On Riesz Transforms Characterization of H1 Spaces Associated with Some Schrödinger Operators

Abstract

Let Lf(x)=-δf(x)+V(x),V≥0,V∈L1loc(ℝd) be a non-negative self-adjoint Schrödinger operator on ℝd. We say that an L1-function f is an element of the Hardy space H1L if the maximal function MLf(x)=supt>0{pipe}e-tLf(x){pipe} belongs to L1(ℝd). We prove that under certain assumptions on V the space H1L is also characterized by the Riesz transforms, associated with L. As an example of such a potential V one can take any V ≥ 0,VεL1loc, in one dimension. © 2010 The Author(s).