Spectral multipliers on nilpotent Lie groups and homogeneous spaces
Final Report Abstract
Let L be the Laplacian on Rn . The investigation of necessary and sufficient conditions for an operator of the form F (L) to be bounded on Lp in terms of “smoothness properties” of the spectral multiplier F is a classical 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 subelliptic. In this context, even the simplest questions related to the Lp-boundedness of operators of the form F (L) are far from being completely understood. For instance, it is possible to prove, under quite general assumptions, a Mihlin–Hörmandertype theorem that yields Lp boundedness of F (L) for 1 < p < ∞ whenever F satisfies a smoothness condition of order s > Q/2, where Q is the “homogeneous dimension” of the ambient space. However, in many particular cases, where detailed information on the geometry of the space and on the spectral resolution of L is available, the condition s > Q/2 turns out not to be optimal, and it is possible to push it down to s > d/2, where d is the “topological dimension”. In the course of the project we studied several subelliptic operators L, particularly sublaplacians, on various Lie groups and homogeneous spaces. Operators acting on differential forms and systems of commuting operators were also considered. Many results and techniques originating from different parts of mathematics (e.g., unitary representation theory of nilpotent and compact Lie groups; estimates for diffusion equations; properties of geodesics on sub-Riemannian o manifolds; Calder´n–Zygmund theory in doubling and nondoubling settings; resolution of singularities) were combined and exploited to obtain Lp -boundedness results for the functional calculus of the studied operators. The obtained results seem to support the “general principle” that s > d/2 rather than s > Q/2 is the optimal condition for large classes of subelliptic operators L. On the other hand, even in the simplest cases, the underlying algebraic and geometric structures turn out to be surprisingly complicated and further investigation and ideas are required to prove sharp results in greater generality.
Publications
- Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups
A. Martini, A. Ottazzi, and M. Vallarino
- Spectral multipliers for the Kohn Laplacian on forms on the sphere in Cn
V. Casarino, M.G. Cowling, A. Martini, and A. Sikora
