Alessio Martini

A Fourier integral operator approach to the sub-Riemannian wave equation

Let L be a sub-Laplacian on a sub-Riemannian manifold of dimension n. We show that the ranges of validity of spectral multiplier estimates of Mihlin-Hörmander type and wave propagator estimates of Miyachi-Peral type for L cannot be wider than the corresponding ranges for the Laplace operator on R^n - despite the lack of ellipticity of L. The proof hinges on a Fourier integral representation for the wave propagator associated with L and nondegeneracy properties of the sub-Riemannian geodesic flow. This is joint work with Detlef Müller and Sebastiano Nicolussi Golo