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

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

2009-10-06

arxiv.org/abs/0910.1017v1

Arkiv for Matematik 48 (2010), no. 2, 301--310

Mathematics

Functional Analysis

6 pages

Scientific paper

10.1007/s11512-010-0121-5

Let L=-\Delta+V be a Schr\"odinger operator on R^d, d\geq 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 L^1(R^d)-function f belongs to the Hardy space H_L^1 associated with L if sup_{t>0} |K_t f| belongs to L^1(R^d). We prove that f\in H_L^1 if and only if R_j f \in L^1(R^d) for j=1,...,d, where R_j= \frac{d}{dx_j} L^{-1/2} are the Riesz transforms associated with L.

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