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.

47 pages

References in corpus (2)

• Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres
• From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

Cited by in corpus (8)

• An optimal multiplier theorem for Grushin operators in the plane, I
• An optimal multiplier theorem for Grushin operators in the plane, II
• Almost everywhere convergence of Bochner-Riesz means on Heisenberg-type groups
• A $p$-specific spectral multiplier theorem with sharp regularity bound for Grushin operators
• Uniform pointwise estimates for ultraspherical polynomials
• An $L^p$-spectral multiplier theorem with sharp $p$-specific regularity bound on Heisenberg type groups
• A sharp multiplier theorem for solvable extensions of Heisenberg and related groups
• Bilinear Bochner-Riesz Means on Métivier groups