation of a neighborhood of ℝ^{n} \ M by the family of hypersurfaces

M_{ϵ} = {x ∈ ℝ^{n} \ M : d(x, M) = ϵ}.

We have

\begin{aligned}T_{x}M_{ϵ} & = (x − p(x))^{⊥} \\ & = T_{p(x)}M. \end{aligned}

Thus, the family {M_{ϵ}}_{ϵ} is the system of the integral leaves of

(2.2.15)  E_{x} = T_{p(x)}M.

In particular, the bundle E defined by (2.2.15) is involutive. Note however that not every bundle E defined by (2.2.15), without some additional assumptions about the “projection” p, that in the previous discussion is “orthogonal”, is involutive. To prove it, we reason backwards. We take a non-integrable (n − 1)-dimensional bundle E in ℝ^{n} and choose M as the unit sphere. We define locally p : ℝ^{n} \ M → M by continuously sending x to either of the two points p(x) ∈ M which satisfies (2.2.15) or equivalently

p(x) − 0 = E_{x}^{⊥}.

Though E is related to a projection p by means of (2.2.15), it is nevertheless non-integrable by the way we chose it.

2.3. Real symplectic spaces—Frobenius-Darboux theorem

Let M be ℝ^{n} or a real submanifold of dimension n in ℝ^{n+l}. Let T M be the tangent bundle to M and T^{∗}M its dual bundle, the cotangent bundle. The fiber T_{x_{o}}^{∗}M at x_{o} ∈ M is the space of differentials in x_{o}, that is, the linear forms on T_{x_{o}}M. Take coordinates x in M and the canonically associated coordinates (x, ξ) in T^{∗}M: the ξ are the coordinates of a form in the canonical basis dx_{1}, ..., dx_{n}. Remember that if the base coordinates change as x̃ = f(x), then the ξ change through ξ̃ = (^{t}f′)^{−1}(x)·ξ (the ξ are here column vectors). We define on T^{∗}M a “symplectic homogeneous structure” by means of the canonical 1-form

ω = ∑^{n} ; j=1ξ_{j}dx_{j}

and the canonical 2-form

σ = dω = ∑^{n} ; j=1dξ_{j} ∧ dx_{j},

respectively. We often write ξdx and dξ ∧dx instead of the above sums.

Exercise 2.3.1. Let π : T^{∗}M → M be the projection, let ^{t}π′ : T^{∗}M ×_{M} T^{∗}M ↪ T^{∗}T^{∗}M be the transposed injection and let δ : T^{∗}M ↪ T^{∗}M ×_{T^{∗}M} T^{∗}M be the diagonal immersion. Prove that

ω(x, ξ) = ^{t}π′ ∘ δ(x, ξ).

(Hint: in coordinates, ^{t}π′ is the mapping

((x, ξ), (x, η)) ↦ (x, ξ; ηdx, 0dξ),

2.3. REAL SYMPLECTIC SPACES—FROBENIUS-DARBOUX THEOREM 91

δ the embedding (x, ξ) ↦ ((x, ξ), (x, ξ)) and hence ^{t}π′ ∘ δ(x, ξ) = ξdx.)

Thus we learned from Exercise 2.3.1 that the forms ω and σ can be intrinsically defined without use of coordinates. A change of coordinates F : (x, ξ) → (x̃, ξ̃), that is, a C^{1} diffeomorphism of T^{∗}M, is said to be “symplectic” when it preserves σ, that is, (F^{−1})^{∗}σ = σ̃.

Note that σ is non-degenerate, that is, if v is a vector of T T^{∗}M such that σ(v, ·) ≡ 0 on T T^{∗}M, then v = 0. Let ⟨·, ·⟩ be the duality between T T^{∗}M and T^{∗}T^{∗}M. Associated to σ, there is therefore a “Hamiltonian” isomorphism which interchanges the duality with the inner product σ:

(2.3.1) H : T^{∗}T^{∗}M → T T^{∗}M, u ↦ Hu, where Hu is defined by σ(v, Hu) = ⟨v, u⟩, for any v ∈ T T^{∗}M.

In coordinates (x, ξ) of T^{∗}M and for u = u′dx + u′′dξ, the isomorphism H is written as

H(u′dx + u′′dξ) = u′′∂_{x} − u′∂_{ξ}.

For a real smooth function f = f(x, ξ) on T^{∗}M, we define its Hamiltonian vector field by

\begin{aligned}H_{f} & = H(df) \\ & = ∂_{ξ}f∂_{x} − ∂_{x}f∂_{ξ}. \end{aligned}

We define the “Poisson bracket” of two functions on T^{∗}M by

\begin{aligned}\{f, g\} & = H_{f}(g) \\ & = ∂_{ξ}f∂_{x}g − ∂_{x}f∂_{ξ}g. \end{aligned}

We have the equality

H_{{f,g}} = [H_{f}, H_{g}],

whose proof is straightforward. In particular, the product {·, ·} enjoys the same properties as [·, ·]: alternancy, additivity, Jacobi identity. For a change of coordinates F : (x, ξ) ↦ (x̃, ξ̃), we can check whether it is symplectic just by inspecting if it preserves the Poisson relations

{x̃_{i}, x̃_{j}} = 0, {ξ̃_{i}, ξ̃_{j}} = 0, {x̃_{i}, ξ̃_{j}} = κ_{ij}.

Note that these relations are trivial for the initial system (x, ξ).

We use the superscript ^{⊥ σ} to denote the orthogonal with respect to σ.

Definition 2.3.2. A submanifold V of T^{∗}M is said to be involutive if and only if T V^{⊥ σ} ⊂ T V.

Remark 2.3.3. Let V be defined by f_{1} = 0, ...., f_{l} = 0 with df_{1}∧...∧ df_{l} ≠ 0 (that is, rank^{t}(f_{1}...f_{l})′ = l). Then T V^{⊥ σ} = Span{H_{f_{1}}, ..., H_{f_{l}}}. In fact, Span{df_{1}, ..., df_{l}} is the orthogonal space to V in the duality ⟨·, ·⟩, and H interchanges this duality with the inner product σ.

The following relates the notion of involutivity of manifolds to the former involutivity of bundles.

Theorem 2.3.4. The submanifold V ⊂ T^{∗}M is involutive if and only if the bundle T V^{⊥ σ} is involutive, that is, closed under [·, ·].

Proof. The submanifold V is involutive iff

H_{f_{i}}(f_{j})|_{V} ≡ 0 for any i, j

iff

{f_{i}, f_{j}}|_{V} ≡ 0 for any i, j

iff

{f_{i}, f_{j}} = ∑^{l} ; h=1c_{ij}^{h}f_{h} for any i, j and for suitable coefficients c_{ij}^{h}

iff

[H_{f_{i}}, H_{f_{j}}] = ∑^{l} ; h=1c_{ij}^{h}H_{f_{h}} for any i, j and for suitable coefficients c_{ij}^{h}

iff

[H_{f_{i}}, H_{f_{j}}] ∈ Span{H_{f_{1}}, ..., H_{f_{l}}}
= T V^{⊥ σ} for any i, j.
□

The above proof shows that involutivity is also characterized by {f_{i}, f_{j}}|_{V} ≡ 0 for f_{i} and f_{j} ranging through a system of equations of V. We can define a symplectic homogeneous structure in an abstract, conic, real even-dimensional manifold Y just by requiring the presence of a closed non-degenerate homogeneous 2-form σ; in this manifold Y, we can define accordingly involutive submanifolds V. But this is only an illusory generalization: locally, Y still turns out to be T^{∗}ℝ^{n} under different coordinates. Before proving it, we want to define the above notions properly . We say that a manifold Y is conic if it is locally parametrized over a cone of ℝ^{m} \ {0} by a mapping Φ and endowed with a “multiplication” M_{t} for t ∈ ℝ̇^{+}, the push-forward via Φ_{∗} of the multiplication by t in ℝ^{m} \ {0}. In this situation, Φ^{−1} is a system of homogeneous coordinates (of degree 1). When we choose Y = T^{∗}M \(M ×{0}), the conic structure consists in the multiplication only along the fibers. Thus, the homogeneous map Φ^{−1} is expressed, in terms of usual coordinates, by (x, ξ) ↦ (x|ξ|, ξ). In a conic manifold there is intrinsically defined a “radial” vector field ρ. Its action on a C^{1} function f : Y → ℝ is expressed by

(2.3.2)  ρ(f) = \frac{d}{dt}M_{t}^{∗}f|_{t=1}.

In particular, in the coordinates of T^{∗}M, we have ρ = ∑_{j} ξ_{j}∂_{ξ_{j}}. It is readily seen that a function is homogeneous of degree μ, that is, it satisfies

M_{t}^{∗}f = t^{μ}f,

if and only if it satisfies the Euler relation

ρf = μf.

In particular, in T^{∗}M, x and ξ are homogeneous of degree 0 and 1, respectively. When Y is symplectic, in the sense that it is endowed with a non-degenerate closed 2-form σ, we ask for σ to satisfy the homogeneity condition M_{t}^{∗}σ = tσ and call Y a conic symplectic manifold. One can then define a homogeneous 1-form ω by putting

(2.3.3)  ω(v) = σ(ρ, v) for any vector field v.

Note that (2.3.3) is equivalent to ω = H^{−1}(ρ). In other words, in a symplectic manifold, a homogeneous structure is equivalently defined by a radial vector field ρ or a canonical 1-form ω. There naturally arises the question if, σ being closed, it is in fact exact and namely satisfies σ = dω. Since this is true for T^{∗}M, one expects that it is also true for a general Y. The question is hard and its answer stated positively is equivalent to saying that Y is symplectically diffeomorphic to T^{∗}M.

Theorem 2.3.5 (Darboux-Frobenius theorem). Let Y be a symplectic conic manifold and V a conic involutive submanifold of Y of codimension l in a neighborhood of p_{o} = (x_{o}, ξ_{o}). We also assume that V is regular, that is, ω|_{V} ≠ 0. Then we can find local homogeneous symplectic coordinates (x, ξ) in a neighborhood of p_{o}, that is, a local symplectic homogeneous diffeomorphism Y →^{∼} T^{∗}ℝ^{n}, such that x_{o} = 0, ξ_{o} = e_{i_{o}} (the i_{o}-th coordinate vector) for i_{o} ≤ n − l, and

(2.3.4)  V = {(x, ξ) : ξ_{n−l+1} = 0, ..., ξ_{n} = 0}.

Cf. Theorem 2.3.7 for the proof.

Remark 2.3.6. (a) In particular, if we choose Y = V, the theorem says that any symplectic homogeneous manifold Y is locally symplectically diffeomorphic to T^{∗}ℝ^{n} (Darboux theorem).

(b) Theorem 2.3.4 in combination with Theorem 2.2.9 asserts that we can find real coordinates y ∈ Y ≃ ℝ^{2n} such that V = {y : y_{2n−l+1} = 0, ..., y_{2n} = 0}. What Theorem 2.3.5 says additionally is that the trivialization can be obtained in a symplectic system of coordinates.

(c) Associated to an involutive bundle such as T V^{⊥ σ}, there is the foliation by its integral leaves {Γ}. In the present situation, this is called the “bicharacteristic” or “Hamiltonian” foliation. The leaves are “isotropic”, that is, satisfy the inclusion T Γ ⊂ T Γ^{⊥ σ}. Under the choice of coordinates (x, ξ) they are “straightened” into the planes x_{1}, ..., x_{n−l} = const, ξ_{1}, ..., ξ_{n−l} = const, ξ_{n−l+1}, ..., ξ_{n} = 0.

When V is involutive of codimension l, then corank(σ|_{V}) ≡ l; for a prescribed codimension l of V, this is the maximal corank. Theorem 2.3.5 has a generalization to manifolds of constant σ-corank.

Theorem 2.3.7. Let V be a regular (in the sense that ω|_{V} ≠ 0), conic submanifold of codimension l in a conic symplectic manifold Y such that corank σ|_{V} ≡ const in a neighborhood of p_{o} = (x_{o}, ξ_{o}); write this constant as l − 2k. Then, there are local symplectic homogeneous coordinates (x, ξ) in Y such that p_{o} = (0, e_{1}) and

(2.3.5) V = {(x, ξ) : ξ_{i} = 0, i ≥ n−(l−k)+1, x_{j} = 0, j ≥ n−k+1}.

Proof. It suffices to show that any partial system of coordinates x_{j}, j ≥ n − k + 1, and ξ_{i}, i ≥ n − (l − k) + 1, in a neighborhood of (0, e_{1}) which satisfy

(2.3.6) {ξ_{i_{1}}, ξ_{i_{2}}} = 0, {x_{j_{1}}, x_{j_{2}}} = 0, {ξ_{i}, x_{j}} = κ_{ij} if i_{1}, i_{2}, i ≥ n − (l − k) + 1 and j_{1}, j_{2}, j ≥ n − k + 1,

can be supplemented to a full symplectic system (x, ξ) which satisfies therefore the analogue of the Poisson relations (2.3.6) for all indices. Suppose we have proved this assertion. Consider first the case k = 0: V is regular involutive and is therefore defined in a neighborhood of p_{o} = (0, e_{1}) by, say, f_{i} = 0, i ≥ n − l + 1, satisfying {f_{i_{1}}, f_{i_{2}}}|_{V} ≡ 0. By linear algebra (cf. the proof of Theorem 2.2.9) we see that we can replace this set of equations by a new one which satisfies {f_{i_{1}}, f_{i_{2}}} ≡ 0 on the whole of Y instead of V. We then just have to rename f_{i} = ξ_{i} and supplement them to full coordinates.

Otherwise, if k ≥ 1, we know that some Poisson bracket, say {f_{i_{o}}, g_{j_{o}}}, does not vanish in a neighborhood of (0, e_{1}). We rename f_{i_{o}} = ξ_{n}; from ∂_{x_{n}}g_{j_{o}}(= {ξ_{n}, g_{j_{o}}}) ≠ 0 and from the implicit function theorem, we can replace g_{j_{o}} by x̃_{n} = x_{n} − h(x_{1}, ..., x_{n−1}, ξ_{1}, ..., ξ_{n}). We have thus achieved {ξ_{n}, x̃_{n}} ≡ 1 and hence we complete these two coordinates to a full system. In these new coordinates, V is in fact a submanifold of T^{∗}ℝ^{n−1} and hence Theorem 2.3.7 follows by the inductive hypothesis.

We have to prove therefore the statement about supplementing symplectic coordinates. To begin with, we state the linear version of the assertion in the tangent plane T_{p_{o}}Y: the vectors v_{j} = ∂_{x_{j}} for j ≥ n−k+1, and ν_{i} = ∂_{ξ_{i}} for i ≥ n−(l−k)+1, can be supplemented to a symplectic basis (v, ν). In fact, we first choose v′′_{i} for n − (l − k) + 1 ≤ i ≤ n − k such that

\begin{cases}σ(ν_{i}, v_{j}) = κ_{ij}, j ≥ n − (l − k) + 1, \\ σ(v_{i_{1}}, v_{i_{2}}) = 0, i_{1}, i_{2} ≥ n − (l − k) + 1, \end{cases}

which is possible since σ is non-degenerate. If w = ∑_{j≥n−(l−k)+1}x_{j}v_{j} +

ξ_{j}ν_{j} = 0, then σ(w, v_{j})(= −ξ_{j}) = 0 and σ(w, ν_{j})(= −x_{j}) = 0, j ≥ n − (l − k) + 1 and hence the extended system of vectors is still linearly independent. Let V_{1} = Span{v_{j}, ν_{j}}_{j≥n−(l−k)+1} and V_{2} = V_{1}^{⊥σ}. What we have to do is to find a symplectic basis for V_{2}. We reason by induction on dim(V_{2}) = 2m. If it is true for m − 1, we pick ν_{i_{o}} ∈ V_{2} and its “companion” v_{i_{o}} related by σ(ν_{i_{o}}, v_{i_{o}}) = 1, which is possible on account of the non-degeneracy of σ. We then set V_{o} = Span{v_{i_{o}}, ν_{i_{o}}} and decompose V_{2} = V_{o} ⊕ V_{o}^{⊥}. By applying the inductive hypothesis to V_{o}^{⊥}, we conclude the discussion of the linear case.

Coming back to the general case, we can supplement the coordinates x_{j}, ξ_{i} to a full system by adding g_{j}, j ≤ n − k, and f_{i}, i ≤ n − (l − k), such that H_{g_{j}} = −∂_{ξ_{j}}, and H_{f_{i}} = ∂_{x_{i}} at p_{o} = (0, e_{1}) where (x, ξ) are symplectic coordinates in T^{∗}ℝ^{n} ≃ T_{p_{o}}Y: this follows from the discussion of the linear case. Let j_{o} ≤ n − k and Y_{1} = {ξ_{j} = 0, j ≥ n − k + 1, and x_{i} = 0, i ≥ n − (l − k) + 1}. We claim that there exists g_{j_{o}} with dg_{j_{o}} = dx_{j_{o}} at 0 and which satisfies (2.3.7)

\begin{cases}H_{g_{j}}(g_{j_{o}}) = 0, j ≥ n − k + 1, H_{f_{i}}(g_{j_{o}}) = κ_{ij_{o}}, i ≥ n − (l − k) + 1, \\ g_{j_{o}}|_{Y_{1}} = x_{j_{o}}. \end{cases}

In fact, from H_{{·,·}} = [·, ·] and from (2.3.6), we know that all vector fields H_{g_{j}}, H_{f_{i}} for i ≥ n − (l − k) + 1, j ≥ n − k + 1 commute. They are also clearly transversal to Y_{1}. Thus, Corollary 2.2.12 applies and gives the solution to (2.3.7). Also, if j_{o} ≥ n − (l − k) + 1, then g_{j_{o}}|_{Y_{1}} = x_{j_{o}}|_{Y_{1}} ≡ 0 which implies, in combination with the first line of (2.3.7), that dg_{j_{o}} = dx_{j_{o}}. If, instead, j_{o} ≤ n − (l − k), then

\begin{aligned}dg_{j_{o}} & = d(g_{j_{o}}|_{Y_{1}}) \\ & = dx_{j_{o}}, \end{aligned}

where the first and second equalities follow from the first and second lines of (2.3.7), respectively.

We can do the same to find f_{i_{o}} for i_{o} ≤ n − (l − k) which satisfies the symplectic relations with the coordinates previously found and such that df_{i_{o}} = dξ_{i_{o}}. The proof is complete.

□

Theorem 2.3.7 says that when rank σ|_{V} ≡ const, then ker σ|_{V} is integrable. It also provides the explicit straightening of its integral leaves: under a suitable choice of local symplectic coordinates, they are reduced to the (l − 2k)-dimensional planes whose tangent bundle is Span{∂_{x_{i}}}_{i=n−(l−k)+1,...,n−k}, that is, the planes defined by x_{i} = const for i ≤ n − (l − k), x_{i} = 0 for i ≥ n − k + 1, ξ_{j} = const for j ≤ n − (l − k), and ξ_{j} = 0 for j ≥ n − (l − k) + 1.

Suggested research. Let V_{1} and V_{2} be conic regular involutive submanifolds of T^{∗}ℝ^{n} which have regular involutive intersection. Then we can find symplectic homogeneous coordinates (x, ξ) such that V_{1} is defined by x′ = 0 and V_{2} by ξ′′ = 0. The suggested research consists in finding the most general conditions under which a pair of involutive submanifolds can be put “simultaneously” into canonical form. This is of interest, for instance, in the study of boundary value problems for a system M of PDE’s over a domain Ω ⊂ ℝ^{n} where V_{1} = T^{∗}ℝ^{n}|_{∂Ω} and V_{2} = char M, the characteristic variety of M.

2.4. Subelliptic estimates and hypoellipticity of systems of vector fields

Let M be a smooth (C^{∞}) manifold, let L_{1}, ..., L_{k} be a system of smooth vector fields of T M and set L^{1} = Span{L_{1}, ..., L_{k}}. Here L^{1} may happen to be closed under Lie bracket . Otherwise, we set L^{2} = Span{L_{i}, [L_{i_{1}}, L_{i_{2}}] for i, i_{1}, i_{2} ∈ {1, ..., k}} and in general

(2.4.1) L^{j} = Span \{L_{i}, [L_{i_{1}}, L_{i_{2}}], ..., \underbrace{[L_{i_{1}}, [L_{i_{2}}, ...[L_{i_{j}−1}, L_{i_{j}}]...]]}_{j}

for any i, i_{1}, ..., i_{j} ∈ {1, ..., k}} .

2.4. SUBELLIPTIC ESTIMATES FOR SYSTEMS OF VECTOR FIELDS 97

Suppose that for a point x_{o} ∈ M and for an integer m_{1}(x_{o}) ≥ 2 we have

L_{x_{o}}^{j} = L_{x_{o}}^{1}, j ≤ m_{1} − 1, L_{x_{o}}^{m_{1}} ⊃_{≠} L_{x_{o}}^{1}.

Let dim\frac{L_{xo}^{m_{1}}}{L^{1}x_{o}} =: l_{1}(x_{o}); in this situation we call m_{1}(x_{o}) the first “Hörmander number” of the system L_{1}, ..., L_{k} at x_{o} and l_{1}(x_{o}) its “multiplicity”. We continue this process and find integers m_{h}(x_{o}) and l_{h}(x_{o}) such that

L_{x_{o}}^{j} = L_{x_{o}}^{m_{h−1}} if m_{h−1} ≤ j ≤ m_{h} − 1, L_{x_{o}}^{m_{h}} ⊃_{≠} L_{x_{o}}^{m_{h−1}}

and

l_{h}(x_{o}) := dim\frac{L_{x_{o}}^{m_{h}}}{L_{x_{o}}^{m_{h−1}}} .

This gives rise to a chain of spaces of iterated commutators which becomes stationary after, say, r steps:

(2.4.2)  L_{x_{o}}^{1} ⊂ L_{x_{o}}^{2} ⊂ ... ⊂ L_{x_{o}}^{m_{r}}.

Notice that the entries of (2.4.2) are not necessarily bundles (constant rank) but just distributions of tangent vectors.

Definition 2.4.1. The system L_{1}, ..., L_{k} is said to be of “finite type” or to satisfy the “Hörmander condition” at x_{o} when the chain ends after finitely many, say r, steps with L_{x_{o}}^{m_{r}} = T_{x_{o}}M.

In the above situation, we say that m_{r} is the “type” of the system. Otherwise, if L_{x_{o}}^{m_{r}} ≠ T_{x_{o}}M, we say that the type is +∞.

Let F be the Fourier transform defined, over functions u in the Schwartz space S of rapidly decreasing C^{∞} functions, by Fu(ξ) ≡ û(ξ) = ∫_{ℝ^{n}} e^{−i⟨x,ξ⟩}u(x)dx. With the notation ∂_{x}^{2} = ∑_{j=1}^{n} ∂_{x_{j}}^{2}, let Λ^{s} = (I − ∂_{x}^{2})^{\frac{s}{2}} be the standard elliptic pseudodifferential operator of (non necessarily integer) order s defined by Λ^{s}u = F^{−1} ((1 + |ξ|^{2})^{\frac{s}{2}} û(ξ)). In particular, if s > 0 is integer and even, then Λ^{s}u = (I − ∂_{x}^{2})^{\frac{s}{2}} u. We introduce a scalar product by

\begin{aligned}⟨u, v⟩_{H^{s}} & = ⟨Λ^{s}u, Λ^{s}v⟩_{H^{0}} \\ & = ⟨(1 + |ξ|^{2})^{s}û, v̂⟩_{H^{0}}, \end{aligned}

and define H^{s} to be the completion of S under the associated norm. We denote by ⟨·, ·⟩_{H^{s}} and || · ||_{H^{s}} the scalar product and the norm in H^{s}, respectively. By the Plancherel theorem we have, for integer s,

H^{s} = {u : D^{α}u ∈ H^{0}, |α| ≤ s} with ||u||_{H^{s}}^{2} ≃ ∑_{|α|≤s}||D^{α}u||_{H^{0}}^{2}. Let

ϵ > 0.

Definition 2.4.2. The system L_{1}, ..., L_{k} is said to be “ϵ-subelliptic” at x_{o} when, for a neighborhood B of x_{o},

(2.4.3) ||u||_{H^{ϵ}}^{2} <_{∼} ∑_{j=1,...,k}||L_{j}u||_{H^{0}}^{2} + ||u||_{H^{0}}^{2} for any u ∈ C_{c}^{∞}(B).

Definition 2.4.3. The system L_{1}, ..., L_{k} is said to be “hypoelliptic” at x_{o} when for a neighborhood B of x_{o} and for any x ∈ B,

(2.4.4)  u ∈ H^{0}, L_{j}u ∈ C_{x}^{∞}, j = 1, ..., k ⇒ u ∈ C_{x}^{∞},

where C_{x}^{∞} denotes the space of germs of C^{∞}-functions at x.

The system is said to be subelliptic, resp. hypoelliptic, when (2.4.3), resp. (2.4.4), is true for any x_{o}. To prove that subellipticity implies hypoellipticity we need a technical preliminary. Let χ_{ν}’s be a sequence of smooth functions which approximate the Dirac measure.

Lemma 2.4.4. (Friedrichs’) Let g ∈ H^{s} have compact support and suppose Lg ∈ H^{s}. Then,

(2.4.5)  L(g ∗ χ_{ν}) → Lg in H^{s}.

Proof. It suffices to prove that

(2.4.6)  ∥L(g ∗ χ_{ν}) − (Lg) ∗ χ_{ν}∥_{H^{s}} <_{∼} ∥g∥_{H^{s}}.

It is immediate to see that (2.4.6) implies (2.4.5). In fact, if we approximate g_{k} →_{H^{s}} g with g_{k} ∈ C_{c}^{∞}, we have, by applying (2.4.6) to the difference g − g_{k},

∥L(g ∗ χ_{ν}) − (Lg) ∗ χ_{ν}∥_{s} ≤ ∥g_{k} − g∥_{s}_{(I)} + ∥L(g_{k} ∗ χ_{ν}) − (Lg_{k}) ∗ χ_{ν}∥_{s}_{(II)},

with (I) → 0 for k → ∞ and (II) → 0 for ν → ∞. We prove now (2.4.6) for g ∈ C_{c}^{∞}; by the density of C_{c}^{∞} in H^{s}, (2.4.6) follows in full generality. We write L = a(x)∂_{x} or, in detail, L = ∑_{j} a_{j}(x)∂_{x_{j}}. We start by proving (2.4.6) for s integer and denote by ∂_{x}^{α} a derivative of order |α| ≤ s. We have

\begin{aligned}∂_{x}^{α}(L(g ∗ χ_{ν}) & − L(g) ∗ χ_{ν}) = L(∂_{x}^{α}g ∗ χ_{ν}) − (L∂_{x}^{α}g) ∗ χ_{ν} + [∂_{x}^{α}, L](g ∗ χ_{ν}) − [∂_{x}^{α}, L](g) ∗ χ_{ν} \\ & = ∑_{j} ∫ ((a_{j}(x) − a_{j}(x − t))∂_{x}^{α}g(x − t)∂_{t_{j}}χ_{ν}(t) \\ & + ∂_{t_{j}}a_{j}(x − t)∂_{x}^{α}g(x − t)χ_{ν}(t))dt_{1} ∧ ... ∧ dt_{n} \\ & + [∂_{x}^{α}, L](g ∗ χ_{ν}) − [∂_{x}^{α}, L](g) ∗ χ_{ν}. \end{aligned} (2.4.7)

Denote by I_{j}^{αν} each integral in the summation on the right of (2.4.7). If M is a bound for ||a||_{C^{1}}, we have

|I_{j}^{αν}| ≤ M ∫ |∂_{x}^{α}g(x − t)|(χ_{ν}(t) + |t| |∂_{t_{j}}χ_{ν}(t)| )dt_{1} ∧ ... ∧ dt_{n}.

Young’s inequality for convolution applied to the above estimate, yields

\begin{aligned}||I_{j}^{αν}||_{0} & ≤ M∥∂_{x}^{α}g∥_{0}∥χ_{ν} + |t| |∂_{t_{j}}χ_{ν}|∥_{L^{1}} \\ & ≤ c∥∂_{x}^{α}g∥_{0} <_{∼} c∥g∥_{s}, \end{aligned} ; (2.4.8)

where c is independent of ν. (This independence follows from the fact that ∥χ_{ν}∥ = 1 and ∫ |t| |∂_{t_{j}}χ_{ν}| dt_{1} ∧ ... ∧ dt_{n} ≤ ∫ |∂_{t_{j}}χ|dt_{1} ∧ ... ∧ dt_{n} for any ν.) On the other hand, [∂_{x}^{α}, L] is of order s and hence the H^{0}- norm of the two terms on the last line of (2.4.7) is controlled by ∥g∥_{s} uniformly in ν.

By taking all choices of |α| ≤ s, we get (2.4.6) for s integer. As it has already been said, this implies (2.4.5), that is, the continuity from H^{s} to H^{s} of the mapping

(2.4.9)  g ↦ L(g ∗ χ_{ν}) − L(g) ∗ χ_{ν}.

Next, by interpolation lemma, the mapping (2.4.9) is continuous also for s fractionary. This concludes the proof of the Lemma.

□

Theorem 2.4.5. If L_{1}, ..., L_{k} is subelliptic, then it is also hypoelliptic.

Proof. (i): Let ψ be a smooth cut-off function such that ψ ≡ 1 at x. We show that ψu ∈ H^{0} and L_{j}(ψu) ∈ H^{0} for any j implies ψu ∈ H^{ϵ}. In fact, let us approximate ψu by u_{ν} := χ_{ν} ∗ (ψu) ∈ C_{c}^{∞}; by Lemma 2.4.4 applied with s = 0 and g = χu, we have that u_{ν} → ψu and L_{j}u_{ν} → L_{j}(ψu) in H^{0}-norm. From

||u_{ν} − u_{μ}||_{H^{ϵ}}^{2} <_{∼} ∑_{j} ||L_{j}u_{ν} − L_{j}u_{μ}||_{H^{0}}^{2} + ||u_{ν} − u_{μ}||_{H^{0}}^{2},

we deduce that {u_{ν}}_{ν} is Cauchy in H^{ϵ} and therefore its limit ψu is also in H^{ϵ}.

(ii): Let L_{j}(u) ∈ H^{s} for any j over supp ψ and suppose that we already know that u belongs to H^{σ} over supp ψ for σ ≤ s: we wish to prove that, in fact, ψu ∈ H^{σ+ϵ}. Let ψ^{1} ∈ C_{c}^{∞} with ψ^{1} ≡ 1 over supp(ψ); be begin by proving that Lψ^{1}Λ^{σ}ψu ∈ H^{0}. In fact, we have the elementary equality

Lψ^{1}Λ^{σ}u_{ν} = Λ^{σ}Lψ^{1}u_{ν} + [L, ψ^{1}Λ^{σ}]u_{ν}.

Now, the first term in the right is H^{0} convergent, for ν → ∞, to Λ^{σ}Lψu; this follows from Lemma 2.4.4. The second is trivially H^{0} convergent to [L, ψ^{1}Λ^{σ}]ψu. Thus, the first is also H^{0} convergent and, by H^{−1} uniqueness, it must converge to Lψ^{1}Λ^{σ}ψu; so this belongs to H^{0}. Hence, we apply (i) to ψ^{1}Λ^{σ}(ψu) and conclude that ψu ∈ H^{σ+ϵ}. Iterated use of this argument, with a gain of Sobolev index ϵ and a shrinkig of supp(ψ) at each step, makes it possible to conclude that ψu ∈ H^{s}. (Note here that the cut off in next step must have support where the former is ≡ 1.)

□

We notice now that if the system L_{1}, ..., L_{r} is not of finite type but the Lie span L has locally constant rank a < n, then in new variables L = Span{∂_{x′_{j}}}_{1≤j≤a}; this follows from the Frobenius theorem. Any function u of x′′_{j}, j ≥ a + 1 only, is then a solution of the system L_{j}u = 0 but it is not necessarily regular. Thus the sufficient condition in next theorem is also essentially necessary.

Theorem 2.4.6. Let L_{1}, ..., L_{k} be of finite type m. Then it is \frac{1}{2^{m−1}} subelliptic (and hence in particular hypoelliptic).

To prove the theorem, we need an auxiliary result which holds in full generality (even for systems of non-finite type).

Lemma 2.4.7. We have

||\underbrace{[L_{1}, [L_{2}, [...L_{m}]]]}_{m}u||_{H^{−1+ϵ}}^{2} (2.4.10)

<_{∼} ∑_{j} ||L_{j}u||_{H^{0}}^{2} + ||u||_{H^{0}}^{2} for ϵ = \frac{1}{2^{m−1}} .

Proof. We first consider the case of a simple commutator [L_{1}, L_{2}]. Recall the pseudodifferential operator Λ^{s} with symbol (1 + |ξ|^{2})^{\frac{s}{2}} . We have

\begin{aligned}||[L_{1}, L_{2}]u||_{H^{−1+ϵ}}^{2} & = ⟨Λ^{−1+ϵ}[L_{1}, L_{2}]u, Λ^{−1+ϵ}[L_{1}, L_{2}]u⟩_{H^{0}} \\ & = ⟨[L_{1}, L_{2}]u, Λ^{−2+2ϵ}[L_{1}, L_{2}]u⟩_{H^{0}}. \end{aligned}

We write in the left side [L_{1}, L_{2}] = L_{1}L_{2} − L_{2}L_{1} and consider only the term L_{1}L_{2} since the other can be handled likewise. We have

|⟨L_{1}L_{2}u, Λ^{−2+2ϵ}[L_{1}, L_{2}]u⟩_{H^{0}} | = |⟨L_{2}u, L_{1}Λ^{−2+2ϵ}[L_{1}, L_{2}]u⟩_{H^{0}} |

≤ |⟨L_{2}u, \underbrace{[L_{1}, Λ^{−2+2ϵ}[L_{1}, L_{2}]]}_{order −1 + 2ϵ}u⟩_{H^{0}}| + |⟨L_{2}u, \underbrace{Λ^{−2+2ϵ}[L_{1}, L_{2}]}_{order −1 + 2ϵ}L_{1}u⟩|

<_{∼} ||L_{2}u||_{H^{−1+2ϵ}}^{2} + \begin{cases}||L_{1}u||_{H^{−1+2ϵ}}^{2} + ||L_{2}u||_{H^{0}}^{2} + ||u||_{H^{0}}^{2} \\ ||L_{2}u||_{H^{−1+2ϵ}}^{2} + ||L_{1}u||_{H^{0}}^{2} + ||u||_{H^{0}}^{2}, \end{cases}

(2.4.11)

where we can choose either of the two alternatives in the third line of (2.4.11). When m = 2, we let ϵ = \frac{1}{2}, choose either of the alternatives and get (2.4.10). Otherwise, we reason by induction and suppose (2.4.10) true for commutators of m − 1 vector fields. When we go on to m, we have to use the second alternative in (2.4.11) with L_{2} replaced by [L_{2}, [L_{3}, [..., L_{m}]]]. Notice that, since we get a factor 2 for each commutator, the choice ϵ = 2^{−(m−1)} is the one needed in the induction. We then conclude

||[L_{1}, [L_{2}, [..., L_{m}]]]u||_{H^{−1+ϵ}}^{2}

<_{∼} ||[L_{2}, [..., L_{m}]]u||_{H^{−1+2ϵ}}^{2} + ||L_{1}u||_{H^{0}}^{2} + ||u||_{H^{0}}^{2}

<_{∼} ∑_{2≤j≤k}||L_{j}u||_{H^{0}}^{2} + ||L_{1}u||_{H^{0}}^{2} + ||u||_{H^{0}}^{2}.

□

End of Proof of Theorem 2.4.6. We are assuming that L_{1}, ..., L_{k} have finite type, say, m. Hence all derivatives of T M can be expressed through commutators involving at most m vector fields L_{j}’s. Hence, (2.4.10) yields the subelliptic estimate for ϵ = 2^{−(m−1)}.

□

Remark 2.4.8. If L_{1}, ..., L_{k} has finite type m, then u “gains” ϵ = 2^{−(m−1)} derivatives with respect to (L_{j}u, j = 1, ..., k). A more efficient proof shows that we have, in fact, the better gain of any ϵ with ϵ < \frac{1}{m} (cf. [75]). However, for application to hypoellipticity, this improvement is irrelevant.

Remark 2.4.9. We spend a few words about the “complex” version of Theorem 2.4.6 and Lemma 2.4.7. If a system of vector fields L_{j} ∈ T^{1,0}ℂ^{n} ⊕ T^{0,1}ℂ^{n} is closed under conjugation then, identified to a system in T ℝ^{2n}, it can enjoy the conclusions of Lemma 2.4.7 and Theorem 2.4.6. Otherwise, Lemma 2.4.7 holds only if an ϵ amount of ∥L̄_{j}u∥_{H^{0}}^{2} norm is added on the right. On the other hand, on account

of the estimate ∥L̄_{j}u∥_{H^{0}}^{2} <_{∼} ∥L_{j}u∥_{H^{0}}^{2} + ∥u∥_{H^{\frac{1}{2}}}^{2} , the norm ϵ∥u∥_{H^{\frac{1}{2}}}^{2} is ab-

sorbed in the left side provided that m = 2 (and taking summation over all bracket of order ≤ 2). This explains why Theorem 2.4.6 stays true in complex version when m = 2 and, instead, it fails for m > 2 (cf. Kohn [60]).

2.5. Miscellanea: foliations—orbits

We have seen that the involutivity of a bundle E over a manifold M characterizes the existence of a local foliation by integral leaves Γ of E, that is, manifolds which satisfy T_{x}Γ = E_{x} for any x ∈ Γ; in particular, the various leaves have the same dimension, equal to the rank of E. When E is not involutive, let L(E) denote the Lie algebra with respect to the product [·, ·] generated by E. Two situations may occur:

• L(E) is a bundle: thus, we cannot apply the Frobenius theorem to E but we can to L(E).

• L(E) is not a bundle: the Frobenius theorem fails. The failure is substantial: there cannot be a system of integral leaves of L(E) with the same dimension.

We can also enlarge our setting and, instead of a bundle E, deal with Span{𝕏}, the span of a set of vector fields 𝕏 = {L_{j}}_{j=1,...,k} whose rank is not necessarily constant (the span meant over the ring of functions on M of the same regularity as 𝕏).

Surprisingly, in case M and 𝕏 are real analytic, there is in any case the local integral leaf of L(𝕏), the Lie algebra engendered by Span(𝕏) (having possibly different dimension leaf by leaf).

Theorem 2.5.1 (Nagano theorem). Let M and 𝕏 be real analytic; then for any x_{o} there is a real analytic leaf Γ through x_{o} such that T_{x}Γ = L_{x}(𝕏) for any x ∈ Γ in a neighborhood of x_{o}.

Thus, the rank of L(𝕏) may vary but it is forced by the real analyticity to remain constant along each Γ. The manifold Γ is referred to as the “Nagano leaf” of 𝕏 (or L(𝕏)). It is unique: we postpone the proof to Remark 2.5.7(1) and (2).

Proof. Let n be the dimension of M and k_{o} the rank of L_{x_{o}}(𝕏). The cases k_{o} = 0, k_{o} = n, n = 1 are trivial. We assume by induction that the theorem is true in dimension n − 1. Let 𝕏 = {L_{1}, L_{2}, ...}; by a change of coordinates x = (x_{1}, x′) in M, we may assume x_{o} = 0 and L_{1} = ∂_{x_{1}}. Let L_{j} = a_{j1}∂_{x_{1}} + a_{j2}∂_{x_{2}} + ... with a_{ji} extending holomorphically to a neighborhood of x_{o}. We can replace L_{j} by L_{j} −a_{j1}∂_{x_{1}} without changing L(𝕏); thus it is not restrictive to assume L_{j} = L_{j}(x, ∂_{x′}) for ∂_{x′} = (∂_{x_{2}}, ∂_{x_{3}}, ...). We write L_{j} = ∑_{ν=0}^{+∞} x_{1}^{ν}L_{jν}(x′, ∂_{x′}), j = 2, 3, .... We have

\underbrace{[L_{1}, [L_{1}, ...[L_{1}}_{μ times}, L_{j}]...]] = μ!L_{jμ} + x_{1}R_{jμ}(x, ∂_{x′}).

