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

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