Seminario di analisi matematica
ore
16:00
presso Aula Enriques
Abstract. Let L be the Laplacian on R^n . The investigation of necessary
and sufficient conditions for an operator of the form F (L) to be bounded
on L^p in terms of smoothness properties of the spectral multiplier F is a
classical and very active research area of harmonic analysis, with long-standing
open problems (e.g., the Bochner–Riesz conjecture) and connections with the
regularity theory of PDEs.
In settings other than the Euclidean, particularly in the presence of a sub-
Riemannian geometric structure, the natural substitute L for the Laplacian
need not be an elliptic operator, and it may be just sub-elliptic. In this context,
even the simplest questions related to the L^p -boundedness of operators of the
form F (L) are far from being completely understood.
I will survey recent results dealing with the case of sub-Laplacians on 2-step
Carnot groups, complex and quaternionic spheres, and Grushin operators.