It follows that, if L′ is the Lie span of the L_{jμ}’s, then L′(x′) ⊂ L(0, x′). Thus, let Γ′ be the integral leaf of L′ in the x′-space ℝ^{n−1} which exists by the inductive hypothesis. Then (−ϵ, +ϵ) × Γ′ is the integral leaf of L in ℝ^{n}.

□

When M or 𝕏 is not C^{ω}, we cannot expect the existence of leaves such that T Γ = L(𝕏)|_{Γ}. Instead, we can look for manifolds Γ of minimal dimension which satisfy

(2.5.1)  T_{x}Γ ⊃ 𝕏_{x} for any x ∈ Γ.

As was already noticed, since T Γ is stable by linear combinations and commutators, (2.5.1) implies

(2.5.2)  T_{x}Γ ⊃ L_{x}(𝕏) for any x ∈ Γ.

It turns out that, if M and 𝕏 are C^{∞}, there are no more Frobenius or Nagano leaves but, nevertheless, there exist locally the manifolds of minimal dimension which satisfy (2.5.2). To describe them, we need new terminology.

Definition 2.5.2. A “piecewise smooth integral curve”, or “polygonal integral path” of 𝕏 in M, is the junction of finitely many integral curves γ_{1}, ..., γ_{s} of smooth sections of 𝕏 with each γ_{j} being issued from the ending point of the former γ_{j−1}.

Definition 2.5.3. We call the set of all ending points of piecewise smooth integral curves of 𝕏 issued from x_{o} the “orbit” of 𝕏 through x_{o} ∈ M and denote it by O(x_{o}, 𝕏, M). We call the set of the ending points of the piecewise smooth integral curves contained in B the orbit of 𝕏 in a neighborhood B of x_{o} ∈ M and denote it by O(x_{o}, 𝕏, B).

Some examples are in order.

Example 2.5.4. Let 𝕏 = Span{L_{1}, L_{2}} in ℝ^{3} where L_{1} = ∂_{x_{1}} and L_{2} = ∂_{x_{2}} + x_{1}x_{3}∂_{x_{3}}. Note that [L_{1}, L_{2}] = x_{3}∂_{x_{3}}. It follows that

rank 𝕏 = 2, (2.5.3)

(2.5.4) rank L(𝕏) = \{ \begin{matrix}3 for x_{3} ≠ 0, \\ 2 for x_{3} = 0. \end{matrix} According to the Nagano theorem, we must have three orbits, the Nagano leaves, of different dimensions. These are in fact

O^{+} = {x ∈ ℝ^{3} : x_{3} > 0}, (2.5.5)

O^{0} = {x ∈ ℝ^{3} : x_{3} = 0}, (2.5.6)

O^{−} = {x ∈ ℝ^{3} : x_{3} < 0}. (2.5.7)

When 𝕏 is no longer C^{ω}, the situation changes strikingly.

Example 2.5.5. For a function χ = χ(x_{1}) in C^{∞} such that

χ = 0 for x_{1} ≤ 0,
χ ≠ 0 for x_{1} > 0,

we define in ℝ^{2} the vector fields L_{1} = ∂_{x_{1}}, L_{2} = χ(x_{1})∂_{x_{2}} and put 𝕏 = {L_{1}, L_{2}}; we have

rank L(𝕏) = rank 𝕏 = \{ \begin{matrix}1 for x_{1} ≤ 0, \\ 2 for x_{1} > 0, \end{matrix}

The Frobenius theorem fails (rank 𝕏 being non-constant), the Nagano theorem fails (𝕏 not being real analytic). In this situation, which are the orbits? We choose as a system of neighborhoods B of x^{o} the rectangles centered at x^{o}. If x_{1}^{o} > 0, we readily have O(x^{o}, 𝕏, B) = B. If x_{1}^{o} < 0 and B does not intersect the half-plane x_{1} > 0, we easily see that O(x^{o}, 𝕏, B) reduces to a horizontal ray (−a, −b) × {x_{2}^{o}}. If x_{1}^{o} < 0 and, instead, B intersects x_{1} > 0, then O(x^{o}, 𝕏, B) = B. This requires a few words of explanation: if we want to reach from x^{o} a point ξ ∈ B with ξ_{1} < 0 and ξ_{2} ≠ x_{2}^{o}, we first move along the horizontal line till we reach the half-plane x_{1} > 0, next we move on the vertical line till we reach x_{2} = ξ_{2}, and finally we move backwards along the horizontal line till we reach x_{1} = ξ_{1}. Of course, the global orbit through x^{o} is always the full plane ℝ^{2} regardless of x_{1}^{o} > 0 or x_{1}^{o} < 0.

On one hand, the conclusion is unpleasant because T_{x}O ≠ L_{x}(𝕏) for x_{1} < 0. On the other hand, we have found that the orbits are manifolds. These are far from being unique, and the dimension itself may vary with B. However, if M and 𝕏 are C^{∞}, this ambiguity disappears in germs: when shrinking B, the corresponding orbit stabilizes and is the germ of a unique, well-defined, manifold.

Theorem 2.5.6 (Sussmann). For any x_{o} ∈ M, there is a unique smooth submanifold Γ ⊂ M in a neighborhood of x_{o}, passing through x_{o} satisfying T Γ ⊃ 𝕏 and such that

(i) if Γ′ is a manifold which satisfies T Γ′ ⊃ 𝕏, then for some neighborhood B of x_{o} we have B ∩ Γ′ ⊃ B ∩ Γ, (ii) for any neighborhood B of x_{o}, there are an index j_{o} and neighborhoods B_{1} ⊂ B_{2} contained in B such that any point x ∈ Γ ∩ B_{1} can be reached by a polygonal path of j_{o} integral curves of 𝕏 contained in Γ ∩ B_{2}.

Moreover, Γ is C^{ω} if M and 𝕏 are C^{ω}.

We refer to [8] for a proof.

Remark 2.5.7. (1) The uniqueness of Γ is explicit from (i). We call the manifold Γ of Theorem 2.5.6 the “local orbit” or “Sussmann orbit” and sometimes denote it by Γ_{S} or O_{loc}.

(2) If 𝕏 is C^{ω}, then the Nagano leaf Γ_{N} coincides with the Sussmann orbit Γ_{S}. In fact, from (i) we know that B ∩Γ_{S} ⊂ B ∩Γ_{N}. On the other hand,

L(E) ⊂ T Γ_{S} ⊂ T Γ_{N} = L(E).

Thus Γ_{S} = Γ_{N}. This identity shows the uniqueness of Γ_{N} (and also makes more explicit the independence of Γ_{S} of the neighborhood where it is defined).

We want to adapt (2.5.1) to general closed subsets Σ.

Definition 2.5.8. A closed subset is said to be stable under 𝕏 if, whenever it contains a point x_{o}, it also contains the orbit O(x_{o}, 𝕏, B) for a suitable neighborhood B of x_{o}.

This condition is the natural substitute of (2.5.1) to which it is indeed equivalent when Σ is a manifold. In this case, it is also obviously equivalent to Γ ⊃ O(x_{o}, L(𝕏), B). But, if 𝕏 is C^{∞}, this equivalence is not specific to a manifold Γ.

Corollary 2.5.9. Let 𝕏 be C^{∞} and Σ be a closed subset of M stable under 𝕏. Then Σ is stable under L(𝕏).

Proof. Let Γ_{S} be the Sussmann orbit: by hypothesis Σ contains Γ_{S}. Since T Γ_{S} ⊃ 𝕏, then T Γ_{S} ⊃ L(𝕏) and thus Σ itself is stable under L(𝕏).

□

The fact is that, if 𝕏 is C^{∞}, the equivalence holds for orbits: O(x_{o}, 𝕏, B) = O(x_{o}, L(𝕏), B).

Corollary 2.5.10. Let 𝕏 be C^{∞}, assume Span(𝕏) involutive of constant rank k, and let Σ be closed and stable under 𝕏. Then for any x ∈ Σ there is an integral leaf Γ of Span(𝕏) through x contained in Σ.

Proof. We know that Σ ⊃ Γ_{S}, the Sussmann orbit through x. We also know from Section 2.2 that there exists the integral leaf Γ of Span(𝕏) through x. We have T Γ_{S} ⊃ 𝕏|_{Γ_{S}}; by (i) of Theorem 2.5.6, Γ = Γ_{S}.

□

We now discuss some examples.

Example 2.5.11. For coordinates x = (x′, x′′) in ℝ^{n}, we consider the transformations

T_{1} = \{ \begin{matrix}x̃′ = x̃′(x), \\ x̃′′ = x̃′′(x′′), \end{matrix} (2.5.8)

T_{2} = \{ \begin{matrix}x̃′ = O(x′) + O(x′)σ(x), \\ x̃′′ = x̃′′(x′′), \end{matrix} (2.5.9)

T_{3} = \{ \begin{matrix}x̃′ = O(x′), \\ x̃′′ = O(x′′). \end{matrix} (2.5.10)

We denote by ∼_{1}, ∼_{2}, ∼_{3} the equivalence relations in the set C of coordinates in ℝ^{n} which identify those which are related by T_{1}, T_{2} and T_{3}, respectively. We consider the foliation x′′ = const, the projection x ↦ x′ and the product ℝ_{x′}^{n_{1}} × ℝ_{x′′}^{n_{2}}, or equivalently the pair of projections x ↦ x′ and x ↦ x′′, and denote by F, P and Π these three structures with all possible choices of coordinates. It is clear that the transformations which preserve the above three structures are those of T_{1}, T_{2} and T_{3}, respectively. This yields

F ≃ \frac{C}{∼_{1}}
↑ p_{1}  p_{1} ↑
P ≃ \frac{C}{∼_{2}}
↑ p_{2}  p_{2} ↑
Π ≃ \frac{C}{∼_{3}} .

We explain p_{1} and p_{1}^{−1}. A projection h : ℝ^{n} → S to a submanifold S induces a foliation defined by its fibers {h^{−1}(x)}_{x∈S}; a foliation and a choice of a transversal manifold of dimension complementary to the fibers give a projection which assigns to each x the point h(x) in which the leaf through x meets S. The mapping p_{2} is clear by itself.

Example 2.5.12. We take two local foliations R = {R} and S = {S} with leaves of dimension l + k and l, respectively. We claim that there is a local diffeomorphism F in ℝ^{n} which transforms S in an ldimensional foliation {F (S)} which is “embedded” in R, that is, for any S there is R such that F (S) ⊂ R. To see this, we choose an n − (l + k)-foliation K = {K} transversal to R and S and a manifold Σ transversal to S of dimension n − l which is a union of K-leaves. Let ξ_{S} be the point of intersection S ∩ Σ = {ξ_{S}}, and let R_{ξ_{S}} be the R-leaf through ξ_{S}. We consider the projection induced by K by the choice of the transversal manifold R_{ξ_{S}}. This is defined by

p_{S} : ℝ^{n} → R_{ξ_{S}}, p_{S}(x) := K_{x} ∩ R_{ξ_{S}},

where K_{x} is the K-leaf through x. Note that by restriction, p_{S} induces a diffeomorphism between S and p_{S}(S), which we denote by F_{S}. We then have to define our diffeomorphism F by gluing the various F_{S}.

Notes to Chapter 2. We refer to the book by Boggess [27] for a general account about real spaces, real manifolds, vector fields and the Frobenius theorem. For the presentation of the symplectic spaces, we refer to the book by Hörmander [51] that we follow closely in the proof of Theorem 2.3.7. Subelliptic estimates and hypoellipticity can be found in many books, as for instance Folland and Kohn [42]. We also refer to Rothschild and Stein [75] for the sharp estimate on the gain of derivatives, that is, \frac{1}{m} instead of \frac{1}{2^{m−1}} . Section 2.5 contains two very relevant results in the integration of vector fields: Theorem 2.5.1 due to Nagano [70] and Theorem 2.5.6 due to Sussmann [84].

CHAPTER 3 Real/Complex Structures

3.1. Complex structures—real underlying structures—complexifications

Summary. We exhibit the identification of the real underlying space ℝ^{2n} to ℂ^{n} with the diagonal ℂ^{n} ×_{ℂ^{n}} ℂ̄^{n}; this is the same as replacing (x, y), the real and imaginary parts of z = x + iy, with (z, z̄). The complexification of ℝ^{2n} in ℂ^{2n} coincides with the embedding of ℂ^{n} ×_{ℂ^{n}} ℂ̄^{n} in ℂ^{n} × ℂ̄^{n}: coordinates are extended from (x, y) to (x^{ℂ}, y^{ℂ}) or (z, z̄) to (z, w̄). We also discuss the intrinsic version, for manifolds, of this operation. We then introduce complex structures and their generalization, the almost complex structures, and state the integrability theorem for the latter when they are involutive.

For coordinates (x, y) in ℝ^{2n}, let z = x + iy be coordinates in ℂ^{n}; here x = (x_{1}, ..., x_{n}), y = (y_{1}, ..., y_{n}), z = (z_{1}, ..., z_{n}). If z̄ = x − iy is the conjugate to z, we identify

(3.1.1) ℝ^{2n} → ℂ^{n} ×_{ℂ^{n}} ℂ̄^{n}, (x, y) ↦ (z = x + iy, z̄ = x − iy),

with inverse

(3.1.2)  ℂ^{n} ×_{ℂ^{n}} ℂ̄^{n} → ℝ^{2n}, (z, z̄) ↦ (\frac{z + z̄}{2}, \frac{z − z̄}{2i}).

Associated to (3.1.1) and (3.1.2), there is the transformation of differentials

(dx, dy) ↦ (dz, dz̄) = (dx + idy, dx − idy).

We define (∂_{z}, ∂z̄) by

(∂_{x}, ∂_{y}) ↦ (∂_{z}, ∂z̄) = (\frac{∂_{x} − i∂_{y}}{2} , \frac{∂_{x} + i∂_{y}}{2} ).

Thus (∂_{z}, ∂z̄) satisfies the chain rule related to (3.1.1); it is the dual basis to (dz, dz̄). This yields an identification for tangent vectors, that is, derivatives,

(3.1.3) T ℝ^{2n} ≃ T ℂ^{n} ×_{T ℂ^{n}} T ℂ̄^{n}, a∂_{x} + b∂_{y} ↦ (a + ib)∂_{z} + (a − ib)∂z̄

and the dual identification for cotangent vectors, that is, forms, (3.1.4)

T^{∗}ℝ^{2n} → T^{∗}ℂ^{n} ×_{T^{∗}ℂ^{n}} T^{∗}ℂ̄^{n}, adx + bdy ↦ \frac{a − ib}{2}dz + \frac{a + ib}{2}dz̄.

Under (3.1.3) and (3.1.4), the duality between T ℝ^{2n} and T^{∗}ℝ^{2n} in the left hand side is interchanged with the real part of the duality between T ℂ^{n} and T^{∗}ℂ^{n} on the right hand side

⟨a∂_{x} + b∂_{y}, a′dx + b′dy⟩ = Re ⟨(a + ib)∂_{z}, (a′ − ib′)dz⟩,

that is,

aa′ + bb′ = Re ((a + ib) · (a′ − ib′)) .

This identifies the euclidean structure of ℝ^{2n} with the real part of the hermitian structure of ℂ^{n}. After we have extracted the real structure ℝ^{2n} underlying ℂ^{n}, we want to complexify it, that is, to go on to ℂ^{2n}. We first revisit the above construction in an intrinsic way and prefer to reason on a complex manifold X, a union of local holomorphic patches {X_{j}} parametrized by means of homeomorphisms U_{j} →^{∼}_{Φ_{j}} X_{j} over open

domains U_{j} ⊂ ℂ^{n} which are holomorphically compatible over overlappings U_{i} ∩ U_{j}. We denote by X̄ the conjugate, the manifold with local holomorphic parametrizations Φ̄_{j} : Ū_{j} → X̄_{j}, and define the real underlying manifold X^{ℝ} through the diagonal identification

(3.1.5)  X^{ℝ} →^{∼}_{δ_{X}} X ×_{X} X̄.

Going on to tangent vectors, we get

(3.1.6)  T X^{ℝ} →^{∼}_{δ′_{X}} T X ×_{T X} T X̄,

and to cotangent vectors,

(3.1.7)  T^{∗}X^{ℝ} →^{∼}_{^{t}δ_{X}^{′−1}} T^{∗}X ×_{T^{∗}X} T^{∗}X̄.

Thus, on the right side of (3.1.6), (3.1.7), we continue to find diagonals, that is, real underlying manifolds: T X ×_{T X} T X̄ = (T X)^{ℝ} and T^{∗}X ×_{T^{∗}X} T^{∗}X̄ = (T^{∗}X)^{ℝ}. We thus have evidence that

δ′_{X} = δ_{T X} : T X^{ℝ} →^{∼} (T X)^{ℝ}, (3.1.8)

tδ_{T^{∗}X}^{′−1} = δ_{T^{∗}X} : T^{∗}X^{ℝ} →^{∼} (T^{∗}X)^{ℝ}. (3.1.9)

We now complexify X^{ℝ} to (X^{ℝ})^{ℂ} by the embedding j : X ×_{X} X̄ ↪ X × X̄. When complexifying in fiber the bundles T X^{ℝ} and T^{∗}X^{ℝ}, we have, instead, two options: either to complexify the coefficients of vector fields and forms or to use the mapping j′ and ^{t}j^{′−1}. The two ways are related by the diagonal correspondences

(3.1.10)  ℂ ⊗_{ℝ} T X^{ℝ} →^{δ^{′ℂ}} T X ×_{X} T X̄

and

(3.1.11)  ℂ ⊗_{ℝ} T^{∗}X^{ℝ} →^{^{t}δ^{′−1ℂ}} T^{∗}X ×_{X} T^{∗}X̄.

If we wish to express (3.1.10) and (3.1.11) in coordinates, we have just to use (3.1.3), (3.1.4) and there replace real coefficients a and b by complex ones. We note that the “diagonal” relations a − ib = \overline{a + ib} correspond precisely to the choices of a and b real.

Definition 3.1.1. A complex structure on X is the morphism J induced on T X^{ℝ} by the multiplication by i on T X:

T X →^{i} T X
||  ||
T X^{ℝ} →^{J} T X^{ℝ}.

Note that we have the identity J^{2} = −id as the real counterpart of i^{2} = −1. Also, in T X^{ℝ}, the complex structure is given in local holomorphic coordinates z = x + iy of X by J ∂_{x} = ∂_{y}, J ∂_{y} = −∂_{x}. With the aid of J we can write (3.1.10) and (3.1.11) as (3.1.12)

X ↦ (\frac{X − iJ X}{2} , \frac{X + iJ X}{2} ) , ξ ↦ (\frac{ξ + i^{t}J^{−1}ξ}{2} , \frac{ξ − i^{t}J^{−1}ξ}{2} ) ,

with inverses

(3.1.13)  (Z, W̄) ↦ Z + W̄ , (ζ, μ̄) ↦ ζ + μ̄.

When the mappings in (3.1.13) are restricted to the diagonal, they take values in T X^{ℝ} and T^{∗}X^{ℝ}, respectively. We denote by “Re′′ the composition of (3.1.13) restricted to the diagonal with (π_{1}|_{T X ×_{T X} T X̄})^{−1} and (π_{1}|_{T^{∗}X ×_{T X} T^{∗}X̄})^{−1}, respectively: (3.1.14) Re : T X → T X^{ℝ}, Z ↦ Z + Z̄, Re : T^{∗}X → T^{∗}X^{ℝ}, ξ ↦ ξ + ξ̄.

We remark that in ℂ ⊗_{ℝ} T X^{ℝ} there are two “distinguished” bundles T^{1,0}X = \{\frac{X−iJ X}{2} : X ∈ T X^{ℝ}\} and T^{0,1}X = \{\frac{X+iJ X}{2} : X ∈ T X^{ℝ}\}, the bundles of holomorphic and antiholomorphic vector fields or of type (1, 0) and (0, 1), respectively. They satisfy

(3.1.15)  T^{1,0}X ∩ T^{0,1}X = {0}, T^{1,0}X ⊕ T^{0,1}X = (T X^{ℝ})^{ℂ}.

They can be characterized as the eigenspaces of the eigenvalues +i and −i for J^{ℂ}, respectively. In coordinates, they are described as T^{1,0}X = Span{∂_{z_{j}}}_{j=1,...,n} and T^{0,1}X = Span{∂z̄_{j}}_{j=1,...,n}, respectively. What we have just seen to be true for T X^{ℝ} continues to hold for a general real vector space V. We start by extending Definition 3.1.1.

Definition 3.1.2. A morphism of complex structure J on V is a morphism which satisfies J^{2} = −id.

We have the analogues of the bundles T^{1,0}X and T^{0,1}X. In fact, given a complex structure J, there remain well-defined eigenspaces of the eigenvalues +i and −i of J^{ℂ} that we denote 𝕃 and 𝕃̄. They satisfy

(3.1.16)  𝕃 ∩ 𝕃̄ = {0} and 𝕃 ⊕ 𝕃̄ = V^{ℂ}.

Conversely,

Theorem 3.1.3. Let 𝕃 be a subspace of an even-dimensional vector space V, let 𝕃̄ be the conjugate and assume that (3.1.16) is satisfied. Then, there is a unique morphism J : V → V such that 𝕃 and 𝕃̄ are the eigenspaces of the J^{ℂ}-eigenvalues +i and −i, respectively.

Proof. Define

(3.1.17)  J^{ℂ}L = iL for L ∈ 𝕃, J^{ℂ}L̄ = −iL̄ for L̄ ∈ 𝕃̄.

It is clear that J^{ℂ} is an endomorphism of V^{ℂ} and that 𝕃 and 𝕃̄ are the eigenspaces of +i and −i. What remains to prove is that J maps V into V. Define

(3.1.18)  \begin{cases}X := L + L̄, \\ Y := i(L − L̄). \end{cases}

We have J^{ℂ}X = Y and J^{ℂ}Y = −X; in particular, J^{ℂ} fixes V.

□

We write after (3.1.18)

(3.1.19) L = \frac{X − iY}{2} = \frac{X − iJ X}{2} , L̄ = \frac{X + iY}{2} = \frac{X + iJ X}{2} ,

which agrees well with (3.1.12). We need to make our abstract setting richer and call into play the involutivity properties of systems of vector fields. So, we leave the abstract vector space V and specify our discussion for V = T M where M is a real manifold, not necessarily embedded in ℂ^{n} nor in a complex manifold X. We start by observing that T^{1,0}X and T^{0,1}X are involutive, that is, closed under Lie bracket. For general bundles 𝕃 and 𝕃̄ this is an additional requirement.

Definition 3.1.4. (i) An “almost complex structure” over a manifold M is a bundle 𝕃 ⊂ ℂ ⊗ T M such that

𝕃 ∩ 𝕃̄ = {0}, 𝕃 ⊕ 𝕃̄ = ℂ ⊗ T M.

(ii) An almost complex structure is said to be “involutive” when the bundle 𝕃 is involutive.

(iii) An involutive almost complex structure 𝕃 is a “complex structure” if it is realized as 𝕃 = T^{1,0}X for a complex manifold X.

The following shows that (ii) implies (iii).

Theorem 3.1.5 (Newlander-Nirenberg). A smooth, involutive, almost complex structure is locally a complex structure.

Proof (Kohn [58]). We prove the existence of local holomorphic coordinates (u_{1}, ..., u_{n}) (for n := \frac{dimM}{2} ), that is, complex functions which satisfy

L̄u_{j} = 0 for any L̄ ∈ 𝕃̄ and du_{1} ∧ ... ∧ du_{n} ≠ 0.

Associated to the splitting of vector fields 𝕃⊕𝕃̄ and to the dual splitting of forms, there is a well-defined ¯∂-operator, the “(0, 1)-component of d”. Now, the condition of integrability of the structure is equivalent to

¯∂^{2} = 0.

This is a classical and instructive exercise. We take now n complex functions w = (w_{1}, ..., w_{n}) vanishing at the reference point z_{o} = 0 and such that (Re w, Im w) are a system of real local coordinates. In these coordinates, the operator ¯∂ is expressed, for suitable coefficients, by

\begin{aligned}¯∂u & = ∑_{k} (∑_{j} a_{k}^{j} \frac{∂u}{∂w_{j}} + ∑_{j} ã_{k}^{j} \frac{∂u}{∂w̄_{j}} ) dw_{k} \\ & + ∑_{k} (∑_{j} b_{k}^{j} \frac{∂u}{∂w_{j}} + ∑_{j} b̃_{k}^{j} \frac{∂u}{∂w̄_{j}} ) dw̄_{k}. \end{aligned} ; (3.1.20)

We can assume that at z_{o} = 0 the structure coincides with the standard structure, that is, a_{k}^{j}(0) = ã_{k}^{j}(0) = b_{k}^{j}(0) = 0 and b̃_{k}^{j}(0) = κ_{jk}, the Kronecker symbol. By means of the dilation ϵ : z ↦ ϵz, we may define a structure on the unit ball 𝔹_{1}^{n} that we denote by ¯∂_{ϵ}; its coefficients are obtained by applying ϵ^{∗} to the coefficients of (3.1.20). We claim that for suitable ϵ we can find u_{j}, j = 1, ..., n, complex such that du_{1} ∧ ... ∧ du_{n} ≠ 0 and ¯∂_{ϵ}u_{j} = 0. We start by remarking that ∂_{ϵ} ¯∂_{ϵ}|z|^{2} is a perturbation of ∂ ¯∂|z|^{2} and therefore it stays positive definite. We then go back to the L^{2} estimates (1.9.25) for the ¯∂-Neumann problem. They hold unchanged and provide a ¯∂-Neumann operator N_{ϵ}. (This is not always the case for a general structure if it is not endowed with a strictly plurisubharmonic function such as |z|^{2}.) This allows us to “correct” the z_{j}’s, which are not necessarily ¯∂_{ϵ}-holomorphic, to

(3.1.21)  u_{jϵ} := z_{j} − ¯∂_{ϵ}^{∗}N_{ϵ} ¯∂_{ϵ}z_{j}.

Moreover, the elliptic regularity in the interior for □_{ϵ} = ¯∂_{ϵ} ¯∂_{ϵ}^{∗} + ¯∂_{ϵ}^{∗} ¯∂_{ϵ} yields

(3.1.22)  ||χN_{ϵ} ¯∂_{ϵ}z_{j}||_{H^{k+2}}^{2} ≤ c_{k}||¯∂_{ϵ}z_{j}||_{H^{k}}^{2} for χ ∈ C_{c}^{∞}(𝔹_{1}^{n}).

Because of (3.1.22) the u_{jϵ}’s stay independent.

□

Alternative proof in a special case. For the reader who does not feel comfortable with the ¯∂-Neumann problem, we give the choice of a simplified proof in case 𝕃 is already known to be the bundle 𝕃 = T^{1,0}Y of (1, 0) vector fields tangent to a real submanifold Y ⊂ X, that is, T^{1,0}Y = (ℂ ⊗_{ℝ} T Y) ∩ T^{1,0}X. Note that in this case 𝕃 is involutive and satisfies well 𝕃 ∩ 𝕃̄ = {0}. The argument is still local. Let us define

T^{ℂ}Y := T Y ∩ iT Y and identify T^{1,0}Y →^{∼}^{Re} T^{ℂ}Y. Hence our hypothesis is

T^{ℂ}Y = T Y,

and our task is to show that

if T Y is complex, then Y is complex.

Since T Y and its orthogonal T Y^{⊥} are complex, that is, J -invariant, of complex dimension, say, n − l and l, respectively, then we can choose for them a pair of real basis in the form

(3.1.23)  T Y = Span{v′_{h}, J v′_{h}}, T Y^{⊥} = Span{v′′_{k}, J v′′_{k}}.

Keeping in mind that for coordinates (x, y) in ℝ^{2n} we have J ∂_{x} = ∂_{y}, we define an isomorphism of tangent vectors at z_{o} = 0

Φ : T_{0}Y ⊕ T_{0}Y^{⊥} →^{∼} T_{0}ℝ^{2n}, v ↦ ∂_{x}, J v ↦ ∂_{y}.

Under this identification, we get ΦT_{0}Y = ℂ_{z′}^{n−l}×{0}. It follows that Y is locally in a neighborhood of z_{o} = 0 the graph of a function z′ ↦ h(z′), that is, Y = {(z′, h(z′)) : z′ ∈ ℂ^{n−l}}. Now, our aim is to show that h is holomorphic. If we define Ψ := id × h and take any vector field L ∈ (T ℂ^{n−l})^{ℝ}, we get

J (Ψ_{∗}L) = J (L, h_{∗}L) = (J L, J h_{∗}L).

We have

(L, h_{∗}L)|_{z=0} ∈ T_{0}Y.

Since T_{0}Y is J -invariant, then (J L, J h_{∗}L)|_{z=0} ∈ T_{0}ΨY and hence there is a vector field W such that

(J L, J h_{∗}L) = (W, h_{∗}W),

which implies W = J L and hence also

h_{∗}J L = J h_{∗}L.
□

Notes. For a general background on complex structures, holomorphic and antiholomorphic vector fields, real underlying structures, complexifications and almost complex structures, the reader may consult the books by Baouendi, Ebenfelt and Rothschild [8], Boggess [27] and Chern [34]. Theorem 3.1.5 was proved by Newlander and Nirenberg [72]. The proof that we present here is an application of the ¯∂-Neumann problem and is due to Kohn [58]. Another proof, which uses integral kernel methods, is due to Webster [97].

3.2. CR manifolds

Summary. We present here embedded CR structures and give the normal form of a generic CR submanifold of ℂ^{n}. We also discuss some celebrated classes of manifolds which fail to be CR and still have a nice classification. In these cases we are in the presence of “isolated complex tangencies”.

Linear models for CR structures are not geometrically significant. We introduce them just to warm up with terminology and notation. Let E be a complex vector space of dimension n. We can think of E as a real vector space E^{ℝ} of dimension 2n endowed with a morphism J : E → E which is an antiinvolution, that is, satisfies J^{2} = −id.

Remark 3.2.1. We can identify ℂ-spaces to ℝ-spaces endowed with such a J. Given an ℝ-space, we can realize it as a ℂ-space by defining iv := J v for any v ∈ E and we can go the other way around by defining J v := iv.

Let F ⊂ E be a real subspace. We have

F ∩ J F ⊂ F ⊂ F + J F,

where the term on the left (resp. the right) is the maximal ℂ-space contained in F (resp. the minimal ℂ-space containing F). We set

dim_{CR}(F) = dim_{ℂ}(F ∩ J F).

It is clear that

(3.2.1)  dim_{ℝ}F = dim_{ℂ}(F ∩ J F) + dim_{ℂ}(F + J F),

which is obtained by dividing by 2 the two sides of

(3.2.2) dim_{ℝ}F + dim_{ℝ}J F = dim_{ℝ}(F ∩ J F) + dim_{ℝ}(F + J F).

On account of (3.2.1) we have

(3.2.3)  dim_{ℝ}F − n ≤ dim_{CR}F ≤ \frac{dim_{ℝ}F}{2} .

We set l = cod_{ℂ^{n}}^{ℝ}F, s = cod_{ℂ^{n}}^{ℂ}(F + JF); one can check that if rk_{ℝ} and rk_{ℂ} are the real and complex rank of a system of (linear) equations for F, respectively, then l = rk_{ℝ} and s = l − rk_{ℂ}, which yields

dim_{CR}F = n − l + s.

There are two extremal cases:

(3.2.4)  \begin{cases}s = \frac{l}{2}, dim_{CR}F = n − \frac{l}{2} = \frac{dimF}{2} , \\ s = 0, dim_{CR}F = n − l. \end{cases}

The first corresponds to F being complex. The second means that F + J F = ℂ^{n} and F is said to be “generic” in that case. In the sequel, we refer to the number s as the “defect of genericity” of F. Note that when F is generic, then dim_{ℝ}F ≥ n.

Lemma 3.2.2. For a pair of linear subspaces F, G ⊂ E the following are equivalent:

(i) dim_{ℝ}F = dim_{ℝ}G and dim_{CR}F = dim_{CR}G.

(ii) There is an isomorphism φ : F →^{∼} G which is ℂ-complex, that is, φ ∘ J = J ∘ φ.

Proof. It is not restrictive to assume

(3.2.5)  \begin{cases}F + J F = G + J G = E, \\ F ∩ J F = G ∩ J G. \end{cases}

We choose a basis e_{1}, …, e_{n−l}, J e_{1}, …, J e_{n−l} of F ∩ J F = G ∩ J G and two completions f_{1}, …, f_{l} and g_{1}, … g_{l} for F and G, respectively. Thus, we can explicitly define our morphism φ by

φ(e_{j}) = e_{j}, φ(J e_{j}) = J e_{j}, φ(f_{j}) = g_{j}.

Such a φ fulfills all requirements of the statement.

□

Definition 3.2.3. (i) We say that F is totally real when F ∩J F = {0}, that is, dim_{CR}F = 0.

(ii) We say that F is totally real maximal when ℂ^{n} = F ⊕ J F.

Note that if F is totally real (resp. totally real maximal), then dimF ≤ n (resp. dimF = n). However, it is not just a question of dimension. For instance, in ℂ^{2}, we have

• ℝ × ℝ totally real maximal,

• ℝ × {0} totally real,

• ℂ × ℝ generic,

• ℂ × {0} complex non-generic.

We now discuss embedded CR structures. Let M be a real submanifold of ℂ^{n}. We set

T_{z}^{ℂ}M = T_{z}M ∩ J T_{z}M, z ∈ M,

and call it the maximal complex tangent space at z.

Definition 3.2.4. The manifold M is said to be a CR manifold when rk(T_{z}^{ℂ}M) ≡ const, that is, when T^{ℂ}M is a bundle.

We refer to the rank of T^{ℂ}M as the CR dimension of M and denote it by dim_{CR}M. We define the CR codimension by codim_{CR}M := dimM − 2dim_{CR}M. There are two distinguished situations:

(3.2.6) T_{z_{o}}^{ℂ}M = {0}, that is, M is totally real at z_{o}, (3.2.7) T_{z_{o}}M + J T_{z_{o}}M = T_{z_{o}}ℂ^{n}, that is, M is generic at z_{o}.

We note that (3.2.6) and (3.2.7) are stable: if they hold at z_{o}, they hold also in a neighborhood of z_{o}. This is clear for (3.2.6), which is a condition of transversality T M ∩ J T M = {0}. As for (3.2.7), we have to remark that this is the condition of ℂ-independence of a system of equations of M: ∂r_{1} ∧ ⋯ ∧ ∂r_{l} ≠ 0. Another distinguished situation is given by

T_{z}^{ℂ}M = T_{z}M for any z close to z_{o},

that is, M is a complex manifold at z_{o}.

(3.2.8)

(Cf. here the Newlander-Nirenberg theorem.) If s is the defect of genericity of M, recall from the discussion of the linear case that cod_{CR}M = 2n − l − 2(n − l + s) = l − 2s. In particular,

“M generic” is equivalent to “cod_{CR}M = l and dim_{CR}M = n − l”.

For a CR manifold M, we define the bundles of vector fields tangential to M of type (1, 0) and (0, 1), respectively, by

T^{1,0}M = T^{1,0}ℂ^{n} ∩ (ℂ ⊗ T M),
T^{0,1}M = T^{0,1}ℂ^{n} ∩ (ℂ ⊗ T M).

The second is also called the bundle of CR vector fields. We note that we have two identifications

j_{1} : T^{ℂ}M → T^{1,0}M, X → \frac{X − iJ X}{2} , (3.2.9)

j_{2} : T^{ℂ}M → T^{0,1}M, X → \frac{X + iJ X}{2} , (3.2.10)

with inverse

(3.2.11)  L → Re L = L + L̄.

We wish to discuss these identifications in coordinates z = x + iy of ℂ^{n} and under a choice of ℂ-independent equations for M, say r_{1} = 0, ..., r_{l} = 0, such that ∂r_{1} ∧ ⋯ ∧ ∂r_{l} ≠ 0. We also write r for (r_{j})_{j}; we have

\begin{aligned}T M & ≃ \{(L, L̄) ∈ T^{1,0}ℂ^{n} ×_{T^{1,0}ℂ^{n}} T^{0,1}ℂ^{n} : ⟨L, ∂r⟩ + ⟨L̄, ¯∂r⟩ = 0\} \\ & ≃ \{a∂_{x} + b∂_{y} ∈ (T ℂ^{n})^{ℝ} : a∂_{x}r + b∂_{y}r = 0 for a, b real\}, \end{aligned}

T^{ℂ}M ≃ {(L, L̄) ∈ T^{1,0}ℂ^{n} ×_{T^{1,0}ℂ^{n}} T^{0,1}ℂ^{n} : ⟨L, ∂r⟩ = 0}

≃ {a∂_{x} + b∂_{y} ∈ (T ℂ^{n})^{ℝ} : a∂_{x}r + b∂_{y}r = 0 and − b∂_{x}r + a∂_{y}r = 0

for a, b real}.

It follows that ℂ ⊗ T M ≃ {(L_{1}, L̄_{2}) ∈ T^{1,0}ℂ^{n} ×_{ℂ^{n}} T^{0,1}ℂ^{n} : ⟨L_{1}, ∂r⟩ + ⟨L̄_{2}, ¯∂r⟩ = 0}

≃ {α∂_{x} + β∂_{y} ∈ ℂ ⊗ (T ℂ^{n})^{ℝ} : α∂_{x}r + β∂_{y}r = 0 for α, β complex},

ℂ ⊗ T^{ℂ}M

≃ {(L_{1}, L̄_{2}) ∈ T^{1,0}ℂ^{n} ×_{ℂ^{n}} T^{0,1}ℂ^{n} : ⟨L_{1}, ∂r⟩ = 0, ⟨L̄_{2}, ¯∂r⟩ = 0}

≃ {α∂_{x} + β∂_{y} ∈ ℂ ⊗ (T ℂ^{n})^{ℝ} : α∂_{x}r + β∂_{y}r = 0, −β∂_{x}r + α∂_{y}r = 0

for α, β complex}.

There is a geometric set which deeply represents the CR system.

Definition 3.2.5. We define the characteristic set of the system of CR vector fields tangential to M by

\begin{aligned}char(T^{0,1}M) & := T^{0,1}M^{⊥} ∩ T^{∗}M \\ & = (\frac{T M}{T^{ℂ}M} )^{∗} , \end{aligned}

with the orthogonal “^{⊥}” taken in ℂ ⊗ T^{∗}M.

It is readily seen that the natural projection T^{∗}ℂ^{n}|_{M} ↠ T^{∗}M combined with J^{∗} induces an injection

T_{M}^{∗}ℂ^{n} →^{J^{∗}} (\frac{T M}{T^{ℂ}M} )^{∗} .

This is an isomorphism precisely when M is generic; thus char(T^{0,1}M) ≃ T_{M}^{∗}ℂ^{n} in this case. Let N ⊂ M; when N is also generic, we have the transversality condition

T_{N}^{∗}M ∩ char(T^{0,1}M) = {0}.

This condition is crucial for taking the restriction or trace to N of a solution of the CR system over M.

We have a preliminary description of generic submanifolds which only uses the basics of linear algebra. Let M ⊂ ℂ^{n} be a submanifold of codimension l.

Proposition 3.2.6. The following are equivalent, in a neighborhood of a point z_{o} ∈ M:

(i) M is generic.

(ii) dim_{CR}M is minimal, that is, equals n − l.

(iii) There is a system of equations for M, r_{1} = 0, ..., r_{l} = 0, such that ∂r_{1}, ..., ∂r_{l} are ℂ-independent.

(iv) dz_{1}|_{M}, ..., dz_{n}|_{M} are independent.

(v) There are local coordinates z = x+iy, z = (z′, z′′), z′ = z_{1}, ..., z_{l}, such that z_{o} = 0 and M is graphed over x′, z′′ by

y′_{j} = h_{j}(x′, z′′) with h_{j}(0) = 0, ∂h_{j}(0) = 0.

(vi) There is N ⊂ M totally real maximal.

Proof. The equivalence of (i), (ii) and (iii) has already been seen. That (i) and (iv) are equivalent follows from the fact that

dz_{1}|_{M}, ..., dz_{n}|_{M} are ℂ-independent

if and only if

dz_{1}|_{T M+J T M}, ..., dz_{n}|_{T M+J T M} are independent

if and only if

T M + J T M = ℂ^{n}.

That (v) implies (iii) is obvious. Conversely, modulo linear complex transformations, we may arrange that z_{o} = 0 and

Span{dr_{1}(z_{o}), ..., dr_{l}(z_{o})} = Span{dy′_{1}, ..., dy′_{l}}.

