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Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials. (English) Zbl 1202.42046
Summary: Let \(L= -\Delta +V\) be a Schrödinger operator on \(\mathbb R^d\), \(d\geq 3\). We assume that \(V\) is a nonnegative, compactly supported potential that belongs to \(L^p (\mathbb R^d)\), for some \(p>d /2\). Let \(K t\) be the semigroup generated by \(- L\). We say that an \(L^1(\mathbb R^d )\)-function \(f\) belongs to the Hardy space \(H^1_L\) associated with \(L\) if \(\sup _{t>0}|K_t f|\) belongs to \(L^1(\mathbb R^d)\). We prove that \(f\in H^1_L\) if and only if \(R_j f\in L^1(\mathbb R^d)\) for \(j=1,\dots ,d\), where \(R_j =(\partial /\partial x_j )L- 1/2\) are the Riesz transforms associated with \(L\).

MSC:
42B35 Function spaces arising in harmonic analysis
35J10 Schrödinger operator, Schrödinger equation
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