Weights of Exponential Growth and Decay for Schr\"{o}dinger-type operators

Abstract

Fix $d \geq 3$ and $1 < p < \infty$. Let $V : \mathbb{R}^{d} \rightarrow [0,\infty)$ belong to the reverse H\"{o}lder class $RH_{d/2}$ and consider the Schr\"{o}dinger operator $L_{V} := - \Delta + V$. In this article, we introduce classes of weights $w$ for which the Riesz transforms $\nabla L_{V}^{-1/2}$, their adjoints $L_{V}^{-1/2} \nabla$ and the heat maximal operator $\sup_{t > 0} e^{- t L_{V}} |f|$ are bounded on the weighted Lebesgue space $L^{p}(w)$. The boundedness of the $L_{V}$-Riesz potentials $L_{V}^{-\alpha/2}$ from $L^{p}(w)$ to $L^{\nu}(w^{\nu/p})$ for $0 < \alpha \leq 2$ and $\frac{1}{\nu} = \frac{1}{p} - \frac{\alpha}{d}$ will also be proved. These weight classes are strictly larger than a class previously introduced by B. Bongioanni, E. Harboure and O. Salinas that shares these properties and they contain weights of exponential growth and decay. The classes will also be considered in relation to different generalised forms of Schr\"{o}dinger operator. In particular, the Schr\"{o}dinger operator with measure potential $- \Delta + \mu$, the uniformly elliptic operator with potential $- \mathrm{div} A \nabla + V$ and the magnetic Schr\"{o}dinger operator $(\nabla - i a)^{2} + V$ will all be considered. It will be proved that, under suitable conditions, the standard operators corresponding to these second-order differential operators are bounded on $L^{p}(w)$ for weights $w$ in these classes.