# Seminario di equazioni differenziali e applicazioni: “Spectral multipliers and wave equation for sub-Laplacians”

## Giovedì 31 Gennaio 2019, ore 12:00 - Aula 2AB40 - Sebastiano Nicolussi Golo

ARGOMENTI: Seminars

Giovedì 31 Gennaio 2019 alle ore 12:00 in Aula 2AB40, Sebastiano Nicolussi Golo (Università di Padova) terrà un seminario dal titolo “Spectral multipliers and wave equation for sub-Laplacians”.

Abstract
Mihlin-Hormander theorem gives the sharp Sobolev order $n/2$ for a spectral multiplier of the Laplacian to define a bounded operator on $L^p(\R^n)$ for all $p\in(1,\infty)$.
We study the same type of statements for sub-Laplacians, which are sub-elliptic operators defined on sub-Riemannian manifolds.
It is known that the homogeneous dimension $Q$ can play the same role as $n$ on certain sub-Riemannian manifolds.
However, there are examples, such as the Heisenberg groups, where $Q>n$ but $n/2$ is again the sharp Sobolev order, where $n$ is the topological dimension.
These results led to conjecture that the sharp Sobolev order for a Mihlin--Hormander theorem is half the topological dimension in a large class of sub-Riemannian manifolds, e.g., Carnot groups.
We have proven that in no sub-Riemannian manifold the sharp Sobolev order can be lower than half the topological dimension.
For the proof, we construct a partial Fourier integral representation of the sub-Riemannian wave propagator.
This is a joint work with Alessio Martini and Detlef Muller.