Abstract
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 L p(ℝd), for some p>d /2. Let K t be the semigroup generated by −L. We say that an L 1(ℝd)-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup t>0|K t f| belongs to L 1(ℝd). We prove that \(f\in H^{1}_{L}\) if and only if R j f∈L 1(ℝd) for j=1,…,d, where R j =(∂/∂ x j )L −1/2 are the Riesz transforms associated with L.
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Supported by the Polish Ministry of Science and High Education—grant N N201 397137, the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability" MTKD-CT-2004-013389.
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Dziubański, J., Preisner, M. Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials. Ark Mat 48, 301–310 (2010). https://doi.org/10.1007/s11512-010-0121-5
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DOI: https://doi.org/10.1007/s11512-010-0121-5