Hardy type estimates for Riesz transforms associated with Schrödinger operators

Abstract

Let $\mathcal {L}=-\Delta +V$ be a Schrödinger operator on $\mathbb {R}^n (n\ge 3),$ where the nonnegative potential V belongs to reverse Hölder class $RH_{q_1}$ for $q_1>\frac{n}{2}.$ Let $H^p_\mathcal {L}(\mathbb {R}^n)$ be the Hardy space related to $\mathcal {L}.$ In this paper, we consider the Hardy type estimates for the Riesz transform $T_\alpha =V^\alpha (-\Delta +V)^{-\alpha }$ with $0<\alpha <n/2.$ We show that $T_\alpha$ is bounded from $H^p_\mathcal {L}(\mathbb {R}^n)$ into $L^p(\mathbb {R}^n)$ for $\frac{n}{n+\delta '}<p\le 1,$ where $\delta '=\min \{1, 2-n/q_0\},$ and $q_0$ is the reverse Hölder index of V. Moreover, we prove that the commutator $[b,T_\alpha ],$ which associated with $T_\alpha$ and a new BMO function b,  maps $H^{1}_\mathcal {L}(\mathbb {R}^n)$ continuously into weak $L^1(\mathbb {R}^n)$.