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.