Hence, by the implicit function theorem, we can find h_{j} with h_{j}(0) = 0 and ∂h_{j}(0) = 0, such that M is defined by y′_{j} = h_{j}, j = 1, ..., l. Finally, for the equivalence of (v) and (vi), it suffices to define N by supplementing the equations y′_{j} = h_{j} by the new equations y′′_{j} = 0. The proof is complete.

□

We now want to discuss the normal form of M analogous to the one stated in Section 1.6 about the “2-jet”.

Theorem 3.2.7. Let M ⊂ ℂ^{n} be generic of codimension l and class C^{k} and let z_{o} be a point in M. By a complex holomorphic change of coordinates we can assume that M is graphed over x′, z′′ at z_{o} = 0 by

y′_{j} = h_{j}(x′, z′′), j = 1, ..., l,

such that

(3.2.12)  ∂_{x^{I}z^{′′J}}^{|I|+|J|}h(z_{o}) = ∂_{x^{I}z̄^{′′J}}^{|I|+|J|}h(z_{o}) = 0 if |I| + |J| ≤ k.

Proof. By a linear change of coordinates, we first arrange that z_{o} = 0 and h(0) = 0, ∂h(0) = 0. For the Taylor expansion of h = (h_{j}) up to order k, we have

h = ∑^{k} ; |I|+|J|+|K|≥2a_{I J K}x^{′I}z^{′′J}z̄^{′′K} + o^{k},

where a = a_{I J K} is an l-vector a = (a_{j}). We “complexify” from x′ ∈ ℝ^{l} to z′ ∈ ℂ^{l} and from (z′′, z̄′′) ∈ ℂ^{n−l} ×_{ℂ^{n−l}} ℂ̄^{n−l} to (z′′, w̄′′) ∈ ℂ^{n−l} ×ℂ̄^{n−l} and consider the polynomial map

h^{k} : (z′, z′′, w̄′′) ↦ ∑^{k} ; |I|+|J|+|K|≥2a_{IJK}z^{′I}z^{′′J}w̄^{′′K}.

By the implicit function theorem, there is a unique map Φ = Φ(z′, z′′) such that

(3.2.13)  z′ = Φ(z′ + ih^{k}(z′, z′′, 0), z′′).

We define a polynomial change of coordinates in a neighborhood of 0 by

(3.2.14)  \begin{cases}z̃′ = z′ − ih^{k}(Φ(z′, z′′), z′′, 0), \\ z̃′′ = z′′. \end{cases}

We notice that if z ∈ M, that is, y′ = h(x′, z′′, z̄′′), then

\begin{aligned}ỹ′ & = y′ − Re h^{k}(Φ(z′, z′′), z′′, 0) \\ & = Re (h(x′, z′′, z̄′′) − h^{k} (Φ (x′ + ih(x′, z′′, z̄′′), z′′) , z′′, 0)) . \end{aligned} ; (3.2.15)

To obtain an equation for the image M̃ of M under the change (3.2.14), we must replace (x′, z′′) by (x′(z̃′, z̃′′), z̃′′), which yields

(3.2.16) ỹ′ = Re (h (x′(z̃′, z̃′′), z̃′′, \overline{z̃}′′)

−h^{k} (Φ (x′(z̃′, z̃′′) + ih(x′(z̃′, z̃′′), z̃′′, \overline{z̃}′′), z̃′′) , z̃′′, 0)) .

We write (3.2.16) as ỹ′ = Re h̃. By the implicit function theorem, we can remove ỹ′ from h̃; thus Re h̃ has become a graphing function for M̃. It remains to show that h̃ satisfies the requirement (3.2.12) in the statement of the theorem. Now, consider the function

(3.2.17)  (x̃′, z̃′′) ↦ x′(x̃′, z̃′′, \overline{z̃′′}).

By taking the k-jet, x^{′k}, of (3.2.17) and complexifying the variables from (x̃′, z̃′′, \overline{z̃}′′) to (z̃′, z̃′′, \overline{w̃}′′), we get for x^{′k} = x^{′k}(z̃′, z̃′′, \overline{w̃}′′)

(3.2.18) h̃(z̃′, z̃′′, \overline{w̃}′′) = h^{k} (x^{′k}, z̃′′, \overline{w̃}′′)

− h^{k} (Φ (x^{′k} + ih^{k} (x^{′k}, z̃′′, \overline{w̃}′′) , z̃′′) , z̃′′, 0) + o^{k}.

To establish (3.2.12), we have to show that

h̃(z̃′, z̃′′, 0) = o^{k}, h̃(z̃′, 0, \overline{w̃}′′) = o^{k}.

We prove the first, the second following by conjugation. On account of (3.2.13), we have, for x′ and z′′ replaced by x^{′k} = x^{′k}(z̃′, z̃′′, 0) and z̃′′, respectively,

(3.2.19)  x^{′k} = Φ (x^{′k} + ih^{k}(x^{′k}, z̃′′, 0), z̃′′) + o^{k}.

Substitution of (3.2.19) into (3.2.18) and evaluation of h^{k} and Φ for \overline{w̃}′′ = 0 yields

\begin{aligned}h̃ & = h^{k}(x^{′k}, z̃′′, 0) − h^{k} (Φ (x^{′k} + ih^{k}(x^{′k}, z̃′′, 0), z̃′′) , z̃′′, 0) + o^{k} \\ & = h^{k}(x^{′k}, z̃′′, 0) − h^{k} (x^{′k} + o^{k}, z̃′′, 0) + o^{k} \\ & = o^{k}. \end{aligned} (3.2.20)

□

Remark 3.2.8. There is a C^{ω} version of the above statement. In that case the equations y′_{j} = h_{j} can be found such that (3.2.12) holds for any I, J, K and not just for those whose sum of lengths is ≤ k.

CR singularities—isolated complex tangencies. We exhibit now some examples of manifolds which are not CR. In some fortunate case, we still have a nice classification, a normal form for the equation, and an efficient theory of attached analytic discs similar to the one that will be developed in Section 3.7 for CR manifolds.

Let M be a 2-codimensional real submanifold of ℂ^{2} with T_{z_{o}}M = ℂ. We first arrange, by a linear change, that z_{o} = 0 and T_{z_{o}}M = {z : z_{2} = 0}. Thus M is defined by z_{2} = h(z_{1}) for h(0) = 0 and ∂h(0) = 0. The points (z_{1}, h(z_{1})) where M has a complex tangency correspond to the zeroes z_{1} of

\begin{aligned}det [\begin{matrix}−∂_{z_{1}}h & 1 \\ −∂_{z_{1}} h̄ & 0 \end{matrix}] & = ∂_{z_{1}} h̄ \\ & = 0. \end{aligned} ; (3.2.21)

The only chance for M to be CR is that ∂_{z_{1}} h̄ ≡ 0, or equivalently, that M is a complex manifold in accordance with Theorem 3.1.5. Otherwise, M is not CR. If other assumptions are not made, M is hard to handle. To have a nicer situation, we first ask for z_{o} to be an isolated singularity. But the condition which is really crucial to develop some serious analysis is that z_{o} be a non-degenerate singularity in the sense that

∂_{z_{1}}∂z̄_{1}h(z_{o}) ≠ 0.

Thus h = a|z_{1}|^{2} + bz_{1}^{2} + cz̄_{1}^{2} + O^{3} for a ≠ 0. By a quadratic change of variables we may arrange that a = 1, b = c and, by a rotation in the z_{1}-plane, we may reduce the common value of b and c to be real positive, say γ ≥ 0. Hence the equation takes the form

(3.2.22)  z_{2} = |z_{1}|^{2} + γ(z_{1}^{2} + z̄_{1}^{2}) + O^{3}.

We call q = q(z_{1}) the quadric in (3.2.22). We have that the sublevel sets \{q ≤ c\} are compact or not according to γ < \frac{1}{2} or γ ≥ \frac{1}{2}. The complex tangent is said to be “elliptic”, “parabolic” and “hyperbolic” when γ < \frac{1}{2}, γ = \frac{1}{2} and γ > \frac{1}{2}, respectively; if γ ≠ \frac{1}{2}, the singularity is isolated. We are mainly concerned with the elliptic case.

We pass now from ℂ^{2} to ℂ^{n} and consider a manifold which is generically totally real maximal and has a complex tangency at z_{o} = 0 that we suppose to be non-degenerate. By combining (3.2.22) with the normal form of Theorem 3.2.7, we get the canonical form for a system of equations

(3.2.23)  \begin{cases}z_{2} = |z_{1}|^{2} + γ(z_{1}^{2} + z̄_{1}^{2}) + O^{3}(x_{3}, ..., x_{n}, z_{1}), \\ y_{3} = O^{3}(x_{3}, ..., x_{n}, z_{1}), \\ ..., \\ y_{n} = O^{3}(x_{3}, ..., x_{n}, z_{1}). \end{cases}

Note that the x_{j}, j = 3, ..., enter into the error term of the first equation because by a holomorphic transformation of type z̃_{1} = z_{1} +z_{j}, j = 3, ..., terms like z_{j}z̄_{1} are absorbed by |z̃_{1}|^{2}. Also, |z_{1}|^{2} enters into the errors of the remaining equations because, by the first,

\begin{aligned}|z_{1}|^{2} & = z_{2} − 2Re (γz_{1}^{2}) + O^{3} \\ & = Re (z_{2} − 2γz_{1}^{2}) + O^{3}. \end{aligned}

In (3.2.23), the first error term O^{3} is complex and the others are real. We write the first line as z_{2} = h and assume γ < \frac{1}{2}. We note that the set of the complex tangencies forms an (n − 2)-dimensional real submanifold. In fact, they are defined as the 0-set of the determinant of the system (3.2.24)

[\begin{matrix}∂_{z}(z_{2} − h) \\ ∂_{z}(z̄_{2} − h̄) \\ ∂_{z}(y_{3} − O^{3}) \\ ... \\ ∂_{z}(y_{n} − O^{3}) \end{matrix}] = [\begin{matrix}−(z̄_{1} + 2γz_{1}) & 1 & ∗ & ∗ & · & ∗ \\ −(z̄_{1} + 2γz̄_{1}) & 0 & ∗ & ∗ & · & ∗ \\ ∗ & 0 & −\frac{i}{2} & ∗ & · & ∗ \\ ... & ... & ... & ... & ... & ... \\ ∗ & 0 & ∗ & ∗ & · & −\frac{i}{2} \end{matrix} ] + error.

This determinant is

(−\frac{i}{2})^{n−2}(z̄_{1} + 2γz_{1}) + error,

and therefore it has only one zero near z_{1} = 0.

We pass to the general case and suppose that M is generic of CR dimension m apart from some CR singularities where the CR dimension grows to m + 1. We divide the variables as (z_{1}, z_{2}, z′, z′′); by (3.2.23) and Theorem 3.2.7, we can put the equations of M in the form

\begin{cases}z_{2} = q(z_{1}, z′′) + O^{3}, \\ y′ = Q(z_{1}, z′′) + O^{3}. \end{cases}

We assume that z_{o} = 0 is a non-degenerate isolated CR singularity in the (z_{1}, z_{2})-plane. By the discussion of the ℂ^{2} case, we have q(z_{1}, 0) = |z_{1}|^{2} + γ(z_{1}^{2} + z̄_{1}^{2}). We point out the following.

• The points of higher-dimensional complex tangencies, that is, the points where dim_{CR}M = m + 1, do not necessarily form a manifold Σ and, if this is a manifold, it does not necessarily satisfy dimΣ = dimM − 2. Note that the Jacobian matrix whose dropping rank gives the extra complex tangency is no longer squared.

• At a higher-dimensional complex tangency, the quadric in the (z_{1}, z_{2})-plane stays non-degenerate, by continuity.

Example 3.2.9. In ℂ^{4}, let M be defined by

\begin{cases}z_{2} = |z_{1}|^{2} + O^{3}(x_{3}, z_{1}, z_{4}), \\ y_{3} = Q(z_{1}, z_{4}) + O^{3}(x_{3}, z_{1}, z_{4}), \end{cases}

where Q is a quadric. Clearly 0 is an elliptic CR singularity with γ = 0. The set of singularities is defined by two complex equations in the plane of the parameters (x_{3}, z_{1}, z_{4}) whose rank may vary or not according to how the O^{3} terms are chosen. When the rank is constant, a regular set parametrized over ℝ or ℝ × ℂ is defined.

Remark 3.2.10. When M is generically totally real maximal (which is always the case in ℂ^{2}) and when the singularities are elliptic, the method of Bishop [24] permits us to attach an (n − 1)-parameter family of discs. In the space of parameters these are defined by |z_{1}| ≤ r, (x_{3}, ..., x_{n}) = const. Their union gives rise to an (n + 1)- dimensional manifold M̃ with boundary M foliated by discs. It is quite easy to prove, with the aid of the implicit function theorem, that this manifold is smooth outside the manifold which carries the CR singularities. But indeed, Kenig and Webster prove in [54] that M̃ is smooth at any point without exceptions.

Suggested research. (1) Let M be generic apart from some non-degenerate elliptic CR singularities which form a manifold Σ of maximal rank, that is, dimΣ = dimM − 2. Is it possible to construct a manifold M̃, with dimM̃ = dimM + 1, foliated by discs attached to M by a similar procedure as in [54] where M is generically totally real maximal? Is this manifold M̃ smooth at any point without exceptions?

(2) We say that a CR submanifold N of a CR manifold M is “relatively generic” when

(3.2.25)  codim_{M}N = dim_{CR}M − dim_{CR}N.

It is clear that if M is already generic, N is relatively generic in M if and only if it is generic in ℂ^{n}. As the terminology itself suggests, the failure of (3.2.25) is quite exceptional. The most pathological case, the occurrence of a submanifold N ⊂ M such that dim_{CR}N = dim_{CR}M, is referred to as “non-minimality” of M and is the main argument of Section 3.10. When M is a hypersurface and N is a submanifold such that dim_{CR}N = dim_{CR}M, then N is necessarily a complex hypersurface. In this situation, if M is defined by r = 0, then, for some s which satisfies s|_{N} ≡ 0, we have

(3.2.26)  ∂r|_{N} ∧ i∂s|_{N} = 0.

By differentiating (3.2.26) along N, we get ¯∂∂r|_{T^{ℂ}N} = i¯∂∂s|_{T^{ℂ}N} and hence, ¯∂∂r being hermitian, it is 0. In particular, if the Levi form L_{M} is not 0 at a point z_{o}, a complex hypersurface cannot pass through z_{o}. For the same reason, if there are no isotropic vectors for L_{M}, there are no complex curves. In particular, this occurs when L_{M} > 0, that is, M is strictly pseudoconvex. Otherwise, if M is only weakly pseudoconvex, that is, L_{M} ≥ 0, the problem becomes much more involved and higherorder tensors than the Levi form might enter into play. The occurrences of complex submanifolds are all instances of failure of genericity: complex is non-generic.

The suggestion for research is to get, in wide generality, a deeper understanding of the obstructions to the existence of submanifolds which are not relatively generic. By the richness of the situations that may occur, this leads in fact to a galaxy of problems and exercises.

Notes. We refer to the books by Berhanu, Cordaro and Hounie [21], Baouendi, Ebenfelt and Rothschild [8] and Boggess [27] for the presentation of CR structures. For the discussion on the normal form of a submanifold, the main references are Baouendi, Jacobowitz and Treves [9], Chern and Moser [35], Jacobowitz [52] and Treves [88]. Finally, we refer again to Chern and Moser [35] for the discussion on complex tangencies.

3.3. CR functions and CR mappings

Summary. We present CR functions and CR mappings. We prove that any CR real analytic submanifold is embedded in a “partial” or “intrinsic” complexification, a complex manifold in which it is generic. In particular, CR real analytic functions holomorphically extend, with uniqueness, to this complexification. This is a consequence of the Cauchy-Kowalevsky theorem. Also, we prove that any CR manifold, regardless of whether it is real analytic or not, is CR isomorphic to a generic submanifold of a complex manifold obtained via projection.

We then present abstract CR structures and prove that, in the C^{ω} setting, they are embeddable, that is, CR diffeomorphic to submanifolds of ℂ^{n}.

Definition 3.3.1. A C^{1} function f : M → ℂ is said to be CR when

(3.3.1)  L̄f = 0 for any L̄ ∈ 𝕃̄.

In case M is a submanifold of ℂ^{n}, defined by r = 0 for r = (r_{j})_{j=1,...,l}, then (3.3.1) means that L̄f = 0 for any L such that Lr = 0. Let us discuss this case in further detail.

Lemma 3.3.2. Let M ⊂ ℂ^{n} be a CR manifold defined by r = 0 for r = (r_{j})_{j=1,...,l}. A C^{1} function f : M → ℂ is CR if and only if for any extension of f to ℂ^{n}, which we still denote by f, we have ¯∂f ∧ ¯∂r_{1} ∧ ... ∧ ¯∂r_{l} = 0.

Proof. Let ¯∂_{M}f be the component of ¯∂f|_{M} which is orthogonal t¡o the ideal generated by {¯∂r_{j}}_{j=1,...,l}. One readily sees that f is CR precisely when ¯∂_{M}f = 0, and the lemma follows.

□

A large class of CR functions on M ⊂ ℂ^{n} is given by the restrictions to M of functions f holomorphic in a neighborhood of M: since ∂z̄_{j}f = 0 for any j, then L̄(f|_{M}) = 0 for any L̄ ∈ T^{0,1}M. In particular, for M = ℂ^{n}, holomorphic and CR functions coincide. We will see that if M and f are C^{ω}, then the converse is true. But, in general, CR functions are a larger family than just restrictions of holomorphic functions. For example, if M = ℝ×ℂ, then any (C^{1}) function f(x_{1}, z_{2}) which satisfies ∂z̄_{2}f = 0 is CR. But the holomorphic extension requires f to be C^{ω}. We want to start our discussion about extension with the basic uniqueness criterion.

Theorem 3.3.3. Let N and M be CR generic submanifolds with N ⊂ M ⊂ ℂ^{n} and let f be CR on M. Then

f|_{N} ≡ 0 implies f|_{N} = 0^{∞},

that is, f vanishes at infinite order along N. In particular, for M = ℂ^{n}, holomorphic functions which vanish on N generic are identically 0.

Proof. Let {L_{j}}_{j=1,...,m}, {L̄_{j}}_{j=1,...,l} be a basis for T^{1,0}N ⊕ T^{0,1}N, let \{X_{j}\}_{j=1,...,a} be a basis for \frac{T N}{T^{ℂ}N} and let \{N_{j} = J X_{j}\}_{j=1,...,b} with b ≤ a be a basis for \frac{T M|_{N} ∩J T N}{T^{ℂ}N}. The vector fields L_{j}, L̄_{j} and X_{j} are tangential to N and the N_{j}’s normal to N in M, respectively. We observe that by our hypothesis

L̄_{j}f = 0 and N_{j}f = iX_{j}f for any j.

We wish to prove that D^{α}f|_{N} = 0 for any differential operator tangential to M—combination of the L_{j}, L̄_{j}, X_{j}, N_{j}. We proceed by double induction on the degree p in the N_{j}’s and the total degree q of D^{α} and suppose that the conclusion holds for any D^{α} with degrees p and q. If we compose D^{α} with a tangential vector field, then we are taking the derivative of the identity to 0 along N which is still 0. Thus, what we need to prove is that N_{j}D^{α}f|_{N} = 0. Now,

\begin{aligned}N_{j}D^{α}f & = D^{α}N_{j}f + [N_{j}, D^{α}]f \\ & = D^{α}iX_{j}f + [N_{j}, D^{α}]f. \end{aligned}

The first term still contains p normal derivatives and hence it is 0. The second still has order q and so is 0 likewise.

□

Let M be a real smooth manifold of dimension d.

Definition 3.3.4. An abstract CR structure on M is a bundle 𝕃 ⊂ ℂ ⊗ T M such that

(i) 𝕃 ∩ 𝕃̄ = {0},

(ii) 𝕃 is involutive.

It is clear that if M ⊂ ℂ^{n} is a CR manifold, then 𝕃 = T^{1,0}M is a CR structure.

Definition 3.3.5. Let (M, 𝕃_{M}) and (N, 𝕃_{N}) be a pair of CR structures and f : M → N a C^{1} map. Then, f is said to be CR if it satisfies

(3.3.2)  f_{∗}𝕃_{M} ⊂ 𝕃_{N}.

In particular, if f : M → ℂ^{n}, then f is CR if and only if each component f_{j}, for j = 1, ..., n, is a CR function. It is more natural to reason directly with T M instead of ℂ ⊗ T M. Thus, let us define

T^{ℂ}M = {L + L̄ : L ∈ 𝕃}.

If we have an embedded structure 𝕃 = T^{1,0}M for M ⊂ ℂ^{n}, then T^{ℂ}M = T^{ℂ}M(= T M ∩ J T M).

Theorem 3.3.6. Let f : M → N be a C^{1} map between CR structures (M, 𝕃_{M}) and (N, 𝕃_{N}). Then, f is CR if and only if

(3.3.3)  \begin{cases}f_{∗}T^{ℂ}M ⊂ T^{ℂ}N, \\ f_{∗} ∘ J_{M} = J_{N} ∘ f_{∗}. \end{cases}

In other words, f is CR if it preserves the maximal tangent complex space and f_{∗} is complex linear on it.

Proof. We begin by proving that f_{∗}𝕃_{M} ⊂ 𝕃_{N} implies (3.3.3). First, if L + L̄ ∈ T^{ℂ}M, then f_{∗}(L + L̄) = f_{∗}L + \overline{f_{∗}L} ∈ T^{ℂ}N. Moreover, since 𝕃_{M} and 𝕃̄_{M} are the +i and −i eigenspaces for J_{M}, then

\begin{aligned}f_{∗}J_{M}(L + L̄) & = f_{∗}(iL − iL̄) \\ & = i(f_{∗}L − \overline{f_{∗}L}) \\ & = J_{N}(f_{∗}L + \overline{f_{∗}L}) = J_{N}f_{∗}(L + L̄). \end{aligned} ; (3.3.4)

Conversely, let us prove that (3.3.3) implies f_{∗}𝕃_{M} ⊂ 𝕃_{N}. Remember that any vector field L ∈ 𝕃_{M} can be written as L = \frac{X−iJ_{M} X}{2} for X := L + L̄. It follows that

\begin{aligned}f_{∗}L & = \frac{f_{∗}X − if_{∗}(J_{M}X)}{2} \\ & = \frac{f_{∗}X − iJ_{N}f_{∗}X}{2} . \end{aligned} ; (3.3.5)

The vector in the second line is certainly a vector of 𝕃_{N}, which proves our claim.

□

Let M be a CR submanifold of ℂ^{n}. We want to discuss the existence of an intermediate complex manifold Y, with M ⊂ Y ⊂ X such that M is generic in Y. A positive answer comes first.

Theorem 3.3.7. Let M ⊂ ℂ^{n} be CR and C^{ω}. Then, there is a complex submanifold Y ⊂ X which contains M and satisfies T Y |_{M} = T M + J T M.

Proof. Let Φ : ℝ^{2n−l} →^{∼} M be a C^{ω} parametrization of M and Φ^{ℂ} : ℂ^{2n−l} →^{∼} M^{ℂ} its complexification. Let δ : M ↪ ℂ^{n} ×ℂ̄^{n} ℂ̄^{n} be the diagonal embedding, δ^{ℂ} : M^{ℂ} ↪ ℂ^{n} × ℂ̄^{n} its complexification, p_{1} : ℂ^{n} × ℂ̄^{n} → ℂ^{n} the first projection, and let j be the embedding p_{1} ∘ δ|_{M} and k the mapping k = p_{1} ∘ δ^{ℂ}|_{M^{ℂ}}. This yields a commuting diagram

(3.3.6)  \begin{aligned}ℝ^{2n−l} & →^{∼}^{Φ} & M & ↪^{δ} & ℂ^{n} ×ℂ̄^{n} ℂ̄^{n} \\ & & & j ↘ & ↓ p_{1} \\ ↓ & & ↓ & & ℂ^{n} \\ & & & k ↗ & ↑ p_{1} \\ ℂ^{2n−l} & →^{∼}_{Φ^{ℂ}} & M^{ℂ} & ↪_{δ^{ℂ}} & ℂ^{n} × ℂ̄^{n}. \end{aligned}

We define

Y := k(M^{ℂ}).

We wish to prove that

T Y |_{M} = k′(T M^{ℂ})|_{M} = T M + iT M;

in particular, T Y having constant (complex) rank, Y is a (complex) manifold and fulfills the requirements in the statement. In fact, we have

(3.3.7)  δ^{ℂ′}T M^{ℂ} = {α∂_{z} + β̄∂z̄ : α∂_{z}r + β̄∂z̄r = 0},
(3.3.8)  k′T M^{ℂ} = {α∂_{z} : for some β̄, α∂_{z}r + β̄∂z̄r = 0},

(3.3.9) T M + iT M = {(σ + ρ)∂_{z} :

σ∂_{z}r + σ̄∂z̄r = 0 and ρ∂_{z}r − ρ̄∂z̄r = 0}.

We denote by (I) the equation for α, β in (3.3.8) and by (II) the system for σ, ρ in (3.3.9). We observe that the correspondence

α = σ + ρ, β = σ − ρ,

with inverse

σ = \frac{α + β}{2} , ρ = \frac{α − β}{2} ,

interchanges the solutions of (I) with those of (II). This implies T Y |_{M} = T M + iT M, which proves our claim.

□

Guided Exercise 3.3.8. Let M be CR and C^{ω}, parametrized by

ℝ^{d} →^{∼}^{Φ} M. Sometimes, we use the notation ¯∂^{b} to denote the system induced by the vector fields T^{0,1}M in the space of parameters, that is, ¯∂^{b} := (Φ^{−1})_{∗}T^{0,1}M. We want to describe ¯∂^{b} explicitly in an example. We choose M to be the strongly pseudoconvex hypersurface in ℂ^{2} defined by −y_{2} + |z_{1}|^{2} = 0 that we also rewrite as r = 0. In this situation, T^{0,1}M is engendred by the single vector field

L̄ = ∂z̄_{2}r∂z̄_{1} − ∂z̄_{1}r∂z̄_{2}

= −\frac{i}{2}∂z̄_{1} − z_{1}∂z̄_{2}.

On the other hand, M is parametrized over ℝ^{3} by Φ : (x_{1}, y_{1}, x_{2}) ↦ (x_{1} + iy_{1}, x_{2} + i|z_{1}|^{2}). This induces a transformation of vector fields

\begin{cases}∂z̄_{1} → \frac{∂Φ̄_{1}}{∂z̄_{1}} ∂z̄_{1} + \frac{∂Φ_{2}}{∂z̄_{1}} ∂_{z_{2}} + \frac{∂Φ̄_{2}}{∂z̄_{1}} ∂z̄_{2} \\ = ∂z̄_{1} + iz_{1}(∂_{z_{2}} − ∂z̄_{2}), \\ ∂_{x_{2}} → ∂_{z_{2}} + ∂z̄_{2}. \end{cases}

We then have

¯∂^{b} = −\frac{i}{2}∂z̄_{1} − \frac{z_{1}}{2} ∂_{x_{2}},

as we can check from

\begin{aligned}Φ_{∗}(¯∂^{b}) = & − \frac{i}{2}∂z̄_{1} + \frac{z_{1}}{2} (∂_{z_{2}} − ∂z̄_{2}) − \frac{z_{1}}{2} (∂_{z_{2}} + ∂z̄_{2}) \\ & = L̄. \end{aligned}

We refer to the manifold Y of the preceding theorem as an “intrinsic complexification” of M. It can be characterized by

Theorem 3.3.9. Let M be C^{ω} and Y be a complex manifold containing M and such that T Y |_{M} = T M + iT M. Then, any f which is CR and C^{ω}, extends uniquely as holomorphic to Y in a neighborhood of M.

Proof. Let f^{ℂ} : M^{ℂ} → ℂ be the complexification of f, defined, e.g., through a parametrization Φ of M and its complexification Φ^{ℂ}. Our hypothesis that f is CR ensures that L̄f^{ℂ} = 0 for any L̄ ∈ T^{0,1}ℂ^{n}∩ T M^{ℂ}. Thus, f^{ℂ} is constant on the fibers of k defined by (3.3.6). Hence, there is a well-defined holomorphic function f̃ on Y = kM^{ℂ} given by

f̃(z) = f^{ℂ}(z̃) for z̃ ∈ M^{ℂ} with k(z̃) = z.

□

In general, if M is not C^{ω}, we cannot expect that there is a complex submanifold Y ⊂ ℂ^{n} in which M is embedded as generic. However, this is true if we allow a projection instead of an immersion. Before discussing this, we recall that we have already treated the inversion problem for holomorphic mappings: the inverse is holomorphic, if it exists. The same is not true for CR mappings. For instance, the mapping ℝ × ℝ → ℂ, (x, y) → z = x + iy is one-to-one and CR but its inverse Re × Im is not CR. However, it is an instructive exercise to verify that if dim_{CR}M = dim_{CR}M′, then any f : M → M′ which is CR and is a C^{1}-diffeomeorphism is in fact a CR diffeomorphism, that is, f^{−1} is also CR. With this preliminary we can state

Theorem 3.3.10. Let M ⊂ ℂ^{n} be CR. Then, there is a complex manifold Y and a complex projection j : ℂ^{n} ↠ Y such that

\begin{cases}j is a CR diffeomorphism between M and M′ := jM, \\ T Y |_{M′} = T M′ + iT M′. \end{cases}

Proof. We choose Y as the complex plane T_{z_{o}}M + iT_{z_{o}}M, identified in a choice of coordinates to a plane of ℂ^{n}, and choose a transversal projection j : ℂ^{n} ↠ Y. We notice that j is one-to-one over M since the fibers of the projection are transversal to M. Also, j is clearly a CR mapping between M and its image since it is the restriction of a complex projection. We have to show that its inverse is still CR: but this is just a consequence of the remark prior to the statement.

□

Remark 3.3.11. The manifold Y, produced by the proof of Theorem 3.3.10, is not a CR hull of M as it was under the assumption of C^{ω} regularity. However, such a kind of hull exists, indeed, though it has a very strange appearance. According to the theory by Tumanov that we will introduce in Section 3.10, if M is “minimal”, then there is a manifold of wedge type W of dimension n − s, with edge M, and which is contained in the polynomial hull of M. (Here s is the “defect of genericity”.) In particular, CR functions f on M extend uniquely as holomorphic in W and their boundary values on M coincide with f. Thus, the unique holomorphic extension to manifolds of dimension n−s generically “takes” place, minimality being a “generic” condition, provided that we restrict the extension to a “wedge type” domain in an (n − s)-dimensional complex manifold.

We have introduced abstract CR structures and almost complex structures. In one way the latter are more restrictive because they must satisfy 𝕃 ⊕ 𝕃̄ = ℂ ⊗ T M. On the other hand, they are not necessarily assumed to be involutive. If this is the case, they are an example of CR structures and, in fact, they are complex structures according to the Newlander-Nirenberg theorem. There naturally arises the question whether a general CR structure 𝕃 is “embeddable”, that is, if it can be realized as 𝕃 = T^{1,0}M for a submanifold M ⊂ ℂ^{n}. We will see that the answer, in general, is no, even if the question is put in an intrinsic way, that is, by allowing “CR diffeomorphisms” of M as it is clarified in what follows. We refer to the number dim_{ℂ}𝕃 as the dimension of the CR structure and to dim_{ℂ} \frac{ℂ⊗T M}{𝕃⊕𝕃̄} as its CR codimension. There is a complex structure morphism J : T M → T M such that the eigenspaces of the extension of J to 𝕃 ⊕ 𝕃̄ are 𝕃 and 𝕃̄ for the eigenvalues +i and −i, respectively (cf. Section 3.1). Abstract, C^{ω} and CR, structures are embeddable.

Theorem 3.3.12. Let (M, 𝕃) be a C^{ω}, CR structure. Then, there is an embedding M ↪ ℂ^{n} which identifies 𝕃 with T^{1,0}M. More precisely, there is a CR diffeomorphism Φ : M →^{∼} M′ ⊂ ℂ^{n} such that

Φ_{∗}𝕃 = T^{1,0}M′.

Proof. We start from the case of a trivial CR structure 𝕃 = {0}. We denote by d the dimension of M: we want to find an embedding of M into a complex manifold X of dimension d. In fact, if U is open in ℝ^{d} and Φ : U →^{∼} M is a local C^{ω} chart, we take a complexification Φ^{ℂ} which gives a holomorphic parametrization of a d-dimensional manifold X: Φ^{ℂ} : U^{ℂ} →^{∼} X. Here, U^{ℂ} is open in ℂ^{d}. We have the commuting diagram

U ↪^{j} U^{ℂ}
Φ ↓  ↓ Φ^{ℂ}
M  X .

The desired embedding is then obtained by the composition

Φ^{ℂ} ∘ j ∘ Φ^{−1} : M ↪ X .

Our purpose is to generalize this elementary construction. We use the notation m = dim_{CR}M and l = cod_{CR}M. In coordinates x = (x_{1}, ..., x_{2m+l}) ∈ ℝ^{2n+l}, we write

L̄_{j} = ∑^{2m+l} ; i=1a_{ji}(x)∂_{x_{i}},

where each a_{ji} is C^{ω} and ℂ-valued. Since rk_{ℂ}{L̄_{j}}_{j=1,...,m} is m, we can assume, by a linear change of coordinates, that the m × m minor A := (a_{ji})_{j,i≤m} is non-degenerate. We decompose the coordinates as

x = (x′, x′′), x′ = (x_{1}, ..., x_{m}), x′′ = (x_{m+1}, ..., x_{2m+l}).

A linear change of coordinates x̃′ = A^{−1}x′ has the effect of multiplying the coefficients of the ∂_{x′}’s in L̄_{j} by A^{−1}, thus giving

L̄_{j} = ∂_{x_{j}} + ∑^{2m+l} ; i=m+1ã_{ji}∂_{x′′_{i}}.

It follows that

[L̄_{j}, L̄_{i}] ∈ Span{∂_{x′′_{i}}}_{i=m+1,...,2m+l}.

On the other hand, we have

[L̄_{j}, L̄_{i}] ∈ Span{L̄_{j}}_{j=1,...,m},

by the assumption of involutivity. Since the two above “Span”’s are transversal one to another, we conclude that

[L̄_{j}, L̄_{i}] = 0 for all i, j,

that is, we have found a basis of vector fields for 𝕃̄ which commute. We now complexify ℝ^{2m+l}, with variable x, to ℂ^{2m+l}, with variable z, and extend holomorphically the coefficients from a_{ji}(x) to a_{ji}(z). We define

L̄_{j}^{ℂ} = ∂_{z_{j}} + ∑_{i} a_{ji}(z)∂_{z_{i}}.

We recall that [L̄_{j}^{ℂ}, L̄_{i}^{ℂ}]|_{ℝ^{2m+l}} ≡ 0 and hence, by analytic identity, we conclude that

[L̄_{j}^{ℂ}, L̄_{i}^{ℂ}] ≡ 0.

Recall the Frobenius theorem and observe that there is a “complex analytic” version of it for vector fields with holomorphic coefficients (cf. [27, pp. 56–58]). This means that there are holomorphic coordinates z̃ = z̃(z), z̃ = (z̃′, z̃′′) in which the vector fields L̄_{j}^{ℂ} are interchanged with ∂\overline{z̃}′_{j}, j ≤ m. More precisely, we have

\begin{cases}L̄_{j}^{ℂ}z̃′′_{i} = 0, j ≤ m, i ≥ m + 1, \\ z̃′′_{i}(0, z′′) = z_{i}. \end{cases}

Let us denote by Φ : U →^{∼} M the parametrization of M and denote by Ψ the mapping z ↦ z̃′′(z); we define

M̃ := Ψ(Φ^{−1}(M)).

There is no difficulty in proving that M̃ is still a real manifold (cf. [27, pp. 171–172]). What is crucial is to observe that, indeed, the CR structure is preserved, that is, z̃′′(z) is a CR diffeomorphism along M. This follows from

L̄_{j}(z̃′′|_{M}) = (L̄_{j}^{ℂ}z̃′′)|_{M} = 0.
□

We point out that the embedding fails for C^{∞}, instead of C^{ω}, structures. Nirenberg has given an example (cf. [27] and the references therein) of a three-dimensional CR structure which cannot be embedded in ℂ^{2}.

Notes. We refer again to the books by Baouendi, Ebenfelt and Rothschild [8] and Boggess [27] for the presentation of CR structures and CR mappings. As for the notion of partial or intrinsic complexification, we refer, among others, to Rossi [76], Tomassini [85], Andreotti and Hill [6]. As for Theorem 3.3.12 (embedding of C^{ω}, CR structures) we refer to the book by Boggess [27] for an alternative proof.

3.4. The Levi form of a submanifold M ⊂ ℂ^{n} and an abstract CR structure

Summary. We present the so-called “intrinsic” Levi form of a real submanifold M ⊂ ℂ^{n} which is defined by means of commutators of tangential holomorphic and antiholomorphic vector fields. We show that it is “tensorial”, that is, independent of the vector field extension, and it measures the failure of involutivity of the bundle T^{ℂ}M. We also present the “extrinsic” Levi form, defined by contraction with a system of differentials of equations of M.

We first consider the Levi form through the so-called “intrinsic approach” which applies to any abstract CR structure (M, 𝕃) as well. By definition, 𝕃 and 𝕃̄ are involutive, that is, [L_{1}, L_{2}] ∈ 𝕃 and [L̄_{1}, L̄_{2}] ∈ 𝕃̄ for any pair of vector fields L_{1}, L_{2} ∈ 𝕃. Also, [L_{1}, L̄_{2}] belongs to ℂ⊗T M but it does not necessarily belong to 𝕃 ⊕ 𝕃̄. The Levi form precisely measures how much 𝕃 ⊕ 𝕃̄ fails to be involutive. Let

p : ℂ ⊗ T M → \frac{ℂ ⊗ T M}{𝕃 ⊕ 𝕃̄}

be the projection. Let z_{o} be a point of M.

Definition 3.4.1. The intrinsic Levi form at z_{o} is defined over a tangent vector u ∈ T_{z_{o}}^{1,0}M as

L_{M}(z_{o})(u) := \frac{1}{i}p[L, L̄]|_{z_{o}},

where L is any section of T^{1,0}M which extends u from z_{o}, that is, L(z_{o}) = u.

The factor \frac{1}{i} is introduced in order to make the form real.

Sometimes, we drop the symbol p. The Levi form is well defined, that is, independent of the choice of the 𝕃-vector field extension L such that L(z_{o}) = u ∈ T_{z_{o}}^{1,0}M.

Lemma 3.4.2. If N is another vector field extension of u, that is, N(z_{o}) = L(z_{o}) = u, then

p[L, L̄]|_{z_{o}} = p[N, N̄]|_{z_{o}}.

3.4. LEVI FORM OF EMBEDDED AND ABSTRACT STRUCTURES 135

Proof. We write in a basis {L_{1}, ..., L_{m}} of 𝕃, L = ∑_{j} a_{j}L_{j} and N = ∑_{j} b_{j}L_{j}. We know that a_{j}(z_{o}) = b_{j}(z_{o}). We have

p[L, L̄]|_{z_{o}} = ∑_{j i} a_{j}(z_{o})ā_{i}(z_{o})p[L_{j}, L̄_{i}]|_{z_{o}}

= ∑_{j i} b_{j}(z_{o})b̄_{i}(z_{o})p[L_{j}, L̄_{i}]|_{z_{o}}

= p[N, N̄]|_{z_{o}}.

□

Proposition 3.4.3. 𝕃 ⊕ 𝕃̄ is involutive if and only if L_{M} ≡ 0.

Proof. The “only if” part is obvious. “If”: we have to show that if [L, L̄] ∈ 𝕃⊕𝕃̄ for any L ∈ 𝕃, then [L_{1}, L̄_{2}] ∈ 𝕃⊕𝕃̄ for any L_{1}, L_{2} ∈ 𝕃. Now, we have

\begin{aligned}[L_{1} + L_{2}, \overline{L_{1} + L_{2}}] & = [L_{1}, L̄_{1}] + [L_{2}, L̄_{2}] \\ & + ([L_{1}, L̄_{2}] − \overline{[L_{1}, L̄_{2}]}) . \end{aligned} ; (3.4.1)

We also have

\begin{aligned}[L_{1} + iL_{2}, \overline{L_{1} + iL_{2}}] & = [L_{1}, L̄_{1}] + [L_{2}, L̄_{2}] \\ & − i([L_{1}, L̄_{2}] + \overline{[L_{1}, L̄_{2}]}). \end{aligned} ; (3.4.2)

By (3.4.1) and (3.4.2), L_{M} ≡ 0 implies [L_{1}, L̄_{2}] − \overline{[L_{1}, L̄_{2}]} ∈ 𝕃 ⊕ 𝕃̄ and [L_{1}, L̄_{2}] + \overline{[L_{1}, L̄_{2}]} ∈ 𝕃 ⊕ 𝕃̄, respectively, and hence in conclusion [L_{1}, L̄_{2}] ∈ 𝕃 ⊕ 𝕃̄.

