Spectral multipliers for sub-Laplacians: Topological versus sub-Riemannian dimension

Abstract: 

In this  survey talk, I shall try to give an idea about what is known on spectral multiplier estimates of Mihlin-Hörmander type for sub-Laplacians and what are the challenges in their still wide open study.

Sub-Laplacians arise as sum of squares  operators L  in the sense of Hörmander, i.e., sums of squares of smooth vector fields on a given, say d-dimensional, manifold M, which are bracket generating. If there are fewer vector fields than d, then L will not be elliptic, however, still hypoelliptic. Moreover, if the manifold is endowed with a measure  such that the operator  L  is essentially self-adjoint  on the Hilbert space of all  square integrable functions on the manifold, then we may  form functions F(L)  by means of spectral resolution of L.

The question that I shall address is to give conditions on a given spectral multiplier  F which ensure that the operator F(L) is still bounded on other Lebesgue spaces on M,  of order p different from 2, in  particular p=1. I shall discuss versions of the classical Mihlin-Hörmander multiplier theorem in this context, and the relations with the underlying sub-Riemannian geometry and associated wave equations.