Weighted inequalities for commutators of Schr\"odinger-Riesz transforms

TL;DR

This paper establishes weighted inequalities for commutators of Schrödinger-Riesz transforms, extending classical results to broader weight and symbol classes under certain conditions on the potential V.

Contribution

It introduces new weighted $L^p$ and weak $L ext{log}L$ estimates for these commutators, enlarging the applicable classes of weights and symbols beyond classical bounds.

Findings

Weighted $L^p$ estimates obtained for $1<p< \\infty$

Weak $L ext{log}L$ estimates established

Broader classes of weights and symbols than classical results

Abstract

In this work we obtain weighted $L^p$, $1<p<\infty$, and weak $L\log L$ estimates for the commutator of the Riesz transforms associated to a Schr\"odinger operator $-\lap+V$, where $V$ satisfies some reverse H\"older inequality. The classes of weights as well as the classes of symbols are larger than $A_p$ and $BMO$ corresponding to the classical Riesz transforms.