On Schrödinger Pseudo-Multipliers and their Commutators

Abstract

In this article, we establish the unweighted and weighted $L^p$-boundedness of pseudo-multipliers associated with a class of Schrödinger operators, this generalizes the result of our first author and Thangavelu [Bagchi \& Thangavelu, J. Funct. Anal. 2015] for Hermite pseudo-multipliers. 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 Hölder's inequality and quantitative weighted estimates for general sparse operators, which are of independent interest. We also study commutators of Schrödinger pseudo-multipliers, establishing their boundedness and compactness results on these weighted $L^p$-spaces.