□

We wish to discuss in further detail L_{M} when M is an embedded CR generic manifold M ⊂ ℂ^{n}. In this case 𝕃 = T^{1,0}M, 𝕃̄ = T^{0,1}M are both involutive bundles and \frac{ℂ⊗T M}{𝕃⊕𝕃̄} = ℂ ⊗ \frac{T M}{T^{ℂ}M} is the complexification of the totally real part of the tangent bundle. The Levi form is the map

\begin{aligned}L_{M}(z_{o}) : & T_{z_{o}}^{1,0}M & → & \frac{T_{zo}M}{T_{zo}^{ℂ}M} , \\ & u & ↦ & \frac{1}{i}p[L, L̄]|_{z_{o}}, \end{aligned}

for L(z_{o}) = u. We want to revisit Proposition 3.4.3 in this situation. What must be said is that T^{1,0}M and T^{0,1}M are involutive whereas T^{ℂ}M is not, in general. Though these bundles can all be identified by the isomorphisms Re : T^{1,0}M →^{∼} T^{ℂ}M and Re : T^{0,1}M →^{∼} T^{ℂ}M, nevertheless Re [·, ·] ≠ [Re ·, Re ·]. In detail, let M be defined by r_{1} = 0, ..., r_{l} = 0 with ∂r_{1} ∧...∧∂r_{l} ≠ 0 and write r = ^{t}(r_{1}, ..., r_{l}). Take a pair of vector fields L_{1}, L_{2} in T^{1,0}M; these are characterized by L_{j}(r_{i}) = 0 for any j = 1, 2 and i = 1, ..., l. Clearly this condition is stable under [·, ·], that is, [L_{1}, L_{2}](r_{i}) = 0 but it is not preserved by “Re ”. In fact, write L_{1} = ∑_{j} a_{j}∂_{z_{j}}, L_{2} = ∑_{j} b_{j}∂_{z_{j}}, put

c_{j} := ∑_{h} ā_{h}∂z̄_{h}b_{j} − b̄_{h}∂z̄_{h}a_{j},

and define R := ∑_{j} c_{j}∂_{z_{j}}. We have

\begin{aligned}[Re L_{1}, Re L_{2}] & = \frac{1}{4}[L_{1} + L̄_{1}, L_{2} + L̄_{2}] \\ & = \frac{1}{4} ([L_{1}, L_{2}] + \overline{[L_{1}, L_{2}]} + R + R̄) . \end{aligned}

What we have to prove is that R(r) = 0: this characterizes the involutivity of T^{ℂ}M. It is evident that Re R(r) = 0, which is equivalent to saying that T M is involutive. However,

\begin{aligned}R(r) & = − ∑_{j i} ∂_{z_{i}}∂z̄_{j}rā_{j}b_{i} + ∑_{j i} ∂_{z_{i}}∂z̄_{j}rb̄_{j}a_{i} \\ & = −2iIm ∑_{j i} ∂_{z_{i}}∂z̄_{j}rā_{j}b_{i}. \end{aligned}

Also, we have R(r) = 0 for any such R if and only if (∂_{z_{i}}∂z̄_{j}r)_{i j}|_{T^{ℂ}M} ≡ 0, that is, if and only if M is “Levi flat”. On the other hand, the condition R(r) = 0 for any R characterizes the involutivity of T^{ℂ}M. We have thus obtained the proof of the following statement which rephrases Proposition 3.4.3.

Theorem 3.4.4. T^{ℂ}M is involutive if and only if M is Levi flat.

Indeed, what we proved is more than we stated. Whether M is Levi flat or not, the vector field R + R̄ obtained by commutation of Re L_{1} ∈ ker L_{M} and Re L_{2} ∈ ker L_{M} still belongs to ker L_{M}.

Theorem 3.4.5. ker L_{M} is involutive.

Thus, Theorem 3.4.4 is a corollary of Theorem 3.4.5 applied for ker L_{M} = T^{ℂ}M. For an embedded submanifold M ⊂ ℂ^{n}, we want to develop further the relation between L_{M} and the Levi form that we already introduced in Section 1.6 through a system of equations of M. We note that, in case M is generic, the natural projection combined with J, i.e.,

\frac{T M}{T^{ℂ}M} → T_{M}ℂ^{n}, [u] mod T^{ℂ}M ↦ [J u] mod T M,

induces an isomorphism that we still denote by J. By the choice of a basis {∂r_{j}}_{j=1,...,l} of T_{M}^{∗}ℂ^{n} and the dual basis of T_{M}ℂ^{n}, we get an identification K : T_{M}ℂ^{n} → T_{M}^{∗}ℂ^{n}. In composition with J this yields

\frac{T M}{T^{ℂ}M} →^{∼}^{J} T_{M}ℂ^{n} →^{K} T_{M}^{∗}ℂ^{n}.

(Here, as always, ∂r_{j} are identified to sections of T_{M}^{∗}ℂ^{n} by the aid of the correspondence 2Re ∂r_{j} →^{∼} dr_{j}.) We define, in analogy to Section 1.6 where the hypersurface case was treated, the “extrinsic Levi” form L_{M} by

(3.4.3)  L_{M}(u, ū) = ∑_{j} ∂ ¯∂r_{j}(u, ū)∂r_{j}.

Note that the coefficients ∂ ¯∂r_{j}(ū, u) are real and so (3.4.3) is identified with a vector field of M × ℝ^{l} ≃ T_{M}^{∗}ℂ^{n}.

Theorem 3.4.6. We have

(3.4.4)  L_{M}(u, ū) = K ∘ J L_{M}(u).

Proof. What we have to prove is that by coupling the right side of (3.4.4) with ∂r_{j} (in the contraction between forms and vector fields), we get ∂ ¯∂r_{j}(u, ū). In the contraction, the projection p can be disregarded. Let L be a vector field continuation of u; we begin by noticing that

\begin{aligned}⟨∂r_{j}, \frac{1}{i}J [L, L̄]⟩ & = \frac{1}{i}⟨J^{∗}∂r_{j}, [L, L̄]⟩ \\ & = ⟨∂r_{j}, [L, L̄]⟩, \end{aligned} ; (3.4.5)

because J^{∗} ∘ ∂ = i∂. Cartan’s formula applied to the 1-form ∂r_{j} yields

⟨∂r_{j}, [L, L̄]⟩ = −⟨¯∂∂r_{j}, L ∧ L̄⟩ + L(⟨∂r_{j}, L̄⟩) − L̄(⟨∂r_{j}, L⟩).

In the right side, we have ⟨∂r_{j}, L⟩ = 0 since L ∈ T^{1,0}M and ⟨∂r_{j}, L̄⟩ = 0 since the coupling of a (1, 0)-form with a (0, 1)-vector field gives 0. Also, by the identity ¯∂∂ = −∂ ¯∂, we get

⟨∂r_{j}, [L, L̄]⟩ = ⟨∂ ¯∂r_{j}, L ∧ L̄⟩ = ∂ ¯∂r_{j}(u, ū).
□

Remark 3.4.7. We have already encountered this circumstance in Section 1.9: the form ∂ ¯∂r is alternant on T^{1,0}⊕T^{0,1}, but, when restricted to (T^{1,0} ⊕ {0}) × ({0} ⊕ T^{0,1}), it becomes hermitian.

Definition 3.4.8. Let ξ ∈ (T_{M}^{∗}ℂ^{n})_{z_{o}} ∼ ℝ^{l} and set r^{ξ} := ∑_{j} ξ_{j}r_{j}; the “microlocal” Levi form is the 1-component hermitian form defined by

\begin{aligned}L_{M}^{ξ}(z_{o})(u, ū) & := ⟨ξ, L_{M}(z)(u, ū)⟩ \\ & = ∂ ¯∂r^{ξ}(u, ū) for u ∈ T_{z_{o}}^{ℂ}M. \end{aligned} ; (3.4.6)

Note that this is a real-valued 1-component function: L_{M}^{ξ}(z_{o}) : T^{1,0}M ×_{T^{1,0}M} T^{0,1}M → ℝ.

Notes. Intrinsic/extrinsic Levi form, involutivity of T^{ℂ}M and Levi flatness of M are classical notions. We refer to the books by Boggess [27] and Baouendi, Ebenfelt and Rothschild [8] for a more detailed account of these topics.

3.5. Real/complex symplectic spaces

Summary. The content of this section is the link between the Levi form L_{M} of a submanifold M ⊂ ℂ^{n} and the symplectic 2-form σ = dζ ∧ dz restricted to the conormal bundle T_{M}^{∗}ℂ^{n}. We show that if M has constant Levi corank s^{0}, or equivalently, T_{M}^{∗}ℂ^{n} is a CR submanifold of T^{∗}ℂ^{n}, there is a complex Hamiltonian foliation of T_{M}^{∗}ℂ^{n}. By a symplectic “blow up”, we show how to interchange a highercodimensional submanifold with a hypersurface preserving s^{0}. Next, we assume in addition that M is C^{ω}. In this case, the foliation is straightenable, that is, T_{M}^{∗}ℂ^{n} takes the form T_{M′}^{∗}ℂ^{n−s^{0}} × ℂ^{s^{0}} for M′ a Levi non-degenerate hypersurface of ℂ^{n−s^{0}}. By an argument of geometric optics, we also show how to “deform” M into a hypersurface passing through M and keeping s^{0} unchanged. Application of this result to the tangential ¯∂-system is presented.

Let z = (z_{1}, ..., z_{n}) be complex variables in ℂ^{n} and (z, ζ) the canonically associated coordinates in T^{∗}ℂ^{n} for the choice of dz_{1}, ..., dz_{n} as a basis of 1-forms. Let

ω = ∑_{j=1}^{n} ζ_{j}dz_{j}

and

σ = dω = ∑_{j=1}^{n} dζ_{j} ∧ dz_{j}

be the canonical 1- and 2-forms, respectively: we also write ω = ζdz and σ = dζ ∧ dz for short. The form σ is “symplectic”, that is, nondegenerate. Decompose σ = σ^{ℝ} + iσ^{𝕀} and note that both σ^{ℝ} and σ^{𝕀} induce a real symplectic—non-degenerate—structure on (T^{∗}ℂ^{n})^{ℝ} ≃ T^{∗}ℝ^{2n}. For the form σ we call a complex submanifold V such that T V^{⊥σ} ⊂ T V (resp. T V ⊂ T V^{⊥σ}) involutive (resp. isotropic). We call a submanifold which is both involutive and isotropic lagrangian: note that its dimension must be n, half of the dimension of T^{∗}ℂ^{n}. We use similar terminology for the forms σ^{ℝ} and σ^{𝕀}. Thus, a real submanifold V is ℝ- (resp. 𝕀-) involutive/isotropic/lagrangian when it is involutive/isotropic/lagrangian for σ^{ℝ} (resp. σ^{I}). Note that the real dimension of an ℝ- or 𝕀-lagrangian submanifold must be 2n, half the real dimension of (T^{∗}ℂ^{n})^{ℝ}. Let M be a real generic submanifold of ℂ^{n} and let T_{M}^{∗}ℂ^{n} be its conormal bundle, the set of the forms whose real part vanishes on T M. Let r = 0 for r = ^{t}(r_{1}, ..., r_{l}) be a system of ℂ-independent equations for M, that is, ∂r_{1} ∧ ... ∧ ∂r_{l} ≠ 0. Through the basis ∂r_{1}, ..., ∂r_{l} of T_{M}^{∗}ℂ^{n}, we have an identification M × ℝ^{l} →^{∼} T_{M}^{∗}ℂ^{n}, (z, ξ) → (z, ∂r^{ξ}(z)) where ∂r^{ξ} := ∑_{j} ξ_{j}∂r_{j}. First note that for any (z, ξ) ∈ T_{M}^{∗}ℂ^{n} we have ω^{ℝ}(z, ξ)|_{T_{z}M} = 0 and therefore, by differentiating,

(3.5.1)  σ^{ℝ}(z, ξ)|_{T_{(z,ξ)}T_{M}^{∗} ℂ^{n}} = 0.

Thus, T_{M}^{∗}ℂ^{n} is ℝ-lagrangian and in particular σ = iσ^{𝕀} over T T_{M}^{∗}ℂ^{n}. Also, note that ω(z, ξ)|_{T_{z}M} ≠ 0 or, in other words, T_{M}^{∗}ℂ^{n} is “regular” since, otherwise, we would have ∂r^{ξ}|_{T M} = 0 which is forbidden by the ℂ-independence of the ∂r_{j}’s, that is, the genericity of M. By (3.5.1) and by ℂ-linearity we get

(3.5.2)  ker σ|_{T T_{M}^{∗} ℂ^{n}} = T^{ℂ}T_{M}^{∗}ℂ^{n}.

Thus, in particular, T_{M}^{∗}ℂ^{n} is totally real (that is, its T^{ℂ} is 0) if and only if σ|_{T T_{M}^{∗} ℂ^{n}}, or equivalently, σ^{𝕀}|_{T T_{M}^{∗} ℂ^{n}}, is non-degenerate. We call T_{M}^{∗}ℂ^{n} “𝕀-symplectic” in this case. More generally, T_{M}^{∗}ℂ^{n} is a CR submanifold of T^{∗}ℂ^{n}, that is, its T^{ℂ} has constant rank, if and only if σ|_{T T_{M}^{∗} ℂ^{n}} or σ^{𝕀}|_{T T_{M}^{∗} ℂ^{n}} have constant rank. We go further with our discussion. We differentiate ξ∂r(z)|_{T T_{M}^{∗} ℂ^{n}} and get

(3.5.3)  σ|_{T_{(zo,ξo)}T_{M}^{∗} ℂ^{n}} = dξ ∧ ∂r|_{ℝ^{l}×T_{zo}M} + ¯∂∂r^{ξ_{o}}|_{T_{zo}M}.

We take a closer look of (3.5.3) which we will also describe in coordinates in Exercise 3.5.1. If the alternant form ¯∂∂r is restricted from T^{1,0} ⊕ T^{0,1} to (T^{1,0} ⊕ {0}) × ({0} ⊕ T^{0,1}), it becomes hermitian, but if, instead, it is restricted to the diagonal(T ℂ^{n})^{ℝ} × (T ℂ^{n})^{ℝ}, it becomes purely imaginary, that is, ¯∂∂r = i¯∂∂r^{𝕀} on vectors (ż + \overline{ż}, ż^{1} + \overline{ż}^{1}). More precisely

¯∂∂r(ż + \overline{ż}, ż^{1} + \overline{ż}^{1}) = 2iL_{r}(\overline{ż}, ż^{1}).

Also, in the first term in the right of (3.5.3), we have ∂r^{ℝ}|_{T M} = 0. Thus (3.5.3) shows once more that σ^{ℝ}|_{T T_{M}^{∗} ℂ^{n}} = 0. Moreover, if we restrict (3.5.3) to vectors which are not only in T M but also in T^{ℂ}M, then the first term on the right is 0 and we get

(3.5.4)  σ|_{T_{(zo,ξo)}T_{M}^{∗} ℂ^{n}∩π^{−1}(T^{ℂ}M)} = 2iL_{r^{ξo}}|_{T_{zo}^{ℂ}M}.

In particular, this restriction applies to ker σ|_{T T_{M}^{∗} ℂ^{n}} which is contained in π^{−1}T^{ℂ}M. Thus (3.5.4) shows that the projection π induces an identification

(3.5.5)  ker σ →^{∼} ker L_{M}.

Before going ahead with our discussion, we want to relax and have an alternative proof of what we have seen so far.

Guided Exercise 3.5.1. Prove (3.5.2) and (3.5.5) in coordinates.

(a) Start by proving

(3.5.6) T_{(z_{o},ξ_{o})}T_{M}^{∗}ℂ^{n} = {(ż, ζ̇) : ż ∈ T_{z_{o}}M, ζ̇ = ξ∂r(z_{o})

+ ∂ż∂r^{ξ_{o}}(z_{o}) + ∂\overline{ż}∂r^{ξ_{o}}(z_{o}) for ξ ∈ ℝ^{l}\}.

(Here (3.5.6) is obtained by differentiating the parametrization M × ℝ^{l} → T_{M}^{∗}ℂ^{n}, (z, ξ) ↦ (z, ξ∂r(z)).)

(b) Describe the relation between σ|_{T T_{M}^{∗} ℂ^{n}} and L_{M}. First prove that

\begin{aligned}σ(z_{o}, ξ_{o}) & = d(ξ∂r)(z_{o}, ξ_{o}) \\ & = dξ ∧ ∂r(z_{o}) + ¯∂∂r^{ξ_{o}}(z_{o}). \end{aligned}

Next, observe that for any ξ, ξ^{1} ∈ ℝ^{l}, ż+\overline{ż} ∈ T_{z_{o}}M and ż^{1}+\overline{ż}^{1} ∈ T_{z_{o}}M, we have

⟨d(ξ∂r)(z_{o}, ξ_{o}) , (ξ, ż+\overline{ż})∧(ξ^{1}, ż^{1}+\overline{ż}^{1})⟩ = 2i(∂r^{ξ𝕀}ż^{1}−∂r^{ξ^{1}𝕀}ż+L_{r^{ξo}}(\overline{ż}, ż^{1})).

Conclude that if ż + \overline{ż} ∈ T_{z_{o}}^{ℂ}M and ż^{1} + \overline{ż}^{1} ∈ T_{z_{o}}^{ℂ}M, then

⟨d(ξ∂r)(z_{o}, ξ_{o}) , (ż + \overline{ż}) ∧ (ż^{1} + \overline{ż}^{1})⟩ = 2iL_{r^{ξo}}(\overline{ż}, ż^{1}).

This proves once more (3.5.4).

(c) Carry out the proof. Show that if

(ż, ζ̇) = i(ż^{1}, ζ̇^{1}) ∈ T T_{M}^{∗}ℂ^{n},

then (3.5.6) implies

(3.5.7)  \begin{cases}ż = iż^{1} ∈ T M, that is, ż ∈ T^{ℂ}M, \\ 2∂\overline{ż}∂r^{ξ_{o}} = (−ξ + iξ^{1})∂r, that is, ż ∈ ker L_{r^{ξo}}. \end{cases}

Thus (3.5.2) and (3.5.5) have been re-proved.

We want to go ahead with our discussion and remark that, after we achieved our goal of relating L_{M} to σ, we have evidence that s_{M}^{0}, the number of the null eigenvalues of L_{M}, is a symplectic invariant. If T_{M}^{∗}ℂ^{n} is transformed to T_{M_{1}}^{∗}ℂ^{n}, then s_{M_{1}}^{0} = s_{M}^{0}. Of course, this is not true for s_{M}^{±}; a symplectic transformation can interchange + with − and also affect the codimension l_{1} ≠ l. We well see through explicit examples how this may occur. We wish to adopt a new notation. We write M −−→ M_{1} when M and M_{1} are related by a symplectic transformation, say χ, which interchanges their conormal bundles, that is, when we have a commutative diagram

T_{M}^{∗}ℂ^{n} →^{χ} T_{M_{1}}^{∗}ℂ^{n}
π ↓  ↓ π
M −−→ M_{1}.

First, we want to show that, under a change of coordinates in T^{∗}ℂ^{n}, “a submanifold is generically a hypersurface”. Let M be a submanifold of ℂ^{n} defined, in coordinates z = (z′, z′′) with z′ = (z_{1}, ..., z_{l}) at z_{o} = 0, by the equations

(3.5.8)  y′ = h_{j}(x′, z′′, z̄′′), h_{j}(0) = 0, ∂h_{j}(0) = 0.

We consider the transformation

(3.5.9)  χ : T^{∗}ℂ^{n} → T^{∗}ℂ^{n}, χ = χ̃ × id,

where

(3.5.10)  χ̃ : T^{∗}ℂ^{l} → T^{∗}ℂ^{l}, (z′, ζ′) ↦ (z′ + ϵ\frac{ζ′}{∑_{j} ζ_{j}^{′2}} ; ζ′).

Here ϵ is a small real parameter. We check readily that χ is (holomorphic and) symplectic. We then define M_{1} := πχ(T_{M}^{∗}ℂ^{n}); since T M_{1} = T M ⊕ T S_{ϵ}^{l−1} where S_{ϵ}^{l−1} is the ϵ-sphere in the plane ℝ^{l} of the y′’s and M_{1} is a hypersurface and remains a hypersurface also under any homogeneous transformation close to χ. As we have already noticed, we must have s_{M_{1}}^{0} = s_{M}^{0}. On the other hand, the “gain” of l − 1 in dimension passing from M to M_{1} is obtained by “adding” the sphere S_{ϵ}^{l−1} (from the outward side). Since this has l − 1 positive Levi eigenvalues, we get for suitable ϵ

s_{M_{1}}^{−} = s_{M}^{−}, s_{M_{1}}^{+} = s_{M}^{+} + (l − 1).

This can be readily checked. When L_{M}, or equivalently, σ|_{T T_{M}^{∗} ℂ^{n}}, have constant rank, then we have developed in Sections 2.3 and 3.4 two different theories of integration of their (involutive) kernels; we want to put them now in a unitary frame. We know on one hand that Ker L_{M} is involutive on account of Theorem 3.4.5. Thus, M is foliated by the integral leaves of Ker L_{M}—the Levi leaves—and these are complex manifolds since their tangent bundle, Ker L_{M}, is a complex bundle. In general, the complex structure of T_{M}^{∗}ℂ^{n} being poorer than the one of M, the complex leaves of M cannot be “lifted” to complex leaves of T_{M}^{∗}ℂ^{n}. However, in the present case of Levi leaves, owing

to T^{ℂ}T_{M}^{∗}ℂ^{n} →^{∼}^{π′} Ker L_{M}, for any leaf Γ on M through z and for any ξ ∈ T_{M}^{∗}ℂ_{z}^{n} there is the unique complex section Γ^{∗} through (z, ζ) over Γ, that is, satisfying πΓ^{∗} = Γ. This is also in accordance with Theorem 2.3.5 which gives in addition the trivialization of the foliation (but under a C^{∞}, not holomorphic, change of coordinates). In either way we have obtained

Theorem 3.5.2. Let M be generic and have constant Levi rank. Then T_{M}^{∗}ℂ^{n} is foliated by the integral leaves of ker σ = π^{′ −1} ker L_{M}.

We come back to the foliation {Γ} on M and notice that it can by no means be trivialized, that is, reduced to a “product” M = M′ × ℂ^{s^{0}} by a complex holomorphic change of coordinates in ℂ^{n}. This is true only for a smooth transformation of ℝ^{2n}. A celebrated example is the “tube”

M = {z ∈ ℂ^{n} : y_{1}^{2} = ∑_{j≥2}y_{j}^{2}}.

We choose one of the two cones with vertex at y = 0, say the positive cone y_{1} > 0, and get what is called the “future tube” (“past tube” corresponding to the complementary choice y_{1} < 0). This is a relevant model of symmetric space in quantum mechanics. Here we have a “radial” Levi foliation given by the system of radii defined for ζ = ξ+iη by Γ_{ζ} = ξ+iℂη. These rays are not straightenable as is shown, e.g., in [41] by an investigation on the 3-jet of M. However they are straightenable at least when M is C^{ω} if we tackle the problem from “upstairs”.

Theorem 3.5.3. Let M be generic, C^{ω}, and have constant Levi rank. Then, the Hamiltonian foliation of T_{M}^{∗}ℂ^{n} is straightenable: there are complex symplectic homogeneous coordinates (z, ζ) in T^{∗}ℂ^{n} such that T_{M}^{∗}ℂ^{n} takes the form

T_{M}^{∗}ℂ^{n} ≃ T_{M′}^{∗}ℂ^{n′} × ℂ^{s^{0}},

where M′ is a Levi non-degenerate hypersurface of ℂ^{n′} for n′ = n − s^{0}.

Proof. Let V_{1} be a partial or intrinsic complexification of T_{M}^{∗}ℂ^{n} in T^{∗}ℂ^{n} (cf. Section 3.3) and let f_{1} = 0, ..., f_{s^{o}} = 0 be a system of holomorphic equations which define V_{1} in T^{∗}ℂ^{n}. We choose ζ̃_{n−s^{o}+1} = f_{1}, ..., ζ̃_{n} = f_{s^{0}} and supplement to a full system of coordinates according to the obvious holomorphic version of Theorem 2.3.5. In the new coordinates, V_{1} is interchanged with T^{∗}ℂ^{n′} × ℂ^{s^{0}} and T_{M}^{∗}ℂ^{n} with Λ′ × ℂ^{s^{0}} where Λ′ is an ℝ-lagrangian submanifold. But it is in fact 𝕀-symplectic because (Λ′×ℂ^{s^{0}})^{⊥} = {0}×ℂ^{s^{0}}. By choosing new coordinates in T^{∗}ℂ^{n′} such that Span{∂_{ζ′_{j}}}_{j=1,...,n−s^{0}} ∩T_{(z_{o},ζ′_{o})}Λ = ζ′_{o}ℝ, the real line engendered by the Euler vector field, we achieve that Λ′ = T_{M′}^{∗}ℂ^{n′} for a Levi nondegenerate hypersurface M′ ⊂ ℂ^{n′}.

□

Remark 3.5.4. The numbers s^{+}, s^{−} and s^{0} are invariant under holomorphic change of coordinates in ℂ^{n}. In T^{∗}ℂ^{n}, as has already been noticed, the only invariant is s^{0}. In fact, we can give s^{+} and s^{−} the values we wish provided that s^{0} is respected. For instance, let M = ∂𝔹^{n} be the boundary of the unit ball in ℂ^{n}, and let us “microlocalize” our attention at (z_{o}, ζ_{o}) = ((1, 0, ...), (−1, 0, ...)); then, s_{M}^{−} = n − 1, that is, M is strongly pseudoconcave. By moving the boundary points along the rays of 𝔹^{n} with a “caustic” at 0, we can reverse the curvature. In detail, let

(3.5.11) χ : (z, ζ) ↦ (z − λ\frac{ζ}{(∑_{j} ζ_{j}^{2})^{\frac{1}{2}}} ; ζ) for (z, ζ) close to (z_{o}, ζ_{o}),

where λ > 1. Though this transformation is not globally defined in T^{∗}ℂ^{n} because of the vanishing of the denominator, however it makes sense along T_{M}^{∗}ℂ^{n}. If M −−→ M̃ is the base correspondence underlying (3.5.11), it transforms s_{M}^{−} = n − 1 into s+ ; M̃ = n − 1. Thus M̃ is strongly pseudoconvex at (z̃_{o}, ζ̃_{o}) = χ(z_{o}, ζ_{o}) whereas M was strongly pseudoconcave at (z_{o}, ζ_{o}). However, we cannot always achieve s+ ; M̃ = n − 1, that is, M̃ strongly pseudoconvex, unless s0 ; M̃ = s_{M}^{0} = 0.

Theorem 3.5.3 is the main tool in solving the problem that we describe now. Let ξ_{o} be a conormal to M at z_{o} and assume rk L_{M}(z, ξ) ≡ const in a neighborhood of (z_{o}, ξ_{o}): is there a way to find a hypersurface S ⊃ M with conormal ξ_{o} at z_{o} such that s_{S}^{0} = s_{M}^{0}? If M is Levi non-degenerate, defined, say, by r_{1} = 0, ..., r_{l} = 0, we may define our “extension” S of M for instance by

S = {z : r^{ξ_{o}}(z) + c(r_{1}^{2}(z) + ... + r_{l}^{2}(z)) = 0},

for large c. Otherwise, if M is degenerate but in the favorable case in which M is a product M = M_{1} × ℂ^{s^{0}}, we can do a similar construction with parameters which leads to S = S_{1} × ℂ^{s^{0}}. In general, it is Theorem 3.5.3 which comes to our rescue.

Theorem 3.5.5. If s^{0} ≡ const in a neighborhood of (z_{o}, ξ_{o}) and M is C^{ω}, then there exists a hypersurface S ⊃ M, with conormal ξ_{o} at z_{o} such that s_{S}^{0} ≡ s_{M}^{0}.

Proof (Cf. [102, Theorem 1]). Here is the scheme of the proof. By a symplectic transformation we interchange M with a hypersurface and reduce it to a product. Next, deform the non-degenerate term so that it is brought back to a hypersurface by the inverse transformation: the deformation, as well as the symplectic transformation, does not affect s^{0} since this is a symplectic invariant. In detail, we choose coordinates such that ξ_{o} ∈ ℝ^{l} (by the genericity of M) and consider the transformation defined by (3.5.9) and (3.5.10) that we denote by χ_{1}. It interchanges M −−→ M_{1} where M_{1} is a hypersurface and satisfies s_{M_{1}}^{+} = s_{M}^{+} + (l − 1). Next, we use the symplectic transformation provided by Theorem 3.5.3, which we denote by χ_{2} and which interchanges M_{1} −−→ M_{2} = M′_{2} × ℂ^{s^{0}} without affecting the numbers s^{±}. We now want to deform M_{2}. We take R′_{2} ⊂ M′_{2} of dimension 2(n − s^{o}) − l transversal to χ_{2}χ_{1}(T_{M}^{∗}ℂ_{z_{o}}^{n}). We take a real analytic hypersurface S′_{2} which intersects M′_{2} along R′_{2} with order of contact 2 and stays in the −χ_{2}χ_{1}(p_{o}) side of M′_{2}. We also set S_{2} = S′_{2}×ℂ^{s^{0}}. We go now back along χ_{2}^{−1} and χ_{1}^{−1} and see that χ_{2}^{−1}(T_{S_{2}}^{∗}ℂ^{n}) is still the conormal bundle to ¡a hypersurface, say T_{S_{1}}^{∗}ℂ^{n}, and χ_{1}^{−1}(T_{S_{1}}^{∗}ℂ^{n}) is still the conormal to a hypersurface, say, T_{S}^{∗}ℂ^{n}. We can describe the scheme of our construction by

M −−→ M_{1} −−→ M′_{2} × ℂ^{s^{0}}
↓
S ←−− S_{1} ←−− S′_{2} × ℂ^{s^{0}}.

It is clear that S still contains M and

s_{S}^{0} = s_{M}^{0},

because of the symplectic invariance of the s^{0}’s. Also, by the choice of our orientations, s_{S}^{+} = s_{M}^{+} + (l − 1); thus, the jump of l − 1 in the dimension results in a full gain for s^{+}.

□

Theorem 3.5.5 is applied in [102] to the study of the tangential ¯∂-system. Let Ω = Ω^{±} be the two local components of ℂ^{n} \ S in a neighborhood of a point z_{o} ∈ S. We denote by K^{j}(Ω̄) the cohomology of ¯∂ on C^{∞}(Ω̄)-forms of degree j and by K^{j}(Ω̄)_{z_{o}} its “germ” at z_{o} (cf. Section 1.9). Assume that s_{∂Ω}^{−} ≡ q in a neighborhood of z_{o}, for the choice of the outward conormal to Ω; then

(3.5.12)  K^{j}(Ω̄)_{z} ≡ 0 for any j ≠ 0, q and z close to z_{o}.

In fact, the vanishing of (3.5.12) for j ≥ q + 1 is proved in Section 1.9. The vanishing for j ≤ q − 1 is easier; this is related to the q-concavity of Ω and can be proved, e.g., by the integral method (cf. [27, Chapter 24]. Conversely, if (3.5.12) holds for any j ≥ 1, then Ω must be pseudoconvex as the implications (iii) ⇒ (iv) ⇒ (i) of Theorem 1.8.13 show. The statement therein concerns the cohomology for C^{∞}(Ω)-, instead of C^{∞}(Ω̄)-, forms. But it is readily seen that non-extendibility for functions in hol(Ω) or in hol(Ω) ∩ C^{∞}(Ω̄) are the same. To see this at z = 0, just use f = e^{− \frac{1}{\sqrt{z}}} instead of f = \frac{1}{z}. Thus, if s^{−} ≡ q for q ≠ 0, since the only non-trivial cohomology may occur in degree q, then this must be ≠ 0:

(3.5.13)  K^{q}(Ω̄)_{z} ≠ 0 for any z close to z_{o}.

We consider now a generic submanifold M ⊂ ℂ^{n} of codimension l > 1 and denote by K^{j}(M) the cohomology in degree j of the tangential, or induced, system ¯∂_{M} over C^{∞}(M)-forms. If S is a hypersurface which passes through M, the natural restriction induces a morphism in the cohomology

(3.5.14)  K^{j}(S) → K^{j}(M).

Again, the restriction from the two half spaces Ω̄^{±} to S induces a decomposition

(3.5.15)  K^{j}(Ω̄^{+}) ⊕ K^{j}(Ω̄^{−}) ↠ K^{j}(S),

which is an isomorphism for any j ≥ 1 (cf. for example [6]). We then have the following consequence of Theorem 3.5.5.

Theorem 3.5.6. Let M be a C^{ω}, generic, submanifold of ℂ^{n}, (z_{o}, ξ_{o}) a point of T_{M}^{∗}ℂ^{n}, and assume

s_{M}^{±}(z, ξ) ≡ q^{±} for (z, ξ) ∈ T_{M}^{∗}ℂ^{n} close to (z_{o}, ξ_{o}).

Then, for any z close to z_{o}

K^{j}(M)_{z} ≠ 0 if j = 0, q^{+}, and q^{−}.

Proof. We prove that K^{j}(M)_{z} ≠ 0 for j = q^{−}. On account of Theorem 3.5.5, we take S which contains M, has conormal ξ_{o} at z_{o} and satisfies s_{S}^{−}(z, ξ) ≡ q^{−} for any (z, ξ) ∈ T_{S}^{∗}ℂ^{n} close to (z_{o}, ξ). According to (3.5.13), we then have K^{q^{−}}(Ω̄^{+})_{z} ≠ 0 and hence also K^{j}(S)_{z} ≠ 0 because of (3.5.15). Finally, owing to [53] or [4], we know that (3.5.14) is injective for j = q^{−} and this implies K^{j}(M)_{z} ≠ 0. The non-vanishing of K^{q^{+}}(M)_{z} can be proved likewise, by switching from ξ_{o} to −ξ_{o} and from Ω̄^{+} to Ω̄^{−}.

□

Suggested research. (1) When s_{M}^{0} is constant, it is not necessary to assume N to be C^{ω} for the Hamiltonian foliation to exist: this is a consequence of Theorem 2.3.7. Is this foliation straightenable in T^{∗}ℂ^{n} if M is only C^{∞}?

(2) Theorem 3.5.5 and its corollary Theorem 3.5.6 are likely to be true if M is not necessarily C^{ω} but just C^{∞} but their proof does not apply to this situation. Try an alternative proof which sustains the case C^{∞}.

Notes. The relation between the symplectic and the Levi forms is classical and very fruitful in the geometric study of PDE’s. We followed here the guidelines of Sato, Kashiwara and Kawai [78] and Kashiwara and Schapira [53]. The result on the straightening of a Levi foliation in the symplectic space (Theorem 3.5.5) follows the presentation by Zampieri in [102]. The non-straightening in the base space is discussed, among others, by Ebenfelt in [41]. The deformation argument contained in Theorem 3.5.3 is developed by Zampieri in [102] as well as the non-vanishing of the tangential cohomology contained in Theorem 3.5.5. The latter was already obtained by Andreotti, Fredricks and Nacinovich [5] in case the Levi form is non-degenerate; note that, in their setting, M need not be C^{ω}.

3.6. Approximation of CR functions by polynomials

Summary. We present here the result of [13]: CR functions f on generic submanifolds M ⊂ ℂ^{n} can be approximated by polynomials uniformly in controlled neighborhoods of any point. The result consists in two steps: an approximation of the restriction of f to a totally real submanifold N ⊂ M by means of convolution with the the heat kernel and a deformation of the integration chain in a family of N’s which sweep out M. The deformation leaves the integral unchanged, because of Stokes’s theorem and since f is CR. This result will play a crucial role in Sections 3.7, 3.8 and 3.10 where the holomorphic extension of CR functions is studied.

We first discuss CR functions which are not C^{1}. In ℝ^{n}, we have an identification of locally integrable functions with distributions given by

(3.6.1)  L_{loc}^{1}(ℝ^{n}) → D′(ℝ^{n}), f ↦ [v ↦ ∫_{ℝ^{n}} fvdV],

where v ranges through the family of C_{c}^{∞} functions and dV denotes the element of volume in ℝ^{n}. Let M be an oriented submanifold of ℂ^{n} of codimension l: the identification analogous to (3.6.1) requires the choice of a (2n − l)-form α (an element of the exterior algebra ⋀_{2n−l}T^{∗}M) with

C^{∞} coefficients everywhere ≠ 0. This yields

L_{loc}^{1}(M) → D′(M), f ↦ [v ↦ ∫ fvα].

Let M be generic of codimension l. We choose a local basis dt_{1}, ..., dt_{n−l} of closed forms in (T^{0,1}M)^{∗} and complete into a full basis of ℂ ⊗ T^{∗}M:

dξ_{1}, ..., dξ_{n}, dt_{1}, ..., dt_{n−l}.

We denote the dual basis of ℂ ⊗ T M by

X_{1}, ..., X_{n}, L̄_{1}, ..., L̄_{n−l};

thus, L̄_{1}, ..., L̄_{n−l} are a system of CR vector fields which span T^{0,1}M. We choose α := dξ_{1} ∧ ... ∧ dξ_{n} ∧ dt_{1} ∧ ... ∧ dt_{n−l} as the 2n − l integration form on M. For the adjoint differential operators we have

(3.6.2)  ^{t}X_{j} = −X_{j}, ^{t}L̄_{j} = −L̄_{j},

in the sense that ∫ X_{j}(u)vα = − ∫ uX_{j}(v)α and ∫ L̄_{j}(u)vα = − ∫ uL̄_{j}(v)α for any u, v ∈ C_{c}^{∞}. This is easily proved by means of Stokes’s formula. For example,

∫ L̄_{j}(u)vα = − ∫ uL̄_{j}(v)α + ∫ L̄_{j}(uv)α

= − ∫ uL̄_{j}(v)α + (−1)^{n+j+1} ∫ d(uv... ∧ dt_{j}^{∨} ∧ ...)

= − ∫ uL̄_{j}(v)α + ∫_{∂(supp(uv))} uv... ∧ dt_{j}^{∨} ∧ ...

= − ∫ uL̄_{j}(v)α.

We have used here the symbol “^{∨}” to mean that the corresponding item is omitted. Since every differential operator P on M is a combination of the X_{j}’s and the L̄_{j}’s, then we can define ^{t}P starting from (3.6.2). With this preliminary, we are in a position to define “weak solutions” of the tangential CR equations.

Definition 3.6.1. We say that u is a CR distribution, and we write u ∈ D′_{CR}, when L̄u = 0, for all L̄ ∈ T^{0,1}M, in the sense that

(3.6.3)  ⟨u,^{t}L̄(v)⟩ = 0 for any v ∈ C_{c}^{∞},

where ⟨·, ·⟩ denotes the duality between D′ and C_{c}^{∞}.

The adjunction being made under a choice of the basis X_{j}, L̄_{j}, the above definition is not invariant. If u ∈ L_{loc}^{1}, then (3.6.3) is equivalent

to ∫ u^{t}L̄(v)α = 0 for any L̄ and v where α = ⋀_{j=1,...,n}dξ_{j} ∧ ⋀_{j=1,...,n−l}dt_{j}

which declares the dependence of the condition on the choice of the density α. In particular, we can characterize u ∈ C_{CR}^{0} by ∫ udβ = 0 for any β with compact support of type (n, n − l − 1), that is, of type

β = ∑_{j} β_{j}...dt_{j}^{∨}....

Proposition 3.6.2. If u is C^{1}, then the notions of “weak solution” and “classical solution” coincide.

