Quantitative Weighted Estimates for Schr\"odinger Pseudo-Multipliers and its Commutators

TL;DR

This paper studies the boundedness of Schr"odinger pseudo-multipliers and their commutators on weighted $L^p$ spaces, introducing new weight classes and quantitative estimates that extend classical results.

Contribution

It introduces new weight classes beyond Muckenhoupt $A_p$ and provides quantitative bounds for Schr"odinger pseudo-multipliers and their commutators.

Findings

Weighted boundedness of pseudo-multipliers established

Quantitative reverse H"older's inequality proved

Boundedness and compactness of commutators demonstrated

Abstract

In this article, we investigate the unweighted and weighted $L^p$-boundedness of pseudo-multipliers associated with a class of Schr\"odinger operators. The weight classes we consider are tailored to this framework and strictly contain the classical Muckenhoupt $A_p$-classes. To establish the weighted boundedness, we prove a quantitative version of reverse H\"older's inequality and quantitative weighted estimates for general sparse operators, which are of independent interest. We also study commutators of Schr\"odinger pseudo-multipliers, establishing their boundedness and compactness results on these weighted $L^p$-spaces.