Spectral multipliers and wave equation for sub-Laplacians : lower regularity bounds of Euclidean type

Abstract

Let 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 Mikhlin–Hörmander type and wave propagator estimates of Miyachi–Peral type for L cannot be wider than the corresponding ranges for the Laplace operator on Rn. 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 L and nondegeneracy properties of the sub-Riemannian geodesic flow.