Proof. Let β = ∑_{j} β_{j}...dt_{j}^{∨}...: we have

∫ udβ = − ∫ du ∧ β + ∫ d(uβ)

= − ∫ du ∧ β (by Stokes’s theorem)

= ∑_{j}(−1)^{j+n} ∫ L̄_{j}(u)β_{j}α.

Thus, L̄_{j}(u) = 0, for all j, is equivalent to ∫ udβ = 0 for any β of type (n, n − l − 1), which proves the proposition.

□

We state the approximation theorem due to Baouendi and Treves [13].

Theorem 3.6.3. Let M be a C^{∞}, CR submanifold of ℂ^{n} and z_{o} a point of M. Then, there exists a relatively compact neighborhood M′ of z_{o} on M such that any u ∈ C_{CR}^{0}(M) (resp. C_{CR}^{k}(M), C_{CR}^{∞}(M), D′_{CR}(M)) is approximated by a sequence of polynomials in the convergence of C_{CR}^{0}(M′) (resp. C_{CR}^{k}(M′), C_{CR}^{∞}(M′), D′_{CR}(M′)).

Proof. The proof is given in several steps throughout the rest of the section. We begin by remarking that M is CR diffeomorphic to a generic submanifold (of some subspace ℂ^{n′} for n′ ≤ n): hence it is not restrictive to assume M generic from the beginning. We declare first the kind of the approximation. We take a germ of a totally real maximal submanifold N ⊂ M. We assume that the equations of M and N are normalized at z_{o} as follows. The manifold M is defined by y′_{j} = h_{j} for any j ≤ l and N is defined by y_{j} = h_{j} for any j ≤ n with h_{j}(0) = 0, ∂h_{j}(0) = 0. In particular, |Im z| < ϵ on suitable neighborhoods M′ and N′ of z_{o} on M and N, respectively. With this information and by taking u with compact support u ∈ E′(M), we define, for a large parameter λ, the convolution of u with the heat kernel:

(3.6.4)  T_{λ}u(z) := (\frac{λ}{π})^{\frac{n}{2}} ⟨u(ξ)|_{N}, e^{−λ(z−ξ)^{2}}⟩, z ∈ N.

If u ∈ L^{1}(N) has compact support, then this definition coincides with the one provided by the integral T_{λ}u(z) := ∫_{N} e^{−λ(z−ξ)^{2}}u(ξ)dξ. We denote by K_{λ}(z, ξ) = (\frac{λ}{π})^{\frac{n}{2}} e^{−λ(z−ξ)^{2}} the heat kernel. We notice that for the CR distribution u, the restriction u|_{N} makes sense. To see this, according to [51], it suffices to show that the wave front set satisfies

W F (u) ∩ T_{N}^{∗}M = {0}.

In fact, on one hand we have that, u being a solution of the system of CR vector fields, W F (u) is contained in the characteristic set char(T^{0,1}M) = T^{0,1}M^{⊥} ∩ T^{∗}M ≃ (\frac{T M}{T^{ℂ}M} )^{∗} (defined in Definition 3.2.5). On the other hand, N being generic, we have T_{N}^{∗}M ∩ (\frac{T M}{T^{ℂ}M} )^{∗} = \{0\}. We now prove that T_{λ}u → u over N. In ℝ^{n}, we use the notation dξ = dξ_{1} ∧ ... ∧ dξ_{n} and ξ^{2} = ξ_{1}^{2} + ... + ξ_{n}^{2}. We claim that it suffices to prove the convergence C^{0}. In fact, for the different types of convergence we have the implications

(i) C^{∞} ⇒ E′: this follows from

\begin{aligned}∫ T_{λ}uvdξ & = ∫ ⟨u, K_{λ}⟩vdξ \\ & = ⟨u, ∫ K_{λ}vdξ⟩ = ⟨u, T_{λ}v⟩. \end{aligned}

(ii) C^{0} ⇒ C^{k} and C^{∞}: in fact, recalling that supp(u) is compact,

T_{λ}(X_{j}u) = ∫ K_{λ}X_{j}(u)dξ
= ∫ X_{j}(K_{λ}u)dξ − ∫ X_{j}(K_{λ})udξ
= − ∫ X_{j}(K_{λ})udξ (by Stokes’s formula)
= X_{j}(∫ K_{λ}udξ) = X_{j}(T_{λ}u).

Thus, what we have to show is that for any totally real maximal submanifold N ⊂ M and for a suitable N′ ⊂ M, we have T_{λ}u ⇒ u uniformly on N′. We first consider the case N = ℝ^{n} (and next deform it by a diffeomorphism). We start from the identity

(3.6.5)  (\frac{λ}{π})^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−λ(x−ξ)^{2}}dξ = 1.

By using (3.6.5) and by means of the coordinate change s := \sqrt{λ}(x−ξ) for x ∈ ℝ^{n}, we get

\begin{aligned}T_{λ}u(x) − u(x) & = (\frac{λ}{π})^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−λ(x−ξ)^{2}} [u(ξ) − u(x)] dξ \\ & = (\frac{1}{π} )^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−s^{2}} [u(x − \frac{s}{\sqrt{λ}}) − u(x)] ds. \end{aligned} ; (3.6.6)

The term between bracket, which we denote by u_{λ}, satisfies u_{λ} → 0 and e^{−s^{2}}|u_{λ}| <_{∼} e^{−s^{2}}; thus, by the Lebesgue dominated convergence theorem, the integral converges to 0 for fixed x. We have to show that this convergence is indeed uniform. In fact,

\begin{aligned}|T_{λ}u − u| & <_{∼} ∫_{|s|≥R} · + ∫_{|s|≤R} · \\ & = ϵ(R^{−1}) + ϵ_{R}(λ^{−1}), \end{aligned}

where the first is infinitesimal uniformly in λ. The uniformity follows and the case N = ℝ^{n} is proved.

We take now a general, totally real maximal, submanifold N with a local, orientation-preserving parametrization over ℝ^{n}:

ℝ^{n} → N, ξ ↦ Φ(ξ).

We decompose the n × n complex Jacobian matrix Φ′ as Φ′ = B + iC and suppose C(0) = 0 in our normalization. We need Lemma 3.6.4. We have the identity

(\frac{λ}{π})^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−λ((B+iC)ξ)^{2}}det(B + iC)dξ = 1.

Proof. The above integral is an entire function of the coefficients of B +iC and, when B +iC is real, that is, C = 0, its value is 1 because

∫_{ℝ^{n}} e^{−λ(Bξ)^{2}}det Bdξ = (\frac{1}{λ})^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−ξ̃^{2}}dξ̃ = (\frac{π}{λ})^{\frac{n}{2}} .

By the identity principle for holomorphic functions, its value is identically 1 for any choice of C.

□

End of proof of Theorem 3.6.3 We start from the equality

(3.6.7) u(Φ(x)) = (\frac{λ}{π})^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−λ(Φ′(x)·(ξ−x))^{2}} det Φ′(x)u(Φ(x))dξ,

which is a consequence of Lemma 3.6.4. For z = Φ(x), we have to estimate

T_{λ}u(z) − u(z) = (\frac{λ}{π})^{\frac{n}{2}} (∫_{ℝ^{n}} e^{−λ(Φ(ξ)−Φ(x))^{2}}u(Φ(ξ)) det Φ′(ξ)dξ (3.6.8)
− ∫_{ℝ^{n}} e^{−λ(Φ′(x)·(ξ−x))^{2}}u(Φ(x)) det Φ′(x)dξ)

= (\frac{λ}{π})^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−λ(Φ(ξ)−Φ(x))^{2}} [u(Φ(ξ)) det Φ′(ξ) − u(Φ(x)) det Φ′(x)] dξ

+ (\frac{λ}{π})^{\frac{n}{2}} ∫_{ℝ^{n}} (e^{−λ(Φ(ξ)−Φ(x))^{2}} − e^{−(λΦ′(x)·(ξ−x))^{2}}) u(Φ(x)) det Φ(x)dξ.

We denote by [∗] the term between brackets in the second to last term of (3.6.8). We can assume that

(3.6.9)  \begin{cases}Re (Φ(x) − Φ(ξ))^{2} ≥ c|ξ − x|^{2} − c_{1}, \\ Re (Φ′(x) · (ξ − x))^{2} ≥ c|ξ − x|^{2} − c_{1}. \end{cases}

It follows that

(3.6.10) \begin{cases}|e^{−λ(Φ(ξ)−Φ(ξ))^{2}}[∗]| <_{∼} e^{−λ|ξ−x|^{2}}E(|ξ − x|), \\ |e^{−λ(Φ(ξ)−Φ(x))^{2}} − e^{−λ(Φ′(x)·(ξ−x))^{2}} | <_{∼} e^{−λ|ξ−x|^{2}}E(|ξ − x|), \end{cases}

where E(|ξ − x|) is infinitesimal (in fact we could replace E(|ξ − x|) by O(|ξ − x|) in the second line). By a similar argument to that which follows (3.6.6), we see that

(3.6.11)  lim_{λ→∞} (\frac{λ}{π})^{\frac{n}{2}} ∫_{ℝ^{n}} e^{−λ|ξ−x|^{2}}Edξ = 0.

To prove it, we just have to set s = \sqrt{λ}|ξ − x| and apply the dominated convergence theorem. By (3.6.8), (3.6.10) and (3.6.11) we have T_{λ}u(z) → u(z) as λ → ∞. Next, by the same argument as for N = ℝ^{n}, we conclude that T_{λ}u ⇒ u uniformly on a suitable N′ ⊂ N. Now, we want to show that this convergence also takes place out of N′ in a controlled neighborhood M′ of N′ in M. For this, we remark that any point z in M′′ ⊂⊂ M′ can be reached by a deformation N′_{z} of N′, parametrized over ℝ^{n} by Φ_{z} such that

\begin{cases}N′ and N′_{z} coincide near ∂M′, \\ |Im Φ′_{z}| < \frac{1}{2}. \end{cases}

If we then define T_{λ}(u) by integration along N′_{z}, we get T_{λ}u → u on N′_{z} ∩ M′. But, in fact, this integration coincides with the former one which was performed along N′. In fact, let W be an (n + 1)- dimensional manifold in M with (oriented) boundary +N − N_{z} and take u ∈ C_{c}^{∞}(M′), CR over W̄, and an integration form α of degree n over W̄. Then

\begin{aligned}∫_{N_{z}} K_{λ}uα − ∫_{N} K_{λ}uα & = ∫_{W} d(K_{λ}uα) \\ & = ∫_{W} K_{λ} ¯∂uα = 0. \end{aligned}

This shows that the C^{0} approximation T_{λ}u → u obtained by integration along a prescribed N′ also extends to M′′. The various kinds of convergences, C^{k}, C^{∞}, E′, follow in the same way as before.

□

Exercise 3.6.5. Suppose that M has some non-degenerate elliptic CR singularities in the sense of Section 3.2 which form a manifold Σ. Prove that there is no local uniformity at points of Σ for the polynomial approximation of CR functions on M \ Σ.

Hint: in the simplest case of a 2-codimensional submanifold M ⊂ ℂ^{2} with a non-degenerate elliptic singularity at 0, take a real function on M. If f is approximated by polynomials on K ⊂⊂ M \ {0}, then it is constant over discs whose boundary is in K.

3.7. ANALYTIC DISCS AND THE EXTENSION OF CR FUNCTIONS 153

Notes. Approximation by polynomials of continuous real functions goes back to Stone-Weierstrass. In the specific case of CR functions (Theorem 3.6.3), the result is due to Baouendi and Treves [13]. Further generalizations can be found in Treves [88].

3.7. Analytic discs and the extension of CR functions: the

“edge of the wedge” theorem, the deformation of discs for manifolds of type 2 and the Levi extension

Summary. We first describe the construction of analytic discs by solving a functional equation, Bishop equation, in the spaces of differentiable functions with fractional regularity. Once this theory is settled, we prove two problems. First we prove the edge of the wedge theorem by [3]: CR functions on M which extend to additional directions v_{j} extend in fact to their convex combinations. This is obtained by a construction of a family of discs, parts of whose boundaries belong to the different regions of extension, and in proving that their centers cover a wedge W of edge M and normal cone Γ = convex hull {v_{j}}. The approximation of CR functions by polynomials “propagates” from the boundary of the discs to their interior and yields the holomorphic extension to W. Second, we prove the CR extension in Levi directions of [29]. For this, we show that if L ∈ T^{1,0}M and \frac{1}{2i}[L, L̄](z_{o}) = v ≠ 0 in \frac{T M}{T^{ℂ}M}, then there is a one-parameter family of discs {A_{η}} with A_{η}|_{η=0} = {z_{o}} and CR components ηL(z_{o})τ, τ ∈ Δ, such that ∂_{η}A_{η} points to a direction whose projection modulo T M is J v. Again, by approximation and by the maximum principle, we get CR extension to the Levi cone.

Analytic discs A are the holomorphic bounded images into ℂ^{n} of the standard disc Δ of ℂ. These are “attached” to a generic submanifold M ⊂ ℂ^{n}, that is, ∂A ⊂ M. The basic idea is that “small” analytic discs attached to M are perturbations of analytic discs in T^{ℂ}M. We first introduce the Hilbert transform. If u : ∂Δ → ℝ is a smooth function, it has a unique harmonic extension to Δ̄: this extension u has a harmonic conjugate v on Δ̄, that is, a function v such that u + iv is holomorphic; this is determined up to a constant. The Hilbert transform is the correspondence T : u|_{∂Δ} ↦ v|_{∂Δ} and is denoted by T_{1} when it is normalized by the condition v(1) = 0. Note that u = −T_{1}v + u(1). It is classical that T_{1} is not a continuous functional over functions with integer regularity C^{k}. Instead, we let it act on the spaces C^{k,α} = C^{k,α}(∂Δ), for 0 < α < 1, of functions f endowed with continuous derivatives up to order k which satisfy

||f||_{k,α} = ||f||_{k} + sup_{θ_{1}, θ_{2}} \frac{|∂_{τ}^{k}f(e^{iθ_{1}}) − ∂_{τ}^{k}f(e^{iθ_{2}})|}{|θ_{1} − θ_{2}|^{α}} < +∞.

Here, ||f||_{k} is the usual C^{k}-norm ||f||_{k} = sup_{j θ} |∂_{τ}^{j}f(e^{iθ})|. Endowed with

the norm || · ||_{k,α}, the space C^{k,α} turns out to be Banach. We want to show that T : C^{k,α} → C^{k,α} is continuous and for this we start by describing through Poisson integrals the harmonic extension of u from values τ = e^{iθ} to z = re^{iφ}, r ≤ 1:

\begin{aligned}u(re^{iφ}) & = \frac{1}{2π} ∫_{0}^{2π} \frac{(1 − r^{2})u(e^{iθ})}{(1 + r^{2} − 2rcos(θ − φ))}dθ \\ & = \frac{1}{2πi} ∫_{\{|τ|=1\}} \frac{1 − |z|^{2}}{|τ − z|^{2}} u(τ)\frac{dτ}{τ} \\ & = Re [\frac{1}{2πi} ∫_{\{|τ|=1\}} (\frac{τ + z}{τ − z} ) u(τ)\frac{dτ}{τ} ] . \end{aligned} ; (3.7.1)

The function between brackets [ ], say F, is holomorphic in z and satisfies Re F = u. Note also that Im F (0) = 0 since u is real. Thus, the harmonic conjugate v = T_{0}u, normalized by v(0) = 0, is the boundary value on ∂Δ of Im F:

T_{0}u|_{z=e^{iθ}} = Im [\frac{1}{2πi} ∫_{\{|τ|=1\}} (\frac{τ + z}{τ − z} ) u(τ)\frac{dτ}{τ} ] |_{z=e^{iθ}}.

Proposition 3.7.1. The functional T : C^{k,α}(∂Δ) → C^{k,α}(∂Δ) is continuous.

Proof. Let K be the Cauchy integral defined by

Ku(z) = \frac{1}{2πi} ∫_{\{|τ|=1\}} \frac{u(τ)}{τ − z} dτ, |z| < 1.

Since \frac{τ+z}{τ} is smooth for |τ| = 1, |z| < 1, it suffices to prove the continuity of K. It is not a big deal to estimate the norm ||Ku||_{k} by ||u||_{k,α}. So, we have to establish the difference estimate for ũ := ∂_{θ}^{k}u. Now, for z_{1} = e^{iθ_{1}} and z_{2} = e^{iθ_{2}} in ∂Δ, we set ϵ = |z_{1} − z_{2}|, B_{2ϵ}(z_{1}) = {τ : |τ − z_{1}| < 2ϵ} and get

Kũ(z_{1}) − Kũ(z_{2}) = \frac{1}{2πi} ∫_{∂Δ} ((\frac{ũ(τ) − ũ(z_{1})}{τ − z_{1}}) − (\frac{ũ(τ) − ũ(z_{2})}{τ − z_{2}})) dτ

+ (ũ(z_{1}) − ũ(z_{2}))

= \frac{1}{2πi} ∫_{∂Δ∩B_{2ϵ}(z_{1})} · + \frac{1}{2πi} ∫_{∂Δ\B_{2ϵ}(z_{1})} · + (ũ(z_{1}) − ũ(z_{2})).

Now,

\frac{1}{2πi} ∫_{∂Δ\B_{2ϵ}(z_{1})} · + (ũ(z_{1}) − ũ(z_{2}))

= \frac{1}{2πi} ∫_{∂Δ\B_{2ϵ}(z_{1})} (ũ(τ) − ũ(z_{1}))

× [\frac{1}{τ − z_{1}} − \frac{1}{τ − z_{2}} ] dτ

+ \frac{1}{2πi} ∫_{∂Δ\B_{2ϵ}(z_{1})} \frac{(−ũ(z_{1}) + ũ(z_{2}))}{τ − z_{2}}dτ + (ũ(z_{1}) − ũ(z_{2})).

Because of the identity \frac{1}{2πi} ∫_{∂Δ} \frac{1}{τ−z_{2}} dτ = 1, the sum of the second and third terms in the right side is ∫_{B_{2ϵ}(z_{1})∩∂Δ} \frac{ũ(z_{1})−ũ(z_{2})}{τ−z_{2}}dτ, and therefore its absolute value is estimated by ||ũ||_{α}|z_{1} − z_{2}|^{α}. The absolute value of the first term on the right side can be estimated by

||ũ||_{α}|z_{1} − z_{2}| ∫_{∂Δ\B_{2ϵ}(z_{1})} 2|θ − θ_{1}|^{−2+α}dθ ∼^{<} ||ũ||_{α}|z_{1} − z_{2}|^{α}.

As for the remaining integral over ∂Δ ∩ B_{2ϵ}(z_{1}), we have the estimate

\begin{aligned}\frac{1}{2πi} ∫_{∂Δ∩B_{2ϵ}(z_{1})} · & ∼^{<} ||ũ||_{α} ∫_{∂Δ∩B_{2ϵ}(z_{1})}(|θ − θ_{1}|^{α−1} + |θ − θ_{2}|^{α−1})dθ \\ & ∼^{<} ||ũ||_{α}|z_{1} − z_{2}|^{α}. \end{aligned}

□

We are ready for the construction of discs. Let M be defined by the system of equations

y′_{j} = h_{j}(x′, z′′), h_{j}(0) = 0, ∂h_{j}(0) = 0.

We fix a point z_{o} = (z′_{o}, z′′_{o}) ∈ M and prescribe the “z′′” component of our discs as a holomorphic function w(τ), τ ∈ Δ, whose boundary value is small in C^{k,α}-norm. We look for a disc A, with A(1) = z_{o}, in the form A = z_{o} + (u(τ) + iv(τ), w(τ)).

Proposition 3.7.2 (Solution of Bishop’s equation). Let M be defined by y′_{j} = h_{j} with h ∈ C^{k+m+2}. Then for any w ∈ C^{k,α}(∂Δ, ℂ^{n−l}) small and normalized by w(1) = 0 and for any z_{o} close to 0, there is a unique u ∈ C^{k,α}(∂Δ) which solves the equation

(3.7.2)  u = −T_{1}h(u, z′′_{o} + w(τ)) + x′_{o}.

Moreover, if w depends on some parameter η ∈ ℝ^{d} so that ℝ^{d} → C^{k,α}, η ↦ w_{η} is C^{m}, then also (η, x′_{o}, z′′_{o}) ↦ u_{η,x′_{o},z′′_{o}}, ℝ^{d} × ℂ^{n−l} → C^{k,α}(∂Δ) is C^{m}.

Proof. We consider the mapping

F : C^{k,α}(∂Δ, ℝ^{l}) × C^{k,α}(∂Δ, ℂ^{n−l}) × ℝ^{l} × ℂ^{n−l} → C^{k,α}(∂Δ, ℝ^{l}), (3.7.3)

(u, w, x′_{o}, z′′_{o}) ↦ u + T_{1}h(u, z′′_{o} + w) − x′_{o}. (3.7.4)

The functional F is C^{1} in its action between function spaces. For the Jacobian ∂_{u}F with respect to u, we have ∂_{u}F : u̇ ↦ u̇ − T_{1}∂_{x}hu̇. In particular, evaluation at (u, w, z_{o}) = (0, 0, 0) implies that ∂_{u}F is invertible since ∂_{x}h(0) = 0. Thus, by the implicit function theorem, there is a unique solution to (3.7.2). If w depends on some parameter η, then u̇ = ∂_{η}u_{η} is a solution to the linearized equation

(3.7.5)  ∂_{η}u_{η} − T_{1}∂_{x}h∂_{η}u = 0.

Let us write (3.7.5) as G(∂_{η}u) = 0. Since ∂_{x}h is C^{k+m+1}, then G is C^{1} as an application on C^{k,α}. In particular, ∂_{η}u_{η} ∈ C^{k,α} and (η, z_{o}) ↦ u_{η,z_{o}} is C^{1}. Iterated derivations in η yield the C^{m} dependence on the parameter η.

□

For a solution u of (3.7.2), we set v = T_{1}u + y′_{o} (for y′_{o} = h(x′_{o}, z′′_{o})) and define A = A_{η,z_{o},w} by

A(τ) = ((u(τ) + iv(τ)), z′′_{o} + w(τ)), τ ∈ ∂Δ.

Since v = h(u, z′′_{o}+w(τ)) on ∂Δ, then ∂A ⊂ M and, since u+iv|_{1} = z′_{o}, then A(1) = z_{o}. Thus, A fulfills all requirements.

We can also handle the case in which M is CR non-generic. In this case we choose a projection ℂ^{n} →^{pr} ℂ^{n′} such that M →_{∼}^{pr} M′ is a CR

diffeomorphism and M′ is generic in ℂ^{n′} (cf. Section 3.2). Note that the projection pr is the identity on the “z′′-component” of a disc. So, by the first part of the proof, we may find a disc A′ attached to M′ with prescribed z′′-component and through a fixed point of M′. By the approximation theorem, the CR mapping (pr|_{M})^{−1} extends holomorphically inside A′. Its holomorphic extension provides the supplementary component of a disc A which projects via pr over A′.

Note also that the solutions of Bishop’s equation describe all the small discs attached to M with prescribed z′′-component through a fixed point of M. By the uniqueness of the inversion in the functional spaces, the solution, hence the disc, is unique: thus, through a point z_{o} ∈ M, the small disc A is uniquely determined by w(τ), its projection on T_{z_{o}}^{ℂ}M. This is true even in case M is CR non-generic: A′ is unique and hence A is also unique because ∂A corresponds in a one-to-one manner to ∂A′.

We need to introduce now a tricky method due to Tumanov (cf. [94]) of also prescribing the so-called “additional normal components”. We consider a manifold of wedge type W with edge M; this is the “twisted” version of a plane wedge T W = T M ⊕Γ where Γ is a cone in the normal plane T_{M}ℂ^{n}. We can describe W by “adding” to M defined, say, by y′_{j} = h_{j}(x′, z′′), the extra directions of Γ, that is, by extending h(x′, z′′) to h(x′, z′′, t) for t belonging to the “quadrant” ℝ_{+}^{d} = {t : t_{i} > 0} and setting

W = {z : y′_{j} = h_{j}(x′, z′′, t), t ∈ ℝ_{+}^{d}}.

Then, we can restate the existence result of Proposition 3.7.2: given z_{o} ∈ M, w(τ) and t(τ) small, there is a unique solution u of u = −T_{1}h(u, z′′_{o} +w(τ), t(τ))+x′_{o}; setting v = T_{1}u+y′_{o} and A(τ) = ((u(τ)+ iv(τ)), z′′_{o} + w(τ)), we have A(1) = z_{o} and y′_{j} = h_{j}(u(τ), z′′_{o} + w(τ), t(τ)) on ∂Δ, that is, ∂A ⊂ W. We are ready to prove the following theorem by Ajrapetyan and Henkin [3]

Theorem 3.7.3 (Edge of the wedge). Let M ⊂ ℂ^{n} be generic and let M_{j}, j = 1, ..., l, be a family of manifolds with boundary M and one extra direction v_{j} ∈ T_{M}ℂ^{n}, that is, T M_{j} = T M ⊕ v_{j}ℝ^{+}, such that Span{v_{j}} = T_{M}ℂ^{n}; let Γ be the convex conic hull of {v_{j}}. Then, there is a wedge W with edge M and directional cone Γ, that is, T W |_{M} = T M ⊕ Γ, such that any f ∈ C_{CR}^{0}(M) which extends as C_{CR}^{0} to each (M_{j}) extends holomorphically to W.

Proof. We can find local coordinates at 0 such that M is defined by y′_{j} = h_{j}(x′, z′′, 0) and M_{j} is defined by y′_{j} = h_{j}(x′, z′′, 0, ..., t_{j}, ..., 0), t_{j} ∈ ℝ^{+}, with ∂_{t_{j}}h = v_{j}. We denote by A(τ) = ((u(τ)+iv(τ)), z′′_{o}+w(τ)) a disc attached to ⋃_{j} M_{j} and we denote by t(τ) its “normal component” defined as the solution of the equation

v(τ) = h(u(τ), z′′_{o} + w(τ), t(τ)).

Note that A is attached to ⋃_{j} M_{j} precisely when t_{j}(τ) = 0 for all but one of the indices j_{o} and t_{j_{o}}(τ) ≥ 0. We decompose ∂Δ into a

union ⋃^{l} ; j=1γ_{j} of arcs of length \frac{2π}{l}, take χ_{j}(τ) ≥ 0 with supp χ_{j} ⊂ γ_{j} and

\frac{1}{2π} ∫ χ_{j}dθ = 1, and define

t_{λ}(τ) = (λ_{1}χ_{1}(τ), ..., λ_{l}χ_{l}(τ)) for all λ ∈ S^{l−1} ∩ ℝ_{+}^{l},

where S^{l−1} is the (l − 1)-dimensional sphere. Take w(τ) ≡ 0, z′′_{o} ∈ ℂ^{n−l}, x_{o} ∈ ℝ^{l}, set y′_{o} = h(x′_{o}, z′′_{o}), and consider Bishop’s equation

u = −T_{0}h(u, z′′_{o}, t_{λ}) + x′_{o}, v = T_{0}u + y′_{o}.

Set A_{x′_{o},z′′_{o},λ} = (u + iv, z′′_{o}) where T_{0} is normalized by the condition T_{0}·|_{0} = 0. By our choice of the function t_{λ}(τ) (whose components are all 0 except for one which is ≥ 0), we have that v = y′_{o} + T_{0}u = h(u, z′′_{o}, t_{λ}) belongs to ⋃_{j} M_{j}. We consider the evaluation mapping of the centers of the discs

\begin{aligned}E : (x′_{o}, λ, z′′_{o}) ↦ A_{x′_{o},λ,z′′_{0}}(0) & = (z′(0), z′′(0)) \\ & = (x′_{o} + iv(0), z′′_{o}). \end{aligned}

For the Jacobian of E at 0 we have

(3.7.6)  JE = (\begin{matrix}id & i∂_{t}h & ∗ \\ 0 & 0 & id \end{matrix}) .

In fact, from v(0) = (2π)^{−1} ∫_{0}^{2π} v(e^{iθ})dθ for v = h, we get

∂_{x′_{o}}v(0) = (2π)^{−1} ∫_{0}^{2π} ∂_{x}h∂_{x′_{o}}udθ.

We also have ∂_{λ}v(0) = (2π)^{−1} ∫_{0}^{2π}(∂_{x}h∂_{λ}u+∂_{t}h∂_{λ}t_{λ})dθ. At λ = 0, x′_{o} = 0, w_{o} = 0, we have ∂_{x}h = 0, ∂_{λ}t_{λ} = id. This implies ∂_{x′_{o}}z′(0) = id, ∂_{λ}z′(0) = i∂_{t}h, which completes the proof of (3.7.6). Thus, when the parameters (x′_{o}, λ, z′′_{o}) describe ℝ^{l} × ℝ_{+}^{l} × ℂ^{n−l}, the union of the centers of the discs covers a wedge W such that T W = T M ⊕ Γ where Γ is the image of the quadrant ℝ_{+}^{l} under the l × l matrix ∂_{t}h.

□

It is a useful exercise to prove the analogous statement of Theorem 3.7.3 when the convex hull of the v_{j}’s is a cone of dimension d < l in T_{M}ℂ^{n}. In this case, the wedge of extension W is no more open in ℂ^{n} but in a (l − d)-codimensional submanifold.

We now treat CR extensions in the Levi normal directions. Let L ∈ T^{1,0}M with L(z_{o}) = w_{o} be such that \frac{1}{2i}p[L, L̄] = v ≠ 0 in \frac{T M}{T^{ℂ}M}. Let z_{o} = 0 and suppose that M is locally defined by y′_{j} = h_{j}(x′, z′′, z̄′′) where h_{j}(0) = 0, ∂h_{j}(0) = 0, ∂_{x^{′I}}∂_{z^{′′J}} h(0) = 0 and ∂_{x^{′I}}∂_{}_{z̄^{′′J}} h(0) = 0 according to the normalization of Theorem 3.2.7. By the basis ∂r_{j} for

r_{j} = −y_{j} + h_{j}, we get an isomorphism \frac{T M}{T^{ℂ}M} →^{∼}_{K∘J} T_{M}^{∗}ℂ^{n} →^{∼} ℝ^{l} which

identifies v ≃ (∂_{w_{o}}∂w̄_{o}h_{j})_{j}.

Theorem 3.7.4 (CR extension in Levi directions). Let M ∈ C^{k}, k ≥ 5, and let v be a normal Levi direction. Then CR functions on M extend to a manifold with boundary M_{1} which points to the additional direction v, that is, T M_{1} = T M ⊕ ℝ^{+}v.

Proof. The argument of the proof consists in finding a family of discs {A} attached to M with A(1) = z for z describing a neighborhood of z_{o} which are transversal to M. More precisely, they satisfy pr ∂_{t}A(1) ∼ −v (where pr is the projection T ℂ^{n} → T_{M}ℂ^{n}) with a uniform bound for the angle that they form with T M. Hence the union of their “radii” A|_{[0,1]}, in a neighborhood of z_{o}, forms the desired manifold M_{1}. We set w_{η} = ηw_{o}(1 − τ) and solve, for z ∈ M close to z_{o}, Bishop’s equation

(3.7.7)  u(τ) = −T_{1}h(u(τ), z′′ + w_{η}(τ)) + x′.

If we denote by u = u_{η} the solution of (3.7.7) and put v = T_{1}u + y′ and A_{z,η}(τ) = (u(τ)+iv(τ), z′′+w_{η}(τ)), then this is a family of analytic discs attached to M, with A_{z,η}(1) = z and such that (z, η) ↦ A_{z,η}, M ×ℝ → C^{1,α}, is C^{2}. We now fix z = z_{o} and consider the Taylor development of ∂_{t}A (for τ = te^{iθ}) up to order 2 with respect to η. Since A_{η}|_{η=0} ≡ {z_{o}}, it follows that ∂_{t}A_{η}|_{η=0} = 0. Also, from

\begin{aligned}∂_{η}v & = ∂_{η}h(u_{η}, w_{η}) \\ & = ∂_{x}h∂_{η}u + ∂_{z′′}h∂_{η}w + ∂_{}_{z̄′′}h∂_{η}w̄, \end{aligned}

evaluation at η = 0 yields ∂_{η}v|_{η=0} = 0. By differentiating in η the identity u = −T_{1}v + x′_{o} and interchanging ∂_{η} with T_{1}, we get

∂_{t}A = −w_{o}η + ∂_{t}∂_{η}^{2}A|_{η=0} \frac{η^{2}}{2} + o^{2}.

We now want to prove that the projection of \frac{1}{2i}∂_{t}∂_{η}^{2}A|_{η=0} in the plane ℝ^{l} ≃ T_{M}^{∗}ℂ^{n} is −2(∂_{w_{o}}∂w̄_{o}h_{j})_{j}. In fact, recalling that our equations are taken in normal forms (3.2.12), we have

∂_{η}^{2}v|_{η=0} = 2(∂_{w_{o}}∂w̄_{o}h_{j})_{j}|1 − τ|^{2} on ∂Δ.

Now, |1 − τ|^{2}|_{∂Δ} = 2 − 2 cos θ = 2Re (1 − τ)|_{∂Δ}. Hence the harmonic extension of ∂_{η}^{2}v|_{η=0} from ∂Δ to Δ is (∂_{w_{o}}∂w̄_{o}h_{j})_{j}4Re (1 − τ) and therefore

\frac{1}{2i}∂_{η}^{2}∂_{t}A_{η}|_{η=0} ∼ −2(∂_{w_{o}}∂w̄_{o}h_{j})_{j}.

Thus, if we fix η_{o} close to 0 and move z near z_{o}, we have succeeded in constructing a family of discs transversal to M which point at z to the direction −∂_{t}A_{z,η_{o}} ∼ 4i(∂_{w_{o}}∂w̄_{o}h_{j})_{j}. We pick up from the discs

their radii I_{z,η_{o}} = A_{z,η_{o}}|_{[−1,1]} and define M_{1} := ⋃_{z∈M}I_{z,η_{o}}. Since the I_{z,η_{o}}’s

are transversal to M with a uniform estimate from below for their angle with M at τ = 1, then M_{1} is a manifold with boundary M and additional direction v. After we managed to fill such a manifold by discs attached to M, we can extend CR functions from M to M_{1} by the approximation technique already exploited in the edge of the wedge theorem. The proof is complete.

□

We define the “Levi cone” Γ ⊂ \frac{T M}{T^{ℂ}M} by

(3.7.8)  Γ_{z} = convex hull_{L∈T^{1,0}M} \{\frac{1}{2i}[L, L̄]|_{z} modulo T^{ℂ}M\}.

This is a convex cone of T_{M}ℂ^{n} that we also identify with a cone of T_{M}^{∗}ℂ^{n} ≃ ℝ^{l} via J combined with the contraction K with the basis ∂r_{1}, ..., ∂r_{l} of T_{M}^{∗}ℂ^{n}. Note that, by the aid of Theorem 3.7.4, for any v ∈ Γ_{z_{o}} we can construct M_{1} such that T M_{1} = T M ⊕ℝ^{+}v_{1}. This is a union of discs which enjoys the property that continuous CR functions on M extend to M_{1}. By the edge of the wedge theorem, we can “combine” all different extensions. Note here that the statement therein can be generalized to the case of dimΓ < l. This yields the following corollary to Theorem 3.7.4

Theorem 3.7.5. Let Γ be the Levi cone of M. Continuous CR functions on M extend to a wedge type domain W with edge M and directional cone Γ, that is, T W = T M ⊕ Γ.

Suggested research. In the statement of Theorem 3.7.3, take M totally real and M_{j} Levi flat. To simplify the situation, even assume M_{j} to be C^{ω} so that it can be reduced indeed to ℝ × ... × ℂ × ... × ℝ with a single occurrence of ℂ in the j-th position. Assume there is a set of points Ξ = {ξ} of M and a set of CR curves {γ_{ξ}^{j}} in M_{j} issued from the points ξ which are excluded from the hypothesis of CR extension, for instance, the rays ξ_{1} × ... × (ξ_{j} + iℝ^{+}) × ... × ξ_{n} in the linear case. How small must the set Ξ be for the conclusion of Theorem 3.7.3 to continue to hold?

Notes. The technique of the analytic discs attached to a real submanifold M ⊂ ℂ^{n} has a long history which begins with the work of Lewy [67] who also established the first example of non-solvable first order PDE. Other early basic work is due to Bishop [24]. Since then, analytic discs have obtained enormous development. In particular, in connection with the forced CR extension, we mention Ajrapetyan and Henkin [3], Bogges and Polking [29] and all the references in Sections 3.8 and 3.10. We also recall here the related theory of stationary/extremal discs developed by Lempert [66] in connection with the study of the Kobayashi metric (cf. also the recent higher-codimensional version due to Tumanov [95]). The edge of the wedge theorem is due to Ajrapetyan and Henkin [3] and the Levi extension theorem was established by Lewy in codimension 1 and next by Ajrapetyan and Henkin and Boggess and Polking [29], in general codimension. For the presentation of the material that we propose in this section, we followed the book by Boggess [27], the book by Baouendi, Ebenfelt and Rothschild [8] and, more closely, the survey by Tumanov [91].

3.8. Iterated commutators, finite type, Bloom-Graham

normal form: deformation of discs for manifolds of higher type

Summary. We discuss here the CR extension in normal directions obtained by iterated commutators. We follow the method of [18], based on analytic discs with Lipschitz boundary: this seems to be the most appropriate to describe the geometry of the problem. For a submanifold M ⊂ ℂ^{n}, we introduce the normal form in the sense of Bloom and Graham that we refine by specifying a CR direction. We show the relation between the normal form and the properties of the commutators of vector fields in T^{1,0}M and T^{0,1}M. “Finite type”, in either sense, is the same. We then present the CR extension theorem which is obtained by introducing a new space of discs, of “sector” type, and by proving that they can be approximated by regular discs. But the approximation for their components which are normal to M occurs in fact in the C^{1}-norm because of the vanishing order of the equations. The CR extension then goes along the same lines as in Section 3.7.

For a generic manifold M ⊂ ℂ^{n}, we have seen in Section 3.4 the relation between the commutators [L, L̄], L ∈ T^{1,0}M, and the complex hessian (∂_{z_{h}}∂z̄_{k}r_{j})_{j=1,...,l} for a system of equations r_{1} = 0, ..., r_{l} = 0 of M. We wish to extend this relation to iterated commutators and higher-order Taylor terms. For this purpose we need to introduce the notion of Hörmander’s numbers of M.

The hypersurface case has already been treated in Section 1.10 to which we refer for notations and terminology. We continue to use the notation T^{ℂ}M = T M ∩iT M for the complex tangent bundle to M and ℂ⊗_{ℝ}T^{ℂ}M = T^{1,0}M ⊕T^{0,1}M for its complexification. As it has already been seen, ℂ⊗_{ℝ}T M is integrable, that is, closed under Lie bracket, but ℂ ⊗_{ℝ} T^{ℂ}M is not, in general. We introduce an interpolation between ℂ ⊗_{ℝ} T^{ℂ}M and ℂ ⊗_{ℝ} T M. We set L_{ℂ}^{1} = ℂ ⊗_{ℝ} T^{ℂ}M and denote by L_{ℂ}^{j} the distribution of vector spaces spanned by Lie bracket of holomorphic and antiholomorphic vector fields of length ≤ j − 1. The commutators of length 0 are the vector fields themselves; those of length 1 are simple commutators defining the Levi form and so on. We describe them in greater detail. We choose an orthonormal system of (1, 0)-vector fields ∂_{ω_{k}}, k = 1, ..., n, with ∂_{ω_{k}}r^{h} = κ_{kh}; thus, ∂_{ω_{1}}, ..., ∂_{ω_{n−l}} and ∂ω̄_{1}, ..., ∂ω̄_{n−l}, when restricted to M, are a basis of T^{1,0}M and T^{0,1}M respectively. With this choice, we have

(3.8.1) L_{ℂ}^{j} = Span \{∂_{ω_{k}}, ∂ω̄_{k}, [∂_{ω_{k_{1}}}, ∂ω̄_{k_{2}}], ..., \underbrace{[∂_{ω_{k_{1}}}^{ϵ_{k_{1}}}, [∂_{ω_{k_{2}}}^{ϵ_{k_{2}}}[...∂_{ω_{k_{j}}}^{ϵ_{k_{j}}}]]]}_{j}\} ,

where the suffix ϵ_{k_{s}} in ∂_{ω_{ks}}^{ϵ_{ks}} denotes either choice of ∂_{ω_{ks}} or ∂ω̄_{ks}. Suppose that for an integer m_{1} ≥ 2 we have

(3.8.2) L_{ℂ p_{o}}^{j} = T_{p_{o}}^{1,0}M ⊕T_{p_{o}}^{0,1}M, j ≤ m_{1}−1, L_{ℂ p_{o}}^{m_{1}} ⊃_{≠} T_{p_{o}}^{0,1}M ⊕T_{p_{o}}^{1,0}M.

