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Potential Analysis

, Volume 35, Issue 1, pp 39–50 | Cite as

On Riesz Transforms Characterization of H 1 Spaces Associated with Some Schrödinger Operators

  • Jacek Dziubański
  • Marcin Preisner
Open Access
Article

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
$$ \mathcal M_{\mathcal L} f(x)=\sup\limits_{t>0}|e^{-t\mathcal L} f(x)| $$
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.

Keywords

Riesz transforms Hardy spaces Schrödinger operators Semigroups of linear operators 

Mathematics Subject Classifications (2010)

42B30 42B35 35J10 

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Instytut Matematyczny Uniwersytet WrocławskiWrocławPoland

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