Title: Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
Abstract: Let $\mathcal{L}$ be a smooth second-order real differential operator in divergence form on a manifold of dimension $n$. Under a bracket-generating condition, we show that the ranges of validity of spectral multiplier estimates of Mihlin-Hörmander type and wave propagator estimates of Miyachi-Peral type for $\mathcal{L}$ cannot be wider than the corresponding ranges for the Laplace operator on $\mathbb{R}^n$. The result applies to all sub-Laplacians on Carnot groups and more general sub-Riemannian manifolds, without restrictions on the step. The proof hinges on a Fourier integral representation for the wave propagator associated with $\mathcal{L}$ and nondegeneracy properties of the sub-Riemannian geodesic flow.
Joint work with Alessio Martini and Sebastiano Nicolussi Golo.
Slides