Control of pseudodifferential operators by maximal functions via weighted inequalities

TL;DR

This paper develops weighted $L^2$ inequalities for pseudodifferential operators, enabling control via maximal functions and extending bounds for various weight classes, with implications for harmonic analysis.

Contribution

It introduces a novel approach to control pseudodifferential operators using maximal functions through weighted inequalities, unifying and extending existing bounds.

Findings

Recovered known Muckenhoupt bounds.

Established new bounds for weights in intersection of Muckenhoupt and reverse H"older classes.

Provided a framework for controlling pseudodifferential operators with maximal functions.

Abstract

We establish general weighted $L^2$ inequalities for pseudodifferential operators associated to the H\"ormander symbol classes $S^m_{\rho,\delta}$. Such inequalities allow to control these operators by fractional "non-tangential" maximal functions, and subsume the optimal range of Lebesgue space bounds for pseudodifferential operators. As a corollary, several known Muckenhoupt type bounds are recovered, and new bounds for weights lying in the intersection of the Muckenhoupt and reverse H\"older classes are obtained. The proof relies on a subdyadic decomposition of the frequency space, together with applications of the Cotlar-Stein almost orthogonality principle and a quantitative version of the symbolic calculus.