PDEs and analysis on manifolds and Lie groups
The interaction between harmonic analysis and the study of partial differential equations turned out to be very important in modern analysis and gave rise to interesting results. An example in this direction is given by the celebrated restriction theorem of the Fourier transform to a paraboloid or a cone, which can be reformulated as an estimate (Strichartz estimate) for the solutions of the linear Schroediner and wave equations.
This kind of estimates have important consequences for the global and local wellpsedness of the corresponding nonlinear equations. Such problems, which were originally studied in the euclidean setting, are object of study also in different contexts, like hyperbolic spaces, symmetric spaces, manifolds and Lie groups, where one considers differential equations associated with Laplacians or subLaplacians related with the geometry of the underlying space.
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 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 functional spaces, like Hardy type spaces adapted to the geometry of the manifold and the group, to study boundedness properties of operators.