Spectral multipliers for maximally subelliptic operators

TL;DR

This work develops a spectral multiplier theory for maximally subelliptic operators on compact manifolds. It establishes that spectral multipliers $m(L)$ are singular integrals under a Mihlin-Hörmander condition and proves that non-isotropic Besov and Triebel-Lizorkin spaces defined via the operator coincide with Carnot-Carathéodory-based spaces, yielding a unified functional-calculus framework. It further provides sharp $L^p$-Sobolev bounds for $m(L)$ on the non-isotropic spaces $NL^p_t(X)$ with smoothness quantified by a localized Sobolev norm of $m$, and employs interpolation to extend results across $p$. Collectively, the results unify operator- and geometry-driven analyses for maximally subelliptic operators and extend Calderón-type multiplier theory to a broad Carnot-Carathéodory setting.

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-Hörmander type condition. We establish the equivalence between non-isotropic Besov and Triebel-Lizorkin spaces adapted to the operator and those adapted to a Carnot-Carathéodory geometry on the manifold. We also give a Mihlin-Hörmander type condition for the boundedness of the spectral multiplier on non-isotropic $L^p$ Sobolev spaces.

Theorems & Definitions (49)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Example 1.5
• Theorem 1.6
• Corollary 1.7
• Definition 1.8
• Remark 1.9
• Theorem 1.10