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

Alessio Martini, Detlef Müller, Sebastiano Nicolussi Golo

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--Hörmander 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.

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

Abstract

Let be a smooth second-order real differential operator in divergence form on a manifold of dimension . 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 cannot be wider than the corresponding ranges for the Laplace operator on . 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 and nondegeneracy properties of the sub-Riemannian geodesic flow.

Paper Structure

This paper contains 27 sections, 30 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. Fourier integral and pseudodifferential operators
  3. Distributions and linear operators
  4. Symbol classes
  5. Pseudodifferential and Fourier integral operators
  6. Fourier integral representation of the half-wave propagator
  7. Eikonal equation
  8. Preliminaries on symplectic geometry
  9. Solution to the eikonal equation
  10. Existence domains and smooth dependence on the initial datum
  11. Definition of the phase function
  12. Critical points of the phase function
  13. The Hessian of the phase function
  14. Construction of an operator phase function
  15. Sub-Laplacians on sub-Riemannian manifolds
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $M$ be a smooth manifold of dimension $n$, $H:T^*M\to[0,+\infty)$ a smooth function that is a positive semidefinite quadratic form on each fiber, and $\mu$ a smooth positive measure on $M$. Let $\mathscr{L}$ be the sub-Laplacian defined by $(M,H,\mu)$ and let us fix a self-adjoint extension of $

Theorems & Definitions (59)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • Proposition 3.4
  • ...and 49 more