Weighted inequalities of Fefferman-Stein type for Riesz-Schr\"odinger Transforms

TL;DR

This paper investigates weighted inequalities of Fefferman-Stein type for Riesz transforms associated with Schr"odinger operators, extending classical results to operators involving potential functions satisfying reverse-H"older conditions.

Contribution

It extends Fefferman-Stein inequalities to Riesz-Schr"odinger transforms, providing new weighted bounds for these operators with potentials satisfying reverse-H"older conditions.

Findings

Established weighted inequalities for Riesz-Schr"odinger transforms

Extended classical Fefferman-Stein results to Schr"odinger operators

Provided bounds involving reverse-H"older potentials

Abstract

In this work we are concerned with Fefferman-Stein type inequalities. More precisely, given an operator $T$ and some $p$, $1<p<\infty$, we look for operators $\mathcal{M}$ such that the inequality $$\int |Tf|^pw\leq C\int |f|^p \mathcal{M}w$$ holds true for any weight $w$. Specifically, we are interested in the case of $T$ being any first or second order Riesz transform associated to the Schr\"odinger operator $L=-\Delta + V$, with $V$ a non-negative function satisfying an appropriate reverse-H\"older condition. For the Riesz-Schr\"odinger transforms $\nabla L^{-1/2}$ and $\nabla^2 L^{-1}$ we make use of a result due to C. P\'erez where this problem is solved for classical Calder\'on-Zygmund operators.