Let dim\frac{L_{ℂ po}^{m_{1}}}{L_{ℂ po}^{1}} = l_{1}; in this situation we refer to m_{1} as the first Hörmander

number of M at p_{o} and to l_{1} as its multiplicity. In case L_{ℂ}^{j} = L_{ℂ}^{1} for any j, we set m_{1} = +∞ with multiplicity l_{1} = l. Next, we look for m_{2} > m_{1} such that

(3.8.3)  L_{ℂ p_{o}}^{j} = L_{ℂ p_{o}}^{m_{1}}, j < m_{2}, L_{ℂ p_{o}}^{m_{2}} ≠ L_{ℂ p_{o}}^{m_{1}},

and set l_{2} = dim (\frac{L_{ℂ po}^{m_{2}}}{L_{ℂ po}^{m_{1}}}); again, m_{2} is possibly +∞. We continue the above process. We say M is of finite type when commutators span the whole of ℂ ⊗_{ℝ} T_{p_{o}}^{ℂ}M. Thus, the above chain ends with a number m_{r} < +∞ or m_{r} = +∞ according to whether the type is finite or not. There is a “real” construction of chains of spaces of iterated commutators underlying the complex ones. We recall from Section 1.10 the identifications Re : T^{1,0}M → T^{ℂ}M and Re : T^{0,1}M → T^{ℂ}M. If ∂_{ω_{k}} = X_{k} − iY_{k}, k = 1, ..., n − l, these are defined by Re : ∑_{k}(a_{k} + ib_{k})∂_{ω_{k}} ↦ ∑_{k} a_{k}X_{k} + b_{k}Y_{k} and Re : ∑_{k}(a_{k} − ib_{k})∂ω̄_{k} ↦ ∑_{k} a_{k}X_{k} + b_{k}Y_{k} respectively. Let us define L_{ℝ}^{j} by means of iterated commutators of the vector fields X_{k}, Y_{k} that we denote by a common symbol L_{k}, i = 1, ..., 2(n−l):

L_{ℝ}^{j} = Span \{L_{k}, [L_{k_{1}}, L_{k_{2}}], \underbrace{[L_{k_{1}}, [L_{k_{2}}, [..., L_{k_{j}}]]]}_{j}\}.

It is an immediate exercice to verify that L_{ℂ}^{j} = ℂ ⊗ L_{ℝ}^{j}; thus the interpolation between ℂ ⊗ T^{ℂ}M and ℂ ⊗ T M by means of the L_{ℂ}^{j}’s induces, via Re , a corresponding interpolation between T^{ℂ}M and T M by means of the L_{ℝ}^{j}’s.

We take a system of equations y_{j} = h_{j}, j = 1, …, l, for M at z_{o} = 0, with h(0) = 0 and ∂h(0) = 0, and also set r_{j} = −y_{j} + h_{j} and r = (r_{j}). We identify \frac{T M}{T^{ℂ}M} →^{∼} T_{M}ℂ^{n} with the aid of the complex structure J and T_{M}ℂ^{n} →^{∼} ℝ^{l} with the aid of the dual basis to ∂r_{j}. We can reason, instead of over commutators, over the equations r = 0. We fix numbers m_{1} < ... < m_{r} (perhaps m_{r} = +∞) and multiplicities l_{i} with ∑_{i} l_{i} = l. We take multiindices I_{1} = (1, …, l_{1}), …, I_{r} = ((∑_{i<r} l_{i} + 1), …, l), divide the variables as (z_{I_{1}}, ..., z_{I_{r}}, w), give weight 1 to the w’s and m_{i} to the x_{I_{i}}’s, and define the weighted vanishing order for a function f = f(..., x_{I_{i}}, ..., w) by putting f = O^{+∞} when m_{r} = +∞ and f contains some monomial in the x_{I_{r}}’s, and, otherwise, putting f = O^{m_{i}} when f(t^{m_{1}}x_{I_{1}}, ..., t^{m_{i−1}}x_{I_{i−1}}, tw) = O(t^{m_{i}}). We suppose that associated to M there are numbers m_{1}, ...., m_{r} and multiplicities l_{1}, ..., l_{r} so that M is described by “normal equations”

(3.8.4)  \begin{cases}y_{I_{1}} = P_{I_{1}}(w) + O^{m_{1}+1}, \\ y_{I_{2}} = P_{I_{2}}(x_{I_{1}}, w) + O^{m_{2}+1}, \\ ... \\ y_{I_{r}} = P_{I_{r}}(x_{I_{1}}, ..., x_{I_{r−1}}, w) + O^{m_{r}+1}, \end{cases}

where the P_{I_{j}}’s are weighted homogeneous polynomials of degree m_{j}. We denote by h_{I_{m_{j}}} the right sides of (3.8.4). We point out that this is not necessarily the normal form in the Bloom-Graham sense. In fact, we are not assuming the ⟨ξ, P_{I_{j}}⟩’s non-M-pluriharmonic for any j and any ξ ∈ ℝ^{l_{i}}. To compare the system (3.8.4) of the equations with the chain L^{j}, j = 1, ..., r, of the commutators, we need to describe a basis {X_{j}} of vector fields for T^{1,0}M. We put r_{I_{i}} = −y_{I_{i}} + h_{I_{i}}, r = ^{t}(r_{1}, …, r_{l}), define an (n − l) × l matrix A = (a_{jh}) by

A = −^{t}(∂_{w}r)^{t}(∂_{z}r)^{−1},

and set X_{j} = ∑_{h=1}^{l} a_{jh}∂_{z_{h}} + ∂_{w_{j}}. We have (3.8.5)

∑_{h} a_{jh}∂_{z_{h}}(r_{I_{i}}) + ∂_{w_{j}}(r_{I_{i}}) = 0 for any i = 1, …, r and j = 1, ..., r.

Derivation of (3.8.5) yields (3.8.6)

\begin{cases}∂_{ww̄}^{β}∂_{x_{I_{1}}}^{α_{1}} … ∂_{x_{I_{i−1}}}^{α_{i−1}}(a_{j,I_{i}}) = 0 for |β| + ∑_{j≤i−1} m_{j}|α_{j}| ≤ m_{i} − 2, \\ ∑_{h} ∂_{ww̄}^{β}∂_{x_{I_{1}}}^{α_{1}} … ∂_{x_{I_{i−1}}}^{α_{i−1}}(a_{j,h}) = −2i∂_{ww̄}^{β}∂_{x_{I_{1}}}^{α_{1}} … ∂_{x_{I_{i−1}}}^{α_{i−1}}∂_{w_{j}}(r_{I_{i}}) \\ for |β| + ∑_{j≤i−1} m_{j}|α_{j}| ≤ m_{i} − 1. \end{cases}

Once the equations are ordered as in (3.8.4), we can introduce for any i ≤ r a diagram

(3.8.7)  \begin{matrix}\frac{T M}{T^{ℂ}M} & →^{φ} & ℝ^{l} \\ ↓ & & ↓ \\ \frac{T M}{L^{m_{i}−1}} & →^{ψ} & ℝ^{l_{i}+...+l_{r}}, \end{matrix}

where φ is defined by [v] ↦ ∂rJ (v) and ψ by [v] ↦ ∂(r_{I_{i}}, ..., ∂r_{I_{r}})J (v) (where we are performing multiplication rows by columns). The fact that ψ is well defined (in which case the diagram (3.8.7) is commutative) is a consequence of the first part of the next proposition. Also, the link between the normal form of the equations (3.8.4) and the chain of the L^{j}’s is expressed by

Proposition 3.8.1. Let X_{o} ∈ T^{1,0}M satisfy X_{o}(p_{o}) = ∂_{w_{1}}, and denote by X_{o}^{ϵ} either choice X_{o} or X̄_{o}. Then (3.8.8) J [X_{o}^{ϵ_{1}}, ..., [X_{o}^{ϵ_{m_{i}−2}}, [X_{o}, X̄_{o}]]...](r_{I_{j}})(p_{o}) = 0 for any j > i and any ϵ.

If moreover ⟨ξ_{o}, h_{I_{i}}⟩ is in the form P + O^{m_{i}+1} for P = P (w) homogeneous of degree m_{i} with m_{i} < +∞, then for some choice of ϵ_{1}, ..., ϵ_{m_{1}−2} with k occurrences of X_{o} and k̄ occurrences of X̄_{o}, we have (3.8.9)

J [X_{o}^{ϵ_{1}}, ..., [X_{o}^{ϵ_{m_{1}−2}}, [X_{o}, X̄_{o}]]...]⟨ξ_{o}, r_{I_{i}}⟩(p_{o}) = −2∂_{w_{1}}^{k} ¯∂k̄ ; w_{1}(⟨ξ_{o}, h_{I_{i}}⟩)(p′_{o}).

We refer to [18, Proposition 3.3] for an account of the proof. We now go on to treat the problem of the CR extension in an additional normal direction v^{o} ∈ T_{M}ℂ^{n}.

Theorem 3.8.2. Let M be a generic manifold of ℂ^{n} and, for a choice of a complex tangent direction w_{1}, let (3.8.4) be a normal system of equations for M̃ = M ∩ (ℂ_{z}^{l} × ℂ_{w_{1}}^{1}). Assume that for some j, for ξ^{o} ∈ ℝ^{l_{j}+···+l_{r}}, and with the notation P := ⟨ξ^{o}, P_{I_{j}}⟩, we have (3.8.10)

P = P (w_{1}) is a homogeneous non-harmonic polynomial of degree

m_{j} which does not depend on x_{I_{1}}, ..., x_{I_{j−1}},

P ≥ 0 for w_{1} in a sector S of width > \frac{π}{m_{j}} .

Then CR functions on M extend to a manifold M_{1} with boundary M which points to an additional direction v^{o} satisfying ⟨ξ^{o}, v^{o}⟩ > 0.

Remark 3.8.3. (i) The first condition of (3.8.10) is a sort of semirigidity in direction w_{1} and codirection ξ^{o}. There are large classes of hypersurfaces M for which, when (3.8.10) is violated, we can find a “barrier” that is a holomorphic function F such that M \{0} ⊂ {Im F < 0} (cf. [18, §4, Proposition 4.2 and Corollary 4.3]). In particular, for these M, there always exist CR functions f ∈ CR_{M} which do not extend to any extra direction v such that ⟨v, d(Im F)⟩ > 0. This shows that the statement in Theorem 3.8.2 is sharp.

(ii) When j = 1, the first condition of (3.8.10) is always fulfilled. Also, since P_{I_{1}} is not harmonic, then it is divisible by |w_{1}|^{2} and therefore it has at most 2m_{1} − 2 zeroes on the unit circle |w_{1}| = 1. In particular, for either of ±P, the second condition of (3.8.10) is satisfied.

(iii) There is a sort of “hierarchy” between the Hörmander numbers m_{j} whose geometric meaning will be fully clear from the proof of Theorem 3.8.2. According to it, (3.8.10) gives the control not of v^{o} itself, but of its projection on ℝ^{l_{j}+...+l_{r}}. The higher j becomes, the weaker the control of the extension gets.

(iv) When M is of finite type and semirigid (in all its arguments w), then our proof provides an alternative proof of the extension of f to a wedge W. The first conclusion in this direction is due to [10] where a description of W is also given. We improve this description by specifying the vanishing order in a precise direction w_{1}. Also, the semirigidity in the first condition of (3.8.10) can be released, as well as the hypothesis that the equations are in canonical form as in (3.8.10). What is indeed essential is the weighted vanishing order.

Proof of Theorem 3.8.2. (a) Preliminaries on F^{α} spaces. Let 0 < α < 1 and denote by τ = re^{iθ} the variable in the standard disc Δ. Let us recall from [93], [18] and [15] some basics about attaching analytic discs to M in the subspace F^{α} of the Hölder space C^{α}. This is the space of real continuous functions σ(θ), θ ∈ [−π, π], which are C^{1,α} out of 0 and for which the following norm is finite:

||σ||_{F^{α}} := ||σ||_{C^{0}} + ||θσ^{(1)}||_{C^{α}}.

(Here ·^{(1)} denotes the first derivative.) We remark that for σ ∈ F^{α} we must have θσ^{(1)}|_{θ=0} = 0 for otherwise θσ^{(1)} → c ≠ 0 which implies |σ| ≥ log \frac{|c|}{2} −log |θ| which contradicts the boundedness of σ. This shows that F^{α} is continuously embedded into C^{α}. It is easy to check that F^{α} is a Banach algebra. Also, if σ_{i} ∈ F^{α_{i}}, i = 1, 2, then σ_{1} · σ_{2} ∈ F^{α_{1}+α_{2}} for α_{1} + α_{2} < 1, resp. σ_{1} · σ_{2} ∈ C^{1,β} with β := (α_{1} + α_{2}) − 1 for α_{1} + α_{2} > 1. In both cases the multiplication is continuous with values in the respective spaces.

Let T_{1} denote the Hilbert transform normalized by the condition T_{1}(·)(1) = 0; it is easy to see that T_{1} is a bounded operator in F^{α} as has already been proved to be true in C^{α}. We come back to our manifold M. We write coordinates in ℂ^{n} ≃ ℂ^{l}×ℂ^{n−l} as (z, w) with z = x+iy, choose a distinguished direction, say w_{1}, and describe M̃ := M ∩ (ℂ_{z}^{l} × ℂ_{w_{1}}^{1}) by the system of equations y_{I_{i}} = h_{I_{i}}(x, w_{1}) with h_{I_{i}} = P_{I_{i}} + O^{m_{i}+1}. We consider in ℂ^{n} analytic discs A(τ) = (z(τ), w(τ)), τ ∈ Δ (where Δ is the standard disc in ℂ), attached to M̃, that is, satisfying A(∂Δ) ⊂ M̃. If we prescribe an analytic function w_{1}(τ), τ = te^{iθ} ∈ Δ, the so-called CR component of the disc, and a point p = (z, w_{1}) with y = h(x, w_{1}) and look for an analytic completion z(τ) for A(τ) = (z(τ), w_{1}(τ)) with A(1) = p, we are lead to Bishop’s equation

(3.8.11)  u(τ) = −T_{1}h(u(τ) + x, w_{1}(τ)), τ ∈ ∂Δ.

We consider equation (3.8.11) in the spaces F^{α}, F^{m_{i}α} and C^{1,β} for which T_{1} is bounded. We also use the composition properties of h_{I_{i}} for i ≥ j, with functions in the above classes as stated in [15]. To take advantage of this composition, we assume m_{j}α > 1 (and, to be sharp, α(m_{j} − 1) < 1). Here is our main technical tool.

Proposition 3.8.4. Let h_{I_{i}} be of class C^{m_{i}+3}, and let it satisfy h_{I_{i}} = O^{m_{i}}. For any ϵ there is δ such that if ||h_{I_{i}}||_{C^{1,α}} < δ, ||w_{1}||_{F^{α}} < δ, |x| < δ, then equation (3.8.11) has a unique solution u ∈ F^{α} with ||u∥_{F^{α}} < ϵ. Moreover, u_{I_{1}} ∈ F^{m_{1}α}, …, u_{I_{j−1}} ∈ F^{m_{j−1}α} and (u_{I_{j}}, …, u_{I_{r}}) ∈ C^{1,β} for β = m_{j}α − 1 and, if w_{1} depends on some parameter λ ∈ ℝ^{d} so that λ ↦ w_{1}_{λ}, ℝ^{d} → F^{α} is C^{k} for k ≤ m_{i}, then (λ, x) ↦ (u_{I_{i}})_{λ x}, ℝ^{d+l} → C^{1,β} is also C^{k}. In particular, there exist mixed derivatives in λ, x and t up to order k and 1, respectively, and they commute, that is,

(3.8.12)  ∂_{t}∂_{λ x}^{k′}u = ∂_{λ x}^{k′}∂_{t}u, for any k′ ≤ k.

Proof. One first solves Bishop’s equation (3.8.11) in the F^{α}- spaces with the aid of the implicit function theorem. To this end one considers the mapping F: (λ, x, w_{1}, u) ↦ u − T_{1}h(u + x, w_{1}), ℝ^{d} × ℝ^{l} × F^{α} × F^{α} → F^{α}. Then, for the partial Jacobian ∂_{u}F with respect to u, one has ∂_{u}F : u̇ ↦ u̇ − T_{1}∂_{x}hu̇. In particular, if we evaluate at (λ, x, w_{1}, u) = (0, 0, 0, 0), then this is invertible since ∂_{x}h|_{0} = 0. The differentiability with respect to the parameters in the space F^{α} is also clear in view of [100, Proposition 11].

We show that the components u_{I_{i}}, i ≥ j, of the solution to Bishop’s equation, as well as their harmonic conjugates v_{I_{i}}, are in fact in F^{m_{i}α} for i < j (resp. C^{1,β} for i ≥ j with β := m_{j}α − 1) and we also prove differentiability in the parameters with values in this space. The key point is that, when φ = O^{m_{i}}, the composition φ((1 − τ)^{α}) and, in greater generality, φ(w_{1}) for w_{1} ∈ F^{α} with w_{1}(1) = 0 belong to F^{m_{i}α} for any i < j (resp. C^{1,β} for any i ≥ j). We put z(τ) = u(τ) + iv(τ) + z with v = T_{1}u and z = x+ih(x, w) and also write τ = e^{iθ} on ∂Δ. We can check that if z_{I_{i}}(τ) ∈ F^{kα} for k ≤ m_{i} − 2, then in fact z(τ) ∈ F^{(k+1)α}. In fact, v gains regularity at each step because h_{I_{i}} = O^{m_{i}} together with the fact that if σ(θ) ∈ F^{kα} and σ(0) = 0, then |θ|^{α}σ(θ) ∈ F^{(k+1)α} because of

\begin{aligned}|(|θ|^{α}σ(θ))^{(1)}| & = ||θ|^{α−1}σ(θ) + |θ|^{α}σ^{(1)}(θ)| \\ & ≤ c|θ|^{(k+1)α−1}. \end{aligned} ; (3.8.13)

But the Hilbert transform transfers the F^{(k+1)α} regularity from v to u and thus z(e^{iθ}) ∈ F^{(k+1)α}. This completes the proof when i < j. On the other hand, when i ≥ j, in order to pass from F^{(m_{i}−1)α} to C^{1,β}, we have to prove that (θ^{α}u)^{(1)} = θ^{α−1}u + θ^{α−1}(θu^{(1)}) belongs to C^{β}. But, in fact, since both u and θu^{(1)} are in C^{(m_{i}−1)α} and are 0 at θ = 0, then their product with θ^{α−1} is in C^{β} as one can easily check by the Hardy-Littlewood principle. It follows that (θ^{α}u)^{(1)} ∈ C^{β} and hence θ^{α}u ∈ C^{1,β}. Thus, u(e^{iθ}) and hence z(e^{iθ}) itself are in C^{1,β}. As for the differentiability on x and on the parameters, it is a variant of [15, Proposition 15] by the same feedback argument as above. The proof of Proposition 3.8.4 is complete.

□

We can think of the family of discs produced by the above statement as a deformation of the disc A(τ) ≡ 0 which is, of course, a solution to Bishop’s equation. By the next statement we show how it is possible to make infinitesimal deformations of discs which are no longer small.

Proposition 3.8.5. Let h_{I_{i}} ∈ C^{m_{i}+3} satisfy h_{I_{i}} = O^{m_{i}}, let w̃_{1}(τ) ∈ C^{1,β} with w̃_{1}(1) = 0 be small in F^{α} (not necessarily in C^{1,β}), and let ũ(τ) ∈ F^{α} be a solution to Bishop’s equation ũ = −T_{1}h(ũ, w̃). In particular, ũ_{I_{i}} ∈ C^{1,β} for any i ≥ j according to Proposition 3.8.4. Then, for any w_{1}(τ) with ||w_{1} − w̃_{1}||_{C^{1,β}} < δ, |x| < δ, there is a unique solution to Bishop’s equation u ∈ F^{α} with u_{I_{i}}(τ) ∈ C^{1,β}, i ≥ j, and ||u_{I_{i}} − ũ_{I_{i}}||_{C^{1,β}} < ϵ, i ≥ j. Moreover, if λ ↦ (w_{1})_{λ} is C^{k}, k ≤ m_{i}, then (λ, x) ↦ (u_{I_{i}})_{λ} is also C^{k}.

Proof. In the present situation we define F: ℝ^{d} × ℝ^{l} × C^{1,β} similarly to as in the proof of Proposition 3.8.4 and wish to prove that ∂_{u}F is still invertible. For this purpose it is enough to show that ∂_{u}h_{I_{i}}(ũ, w̃) is small in C^{1,β}-norm. But, in fact, recall that |∂_{x}h_{I_{i}}(u, w)| = O(|w|^{2}) and therefore

\begin{aligned}||∂_{x}h_{I_{i}}(ũ, w̃)^{(1)}||_{C^{β}} & ≤ c||w̃_{1}||_{C^{β}}||w̃_{1}^{(1)}||_{C^{β}} \\ & ≤ c||w̃_{1}||_{F^{α}}||w̃_{1}^{(1)}||_{C^{β}}, \end{aligned}

where the last inequality follows from β < α. Since we are assuming ||w̃_{1}||_{F^{α}} small, the conclusion follows.

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(b) Construction of a singular disc attached to M with controlled normal component. Let us suppose that (3.8.10) is fulfilled. It is not restrictive to assume that the sector of the plane ℂ_{w_{1}} where g ≥ 0 contains (1 − τ)^{α}ie_{l+1}, τ ∈ Δ. (Here e_{l+1} is the unit vector of the w_{1}-plane.) Let α satisfy αm_{j} > 1, α(m_{j} − 1) < 1. We define, for a small real parameter η > 0,

(3.8.14)  w_{1}(τ) = (w_{1})_{η}(τ) := η(1 − τ)^{α}ie_{l+1}.

We attach to M̃ a family of F^{α}-discs A(τ) = A_{η}(τ) whose “w_{1}- component” is w_{1}(τ). We recall from (a) that for i ≥ j we have that η ↦ (z_{I_{i}})_{η}(τ), ℝ → C^{1,β} is C^{m_{i}}. We also write z_{I_{i}}(τ) instead of (z_{I_{i}})_{η}(τ), z_{I_{i}}(τ) = u_{I_{i}}(τ) + iT_{1}u_{I_{i}}(τ), and finally A(τ) = (z(τ), w(τ)). We note that we have

(3.8.15)  ∂_{η}^{s}v_{I_{i}}|_{η=0} ≡ 0, ∂_{η}^{s}u_{I_{i}}|_{η=0} ≡ 0, s ≤ m_{i} − 1.

This is clear for s = 0, 1. If it is true for any s ≤ m_{i} − 2, then it is also true for s = m_{i} − 1 because of h_{I_{i}} = O^{m_{i}} by a “feedback” procedure. If we then take the Taylor expansion of ∂_{r}v_{I_{i}} at η = 0, we get

(3.8.16)  ∂_{r}v_{I_{i}} = \frac{∂_{η}^{m_{i}}∂_{r}v_{I_{i}}|_{η=0}}{m_{i}!} η^{m_{i}} + o(η^{m_{i}}).

By a feedback argument we can also prove that

(3.8.17)  |v_{I_{i}}| ≤ c|w_{1}|^{m_{i}}, |u_{I_{i}}| ≤ c|w_{1}|^{m_{i}}.

In fact, in the classes F^{kα}, regularity and vanishing order are coincident; thus, the equation v_{I_{i}} = h_{I_{i}} gives control from below of the vanishing order of v_{I_{i}} which is transferred as regularity to u_{I_{i}} through Hilbert transform and again as vanishing order to v_{I_{i}}. In this way, we can prove that each v_{I_{i}} and u_{I_{i}} belongs to F^{m_{i}α} (and also to C^{[m_{i}α],{m_{i}α}} where [m_{i}α], resp. {m_{i}α}, is the integer, resp. fractional, part of m_{i}α). Recall that if ξ_{o} is, say, the unit vector in the l′ := (l_{1} + ... + l_{j−1} + 1)- direction, we have ⟨ξ_{o}, h⟩ ≥ 0 if w_{1} is in a sector S of width > \frac{m_{j}}{π} and if |x_{I_{i}}| ≤ c|w_{1}|^{m_{i}}. We first observe that this latter condition, |x_{I_{i}}| ≤ c|w_{1}|^{m_{i}}, is automatically fulfilled by the components x_{I_{i}} = u_{I_{i}} of our discs A(τ) because of (3.8.17). We show now that ∂_{r}v_{l′} < 0. In fact, in our situation we have

(3.8.18)  ⟨ξ_{o}, ∂_{η}^{m_{j}}v_{l′}⟩|_{η=0} ≥ 0 for any τ ∈ Δ̄.

Hence (3.8.18) yields, through Hopf’s lemma,

(3.8.19)  ⟨ξ_{o}, ∂_{r}∂_{η}^{m_{j}}v_{l′}⟩|_{τ=1, η=0} = −c < 0.

By (3.8.16) we conclude ⟨ξ_{o}, ∂_{r}v_{l′}⟩|_{τ=1} = −c′η^{m_{j}} < 0, for any η sufficiently small.

(c) Polynomial approximation of (1 − τ)^{α} in F^{γ}(Δ̄) for γ < α and construction of a smooth disc transversal to M and of its infinitesimal deformation.

We have the Taylor expansion

\begin{aligned}(1 − τ)^{α} & = 1 − ατ − \frac{α(1 − α)}{2!}τ^{2} − \frac{α(1 − α)(2 − α)}{3!}τ^{3} + … \\ & = 1 − ∑^{+∞} ; j=1 |\binom{α}{j}| τ^{j}. \end{aligned} ; (3.8.20)

We call S_{N} = S_{N}(τ) the partial sum of the power series (3.8.20) for 1 ≤ j ≤ N. We have the following approximation result for whose proof we refer to [18, Theorem 2.3]:

(3.8.21)  S_{N}(τ) → (1 − τ)^{α} in F^{γ}(Δ̄) for any γ < α.

We now put w_{N}(τ) = S_{N}(τ) − S_{N}(1), let u_{N} be the solution in F^{γ} to Bishop’s equation u_{N} = −T_{1}h(u_{N}, w_{N}), and let z_{N} = u_{N} + iv_{N} for v_{N} = T_{1}u_{N}. Let u be the solution to u = −T_{1}h(u, (1 − τ)^{α}), and set z = u + iv for v = T_{1}u. Since

w_{N}(τ) → (1 − τ)^{α} in F^{γ}(Δ̄)

and since w_{N}(1) ≡ 0 for any N, then (z_{I_{j},...,I_{r}})_{N}(τ) → z_{I_{j},...,I_{r}}(τ) in C^{1,β′}(Δ̄) by Proposition 3.8.4. (Clearly, we are supposing γ close enough to α so that β′ := m_{j}γ − 1 > 0.) In particular, for any ϵ and for large N, the discs A_{Nη} = (z_{Nη}, w_{Nη}) for w_{Nη} := ηw_{N} are in C^{1,β′} and satisfy (cf. (3.8.19))

(3.8.22)  |∂_{r}(v_{I_{j},...,I_{r}})_{Nη}(1) − ∂_{r}(v_{I_{j},...,I_{r}})| < ϵη^{m_{j}}.

To explain (3.8.22), we still denote by l′ any index satisfying l′ ≥ ∑_{i≤j−1}l_{i} + 1 and notice that

||v_{l′} − v_{l′N}||_{C^{1,γ}} = ||h_{l′}(u_{η}, ηw) − h_{l′}(u_{Nη}, S_{Nη})||_{C^{1,γ}}

= η^{m_{j}}||h^{(m_{j})}(0)w^{m_{j}} − h^{(m_{j})}(0)S_{N}^{m_{j}}||_{C^{1,γ}}

<_{∼} η^{m_{j}}||w^{m_{j}} − S_{N}^{m_{j}}||_{F^{α′}}.

This proves (3.8.22). Also, once N has been fixed so that (3.8.22) holds, one chooses η such that ||ηw_{N}||_{C^{1,γ}} << 1 so that the hypotheses of Proposition 3.8.5 are fulfilled. We call Ã = (z̃, w̃) one of these discs for the choices of the parameters N and η which have been pointed out, and use the notation ṽ^{o} := ∂_{r}(ṽ_{I_{j},...,I_{r}}). With a smooth transversal disc in our hands, we are in the same situation as for the Levi extension discussed in Section 3.7 (where the discs were smooth from the beginning since we had α = 1 therein). By a deformation of the disc Ã, we are ready to construct a manifold M_{1} with boundary M which gains one more direction and has the property that CR functions extend from M to M_{1}. For this, we consider Bishop’s equation

(3.8.23)  u = −T_{1}h(u + x, w + w̃),

for x ∈ ℝ^{l}, w ∈ ℂ^{n−l} with |x| < δ, |w| < δ. According to Proposition 3.8.5, for any ϵ and for suitable δ = δ_{ϵ}, there is a unique solution u which satisfies ∥u − ũ∥_{C^{1,β′}} < ϵ for β′ < β := kα − 1. We write p = (x + ih(x, w), w) with v = T_{1}u and define A_{p}(τ) = p + (u(τ) + iv(τ), w̃(τ)). We also write I_{p} = A_{p}|_{[−1,+1]} and define

(3.8.24)  M_{1} = ⋃_{p}I_{p}([1 − ϵ, 1]).

In the same way as in Section 3.7 we can see that M_{1} is a manifold with boundary M which satisfies T M_{1} = T M +ℝ^{+}ṽ^{o}. By the approximation Theorem 3.6.3 and by the maximum principle, we can extend f from the boundaries ∂A_{p} ⊂ M to the “radii” I_{p} which fill the manifold M_{1}.

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Corollary 3.8.6. Let M be a manifold of finite type and “semirigid” in the sense that it has normal equations of type (3.8.4) with m_{r} < ∞ and with each P_{I_{j}} non-pluriharmonic and independent of the x_{I_{i}}’s (cf. [10]). Then CR functions extend from M to an open wedge W with edge M.

Proof. We start from j = 1 and consider all related directions of extension v^{o} given by Theorem 3.8.2 (cf. Remark 3.8.3(ii)). By taking their convex hull according to the edge of the wedge theorem, we obtain a first approximate extension open cone Γ_{1} ⊂ ℝ^{l_{1}}. Next, we apply Theorem 3.8.2 for j = 2 and obtain a perturbation of Γ_{1} in the form Γ_{1} + ϵ_{2}Γ_{2} as a second approximate cone of extension. We continue this process and end up with a full-dimensional approximate extension cone in the form Γ = Γ_{1} + ϵ_{2}Γ_{2} + ... + ϵ_{r}Γ_{r}.

□

Notes. The CR extension in directions shown by iterated bracket up to the first Hörmander number goes back to Boggess and Pitts [28] and to Baouendi and Rothschild [10]. The method of [10] also applies to further Hörmander numbers but under an additional hypothesis of “semirigidity” which is required by the technique based upon Fourier operators and microlocal analysis. Theorem 3.8.2 is stated in Baracco and Zampieri [18]. More recent work by Baracco, Zaitsev and Zampieri [15] treats the CR extension from a wedge to a bigger wedge. We refer to the latter paper for a further study of the “sectors” which develop the “thin discs” originally introduced by Tumanov in [94].

3.9. Partial lifts of analytic discs and CR curves

Summary. We prepare the geometric background for the problem that we will shortly anticipate here and that will be completely solved in Section 3.11. We introduce “partial lifts” of CR curves and analytic discs. We show that any CR curve γ in M ⊂ ℂ^{n} is endowed with partial lifts, CR curves γ^{∗′} in (T_{M}^{∗}ℂ^{n})′ (the “orthogonal” projection of the conormals) such that π(γ^{∗′}) = γ. Also, any small analytic disc A ⊂ ℂ^{n} with ∂A ⊂ M is endowed with partial lifts A^{∗′}, holomorphic sections of (T^{∗}ℂ^{n})′ with ∂A^{∗′} ⊂ (T_{M}^{∗}ℂ^{n})′ and π(A^{∗′}) = A. On the other hand, the uniqueness of the lift through a fixed conormal, both for curves and discs, is obvious. For this reason, the lifts provide a “connection”, a morphism of T_{M}^{∗}ℂ^{n} between the fibers over different points of γ and ∂A, and a dual-inverse connection of T_{M}ℂ^{n}. Moreover, the approximation of CR curves γ^{∗′} by discs A^{∗′}, induces an approximation for the respective connections. We end the section with a geometric application of the lifts: “microlocally pseudoconvex” manifolds “absorb” tangent analytic discs.

Let f be a CR function on a generic CR manifold M ⊂ ℂ^{n} which has CR extension at some point z_{o} ∈ M to some “additional direction” v_{o} normal to M in ℂ^{n}, that is, belonging to (T_{M}ℂ^{n})_{z_{o}}. This means that there is a germ of a manifold M′ with boundary M of dimension 2n − l + 1, that is, dimM′ = dimM + 1, which “points to” v_{o} in the sense that T_{z_{o}}M′ = T_{z_{o}}M ⊕ ℝ^{+}v_{o}, to which f extends as CR. Let γ be a CR curve in M, that is, a curve which runs in a complex tangent direction, i.e., T γ ⊂ T^{ℂ}M|_{γ}, let z_{o} = γ(0), and let z_{1} = γ(1) be another point of γ. Is there a rule to find a “connection” Φ : (T_{M}ℂ^{n})_{z_{o}} → (T_{M}ℂ^{n})_{z_{1}}, v_{o} ↦ v_{1} so that if f extends to v_{o} at z_{o}, then it extends to v_{1} at z_{1}? In the identifications (T_{M}ℂ^{n})_{z} →^{∼} ℝ^{l} through the dual basis to ∂r_{1}(z), ..., ∂r_{l}(z), what is the l×l matrix B associated to Φ, that is, such that v_{1} = Bv_{o}? In practice, there is no natural connection in T_{M}ℂ^{n}. But there is in fact in T_{M}^{∗}ℂ^{n} and it turns out that its dual-inverse is the connection in T_{M}ℂ^{n} we are looking for (cf. Section 3.11). Let us start the construction of^{t}Φ^{−1} and^{t}B^{−1}. Let z and (z; ζ) be coordinates in ℂ^{n} and T^{∗}ℂ^{n}, respectively. Let M be described in a neighborhood of 0 as a graph y′_{j} = h_{j}(x′, z′′) with h_{j}(0) = 0 and ∂h_{j}(0) = 0 for j = 1, ..., l. Let pr_{ℂ^{n−l}} be the projection in the fibers of T^{∗}ℂ^{n} modulo ℂ^{n−l}, given by ℂ^{n} → ℂ^{l}, ζ → ζ′, and define

(3.9.1)  (T_{M}^{∗}ℂ^{n})′ := pr_{ℂ^{n−l}}(T_{M}^{∗}ℂ^{n}).

We denote by T^{ℂ}M := T M ∩ iT M and T^{ℂ}(T_{M}^{∗}ℂ^{n})′ := T (T_{M}^{∗}ℂ^{n})′ ∩ iT (T_{M}^{∗}ℂ^{n})′ the complex tangent bundles to M and (T_{M}^{∗}ℂ^{n})′, respectively.

Theorem 3.9.1. The natural projection π : T^{∗}ℂ^{n} → ℂ^{n} induces an identification

(3.9.2)  T^{ℂ}(T_{M}^{∗}ℂ^{n})′ →^{∼}_{π′} T^{ℂ}M.

Proof. To begin our proof, we need to describe tangent and complex tangent vectors to (T_{M}^{∗}ℂ^{n})′. We put r_{j} := −y_{j} + h_{j}, r = (r_{j}), and observe that ∂r_{j} is a basis for the conormal bundle T_{M}^{∗}ℂ^{n}. We fix a base point z_{o} ∈ M and a conormal t_{o}∂r for t_{o} ∈ ℝ^{l} that we also denote by ∂r_{o}, write p = (z_{o}; ∂r_{o}), and denote by p′ := (z_{o}; ∂′r_{o}) its projection under pr_{ℂ^{n−l}}. We have (cf. (3.5.6))

(3.9.3) T_{p′}(T_{M}^{∗}ℂ^{n})′ = {(u; t∂′r+∂_{u}∂′r_{o}+ ¯∂_{u}∂′r_{o}) : u ∈ T_{z_{o}}M, t ∈ ℝ^{l}}.

It follows that

T_{p′}^{ℂ}(T_{M}^{∗}ℂ^{n})′ ={(u; t∂′r + ∂_{u}∂′r_{o} + ¯∂_{u}∂′r_{o}) : u ∈ T^{ℂ}M,

¯∂_{u}∂′r_{o} = \frac{1}{2}(−t + is)∂′r for some t and s in ℝ^{l}\}.

(3.9.4)

Now, since ∂′r is non-degenerate, for any u ∈ T^{ℂ}M it is possible to find a unique λ = \frac{1}{2}(−t + is) ∈ ℂ^{l} which solves the equation ¯∂_{u}∂′r_{o} = λ∂′r; hence, over any vector u ∈ T^{ℂ}M, there is a vector of T^{ℂ}(T_{M}^{∗}ℂ^{n})′. Also, as already mentioned, this vector is unique.

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For a CR curve γ in M, we call a CR curve in T_{M}^{∗}ℂ^{n}, resp. (T_{M}^{∗}ℂ^{n})′, which projects over γ a “lift”, resp. a “partial lift”, of γ. We denote by γ^{∗} and γ^{∗′} a lift and a partial lift of γ, respectively.

Corollary 3.9.2. Let γ be a CR curve in M and p′ a point in (T_{M}^{∗}ℂ^{n})′. Then, there is an unique partial lift γ^{∗′} such that γ^{∗′}(1) = p′.

Let z_{o} and z_{1} be two points of M connected by a CR curve γ with

γ(0) = z_{o}, γ(1) = z_{1}.

Associated to γ, there is naturally defined the connection

tΦ^{−1} : (T_{M}^{∗}ℂ^{n})_{z_{o}} → (T_{M}^{∗}ℂ^{n})_{z_{1}},

defined as follows. For any p_{o} ∈ T_{M}^{∗}ℂ^{n} with π(p_{o}) = z_{o} we take the unique “partial lift” γ_{p_{o}}^{∗′} such that γ_{p_{o}}^{∗′}(0) = p_{o} and define

tΦ^{−1} : p_{o} ↦ p_{1} where p_{1} = pr_{ℂ^{n−l}}^{−1}(γ_{p_{o}}^{∗′})(1).

(Recall here that pr_{ℂ^{n−l}} is one-to-one when restricted to T_{M}^{∗}ℂ^{n}.) There is no doubt that this definition contains a great deal of geometric sense: the section γ^{∗′} of (T_{M}^{∗}ℂ^{n})′ that we have chosen is a distinguished one. It is the unique CR. Note also that the partial lifts of γ are a vector space. In fact, the equation λ∂′r = t_{o}∂ū∂′r which defines the vector ũ such that π′ũ = u is linear in t_{o}. Thus, ^{t}Φ^{−1} is also linear. We denote by ^{t}B^{−1} the matrix associated to ^{t}Φ^{−1} in the basis ∂r_{1}, ..., ∂r_{l} (with coordinates of differentials being column vectors). We can summarize the aforementioned construction by

(3.9.5)  \begin{matrix}(T_{M}^{∗}ℂ^{n})_{z_{o}} & →^{^{t}Φ^{−1}} & (T_{M}^{∗}ℂ^{n})_{z_{1}} \\ || & & || \\ ℝ^{l} & →_{^{t}B^{−1}} & ℝ^{l}. \end{matrix}

Guided Exercise 3.9.3. Let us come back to our former transformation χ : (z; ζ) ↦ (z′+ϵ\frac{ζ′}{(∑_{j≤l} ζ_{j}^{′2})^{\frac{1}{2}}} , z′′; ζ) already introduced in (3.5.9)

and (3.5.10); it interchanges M −−→ M̃, that is, T_{M}^{∗}ℂ^{n} → T ∗ ; M̃ℂ^{n}. Prove that χ interchanges CR curves γ −−→ γ̃.

