On Riesz transforms characterization of H^1 spaces associated with some Schr\"odinger operators

TL;DR

This paper characterizes the Hardy space associated with certain Schrödinger operators using Riesz transforms, extending classical harmonic analysis results to operators with non-negative potentials.

Contribution

It establishes that under specific conditions on the potential V, the Hardy space H^1_L can be characterized by Riesz transforms linked to the Schrödinger operator.

Findings

H^1_L characterized by Riesz transforms under certain conditions

Applicable to potentials V ≥ 0 in L^1_{loc} in one dimension

Extends classical harmonic analysis to Schrödinger operators

Abstract

Let Lf(x)=-\Delta f(x) + V(x)f(x), V\geq 0, V\in L^1_{loc}(R^d), be a non-negative self-adjoint Schr\"odinger operator on R^d. We say that an L^1-function f belongs to the Hardy space H^1_L if the maximal function M_L f(x)=\sup_{t>0} |e^{-tL} f(x)| belongs to L^1(R^d). We prove that under certain assumptions on V the space H^1_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 L^1_{loc}, in one dimension.