On weighted compactness of commutators of Schr\"{o}dinger operators

TL;DR

This paper investigates the weighted compactness properties of commutators associated with Schrödinger operators, including Riesz transforms and Calderón-Zygmund operators, extending existing results in harmonic analysis.

Contribution

It provides new generalizations of weighted compactness results for commutators of Schrödinger operators, encompassing various operators like Riesz transforms and Littlewood-Paley functions.

Findings

Weighted compactness results established for a broad class of Schrödinger operator commutators.

Generalization of classical results to operators with potentials in reverse Hölder classes.

Extension of compactness properties to weighted function spaces.

Abstract

Let $\mathcal{L}=-\Delta+\mathit{V}(x)$ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse H\"{o}lder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schr\"{o}dinger operators, which include Riesz transforms, standard Calder\'{o}n-Zygmund operatos and Littlewood-Paley functions. These results generalize substantially some well-know results.