Hardy spaces related to Schr\"odinger operators with potentials which are sums of L^p-functions

TL;DR

This paper studies Hardy spaces associated with Schr"odinger operators with potentials that are sums of functions in L^p spaces, establishing isomorphisms with classical Hardy spaces and characterizations via Riesz transforms.

Contribution

It introduces new isomorphisms between Hardy spaces related to Schr"odinger operators with sum potentials and classical Hardy spaces, along with atomic and Riesz transform characterizations.

Findings

Existence of two distinct isomorphisms with classical Hardy space.

Atomic characterization of the Hardy space H^1_L.

Description of H^1_L via Riesz transforms R_{L,i}.

Abstract

We investigate the Hardy space H^1_L associated to the Schr\"odinger operator L=-\Delta+V on R^n, where V=\sum_{j=1}^d V_j. We assume that each V_j depends on variables from a linear subspace VV_j of \Rn, dim VV_j \geq 3, and V_j belongs to L^q(VV_j) for certain q. We prove that there exist two distinct isomorphisms of H^1_L with the classical Hardy space. As a corollary we deduce a specific atomic characterization of H_L^1. We also prove that the space H_L^1 is described by means of the Riesz transforms R_{L,i} = \partial_i L^{-1/2}.