Hardy spaces for Bessel-Schr\"odinger operators

TL;DR

This paper characterizes Hardy spaces associated with Bessel-Schrödinger operators using atomic decompositions, providing new insights into their structure under certain potential conditions.

Contribution

It introduces a novel atomic decomposition characterization of Hardy spaces for Bessel-Schrödinger operators, especially for the case a in (0,1) without extra potential assumptions.

Findings

Characterization of H^1(L) via atomic decompositions.

Applicable to a in (0,1) with minimal potential restrictions.

Provides a framework for analyzing Bessel-Schrödinger Hardy spaces.

Abstract

Consider the Bessel operator with a potential on L^2((0,infty), x^a dx), namely Lf(x) = -f"(x) - a/x f'(x) + V(x)f(x). We assume that a>0 and V\in L^1_{loc}((0,infty), x^a dx) is a non-negative function. By definition, a function f\in L^1((0,infty), x^a dx) belongs to the Hardy space H^1(L) if sup_{t>0} |e^{-tL} f| \in L^1((0,infty), x^a dx). Under certain assumptions on V we characterize the space H^1(L) in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to L for a \in (0,1) with no additional assumptions on the potential V.