Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Abstract

Let L=−Δ+V be a Schrödinger operator on ℝ^d, d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L^p(ℝ^d), for some p> d/2. Let K_t be the semigroup generated by −L. We say that an L¹(ℝ^d)-function f belongs to the Hardy space $H^{1}_{L}$ associated with L if sup _t>0|K_tf| belongs to L¹(ℝ^d). We prove that $f\in H^{1}_{L}$ if and only if R_jf∈L¹(ℝ^d) for j=1,…, d, where R_j=(∂/∂x_j)L^−1/2 are the Riesz transforms associated with L.