Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials
Let L=−Δ+V be a Schrödinger operator on ℝd, d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to Lp(ℝd), for some p>d/2. Let Kt be the semigroup generated by −L. We say that an L1(ℝd)-function f belongs to the Hardy space $H^{1}_{L}$ associated with L if sup t>0|Ktf| belongs to L1(ℝd). We prove that $f\in H^{1}_{L}$ if and only if Rjf∈L1(ℝd) for j=1,…,d, where Rj=(∂/∂xj)L−1/2 are the Riesz transforms associated with L.