Hint: let z_{o} be a point of γ and γ^{∗′} = γ_{p′_{o}}^{∗′} its partial lift through p′_{o} ∈ (T_{M}^{∗}ℂ^{n})′; define γ̃^{∗′} := χ(γ^{∗′}) and γ̃ := π(γ̃^{∗′}). Then, γ̃^{∗′} is a CR curve in (T ∗ ; M̃ℂ^{n})′ and therefore γ̃ a CR curve in M̃. The situation is described by the diagram

(3.9.6)  \begin{matrix}T_{M}^{∗}ℂ^{n} & →^{χ} & T ∗ ; M̃ℂ^{n} & γ^{∗′} & →^{χ} & γ̃^{∗′} \\ π ↓ & & ↓ π & π ↓ & & ↓ π \\ M & −−→ & M̃, & γ & −−→ & γ̃. \end{matrix}

Thus, by means of the partial lift, we have explained that χ interchanges CR curves.

We go on to discuss lifts of analytic discs. We denote by A both an analytic disc and its parametrization over the standard disc Δ. The disc is said to be “attached” to M when ∂A ⊂ M. A lift of A is an analytic section of T^{∗}ℂ^{n} over A. If A is attached to M, we require its lift to be attached to T_{M}^{∗}ℂ^{n}. A disc endowed with analytic lifts is said to be “defective” and its “defect” is the dimension of the space of its lifts. The presence of defective discs is related to the “non-minimality” of M in the sense of Tumanov that will be discussed in Section 3.10. Let us recall here that the minimality of M is equivalent to the existence of small non-constant discs of defect 0 near any point of M. Generically, CR manifolds are minimal and therefore defective discs are quite exceptional. However, any disc has “partial lifts” and the dimension of their space is maximal (equal to the codimension l of M).

Definition 3.9.4. A^{∗ ′} is said to be a partial lift of A when A^{∗ ′} is a holomorphic section of (T^{∗}ℂ^{n})′ over A and is attached to (T_{M}^{∗}ℂ^{n})′, that is, ∂A^{∗_{′}} ⊂ (T_{M}^{∗}ℂ^{n})′.

The family of all partial lifts of a disc can be found by solving a Riemann-Hilbert problem as described by the following lemma.

Lemma 3.9.5. Let M be defined by a family of “normalized” equations of class C^{2,α}

y_{j} = h_{j}(x′, z′′), j = 1, ...l, with h_{j}(0) = 0 and ∂h_{j}(0) = 0 for any j.

We also set r_{j} := −y_{j} + h_{j}. Let A be a disc attached to M which is small in C^{1,α}-norm. Then, there is a real l×l matrix G = G(τ) of class C^{1,α} on ∂Δ close to id_{l×l} such that

G · ∂′r(A) extends holomorphically from ∂Δ to Δ and G(1) = id_{l×l} .

Proof. We denote by T_{1} the Hilbert transform normalized by the condition T (·)(1) = 0. We then have to find G such that G(i · id_{l×l} + ∂_{x}h(A)) extends holomorphically and G(1) = id_{l×l}, which is equivalent to solving the linear Bishop equation

G = T_{1} (G · ∂_{x}h(A)) .

This equation can be solved, by the implicit function theorem, in the spaces with fractional continuity such as C^{1,α}, because ∂_{x}h(A) is small (and because T_{1} is continuous in these spaces).

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Remark 3.9.6. The system r_{1}, ..., r_{l} is normalized in ℂ^{n} but ∂r_{1}, ..., ∂r_{l} is not in T^{∗}ℂ^{n}. The matrix G serves as a renormalization: the lines of G · ∂r(A) are a basis of T_{M}^{∗}ℂ^{n} whose “prime” components extend holomorphically from ∂Δ to Δ.

Let z_{1} = A(1) and take ξ_{1} ∈ (T_{M}^{∗}ℂ^{n})_{z_{1}}. Then, for any A attached to M with A(1) = z_{1}, there is a unique partial lift A^{∗ ′} with A^{∗ ′}(1) = ξ′_{1}; in fact, the disc

(3.9.7)  A^{∗ ′} = (A;^{t}∂′r(A)^{t}Gξ′_{1})

clearly serves the purpose. Also, if ζ′(τ) = ^{t}∂′rξ′(τ) is another lift, then if we define g(τ) := (^{t}∂′r(A)^{t}G)^{−1}ζ(τ), we have

\begin{cases}g(τ) ∈ ℝ for any τ ∈ ∂Δ, \\ g(τ) extends holomorphically. \end{cases}

It follows that g(τ) ≡ ξ′_{1} (constant) and therefore ζ′ = ^{t}∂′r^{t}Gξ′_{1}; thus, ζ′(τ) is uniquely determined by ζ′(1) = ξ′_{1}. Also, through the partial lifts of A, we can define a connection Φ over ∂A similar to that obtained over γ. Let z_{o} = A(−1) and z_{1} = A(1), take ξ′_{o} ∈ (T_{M}^{∗}ℂ^{n})′_{z_{o}}, and let A_{(z_{o},ξ′_{o})}^{∗′} be the unique partial lift such that A_{(z_{o},ξ_{o})}^{∗′}(−1) = ξ′_{o}. We define a morphism (T_{M}^{∗}ℂ^{n})′_{z_{o}} → (T_{M}^{∗}ℂ^{n})′_{z_{1}} by

(3.9.8)  ^{t}Φ^{−1} : ξ′_{o} ↦ ξ′_{1} where ξ′_{1} = pr^{−1}(A_{(z_{o},ξ_{o})}^{∗′}(1)).

Since A_{(z_{o},ξ_{o})}^{∗′} is in the form ^{t}∂′r(A)^{t}Gξ′_{1}, then if we want it to pass through ξ′_{o} at −1, we must have ξ′_{1} = ^{t}G^{−1}(−1)ξ′_{o}. In other words, we have ^{t}B^{−1} = ^{t}G^{−1}(−1). Again, the definition contains a great deal of geometric foundation. We have chosen a distinguished section A^{∗′} of (T_{M}^{∗}ℂ^{n})′ over ∂A: the one uniquely endowed with holomorphic extension to Ā. Since CR curves γ^{∗′} can be approximated by chains of analytic discs A^{∗′}, then the matrix B associated to Φ is approximable by a composition of matrices G(−1)’s.

Guided Exercise 3.9.7 (Twin of Exercise 3.9.3). Let χ be the transformation of Exercise 3.9.3 which interchanges M −−→ M̃. Prove that it interchanges attached analytic discs A −−→ Ã.

Hint: take a partial lift A^{∗′}(τ) = (A, ζ′(τ)) for ζ′(τ) = ^{t}∂′_{z}r(A)·^{t}Gξ_{o}, and define Ã^{∗′} = χ(A^{∗′}) and Ã = π(Ã^{∗′}). Now, in contrast to full lifts, we have that the partial lifts A^{∗′} and Ã^{∗′} are holomorphic. Thus Ã = π(Ã^{∗′}) is holomorphic. Also, ∂Ã ⊂ M̃ since ∂Ã^{∗′} ⊂ (T_{M}^{∗}ℂ^{n})′. The conclusion is illustrated by the diagram

(3.9.9)  \begin{matrix}A^{∗′} & →^{χ} & Ã^{∗′} \\ π ↓ & & ↓ π \\ A & −−→ & Ã. \end{matrix}

We end this section with a geometric application of the construction of lifts. This part is not essential for what follows.

For z_{o} = A(1), z̃_{o} = Ã(1), assume that

⟨ξ_{o}, T_{z_{o}}A⟩ = 0;

it follows that

⟨ξ_{o}, Tz̃_{o} Ã⟩ = 0.

In fact, for the 1-form ω, we have ω|_{T_{zo}A} = 0 if and only if χ_{∗}ω|_{Tz̃o Ã} = 0. This yields the following higher-codimensional version of the fact that pseudoconvex boundaries “absorb” tangent analytic discs (cf. [12]).

Theorem 3.9.8. Let s_{M}^{−} ≡ 0 in a neighborhood of p_{o} = (z_{o}, ξ_{o}), and let A be a small analytic disc attached to M and orthogonal to ξ_{o} at z_{o}. Then A ⊂ M and, moreover, there is an analytic lift A^{∗} through (z_{o}, ξ_{o}).

Proof. By means of the transformation χ, we have transformed M into a pseudoconvex hypersurface M̃ (because of the symplectic invariance of s^{0}) and A into Ã such that ∂Ã ⊂ M̃ and Tz̃_{o} Ã ⊂ T_{z_{o}}^{ℂ}M̃. We now prove that Ã ⊂ M̃. In fact, if we denote by Ω̃ the open halfspace with boundary M̃ and outward conormal ξ_{o}, it is clear that Ã ⊂ \overline{Ω̃} because otherwise pseudoconvexity is violated. Next, we reason by contradiction and suppose that Ã is not contained in M̃. If ψ̃ is a bounded plurisubharmonic exhaustion function for Ω̃ which satisfies ψ̃ < 0 in Ω̃ and ψ̃ = 0 in M̃ (cf. Diederich and Fornaess [40]), we have

\begin{cases}ψ̃ ∘ Ã ≤ 0 all over ∂Δ, \\ ψ̃ ∘ Ã < 0 somewhere on ∂Δ. \end{cases}

But then, Hopf’s lemma yields

∂_{τ}(ψ̃ ∘ A)|_{1} > 0,

which contradicts Tz̃_{o} Ã ⊂ T ℂ ; z̃_{o} M̃. Thus, Ã ⊂ M̃. Also, by Bedford and Fornaess [22], M̃ being pseudoconvex, Ã is endowed with a holomorphic lift. Since its “prime” components coincide with Ã^{∗ ′}, this lift is Ã^{∗}. Thus, A^{∗} = χ^{−1}Ã^{∗} is the holomorphic lift of A contained in T_{M}^{∗}ℂ^{n}.

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Remark 3.9.9. According to Remark 3.5.4, we do not need to assume s_{M}^{−} ≡ 0 in order to interchange M −−→ M̃ where M̃ is a pseudoconvex hypersurface: s_{M}^{−} ≡ const suffices. However, the symplectic transformation that we need in this case is not as simple as the one defined by (3.5.9), (3.5.10): we need to take z̃ = z̃(z, ζ) instead of z̃ = z̃(z, ζ′). For this reason, the “transformed” disc Ã is not necessarily holomorphic unless A is endowed with a full, not only partial, lift.

Notes. Lifts of discs are discussed by, among others, Tumanov [89], [90], Trépreau [87] and Baouendi, Rothschild and Trépreau [12] (cf. also the book [8] for a survey). A connection related to the lifts of CR curves is introduced by Trépreau in [87] and Tumanov [90] where the relation to the connection by discs is explained. Partial lifts are introduced by Baracco and Zampieri in [16] and [17], which also present the related connections. Theorem 3.9.8 for the hypersurface case can be found in Baouendi, Rothschild and Trépreau [12] and in Zampieri [103] in general codimension.

3.10. Defect of analytic discs—deformation of non-defective discs—wedge extension from minimal manifolds

Summary. We present here the general theory by [89] of wedge extendibility of CR functions from a submanifold M ⊂ ℂ^{n}. This is founded on the notion of “minimality”, that is, the absence of proper submanifolds S ⊂ M with the same CR structure. Note that if M has finite type, then it is minimal: otherwise, the iterated commutators of (1, 0) and (0, 1) tangential vector fields would be forced to belong to ℂ ⊗ T S. But the converse is true only in some particular cases such as if M ∈ C^{ω}: if the type is ∞, then the Nagano leaf is the proper submanifold S, which violates minimality. When M is minimal, there exist arbitrarily small discs of defect 0. These are endowed with infinitesimal deformations which span all normal directions to M. Combining all these directions by the edge of the wedge theorem, we get a common wedge to which CR functions are forced to extend. In contrast to the theory developed in Section 3.8, there is no explicit description for the wedge W.

Let M be a CR manifold of ℂ^{n}; we have already explained the trick for making M generic up to a CR diffeomorphism. So, we assume M generic from the beginning.

Definition 3.10.1. A disc A attached to M is said to be “defective” when there exists a holomorphic section A^{∗} which projects via π over A and which is attached to T_{M}^{∗}ℂ^{n}.

Small defective discs are quite exceptional. For example, if M is Levi non-degenerate, we saw in Section 3.5 that T_{M}^{∗}ℂ^{n} is totally real; on the other hand, the construction of small discs through Bishop’s equation reveals that there is a one-to-one correspondence induced by π : T^{∗}ℂ^{n} → T^{ℂ}T_{M}^{∗}ℂ^{n} between discs on T^{∗}ℂ^{n} attached to T_{M}^{∗}ℂ^{n} and discs on T^{ℂ}T_{M}^{∗}ℂ^{n}. But the latter is {0} so no small disc is defective unless it shrinks to a point. The section A^{∗} = (z(τ), ζ(τ)) of T^{∗}ℂ^{n}

with A^{∗} →^{∼}^{π} A is said to be a “lift” of A. Let M be defined by y′_{j} = h_{j}(x′, z′′, z̄′′) with h(0) = 0, ∂h(0) = 0, and also write r_{j} = y_{j} − h_{j}. Associated to a disc A attached to M, there exists a real l × l matrix G over ∂Δ, a solution of the linear Bishop equation

G = T_{1}(^{t}∂_{x}h(A)G),

such that ^{t}∂_{z′}h(A)G extends holomorphically to Δ (cf. Lemma 3.9.5). In particular, given ξ_{o} ∈ ℝ^{l} ≃ (T_{M}^{∗}ℂ^{n})_{z_{o}}, we can find a partial lift A^{∗′} = (z(τ), ζ′(τ)) with ζ′(τ) = ^{t}∂_{z′}r(A)^{t}Gξ′_{o}. We have seen that ζ′(τ) is unique under the condition ζ′(1) = ξ′_{o}. Thus, A is defective if and only if ζ′(τ) can be supplemented to a full holomorphic lift ζ(τ) = (ζ′(τ), ζ′′(τ)) which, by uniqueness, is forced to be in the form ζ(τ) = t∂_{z}r(A)^{t}Gξ′_{o}. The lifts form a linear space that we denote by V_{A}; by the uniqueness of the (partial) lift through a prescribed conormal point, the dimension of the space of lifts is constant along ∂A.

Definition 3.10.2. We call dim(V_{A}) the “defect” of A.

We call any smooth family of discs {A_{η}} passing through a fixed point A_{η}(1) = z_{o} such that A = A_{η}|_{η=0} the “deformation” of a disc. We denote by Ȧ the derivative with respect to η evaluated at η = 0. The existence of a lift for A is a constraint for its deformations.

Proposition 3.10.3. If A has a holomorphic lift A^{∗}, then ⟨A^{∗}, Ȧ⟩ ≡ 0 (all over A).

Proof. Since Ȧ(τ) ∈ T M_{A(τ)} and A^{∗}(τ) ∈ (T_{M}^{∗}ℂ^{n})_{A(τ)}, then we have

(3.10.1)  \begin{cases}Re ⟨A^{∗}, Ȧ⟩|_{∂Δ} ≡ 0, \\ ⟨A^{∗}, Ȧ⟩ extends holomorphically from ∂Δ to Δ. \end{cases}

Hence, ⟨A^{∗}, Ȧ⟩ is constant. But since Ȧ(1) = 0, this is in fact 0.

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We have just seen that Ȧ ⊂ V_{A}^{⊥} at any point τ ∈ Δ; in particular, if we denote by ∂_{t}Ȧ(1) the radial derivative at 1, then ∂_{t}Ȧ(1) ⊂ V_{A}^{⊥}(1). Now, denote by F_{A} the space of deformations of A at −1, that is, F_{A} = {Ȧ(−1)} in T M and denote by G_{A} the equivalence classes modulo T M of their radial derivatives at +1: G_{A} = {[∂_{t}Ȧ(1)]} in T_{M}ℂ^{n}. One can prove the converse of the inclusions which have followed from Proposition 3.10.3.

Proposition 3.10.4 ([89], [8], [12]). We have

F_{A} = V_{A}^{⊥}(−1) (in T M), G_{A} = V_{A}^{⊥}(1) (in T_{M}ℂ^{n}).

(We refer to [91, pp. 137–140] for a (short) proof.) With Proposition 3.10.4 in hand, we can state the main theorem of wedge extendibility due to Tumanov. We say that M is “minimal” at z_{o} if there does not exist any proper CR submanifold S ⊂ M through z_{o} with dim_{CR}S = dim_{CR}M. Note that if M has finite type, then it is minimal. Otherwise, all spaces L_{M}^{j} made up of iterated commutators, would be contained in ℂ ⊗ T S and therefore could not reach the full tangent space ℂ ⊗ T M. When M is real analytic, the converse is also true. In fact, if M is not finite type, then its Nagano leaf through z_{o} would be proper and would carry the same CR structure as M, thus violating its minimality.

Theorem 3.10.5 (Tumanov). Let M be minimal at z_{o}. Then, there exists an open wedge W of ℂ^{n} with edge M such that continuous CR functions on M extend holomorphically to W.

The converse is also true (cf. [11]). Theorem 3.10.5 and its converse generalize [86]. We remark here that in contrast to the theory developed in Section 3.8, the directional cone Γ ⊂ T_{M}ℂ^{n} of W is not explicitly described from the minimality.

Proof. (Cf. [93].) We start by a complementary result: the existence of a family of small transversal discs A_{j} with ∂_{t}A_{j} = v_{j} implies the CR extension to the directions v_{j} and hence, by the edge of the wedge theorem, to their convex hull. More precisely, let {A_{j}}_{j=1,...,m} be attached to M with A_{j}(1) = z_{o} and let ∂_{t}A_{j} = v_{j} ≠ 0 in T_{M}ℂ^{n}. Define

Γ = convex hull {v_{j}}_{j=1,...,m}.

We claim that continuous CR functions on M extend to a wedge W with T_{z_{o}}W = T_{z_{o}}M ⊕ Γ. As already remarked, it suffices to prove the result for a single disc A_{1} with radial direction v_{1}. Let M be defined by y′_{j} = h_{j}(x′, z′′, z̄′′) and let w(τ) be the projection of A_{1} on T_{z_{o}}^{ℂ}M. Then, for any z = (z′, z′′) close to z_{o} we can attach a small disc with A_{z}(1) = z and whose projection on T^{ℂ}M is w(τ). By moving z ∈ M, for these discs A_{z} we have that ∂_{t}A_{z}(1) is close to ∂_{t}A_{1}(1) and hence through them we extend f to a manifold M_{1} with additional direction ∂_{t}A_{1}(1).

We now show that there exists A small with defect 0 through z_{o}; hence Γ := convex hull G_{A} has non-empty interior by Proposition 3.10.4, and the conclusion follows from the above argument. In fact, let d = lim inf def(A) for A → {z_{o}} with A ∋ z_{o} and let A_{o} be a small disc with defect d. It is not restrictive to assume A_{o}(−1) = z_{o} (for otherwise we just have to reparametrize A_{o}(τ^{2})). Note that the discs close to A_{o} have the same defect d as A_{o}. It follows that rk F_{A} ≡ dim(M) − d for A close to A_{o}; thus, the evaluation mapping A ↦ A(−1) transforms a neighborhood of A_{o} into a manifold S ⊂ M of codimension d. On other hand, for any disc A close to A_{o} we have T_{A(−1)}^{ℂ}M ⊂ Range F′_{A} = T_{A(−1)}S. Thus, S has the same CR dimension as M and, M being minimal, this can happen only if S = M. Thus, d = 0, which completes the proof.

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Suggested research. Describe the wedge of holomorphic extendibility obtained by minimality. This is a hard problem which does not yet have a complete answer. A partial conclusion is provided by Section 3.8.

Notes. Minimality was introduced by Tumanov in [89] who explained its relation to the defect of discs and obtained the ultimate result on (unspecified) wedge extension (Theorem 3.10.5). The necessity of minimality was proved by Baouendi and Rothschild (reported in the book [8]). The one-side holomorphic extension from a hypersurface, that is, the version of Theorem 3.10.5 in codimension 1, was obtained by Trépreau in [86].

3.11. Propagation of CR extendibility

Summary. In Section 1.7, we saw that analytic discs which lie in a hypersurface M ⊂ ℂ^{n} are “propagators” of holomorphic extendibility from either side of M. We see now that the same is true in higher codimension, for CR functions instead of holomorphic functions from one side, and for discs that are no longer contained but just attached to M provided that they are tangent. Also, the connection that we prepared in Section 3.9 by means of partial lifts of analytic discs and which was represented by a matrix G(−1) describes how the directions of CR extension evolve along the boundary of the disc. The analytic argument for proving the result on propagation consists in coupling deformations of discs, that is, discs in T ℂ^{n}, with lifts, that is, discs in T^{∗}ℂ^{n}. The deformation takes the boundary of the disc out of M in the direction of the initial extension; in particular it can be chosen with null CR components. In this way, instead of the full lift, it only interacts with the partial lift which is holomorphic. At the end point, by Hopf’s lemma, the deformation points in a new normal direction. The evolution of this direction is ruled by the dual inverse connection to that of the lift (by effect of the coupling). The passage from propagation along boundaries of (tangent) discs to CR curves is obtained by approximation. We end this section by explaining the relation with the construction of [87] and [90] where the connection and its dual-inverse are given over a subbundle of T_{M}^{∗}ℂ^{n} and a projection of T_{M}ℂ^{n}, respectively. In this case the “tangency” of the discs, that is, the orthogonality to the subbundle, is automatic.

Let γ be a CR curve in M with γ(0) = z_{o} and γ(1) = z^{1}. We approximate γ by a chain of discs {A^{ν}} attached to M. We go backwards. We start from A^{1} with A^{1}(1) = z^{1} and whose projection on T_{z^{1}}^{ℂ}M is z^{1}′′ + γ̇(1)(1 − τ); we notice that the distance of A^{1}(−1) to some point z^{2} = γ(t_{2}) in γ is o(diam(A^{1})). Next, we take A^{2} with A^{2}(1) = A^{1}(−1) and whose projection in T_{z^{2}}^{ℂ}M is z^{2}′′ + γ̇(t_{2})(1 − τ). In this way we find a chain ending at A^{m}(−1) with |A^{m}(−1) − z_{o}| < ϵ. Associated to any partial lift of γ ⊂ M to γ^{∗′} ⊂ (T_{M}^{∗}ℂ^{n})′ passing through (z^{1}, ξ^{1′}) at t = 1, there is a unique chain of lifts of the A^{ν}’s, {A^{∗′ν}}_{ν=1,...,m}, attached to (T_{M}^{∗}ℂ^{n})′. Moreover, we have

|A^{∗′m}(−1) − γ^{∗′}(0)| < cϵ uniformly over ξ^{1} ∈ (T_{M}^{∗}ℂ^{n})_{z^{1}}.

In fact, according to Section 3.9, for the vectors ũ ∈ T_{(z^{1},ξ^{1′})}^{ℂ}(T_{M}^{∗}ℂ^{n})′ such that π′ũ = u, we have |ũ| ≤ c|u| for c independent of ξ^{1}. In this way, we approximate the connection obtained by means of the partial lifts γ^{∗′} with the composition of those associated to the lifts {A^{∗′ν}}_{ν=1,...,m}. In other words,

B is approximated by G^{1}(−1) · ... · G^{m}(−1).

Let E^{∗} be a subbundle of T_{M}^{∗}ℂ^{n} and denote by (E^{∗})′ its projection similarly as to (3.9.1). Note that (E^{∗})′ is a CR bundle which has the same CR structure as (T_{M}^{∗}ℂ^{n})′, identified through π′ with the CR structure of M. In particular, (E^{∗})′ and (T_{M}^{∗}ℂ^{n})′ have the same families of CR curves and of attached analytic discs once they share a common point.

By this, it is clear that ^{t}Φ^{−1} induces a connection over E^{∗}. Define

Σ := (E^{∗})^{⊥},

where ^{⊥} denotes the orthogonal. Here Σ is a bundle which contains T M and satisfies

E = \frac{T ℂ^{n}}{Σ} |_{M} .

Theorem 3.11.1. Let γ be a CR curve in M. We assume that the chain of discs whose boundaries {∂A^{ν}}_{ν} approximate γ satisfies

(3.11.1)  T A^{ν}(1) ⊂ Σ ∩ iΣ.

Then, if f is a CR function on M which extends at z_{o} to the direction v_{o} ∈ T_{M}ℂ^{n} with v_{o} ∉ Σ, it follows that f extends at z_{1} to the direction v_{1} ∈ T_{M}ℂ^{n}, such that

pr_{Σ}(v_{1}) is close to pr_{Σ}(Φv_{o}),

where pr_{Σ} is the projection (\frac{T ℂ^{n}}{T M} ) →^{pr_{Σ}} (\frac{T ℂ^{n}}{Σ} ).

(Different chains of discs are needed according to the accuracy in the approximation of the direction Φv_{o} that we desire.)

Prior to the proof, we make some comments and give an example. The above theorem applies in particular when there is a CR submanifold N properly contained in M with T N + iT N ⊂ T M|_{N}. In fact, if we approximate a curve γ^{∗′} by a family of discs {A^{∗ ′ν}}, attached to (T_{M}^{∗}ℂ^{n})′ whose CR components are prescribed in (π′)^{−1}(T^{ℂ}N) (where π′ : T^{ℂ}(T_{M}^{∗}ℂ^{n})′ →^{∼} T^{ℂ}M), then in particular we have T A^{ν}|_{∂A^{ν}} ⊂ T N + iT N|_{∂A^{ν}} ⊂ T M|_{∂A^{ν}}, which makes it possible to apply Theorem 3.11.1 for E^{∗} = T_{M}^{∗}ℂ^{n}. The above remark applies in particular when N is a complex curve and γ ⊂ N; this generalizes to highercodimensional submanifolds the Hanges-Treves propagation theorem of [47] for hypersurfaces. But it holds, in fact, in greater generality:

Example 3.11.2. Let N and M be defined in ℂ^{3} by

N = {z : z_{1} = 0, y_{3} = y_{2}^{2}}, M = {z : y_{1} = (y_{3} − y_{2}^{2})^{2}}.

We have T N + iT N ⊂ T M|_{N} = ℝ × ℂ^{2} and thus Theorem 3.11.1 applies.

Proof. Propagation of CR extendibility is reduced to propagation along the boundaries {∂A^{ν}} of the projections of the discs which provide the approximation of γ^{∗′}. For these discs, which we denote by A, we are assuming condition (3.11.1) of complex tangency at b := A(1); we also set a := A(−1). Let M be defined by y′_{j} = h_{j}(x′, z′′, z̄′′) normalized at a = 0, and let its extension M_{1} be defined, e.g., by introducing a new parameter t ∈ ℝ^{+} ∪ {0} and extending the domain of h from ℝ^{l} ×ℂ^{n−l} ×ℂ̄^{n−l} to ℝ^{l} ×ℂ^{n−l} ×ℂ̄^{n−l} ×(ℝ^{+} ∪{0}) with ∂_{t}h(z_{o}) = v_{o}. We write A(τ) = (u(τ) + iv(τ), w(τ)) and call w(τ) the “w components” of A. We deform A to A_{η} by prescribing “t-components” t_{η}(τ) = ηχ(τ) where η is a small parameter and χ is C_{c}^{∞}, χ ≥ 0 and supp(χ) is contained in a neighborhood of 1. We assume w(1) = 0, take w_{o} ∈ ℂ^{n−l} and s ∈ ℝ^{l} small, and find a unique solution u = u_{η w_{o} s}(τ) in ℂ^{k,α}(∂Δ) of Bishop’s equation

(3.11.2)  u = −T_{1}(h(u, w + w_{o}, t_{η}) + s).

Moreover, if we define v = v_{η w_{o} s} by v = T_{1}(u) + h(s, w_{o}) and set A_{η w_{o} s}(τ) = (u(τ) + iv(τ), w(τ) + w_{o}), we have that A_{η w_{o} s} is C^{k,α′} and ∂_{τ}A_{η w_{o} s} is C^{k−1,α′} with respect to (η, w_{o}, s) for any α′ < α (cf. Section 3.8). We denote by Ȧ the derivative with respect to η at η = 0 and begin by fixing w_{o} = 0, s = 0 and η = 0. Let G be the real l × l real matrix on ∂Δ such that G∂_{z′}r(A) extends holomorphically. We note that G∂_{z}r(A) · Ȧ also extends holomorphically since the w-components of A are constant in η and hence those of Ȧ are 0. We have

(3.11.3)  GRe ∂_{z}r(A) · Ȧ = G∂_{t}h(u, w + w_{o})χ.

Recall that ∂_{t}h|_{a} = v_{o} and that χ has support in a neighborhood of a. We evaluate the derivative ∂_{τ} of the harmonic vector function (3.11.3) at τ = 1, recall that G(1) = id and conclude by Hopf’s lemma that

Re ∂_{z}r(z_{1}) · ∂_{τ} Ȧ|_{τ=1} ∼ G(−1)v_{o},

where ∼ means equality up to a small error. Thus, ∂_{τ} Ȧ ∼ G(−1)v_{o} in the basis of (T_{M}ℂ^{n})_{b} dual to ∂_{z}r_{j}, j = 1, ..., l. Let pr_{M} and pr_{Σ} be the projections modulo T M and Σ, respectively. On one hand, pr_{M}∂_{τ} Ȧ|_{1} ∼ G(−1)v_{o}; on the other hand, pr_{Σ}∂_{τ}A|_{1} = 0 (because ∂_{τ}A|_{1} ∈ Σ ∩ iΣ). Hence, for the Taylor expansion of ∂_{τ}A_{η} in η, (3.11.3) yields

pr_{Σ}∂_{τ}A_{η}|_{1} ∼ (G(−1)v_{o} + ϵ)η + o(η),

where ϵ is infinitesimal with the diameter of supp χ. So far, w_{o} and s were frozen to 0. We now take a 2n−l−1 plane Σ in T M_{z_{o}} = ℝ^{l} ×ℂ^{n−l} transversal to ∂_{θ}A_{η}|_{1}, solve (3.11.2) for all values of the parameters (w_{o}, s) ∈ Σ and denote by A_{η w_{o} s} the corresponding solutions. In particular A_{η w_{o} s}|_{w_{o}=0,s=0} coincides with A = A_{η} considered above. We set

M_{2} := ⋃_{w_{o} s}A_{η w_{o} s} and parametrize M_{2} by D : Σ×Δ → M_{2}, (s, w_{o}, τ) ↦

A_{η w_{o} s}(τ). We can see easily that

rk ∂_{s w_{o} τ}D|_{s=0, w_{o}=0, τ=1} = 2n − l + 1.

Thus, M_{2} is a manifold at z^{1} which contains M and “points” to an extra direction v_{1} such that pr_{Σ}(v_{1}) is close to pr_{Σ}(G(−1)v_{o}). From now on, we can carry out the proof in the same way as in Sections 3.7, 3.8 and 3.10. We approximate a CR function f by polynomials in a neighborhood of M ∪ M_{1} (localized where approximation takes place). The approximating sequence is Cauchy over ⋃ ∂A_{η w_{o} s}, hence over ⋃ A_{η w_{o} s}, by the maximum principle. Thus, it converges over M_{2}, the union of the discs, to a CR function, the extension of f.

□

We want to show now how the theory of the propagation of CR extendibility by [87] and [90] can be reduced to Theorem 3.11.1. Let S be the local CR orbit of M at z_{o}, that is, the union of all piecewise smooth small CR curves issued from z_{o}; note that we have T^{ℂ}M|_{S} = T^{ℂ}S. In what follows, we assume that M is non-minimal, that is, it properly contains S. Set Σ|_{S} := T M|_{S} + iT S and E|_{S} := \frac{T ℂ^{n}|_{S}}{T M|_{S}+iT S};

remark that we have an identification \frac{T ℂ^{n}|_{S}}{T M|_{S}+iT S} ≃ \frac{T M|_{S}}{T S}. Extend Σ|_{S} and E|_{S} from S to M as smooth bundles. We are then in the situation described before Theorem 3.9.1: T^{ℂ}(E^{∗})′ may be identified with T^{ℂ}M via π′. But in this quite exceptional case the identification holds in fact for the whole of E^{∗}|_{S} and not only its projection (E^{∗})′|_{S} as was already remarked in [87] and [90]:

Proposition 3.11.3. Let p = (z; ζ) ∈ (E^{∗})|_{S}; then (E^{∗})|_{S} is a CR manifold such that

T_{p}^{ℂ}E^{∗} →^{∼}_{π′} T_{z}^{ℂ}M.

Proof. We have to modify the proof of Theorem 3.9.1. We have a situation analogous to (3.9.4) but with full vectors in ℂ^{n} = T_{z}ℂ^{n} and not only their components in ℂ^{l}. We are thus reduced to prove that (3.11.4)

for any u ∈ T^{ℂ}M there is a unique λ ∈ ℂ^{l} s.t. ¯∂_{u}∂r_{o} = λ∂r.

Since p ∈ E^{∗} = T_{M}^{∗}ℂ^{n} ∩ iT_{S}^{∗}ℂ^{n}, then from the identity

∂r_{o} = i∂s_{o} for suitable s_{o},

we get, by differentiation along S,

\begin{aligned}¯∂∂r_{o}|_{T^{ℂ}S} & = i¯∂∂s_{o}|_{T^{ℂ}S} \\ & = 0, \end{aligned} ; (3.11.5)

where the last equality is a consequence of the fact that ¯∂∂r_{o}|_{T^{ℂ}S} is Hermitian. Thus, for any u ∈ T^{ℂ}S, we conclude that ¯∂_{u}∂r_{o} is a combination of differentials ∂r_{j}, or, in other words, (3.11.4) is satisfied.

□

In the present situation, we can “lift” CR curves and analytic discs from S to the whole of E^{∗}|_{S} and not only (E^{∗})′|_{S}. Also, if we denote lifts of analytic discs attached to E^{∗} by (A(τ); ζ(τ)) for τ ∈ Δ, we have Re ⟨iτ∂_{τ}A, ζ(τ)⟩|_{∂Δ} ≡ 0. We denote by Re F the function on the left side of the above equality. Since F extends holomorphically from ∂Δ to the whole of Δ, then the equality implies that F is constant and, since its value at τ = 0 is 0, then in fact F ≡ 0 on the whole of Δ̄. From this, we deduce that any disc A attached to M is orthogonal to E^{∗} along its boundary and therefore T A ⊂ T M + iT S. Condition (3.11.1) is thus fulfilled by any disc which approximates a CR curve. Theorem 3.11.1 can therefore be applied and it yields

Corollary 3.11.4 (Cf. [87] and [90]). Let M be non-minimal at z_{o}, let S be the local CR orbit, and set E = \frac{T ℂ^{n}|_{S}}{T M|_{S}+iT S}. For any γ with z_{1} = γ(1) and z_{o} = γ(0) we get a partial connection

Φ : E_{z_{o}} → E_{z_{1}},

in such a way that if a CR function f extends at z_{o} to the direction v_{o}, then it also extends at z_{1} to v_{1} such that

pr_{T M+iT S}(v_{1}) is close to pr_{T M+iT S}(Φv_{o}).

Notes. Propagation of holomorphic extendibility goes back to the celebrated Hanges-Treves theorem [47], which was obtained by means of the microlocal theory of PDE’s. A more geometric approach is contained in Section 1.7 of the present book. Propagation in general codimension was described by Trépreau [87] and Tumanov [90] and was later refined by Merker [68]. In particular, Trépreau was the first to relate, at the points of a subbundle E^{∗} ⊂ T_{M}^{∗}ℂ^{n}, the CR structure of T_{M}^{∗}ℂ^{n} to that of M and to show that the wave front set of a CR function is a union of CR orbits in E^{∗}. Tumanov described propagation of CR extendibility by using explicitly the pairing between lifts and infinitesimal deformations of discs. Baracco and Zampieri introduced in [16] and [17] partial lifts and pointed out that they suffice for propagation. It is worth noticing that lifts to E^{∗} are not partial but full. In general, existence of full lifts imposes the constraint of the vanishing of the Levi form of M. However, according to [87] and [90], this occurs when M is not minimal. If S ⊂ M with T^{ℂ}S = T^{ℂ}M, then L_{M}(p) = 0 for any p ∈ E^{∗} := T_{M}^{∗}ℂ^{n} ∩ J T_{S}^{∗}ℂ^{n}.

3.12. Separate real analyticity

Summary. We end our review by discussing separate real analyticity in the framework of CR geometry. This is the real counterpart of the problem encountered in Section 1.5. We first present the problem in ℂ^{2}; the general case follows next. In ℂ^{2} = ℝ^{2} + iℝ^{2} with coordinates z = (z_{1}, z_{2}), z = x + iy, we consider a separately real analytic function f: this means that f is real analytic in x_{1} (resp. x_{2}) when x_{2} (resp. x_{1}) is kept fixed. We make, indeed, the additional assumption that f is “CR extendible” in one of the two arguments, say x_{2}, which means that for x_{1} fixed, f extends holomorphically in z_{2} for y_{2} < ϵ with ϵ locally uniform in (x_{1}, x_{2}). We show that f is real analytic. The result is classical for a function f which satisfies the estimate |\frac{∂_{x_{h}}^{ν}f}{ν!} | ≤ ϵ_{o}^{−ν} for h = 1, 2 and ν ∈ ℕ. In this case f is CR extendible in both arguments and, moreover, all of its derivatives ∂_{x_{h}}^{ν}f are uniformly bounded; in particular, (∂_{x_{1}}^{2ν} + ∂_{x_{2}}^{2ν})f are bounded for any ν. But then, the elliptic regularity of the operators ∂_{x_{1}}^{2ν} + ∂_{x_{2}}^{2ν} yields f ∈ C^{∞}. In contrast, we only assume that f is C^{ω} in x_{1} and CR extendible in x_{2}; thus our

assumptions are |\frac{∂_{x_{1}}^{ν}f}{ν!} |^{\frac{1}{ν}} ≤ c_{x_{2}}ϵ_{x_{2}}^{−ν} and |\frac{∂_{x_{2}}^{ν}f}{ν!} |^{\frac{1}{ν}} ≤ c_{x_{1}}ϵ_{o}^{−ν}.

We stress paying attention to the fact that CR extension in at least one argument is needed as the example of f(x_{1}, x_{2}) = x_{1}x_{2}exp(−\frac{1}{x_{1}^{2}+x_{2}^{2}} ) shows. This function is C^{∞}, separately C^{ω}, not C^{ω}: the point is that it is not CR extendible in either argument. Also notice that no assumption is made on uniform continuity or uniform boundedness for the different extensions: these come as consequences of the statement. In any case, uniform boundedness for the different extensions is the main issue: once this is proved, continuity follows by an argument of normal family similar to that of Lemma 1.5.1. Holomorphic extension is then a consequence of the edge of the wedge theorem of Section 3.7.

As for the proof, the main technical tool is a refined version of the Hartogs lemma which is contained in Theorem 3.12.1: it is a combination of Fatou’s lemma and the Phragmén-Lindelöf principle. For the rest, we exploit the power of CR theory. There is an open set in ℝ^{2} from which the separate holomorphic extensions to y_{1} ≠ 0 and y_{2} ≠ 0 are bounded and hence continuous. By the edge of the wedge theorem there is a holomorphic extension to ℂ^{2}. Next, by the theorem of Hanges and Treves we propagate this extension along the planes y_{2} = const. At this point we apply our generalized Hartogs lemma to the sequence of the ν-th roots of the Taylor coefficients of the expansion of f in z_{2} to fill all missing values of x_{2} in the domain of f.

We begin by some preliminary result on functions of one complex variable. Let ℂ be the complex plane with coordinate τ, Δ = {τ ∈ ℂ : |τ| < 1} the unit disc, Δ^{+} = {τ ∈ Δ : Im τ > 0} the upper half-disc, I = {t ∈ ℝ : |t| < 1} the unit real interval. We recall that a subharmonic function is a function which is upper semicontinuous and whose value at any point τ is dominated by the mean value on the boundary of any disc centered at τ. The following generalization of the Hartogs lemma plays a crucial role in what follows.

Theorem 3.12.1. Assume given a sequence {φ_{ν}}_{ν} of functions upper semicontinuous and uniformly bounded on Δ̄^{+}, subharmonic on Δ^{+} and which satisfy for some positive constant l

(3.12.1)  \begin{cases}lim sup_{ν} φ_{ν}(τ) ≤ l for any τ ∈ ∂Δ^{+}, \\ lim sup_{ν} φ_{ν}(t) ≤ 0 for any t ∈ I. \end{cases}

Then, for any η, there is ν_{η} such that for all ν ≥ ν_{η}

(3.12.2)  φ_{ν}(τ) ≤ lκ Im τ for Im τ ≥ η, |τ| ≤ 1 − β,

where κ is a uniform constant and β is fixed.

