Riesz transform characterization of Hardy spaces associated with Schr\"odinger operators with compactly supported potentials

TL;DR

This paper characterizes Hardy spaces linked to Schrödinger operators with compactly supported potentials using Riesz transforms, establishing an equivalence between membership in the Hardy space and the integrability of these transforms.

Contribution

It provides a new Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials.

Findings

Hardy space H_L^1 characterized by Riesz transforms R_j

Equivalence between Hardy space membership and Riesz transform integrability

Extension of classical Hardy space theory to Schrödinger operators

Abstract

Let L=-\Delta+V be a Schr\"odinger operator on R^d, d\geq 3. We assume that V is a nonnegative, compactly supported potential that belongs to L^p(R^d), for some p>d/2. Let K_t be the semigroup generated by -L. We say that an L^1(R^d)-function f belongs to the Hardy space H_L^1 associated with L if sup_{t>0} |K_t f| belongs to L^1(R^d). We prove that f\in H_L^1 if and only if R_j f \in L^1(R^d) for j=1,...,d, where R_j= \frac{d}{dx_j} L^{-1/2} are the Riesz transforms associated with L.