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On Riesz transforms characterization of H^1 spaces associated with some Schrödinger operators (1006.5189v1)

Published 27 Jun 2010 in math.FA

Abstract: Let Lf(x)=-\Delta f(x) + V(x)f(x), V\geq 0, V\in L1_{loc}(Rd), be a non-negative self-adjoint Schr\"odinger operator on Rd. We say that an L1-function f belongs to the Hardy space H1_L if the maximal function M_L f(x)=\sup_{t>0} |e{-tL} f(x)| belongs to L1(Rd). We prove that under certain assumptions on V the space H1_L is also characterized by the Riesz transforms R_j=\frac{\partial}{\partial x_j} L{-1/2}, j=1,...,d, associated with L. As an example of such a potential V one can take any V\geq 0, V\in L1_{loc}, in one dimension.

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