Proof. We denote by χ the function on ∂Δ^{+} which is 0 for Im τ = 0 and l for |τ| = 1. Let P_{z}(ζ), z ∈ Δ^{+}, ζ ∈ ∂Δ^{+}, be the Poisson kernel of Δ^{+}. For any τ ∈ Δ^{+} we have

lim sup_{ν} φ_{ν}(τ) ≤ ∫_{∂Δ^{+}} P_{τ}(ζ)lim sup_{ν} φ_{ν}(ζ)\frac{dζ}{iζ}

≤ ∫_{∂Δ^{+}} P_{τ}(ζ)χ(ζ)\frac{dζ}{iζ} ,

(3.12.3)

where the first inequality comes from the subharmonicity of each φ_{ν} and the second from Fatou’s lemma. Let h(τ) = ∫_{∂Δ^{+}} P_{τ}(ζ)χ(ζ)\frac{dζ}{iζ}; h is harmonic, h = \frac{2l}{π} arg \frac{1+τ}{1−τ}, and in particular h ≤ κlIm τ for |Re τ| > 1−β. Since lim sup_{ν} φ_{ν} is estimated by h over ∂Δ^{+}, then it is also estimated

over Δ^{+}. Thus

(3.12.4)  lim sup_{ν} φ_{ν}(τ) ≤ κlIm τ.

To make the estimate (3.12.4) uniform in τ, we need to make another use of Fatou’s lemma: in the present case inside double integrals. For τ_{o} ∈ Δ^{+}, we decompose its distance to ∂Δ^{+} as r + δ with δ << r. By the submean property, we have for any τ in Δ_{δ}(τ_{o}), the disc of center τ_{o} and radius δ,

\begin{aligned}φ_{ν}(τ) & ≤ (πr^{2})^{−1} ∫_{0}^{2π} ∫_{0}^{r} tφ_{ν}(τ + te^{iθ})dt dθ \\ & ≤ (π(r + δ)^{2})^{−1} ∫_{0}^{2π} ∫_{0}^{r+δ} tφ_{ν}(τ_{o} + te^{iθ})dt dθ + O(δ), \end{aligned} ; (3.12.5)

where the error can be controlled by O(δ) because of the uniform boundedness of the φ_{ν}’s. On the other hand, by (3.12.4), lim sup_{ν} φ_{ν}(τ_{o} + te^{iθ}) ≤ κl(Im τ_{o} + r + δ) ≤ 2κlIm τ_{o} (because

Im τ_{o} > r + δ, the distance to ∂Δ^{+}). It follows by (3.12.5) that

(3.12.6)  lim sup_{ν} sup_{τ∈B(τ_{o},δ)}φ_{ν}(τ) ≤ 2κlIm τ_{o} + O(δ).

We now cover K_{η}^{+} = {τ ∈ Δ^{+} : d(τ, ∂Δ^{+}) > η, |τ| ≤ 1 − β} by δ-discs centered at various points τ_{o}. For these discs to intersect the set K_{η}^{+}, we must have Im τ_{o} > η − δ. In the δ-discs centered at these points we get therefore inf_{τ∈B(τ_{o},δ)}Im τ ≥ (1 − \frac{δ}{η−δ})Im τ_{o}. Thus, we can conclude that

for some ν_{η} and for a new κ, we have (3.12.7)

φ_{ν}(τ) ≤ κlIm τ uniformly for τ ∈ K_{η}^{+} and for any ν ≥ ν_{η}.

□

Remark 3.12.2. We get the same conclusion as (3.12.2) if we replace the half-disc Δ^{+} by the strip U_{δ}^{+} = I + i(0, δ); in this case Im τ needs an extra factor δ^{−1} because of rescaling.

We go on to the discussion in general dimension. Let z = x + iy be the coordinates in ℂ^{n}, Ω an open domain of ℝ^{n} and f a function on Ω.

Definition 3.12.3. We adopt the following terminology.

• We say that f is separately real analytic in x_{j} if its restriction to the section of Ω with each line parallel to the x_{j}-axis is real analytic. Thus, when all the other coordinates are fixed, f extends to |y_{j}| < ϵ_{x}.

• We say that f is separately CR extendible to y_{j} if it is separately real analytic in x_{j} and, moreover, it has holomorphic extension to |y_{j}| < ϵ with ϵ having positive lower bound locally uniform in x.

Let ℂ^{n} = ℂ^{n_{1}} × ℂ^{n_{2}} with coordinates z = (z′, z′′) = (x′ + iy′, x′′ + iy′′). Separate analyticity in the group of variables x′′ means analyticity in x′′ when the other variables are fixed. CR extendibility to all the directions of the y′′-plane ℝ^{n_{2}} means holomorphic extendibility to |y′′| < ϵ with ϵ locally uniform in x. CR extendibility to a cone Γ_{2} ⊂ ℝ^{n_{2}} means that f extends holomorphically for y′′ ∈ Γ_{2}, |y′′| < ϵ with ϵ locally uniform in x; we also assume that, for fixed x′, f is locally uniformly continuous up to y′′ = 0. Clearly real analyticity or CR extendibility in a group of variables x′′ is more restrictive than the combination of real analyticity or CR extendibility in any single x′′_{j}. We now state the main result of the section; the proof will follow next.

Theorem 3.12.4. Let f be a function on Ω which is separately real analytic in x′ and CR extendible to y′′. Then f is real analytic.

By iteration, Theorem 3.12.4 implies that if f is separately real analytic in x_{1} and CR extendible to each single variable y_{2}, ...,y_{n}, then it is in fact real analytic.

Remark 3.12.5. Let Ω = Ω_{1} ×Ω_{2} ⊂ ℝ^{n} = ℝ^{n_{1}} ×ℝ^{n_{2}}. We denote by Γ an open convex cone in the y-space ℝ^{n} and by Γ_{ϵ} its truncation by |y| < ϵ; we denote by Γ_{2} or Γ_{2 ϵ} the analogous open cones in the y′′-space ℝ^{n_{2}}. The argument of our following proof can be adapted (cf. [18]) to prove that if f is a continuous function on Ω which is real analytic in x′ and CR extendible to Γ_{2}, uniformly continuous, for fixed x′, up to y′′ = 0, then it extends holomorphically to a wedge W = Ω+iΓ_{ϵ} where Γ is a conic neighborhood of an arbitrarily large proper subcone of Γ_{2} and is uniformly continuous up to y = 0.

Proof of Theorem 3.12.4. We can assume, without loss of generality, that n_{1} = 1. Also, by using iteration, we may assume that n_{2} = 1. We use the notation I = (−1, 1), I^{2} = I × I, I_{ϵ} = (−ϵ, ϵ), Δ = {τ : |τ| < 1}, Δ_{ϵ} = {τ : |τ| < ϵ}, U_{ϵ} = {τ ∈ ℂ : |Re τ| < 1, |Im τ| < ϵ}, U_{ϵ}^{+} = {τ ∈ U_{ϵ} : Im τ > 0}. The statement being local, we can suppose that f is defined and continuous in I^{2}, extends to Δ_{ϵ_{x_{2}}} × {x_{2}} and {x_{1}} × Δ, and is holomorphic with respect to τ ∈ Δ and τ ∈ Δ_{ϵ_{x_{1}}}, respectively. We may assume that f extends to discs of radius slightly bigger than 1 or ϵ_{x_{2}}.

(a) We first prove that there are an open interval I_{δ} ⊂ I and an open strip U_{δ} = I + iI_{δ} such that f is continuous in I_{δ} × Δ and in U_{δ} × I_{δ}. Hence it is a continuous CR function therein. We start from the proof of the continuity on I_{δ} × Δ. Let

K_{l} = {x_{1} ∈ I : sup_{z_{2}∈Δ}|f(x_{1}, z_{2})| ≤ l}.

We note that K_{l} ⊂ K_{l+1} and that ⋃_{l} K_{l} = I since, for each x_{1}, sup_{z_{2}∈Δ}|f(x_{1}, z_{2})| < +∞. We claim that

(3.12.8)  \begin{cases}K_{l} is closed, \\ f is continuous on K_{l} × Δ. \end{cases}

In fact, let x_{1}^{ν} → x_{1}^{o} with x_{1}^{ν} ∈ K_{l} and use the notation F_{ν}(z_{2}) := f(x_{1}^{ν}, z_{2}) − f(x_{1}^{o}, z_{2}). The sequence {F_{ν}}_{ν} is equicontinuous in a neighborhood of Δ. In fact, remember that f was supposed to be holomorphic for |z_{2}| slightly bigger than 1; so the conclusion follows from the uniform bound |F_{ν}| ≤ 2l in addition to Cauchy’s inequalities. We claim that F_{ν} → 0. Otherwise, by the equicontinuity and on account of Arzela’s theorem, there is a subsequence {F_{ν_{k}}}_{k} which converges to a limit F ≠ 0. But this limit is holomorphic in Δ and 0 in I, a contradiction. This proves the claim and thus (3.12.8) follows. The union ⋃_{l} K_{l} being the whole of I, it follows from Baire’s theorem that the sets K_{l} must contain an open interval for large l (cf. also [50] and [80] for a similar use of Baire’s theorem). Also, such an interval can be found centered at points x_{1}^{o} arbitrarily close to 0. We can even assume that x_{1}^{o} = 0 (by the propagation in z_{1}-direction contained in the point (b) which follows). Thus, we can assume that f extends to I_{δ} ×Δ as a continuous function, holomorphic in z_{2}, that is, a continuous CR function.

We now go on to prove that f is a continuous CR function on U_{δ}×I_{δ}. For this purpose, we define J_{l} = \{x_{2} : f(·, x_{2}) extends to |y_{1}| < \frac{1}{l} and |f(·, x_{2})| ≤ l}. If x_{2}^{ν} → x_{2}^{o} with x_{2}^{ν} ∈ J_{l}, then by boundedness, there is a subsequence which converges to a holomorphic function on U_{\frac{1}{l}}; this must be f(·, x_{2}^{o}). As before, we have |f(·, x_{2}^{o})| ≤ l and hence J_{l} is closed and f|_{U_{\frac{1}{l}} ×J_{l}} is continuous. By Baire’s theorem we conclude again that for large l, the set J_{l} contains an open interval that we can suppose to be centered at x_{2}^{o} = 0 (because otherwise we just need to center the following series (3.12.9) at x_{2}^{o} instead of 0). Thus, f is continuous on U_{δ} × I_{δ} for small δ. This concludes the proof of the claim.

(b) At this point we apply the Ajrapetyan-Henkin edge of the wedge Theorem 3.7.3 and conclude that f extends holomorphically to a domain of type Δ_{δ} × Δ_{δ}. We notice now that

• f is continuous and CR on U_{δ} × I_{δ},

• U_{δ} × I_{δ} is foliated by the complex leaves Σ_{x_{2}} := U_{δ} × {x_{2}} for

x_{2} ∈ I_{δ},
• f extends to Δ_{δ} × Δ_{δ},
• each leaf Σ_{x_{2}} intersects Δ_{δ} × I_{δ}.

But then, the propagation of the holomorphic extendibility of CR functions along complex leaves yields the extension of f to an open domain U_{δ} × Δ_{δ} of ℂ^{2}, for small δ. We now make a short digression which is crucial for adapting the proof of the present theorem to the statement of Remark 3.12.5 about the boundary values from wedges. Notice that Δ_{δ} × Δ_{δ} is swept by discs with boundary in the region where f is bounded and U_{δ} × Δ_{δ} is swept by discs with boundary in the union of U_{δ} × I_{δ} and Δ_{δ} × Δ_{δ} where f is also bounded. Hence, by the maximum modulus principle, f is bounded in U_{δ} × Δ_{δ}. Since by such an intervals I_{δ}, where f has (different) bounds, we cover an open dense set D ⊂ I, we conclude that f is continuous up to y = 0 over I × D. By this, in the case of wedge extendibility for y belonging to cones, and for f ∈ C^{0}(I × I), we can see that the “extension” is uniformly continuous up to y = 0 with limit f (cf. [18]). We come back to the main subject.

(c) We consider now the Taylor series of f with respect to z_{2} centered at z_{2} = 0, i.e.,

(3.12.9)  ∑_{ν} \frac{∂_{z_{2}}^{ν}f(z_{1}, 0)}{ν!}z_{2}^{ν}.

This represents a holomorphic function on U_{δ} × Δ_{δ}; when x_{1} is fixed in I, this extends holomorphically for z_{2} ∈ Δ. We write φ_{ν}(z_{1}) := \frac{1}{ν}log \frac{|∂_{z_{2}}^{ν}f(z_{1},0)|}{ν!} and note that these are subharmonic functions of z_{1}. We have

(3.12.10)  lim sup_{ν} sup_{z_{1}∈U_{δ}}φ_{ν}(z_{1}) ≤ −log δ,

and hence, in particular, the φ_{ν}’s are uniformly bounded in U_{δ}. We also have

(3.12.11)  lim sup_{ν} φ_{ν}(x_{1}) ≤ 0 for x_{1} ∈ I.

(In (3.12.10) and (3.12.11) the domains Δ and I should be shrunk in an arbitrarily small fashion.) But then Theorem 3.12.1 applies to the sequence of the φ_{ν}’s over the pair of half-strips x_{1} +U_{δ}^{±}. It implies that for any η > 0, for suitable ν_{η} and for any ν ≥ ν_{η}, we have

φ_{ν}(z_{1}) ≤ δ^{−1}κlog δ|y_{1}| when x_{1} ∈ I and y_{1} ≥ η.

Here the factor δ^{−1} comes from covering U_{δ} by δ-discs. In other words, we have

\frac{|∂_{z_{2}}^{ν}f(z_{1}, 0)|^{\frac{1}{ν}}}{ν!} ≤ δ^{−κ|y_{1}|δ^{−1}}.

Let ϵ = ϵ_{α δ} satisfy δ^{−κϵδ^{−1}} < (1 − α)^{−1}; then, the power series (3.12.9) converges normally for (z_{1}, z_{2}) ∈ U̇_{ϵ} × Δ_{1−α}, where U̇_{ϵ} stands for I + i(I_{ϵ} \ {0}), and its sum is therefore a holomorphic function on U̇_{ϵ} × Δ_{1−α}. Since we already know that this function extends to y_{1} = 0 when z_{2} ∈ Δ_{δ}, then it is in fact holomorphic on U_{ϵ} × Δ_{1−α}. Note that if the initial strip of analyticity is not centered at z_{2} = 0 but, say, at z_{2} = z_{2}^{o}, then we have to replace 0 by z_{2}^{o} in the expansion (3.12.9). The proof goes through without change and yields the analyticity of f in Δ × Δ_{1−|z_{2}^{o}|−α}. The conclusion follows since |z_{2}^{o}| and α are arbitrarily small. The proof is complete.

□

Suggested research. Let f = f(x_{1}, x_{2}) be a continuous function in I × I which is separately C^{ω} in x_{1} and extends separately CR in y_{2} but only for positive values y_{2} > 0. According to Remark 3.12.5 a refinement of the argument of the proof of Theorem 3.12.4 shows that f extends to a wedge defined by y_{2} > c|y_{1}| for suitable c. Does the wedge of maximal extension have the half-space y_{2} > 0 as asymptotic at 0? This is likely to be true.

However, the general challenging problem is to get the extension to a “quadrant”. Thus, let f extend to 0 ≤ y_{1} < ϵ for fixed x_{2} and to 0 ≤ y_{2} < ϵ for fixed x_{1}. Does f extend to a wedge W such that T W |_{I×I} coincides with the quadrant y_{1} ≥ 0, y_{2} ≥ 0? Of course, one can then also try the cases where one of the two lengths of extension ϵ is non-uniform.

Recall once more that if the extensions are continuous or bounded, which turns out, by normal family argument, to be the same, the answer is affirmative because of the edge of the wedge theorem

Notes. The problem of separate analyticity goes back, in the complex frame, to Hartogs [48]. As for the real setting, holomorphic extension to the quadrant y_{1} > 0, y_{2} > 0 for separately CR extendible functions, with uniformly continuous extension, was first obtained by Malgrange and Zerner. A more refined version of this result was next established by Kashiwara under the name of “local Bochner tube theorem” (cf. the presentation by Komatsu in [62]). The real analyticity of functions which are separately CR extendible in all arguments and which have, moreover, all derivatives ∂_{x_{h}}^{ν}f, h = 1, 2, ν ∈ ℕ bounded, is due to Browder [30] and Lelong [65]. The same result, without the assumption of bounded derivatives, is due to Siciak [80]. For the proof of Theorem 3.12.4 we refer to Baracco and Zampieri [?] whose presentation we follow closely in this section.

Bibliography

[1] H. Ahn—Global boundary regularity of the ¯∂-equation on q-pseudoconvex domains, Math. Reich. 280 (2007), 343–350

[2] H. Ahn, L. Baracco and G. Zampieri—Subelliptic estimates and regularity of ¯∂ at the boundary of a Q-pseudoconvex domain of finite type, Math. Reich. (2009)

[3] R. Ajrapetyan and G. Henkin—Analytic continuation of CR functions through the “edge of the wedge”, Dokl. Acad. Nauk. SSSR 259 (1981), 777–781

[4] R. Ajrapetyan and G. Henkin—Integral representation of differential forms on Cauchy-Riemann manifolds and the theory of CR functions, II, Math. USSR-Sb 55 (1986), 91–111

[5] A. Andreotti, G. Fredricks and M. Nacinovich—On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. S.N.S. Pisa 27 (1981), 365–404

[6] A. Andreotti and C.D. Hill—Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. S.N.S. Pisa, 26 (1972), 299– 324

[7] S. Baouendi, C.H. Chang and and F. Treves—Microlocal analyticity and extension of CR functions, J. Diff. Geom. 18(1983), 331–391

[8] M.S. Baouendi, P. Ebenfelt and L.P. Rothschild—Real Submanifolds in Complex Space and Their Mappings, Princeton Math. Series, Princeton Univ. Press (1999)

[9] M.S. Baouendi, H. Jacobowitz and F. Treves—On the analyticity of CR mappings, Ann. Math. 122 (1985), 365–400

[10] M.S. Baouendi and L.P. Rothschild—Normal forms for generic manifolds and holomorphic extension of CR functions, J. Diff. Geom. 25 (1987), 431–467

[11] M.S. Baouendi and L.P. Rothschild—Cauchy Riemann functions on manifolds of higher codimension, Invent. Math. 101 (1990), 45–56

[12] M.S. Baouendi, L.P. Rothschild and J.M. Trépreau—On the geometry of analytic discs attached to real manifolds, J. Diff. Geometry (1994), 379–405

[13] M.S. Baouendi and F. Treves—A property of the functions and distributions annihilated by a locally integrable system of complex vector fields. Ann. Math. (2) 114 (1981), 387–421

[14] M.S. Baouendi and F. Treves—About holomorphic extension of CR functions on real hypersurfaces in complex space, Duke Math. J. 51 (1984), 77–107 [15] L. Baracco, D. Zaitsev and G. Zampieri—Rays condition and exten-

sion of CR functions from manifolds of higher type, J. Anal. Math. 101 (2007), 95–121

[16] L. Baracco and G. Zampieri—Analytic discs and extension of CR functions, Compositio Math. 127 (2001), 289–295

[17] L. Baracco and G. Zampieri—Propagation of CR extendibility along complex tangent directions, Compl. Var. Theory Appl. 50 (12) (2005), 967– 975

[18] L. Baracco and G. Zampieri—CR extension from manifolds of finite type, Canad. J. Math. 60 (6) (2008), 1219–1239

[19] L. Baracco and G. Zampieri—Separate real analyticity and CR extendibility, Forum Math. 21 (3) (2009), 519–531

[20] C.A. Berenstein and R. Gay—Complex variables: an introduction, Graduate Texts in Math., Springer Verlag, N.Y., 125 (1991)

[21] S. Berhanu, P.D. Cordaro and J. Hounie—Introduction to involutive structures, Cambridge Univ. Press (2007)

[22] E. Bedford, J. Fornaess—Complex manifolds in pseudoconvex boundaries, Duke Math. J. 48, n. 1 (1981), 279–288

[23] S. Bell and E. Ligocka—A simplification and extension of Fefferman’s theorem on biholomorphic mapings, Invent. Math. 57 (1980), 283–289

[24] E. Bishop—Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–22

[25] T. Bloom and I. Graham—On type conditions for generic real submanifolds of ℂ^{n}, Invent. Math. 40 (1977), 217–243

[26] S. Bochner and W.T. Martin—Several complex variables, Princeton (1948)

[27] A. Boggess—CR manifolds and the tangential Cauchy-Riemann complex, CRC Press, Boca Raton, FL (1991)

[28] A. Boggess and J. Pitts—CR extension near a point of higher type, Duke Math. J. 52 (1985), 67–102

[29] A. Boggess and J.C. Polking—Holomorphic extension of CR functions, Duke Math. J. 49 (1982), 757–784

[30] F.E. Browder—Real analytic functions on product spaces and separate analyticity, Canadian J. of Math. 13 (1961), 650–656

[31] H. Cartan and P. Thullen—Regularitäts und Konvergenzbereiche, Math. Ann. 106 (1932), 617–647

[32] D. Catlin—Necessary conditions for subellipticity of the ¯∂-Neumann problem, Ann. of Math. 117 (1983), 147–171

[33] D. Catlin—Subelliptic estimates for the ¯∂-Neumann problem on pseudoconvex domains, Ann. of Math. 126, n. 1 (1987), 131–191

[34] S.S. Chern—Complex manifolds without potential theory, 2nd ed., Springer-Verlag, New York (1979)

[35] S.S. Chern and J.K. Moser—Real hypersurfaces in complex manifolds, Acta. Math. 133 (1974), 219–271

[36] S.C. Chen and M.C. Shaw—Partial differential equations in several complex variables, Studies in Adv. Math., AMS and Int. Press, 19 (2001)

[37] J. D’Angelo—Real hypersurfaces, order of contact, and applications, Ann. of Math. 115 (1982), 615–637 [38] J.E. Fornaess—The disc method, Math. Z. 227, n. 4 (1998), 705–709 [39] J.E. Fornaess and C. Rea—Local holomorphic extendability of CR func-

tions, Ann. Sci. SNS Pisa 12 (1985), 491–502

[40] K. Diederich and J. Fornaess—Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), 129–141

[41] P. Ebenfelt—Uniformly Levi degenerate CR manifolds; the 5 dimensional case, Duke Math. J. 110 (2001), no. 1, 37–80

[42] G.B. Folland and J.J. Kohn—The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Studies, Princeton Univ. Press, Princeton, N.J., 75 (1972)

[43] L. Garding and L. Hörmander—Strongly subharmonic functions, Math. Scand. 15 (1964), 93–96

[44] P.R. Garabedian and D.C. Spencer—Complex boundary value problems, Trans. AMS 73 (1952), 223–242

[45] R.C. Gunning—Introduction to holomorphic functions of several vaiables, Vols. I, II, II, Wadsworth & Brooks/Cole Adv. Books & Software, Pacific Grove, CA, 1990

[46] R.C. Gunning and H. Rossi—Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, NJ, 1965

[47] J. Hanges and F. Treves—Propagation of holomorphic extendability of CR functions, Math. Ann. 263 (1983), no. 2, 157–177

[48] F. Hartogs—Zur Theorie der analytischen Funktionen mehererer Veränderlichen, Math. Ann. 62 (1906), 1–88

[49] L. Hörmander—L^{2} estimates and existence theorems for the ¯∂ operator, Acta Math. 113 (1965), 89-152

[50] L. Hörmander—An intoduction to complex analysis in several complex variables, Van Nostrand, Princeton, N.J. (1973)

[51] L. Hörmander—The analysis of linear partial differential operators I, II, III and IV, Grundlehren, Springer-Verlag, 256 (1984)

[52] H. Jacobowitz—An introduction to CR structures, Math. Surveys and Monographs AMS, 32, Providence, RI, 1990

[53] M. Kashiwara and P. Schapira—Sheaves on manifolds, Grundlehren mat., Springer-Verlag, 292 (1990)

[54] C. Kenig and S. Webster—On the hull of holomorphy of an n-manifold in ℂ^{n}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11-2 (1984), 261–280

[55] J.J. Kohn—Harmonic integrals on strongly pseudoconvex manifolds, I, Ann. Math. 78 (1963), 112–148; II, Ann. Math. 79 (1964), 450–472

[56] J.J. Kohn—Boundary behavior of ¯∂ on weakly pseudo-convex manifolds of dimension two, J. Diff. Geom. 6 (1972), 523–542

[57] J.J. Kohn—Global regularity for ¯∂ on weakly pseudo-convex manifolds, Transactions of the A.M.S. 181 (1973), 273–292

[58] J.J. Kohn—Methods of partial differential equations in complex analysis, Proceedings of Symposia in Pure Mathematics 30 (1977), 215–237

[59] J.J. Kohn—Subellipticity of the ¯∂-Neumann problem on pseudoconvex domains: sufficient conditions, Acta Math. 142 (1979), 79–122

[60] J.J. Kohn—Hypoellipticity and loss of derivatives, Annals of Math. 162 (2005), 943–286 [61] J.J. Kohn and L. Nirenberg—Non-coercive boundary value problems,

Comm. Pure Appl. Math. 18 (1965), 443–492

[62] H.Komatsu—A local version of the local Bochner tube theorem, J. Fac. Sci. Univ. Tokyo (1972), 201–214

[63] S.G. Krantz—Function theory of several complex variables, J. Wiley & Sons Inc., New York, 1982

[64] S.G. Krantz—Partial differential equations and complex analysis, Wadsworth, Belmond, CA, 1992

[65] L. Lelong—Fonctions plurisousharmoniques et fonctions analytiques de variables reélles, Ann. Inst. Fourier 11 (1961), 515–562

[66] L. Lempert—La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. SMF 109 (1981), 427–474

[67] H. Lewy—On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. 64 (1956), 514–522

[68] J. Merker—Global minimality of generic manifolds and holomorphic extendibility of CR functions, Int. Math. Res. Notices 1 (1997), 21–56

[69] C.B. Morrey—The analysis embedding of abstract real analytic manifolds, Ann. of Math. 40 (1958), 62–70

[70] T. Nagano—Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398–404

[71] R. Narasimhan—Several complex variables, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, Ill., London (1971)

[72] A. Newlander and L. Nirenberg—Complex coordinates in almost complex manifolds, Ann. Math. 64 (1957), 391–404

[73] K. Oka—Sur les fonctions de plusieurs variables, J. Sci. Hiroshima Univ., (I) 6 (1936), 245–255, (II) 7 (1937), 115–130, (III) 9 (1939), 7–19

[74] R.M. Range—Holomorphic functions and integral representations in several complex variables, Graduate Texts in Math., Springer-Verlag, N.Y., 108 (1986)

[75] L.P. Rothschild and E.M. Stein—Hypoelliptic differential operators and nilpotent groups, Acta Math. 137, no. 3-4 (1976), 247–320

[76] H. Rossi—Differentiable manifolds in complex euclidean space, Proc. Int. Congr. Math., Moscow, Izdat. Mir (1966), 512–516

[77] W. Rudin—Real and complex analysis, McGraw-Hill (1987)

[78] M. Sato, M. Kashiwara and T. Kawai—Hyperfunctions and pseudodifferential operators, Springer Lect. Notes in Math. 287 (1973), 265–529

[79] B. Shiffman—Separately meromorphic functions and separately holomorphic mappings, Proc. Symp. Pure Math. 52 (1991), 191–198

[80] J. Siciak—Analyticity and separate analyticity of functions defined on lower dimensional subsets of ℂ^{n}, Zeszyty Nauk. U.J. 13 (1969), 53–70

[81] J. Siciak—Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of ℂ^{n}, Ann. Pol. Math. 22 (1969), 145–171

[82] L. Stout—A domain whose envelope of holomorphy is not a domain, Ann. Polon. Math. 89, n. 2 (2006), 197–201

[83] L. Stout—Polynomial convexity, Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 261 (2007)

[84] H.J. Sussmann—Orbits of families of vector fields and integrability of distributions, Trans AMS 180 (1973), 171–188

[85] G. Tomassini—Tracce delle funzioni olomorfe sulle sottovarietà analitiche reali d’una varietà complessa, Ann. S.N.S. Pisa 20 (1966), 31–44

[86] J.M. Trépreau—Sur le prolongement holomorphe des fonctions CR definies sur une hypersurface réelle de classe C^{2} dans ℂ^{n}, Invent. Math. 83 (1986), 583–592

[87] J.M. Trépreau—Sur la propagation des singularités dans les varietés CR, Bull. Soc. Math. de France 118 (1990), 403–450

[88] F. Treves—Hypo-analytic structures, Local theory, Princeton Univ. Press, Princeton (1992)

[89] A. Tumanov—Extending CR functions on a manifold of finite type over a wedge, Mat. Sb. 136 (1988), 129–140

[90] A. Tumanov—Connection and propagation of analyticity of CR functions, Duke Math. J. 71, n. 1 (1994), 1–24

[91] A. Tumanov—Analytic discs and the extendibility of CR functions, Springer Lect. Notes in Math. 1684, Berlin (1998)

[92] A. Tumanov—Extending CR functions from manifolds with boundaries, Math. Res. Lett. 2 (5) (1995), 629–642

[93] A. Tumanov—Analytic discs and the extendibility of CR functions, Integral geometry, Radon transforms and complex analysis (Venice, 1996) Lecture Notes in Math., Springer, Berlin, 1684 (1998), 123–141

[94] A. Tumanov—Thin discs and a Morera theorem for CR functions, Math. Z. 226 (2) (1997), 327–334

[95] A. Tumanov—Extremal discs and the regularity of CR mappings in higher codimension, American J. of Math. 123 (2001), 445–473

[96] V.S. Vladimirov—Metody teorii funktsii mnogikh kompleknykh peremennykh, Nauca, Moscow (1964)

[97] S. Webster—A new proof of the Newlander-Nirenberg theorem, Math. Z. 201 (1989), 303–316

[98] R.O. Wells Jr.—Differential analysis on complex manifolds, Grad. Texts in Math. 65, Springer Verlag, New York, Berlin, 1980

[99] D. Zaitsev and G. Zampieri—Extension of CR functions on wedges, Math. Ann. 326 (2003), no. 4, 691–703

[100] D. Zaitsev and G. Zampieri— Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1443–1462

[101] G. Zampieri—q-Pseudoconvexity and regularity at the boundary for solutions of the ¯∂-problem, Compositio Math. 121 (2000), 155–162

[102] G. Zampieri—q-Pseudoconvex hypersurfaces through higher codimensional submanifolds of ℂ^{n}, J. Reine Angew. Math. 544 (2002), 83–90

[103] G. Zampieri—Analytic discs in pseudoconvex submanifolds of ℂ^{N} of higher codimension, Manuscripta Math. 116 (4) (2005), 397–400

Subject Index

Almost complex structure, 114

involutive, 115 Analytic disc, 155

attached, 155 defective, 175, 179

Bishop equation, 157, 167

linearized, 158 Bundle, 85

Canonical 1-form, 92, 140 2-form, 92, 140 transformation, 143

Cauchy formula in polydiscs, 5 inequalities, 10

Complex Laplacian, 61 Complex structure, 113

morphism of, 114 Connection, 174 Continuity principle, 34 CR

abstract structure, 129 distribution, 150 embeddable structure, 133 extension, 160 extension in Levi directions, 160 function, 128 hull, 133 manifold, 119 map, 129

Defect (of a disc), 180 Deformation (of a disc), 180 Domain

q-pseudoconvex, 58 holomorphically convex, 42 logarithmically convex, 12 of convergence, 11 of holomorphy, 40 pseudoconvex, 45 Reinhardt, 12

Feedback, 170

Finite type, 69, 102 Foliation, 87, 104 Hamiltonian, 144

Function of exhaustion, 45 separately CR extendible, 190 separately real analytic, 190 subharmonic, 15

Gain of derivatives, 103

Hörmander condition, 99 Hörmander number, 164 Hamiltonian

isomorphism, 93 vector field, 93 Heat kernel, 151 Hilbert transform, 155 Holomorphic polyhedron, 44 Hypoelliptic system, 100

Identity principle for generic submanifolds, 128

Integral leaf, 87

Intrinsic complexification, 132 Involutive bundle, 86 submanifold, 93

Iterated commutators, 69, 164

Lemma of Hartogs, 188 of Hopf, 178 Levi cone, 162 Levi flat, 138, 186 Levi form

of a function, 24

of a hypersurface, 26

of a submanifold (extrinsic), 139 of a submanifold (intrinsic), 136

Lie algebra, 104 bracket, 85

Lift of a CR curve, 174 of a disc, 179

Logarithmic superharmonicity, 30

Manifold generic, 119 totally real, 119

Maximum modulus principle, 8 Minimal submanifold, 180

Normal coordinates

of a hypersurface, 26 Normal form

of Bloom and Graham, 164

Operator

¯∂-Neumann, 61 Orbit

global, 105

local, 107

Partial connection, 186 Partial lift

of a CR curve, 174

of a disc, 175 Poisson bracket, 93 Polynomial approximation of a disc,

170 Polynomial approximation of CR

functions, 150 Propagation of CR extendibility, 183 Propagation of holomorphic

extendibility, 36 Push forward (of a vector field), 84

Riemann-Hilbert problem, 175

Standard elliptic pseudodifferential

operator, 99 Subelliptic estimates, 100 Submanifold

𝕀-symplectic, 141 ℝ-lagrangian, 141 involutive, 141 isotropic, 141 lagrangian, 141 regular, 141 Submean, 15 Symplectic homogeneous structure,

92

Theorem edge of the wedge, 159 implicit function, 82 of Baire, 191 of Baouendi and Treves, 150 of Darboux and Frobenius, 95 of Frobenius, 87 of Hanges and Treves, 38 of Hartogs, 23 of Hartogs of extension, 29 of Nagano, 104 of Newlander and Nirenberg, 115 of separate real analyticity, 190 of Sussmann, 106 of Tumanov, 181

Weak solution (of the tangential ¯∂),

150 Wedge, 159 Weighted H^{0} estimates, 51

Symbols Index

H_{f} := H(df) Hamiltonian vector

field of f, 93 r_{f}^{v} separate Taylor radius of f in

direction v, 31 (T_{M}^{∗}ℂ^{n})′ partial conormal bundle,

173 (r_{ij})_{ij} and (φ_{ij})_{ij} the matrix of

∂ ¯∂r and ∂ ¯∂φ, 55, 56 A analytic disc, 29, 155 A^{∗} lift of A, 177, 180 A^{∗′} partial lift of A, 175 B, G matrices associated to the

connection Φ, 177 C^{α} and C^{k,α} Hölder space, 155,

167 F′ or F_{∗} tangent map or

push-forward by F, 80, 84 H Hamiltonian isomorphism, 93 H^{s} or H_{φ_{t}}^{s} s-Sobolev space

(weighted by φ_{t}), 59 JF Jacobian of F, 80 J_{x}x̃ Jacobian of x̃(x), 80 K_{λ} heat kernel, 151 L_{loc}^{1} locally integrable functions,

149 L_{M} extrinsic Levi form of M, 26 L_{M}^{ξ} microlocal Levi form, 140 L_{r} the Levi form of the function

r, 24 N_{t} := □_{t}^{−1} the ¯∂-Neumann

operator, 62 P, P_{z^{o} r} polydisc in ℂ^{n}, 5 T ¡, T_{1}, T_{0} Hilbert transform, 155 T X^{ℝ}, resp. T^{∗}X^{ℝ} tangent, resp.

cotangent, bundle to X^{ℝ}, 112 T X^{ℝ}, resp. T^{∗}X^{ℝ} tangent, resp.

cotangent, bundle to X^{ℝ}, 112

T^{∗}ℂ^{n} cotangent bundle to ℂ^{n}, 38,

140 T_{M}^{∗}ℂ^{n} conormal bundle to M in

ℂ^{n}, 121, 140 T^{ℂ}M complex tangent bundle to

M, 26, 119 T^{1,0}X and T^{0,1}X (0, 1) and (0, 1) vector field tangent to X, 114 T_{M}ℂ^{n} normal bundle to M in ℂ^{n},

139 T_{λ}u convolution of u with the

heat kernel, 151 W wedge, 159

□_{t} := ¯∂¯∂_{t}^{∗} + ¯∂_{t}^{∗} ¯∂ complex

Laplacian, 61 ℂ ⊗_{ℝ} T X^{ℝ} complexification in

fibers of T X^{ℝ}, 113 ℂ ⊗_{ℝ} T^{∗}ℂ^{n} complexification in

fibers of T^{∗}ℂ^{n}, 51 ℂ ⊗_{ℝ} T^{∗}ℂ^{n} = (T^{1,0}ℂ^{n})^{∗} ⊕ (T^{0,1}ℂ^{n})^{∗} decomposition in (1, 0) and (0, 1) components, 51 ℂ ⊗_{ℝ} T^{∗}X^{ℝ} complexification in

fibers of T^{∗}X^{ℝ}, 113 Δ, Δ_{z_{j}^{o} r_{j}} disc in ℂ, 5 Γ Levi cone, 162 J complex structure, 113 SH(Ω) the space of subharmonic

functions on Ω, 15 ∥ · ∥_{φ} or ∥ · ∥_{H_{φ}^{0}} weighted L^{2}-norm,

52 ∥ · ∥_{s} or ∥ · ∥_{H^{s}} or ∥ · ∥_{H_{φ_{t}}^{s}}

H^{s}-norm (weighted), 59 ∥ · ∥_{k,α} the norm in C^{k,α}, 156

α!, |α|, α − β operations on

multiindices, 8 ¯∂ Cauchy-Riemann complex, 52 ¯∂^{∗} the adjoint complex of ¯∂, 53 δ_{ω_{j}} the H_{φ}^{0}-adjoint of −∂ω̄_{j}, 52 dim_{CR} CR dimension, 118 γ^{∗′} partial lift of γ, 174 K̂ convex hull, 41 K̂_{Ω} hol(Ω)-hull, 41 hess r the hessian of the function

r, 25 λ_{j} the ordered Levi eigenvalues,

58 B Bergman projection, 63 D′ distributions, 149 F^{α}, 167 K^{k} k-th cohomology of ¯∂, 45 L(E) Lie span of E, 104 L^{j}, L_{ℂ}^{j}, L_{ℝ}^{j}, 68, 70, 99 L_{M} intrinsic Levi form of M, 136 O orbit, 105 ω canonical 1-form, 92 ω_{j} and ω̄_{j}, (1, 0)- and (0, 1)-forms,

51 r^{∨}_{f}^{v} Taylor radius of f in direction

v, 34 −−→^{ext} forced extension of

holomorphic functions, 14, 40 ∂^{α}f, f^{(α)} multiderivative, 8 ∂_{0}P distinguished boundary of P,

5 ∂_{t}A, resp. ∂_{η}A derivative of the

disc A in the radius t, resp. in the parameter η, 161

∂_{v} derivative along v, 81 ∂_{z} holomorphic derivative, 2 ∂z̄ antiholomorphic derivative, 2 ∂_{ω_{j}} and ∂ω̄_{j} (1, 0) and (0, 1) vector

fields, 51 ρ radial vector field, 95 ↘ or ↗ monotonic convergence σ := dω canonical 2-form, 92 σ^{ℝ}, σ^{𝕀} the real, imaginary part of

σ, 141 <_{∼} or >_{∼} estimate up to

multiplicative constant Ã symplectic transform of A , 177 ∧ wedge product, 55 {·, ·} Poisson bracket, 93 df = ∂f + ¯∂f decomposition of

the differential df in (1, 0) and (0, 1) components, 3 r^{α} multipower, 8 s^{+}, s^{−}, s^{0} numbers of eigenvalues

of the Levi form, 26 t or ^{∗} transposed or adjoint tF′ or F^{∗} cotangent map or pull-back by F, 80, 84 tΦ^{−1}, resp. Φ connection on

T_{M}^{∗}ℂ^{n}, resp. T_{M}ℂ^{n}, obtained by partial lifts, 174

[·, ·] Lie bracket or commutator X^{ℝ} real underlying manifold to

X, 112 ⊥ orthogonal ℝ real underlying structure ⊂⊂ relatively compact subset

ℂ complexification

d, d(z), d(z, ∂Ω) euclidean distance

κ_{ij} Kronecker symbol