Seminario: “Functional Calculus on Homogeneous Groups”

Lunedì 25 Febbraio 2019, ore 12:00 - Aula 2BC30 - Mattia Calzi


Lunedì 25 Febbraio 2019 alle ore 12:00 in Aula 2BC30, Mattia Calzi (Scuola Normale Superiore di Pisa) terrà un seminario dal titolo “Functional Calculus on Homogeneous Groups”.

Let $\mathcal{L}_1, . . . , \mathcal{L}_k$ be left-invariant differential operators on a Lie group $G$ which induce essentially self-adjoint operators on $L^2(G)$ (with initial domain $C_c^\infty(G)$) with commuting self-adjoint extensions. Then, by means of the spectral theorem it is possible to define $m(\mathcal{L}_1, . . . , \mathcal{L}_k)$ for every Borel measurable function $m: \mathbb{R}^k \rightarrow \mathbb{C}$. When $m$ is bounded, $m(\mathcal{L}_1, . . . , \mathcal{L}_k)$ is a bounded operator of $L^2(G)$ which commutes with left translations, so that it has a (right) convolution kernel $\mathcal{K}(m) \in \mathcal{D}'(G)$.
When $G = \mathbb{R}^n$, we have $\mathcal{L}_j = P_j (-i\partial)$ for some polynomial $P_j , j = 1, . . . , k$, so that $\mathcal{K}(m) = \mathcal{F}^{-1} (m \circ P)$, where $\mathcal{F}$ is the Euclidean Fourier transform. Given the analogy between $\mathcal{K}$ and $\mathcal{F}^{-1}$, one may wonder whether some of the basic properties of $\mathcal{F}$ hold, suitably interpreted, for $\mathcal{K}$. On the one hand, A. Martini proved in 2011 that, if $\mathcal{L}_1, . . . , \mathcal{L}_k$ form a weighted subcoercive system of operators, then $\mathcal{K}$ induces an isometry from $L^2(\beta)$ into $L^2(G)$ for some (‘Plancherel’) measure $\beta$ on $\mathbb{R}^k$; furthermore, when $G$ is a group of polynomial growth, then $\mathcal{K}$ maps $\mathcal{S}(\mathbb{R}^k)$ into $\mathcal{S}(G)$. On the other hand, it is not hard to run into reasonably well-behaved examples where $\mathcal{K}(m) \in L^1(G)$ but $m$ is not continuous, or where $\mathcal{K}(m) \in \mathcal{S}(G)$ and $m$ is not a Schwartz function (or even continuous).
I will present some results concerning $\mathcal{K}$ in various contexts, focusing on homogeneous groups.