Some estimates for Riesz transforms associated with Schrödinger operators

Abstract

Let L = −Δ+V be the Schr¨odinger operator on Rn, where n ≥ 3, and nonnegative potential V belongs to the reverse H¨older class RHq with n/2 ≤ q < n. Let Hp L(Rn) denote the Hardy space related to L and BMOL(Rn) denote the dual space of H1L (Rn). In this paper, we show that Tα,β = V α∇L−β is bounded from Hp1 L (Rn) into Lp2 (Rn) for n n+δ′ < p1 ≤ 1 and 1 p2 = 1 p1 − 2(β−α) n , where δ′ = min{1, 2−n/q0}, and q0 is the reverse H¨older index of V. Moreover, we prove T∗ α,β is bounded on BMOL(Rn) when β − α = 1/2.