Abstract
Let \(\mathcal Lf(x)=-\Delta f (x)+V(x)f(x)\), V ≥ 0, \(V\in L^1_{loc}(\mathbb R^d)\), be a non-negative self-adjoint Schrödinger operator on \(\mathbb R^d\). We say that an L 1-function f is an element of the Hardy space \(H^1_{\mathcal L}\) if the maximal function
belongs to \(L^1(\mathbb R^d)\). We prove that under certain assumptions on V the space \(H^1_{\mathcal L}\) is also characterized by the Riesz transforms \(R_j=\frac{\partial}{\partial x_j}\mathcal L^{-1\slash 2}\), j = 1,...,d, associated with \(\mathcal L\). As an example of such a potential V one can take any V ≥ 0, \(V\in L^1_{loc}\), in one dimension.
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In memory of Andrzej Hulanicki.
The research partially supported by Polish Ministry of Science and High Education—grant N N201 397137.
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Dziubański, J., Preisner, M. On Riesz Transforms Characterization of H 1 Spaces Associated with Some Schrödinger Operators. Potential Anal 35, 39–50 (2011). https://doi.org/10.1007/s11118-010-9202-0
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DOI: https://doi.org/10.1007/s11118-010-9202-0