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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
##### Keywords:
Hardy space; Riesz transform; Schrödinger operator
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##### References:
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