Classes of weights and second order Riesz transforms associated to Schrödinger operators

Abstract

We consider the Schrödinger operator $-\Delta+V$ on $\mathbb{R}^{n}$ with $n\ge 3$ and $V$ a member of the reverse Hölder class $\mathcal{B}_s$ for some $s$ > $n/2$. We obtain the boundedness of the second order Riesz transform $\nabla^2 (-\Delta+V)^{-1}$ on the weighted spaces $L^p(w)$ where $w$ belongs to a class of weights related to $V$. To prove this, we develop a good-$\lambda$ inequality adapted to this setting along with some new heat kernel estimates.