Spectral multipliers for maximally subelliptic operators

TL;DR

This paper proves spectral multipliers for maximally subelliptic operators are singular integral operators under certain conditions, and establishes equivalences between function spaces adapted to the operator and Carnot-Carathéodory geometry.

Contribution

It introduces a Mihlin-Hörmander type condition ensuring spectral multipliers are singular integrals and links non-isotropic function spaces to geometric structures.

Findings

Spectral multipliers are singular integral operators under the Mihlin-Hörmander condition.

Equivalence between non-isotropic Besov and Triebel-Lizorkin spaces and those adapted to Carnot-Carathéodory geometry.

Boundedness of spectral multipliers on non-isotropic $L^p$ Sobolev spaces established.

Abstract

Consider a non-negative, self-adjoint, maximally subelliptic operator on a compact manifold. We show that the spectral multiplier is a singular integral operator under an appropriate Mihlin-H\"ormander type condition. We establish the equivalence between non-isotropic Besov and Triebel-Lizorkin spaces adapted to the operator and those adapted to a Carnot-Carath\'eodory geometry on the manifold. We also give a Mihlin-H\"ormander type condition for the boundedness of the spectral multiplier on non-isotropic $L^p$ Sobolev spaces.