Weighted inequalities of Fefferman-Stein type for Riesz-Schrödinger Transforms

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 $$\int |Tf{|}^{p}w\le C\int |f{|}^p\mathcal{M}w$$ 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 Schrödinger operator $L=-\Delta +V$, with $V$ a non-negative function satisfying an appropriate reverse-Hölder condition. For the Riesz-Schrödinger transforms $\nabla L^{-1/2}$ and ${\nabla }^2{L}^{-1}$ we make use of a result due to C. Pérez where this problem is solved for classical Calderón-Zygmund operators.

28 pages