Analysis on manifolds and Lie groups
We are interested in the study of Laplacians and subLaplacians on noncompact Riemannian and subRiemannianmanifolds, such as hyperbolic spaces and symmetric spaces, and on noncompact Lie groups. In particular, we introduce Sobolev and Besov spaces associated to such differential operators and study their properties. Such function spaces play a role in the study of well-posedness results for nonlinear differential equations associated to the Laplacians and subLaplacians mentioned above.
Another interesting research field concerns the study of singular integrals and Fourier integral operators on Riemannian and subRiemannian manifolds and noncompact Lie groups. Riesz transforms, spectral multipliers of Laplacians and subLaplacians and Fourier integral operators related with the wave equation are of particular interest. The classical techniques used in the euclidean setting do not apply in such contexts and one needs to introduce new function spaces, such as Hardy type spaces adapted to the geometry of the manifold and the group, to study boundedness properties of the aforementioned operators.
Singular integral and Fourier integral operators on manifolds can be applied to the study of differential equations in geometric contexts, such as the wave equation driven by a Laplacian or a subLaplacian. Quantifying the optimal loss of derivatives in Lp estimates for the subRiemannian wave propagator is a particularly relevant problem in this area, which is closely connected to other fundamental problems of geometric harmonic analysis, such as determining the optimal regularity thresholds for Lp boundedness of spectral multipliers and Bochner-Riesz means associated with subLaplacians. The investigation of these problems involves a substantial interaction of analytic and geometric techniques, including the study of the properties of the subRiemannian geodesic flow and of estimates for eigenvalues and eigenfunctions of Schrödinger operators.