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

## TL;DR

This paper establishes that spectral multiplier and wave propagator estimates for sub-Laplacians on sub-Riemannian manifolds are limited to the same regularity bounds as the Euclidean Laplacian, regardless of the step of the structure.

## Contribution

It proves that the regularity bounds for spectral multipliers and wave equations for sub-Laplacians match those of the Euclidean case, extending to all sub-Riemannian manifolds without step restrictions.

## Key findings

- Spectral multiplier estimates are bounded by Euclidean ranges.
- Wave propagator estimates are limited to Euclidean regularity bounds.
- Results apply to all sub-Laplacians on Carnot groups and sub-Riemannian manifolds.

## Abstract

Let $\mathscr{L}$ 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--H\"ormander type and wave propagator estimates of Miyachi--Peral type for $\mathscr{L}$ 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 $\mathscr{L}$ and nondegeneracy properties of the sub-Riemannian geodesic flow.