Sparse bounds for pseudodifferential operators

TL;DR

This paper establishes sharp sparse bounds for pseudodifferential operators within H"ormander classes, enabling new weighted estimates and applications to oscillatory Fourier multipliers and dispersive equations.

Contribution

It introduces sharp sparse bounds for pseudodifferential operators, utilizing a single scale analysis, and extends these results to weighted estimates and oscillatory Fourier multipliers.

Findings

Sharp sparse bounds up to the endpoint

Weighted estimates for pseudodifferential operators

Applications to oscillatory Fourier multipliers

Abstract

We prove sparse bounds for pseudodifferential operators associated to H\"ormander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.