Riesz Transform Characterization of Weighted Hardy Spaces Associated with Schrödinger Operators

Hua Zhu

doi:10.1155/2016/2856103

Abstract

We characterize the weighted local Hardy spaceshρ1(ω)related to the critical radius functionρand weightsω∈A1ρ,∞(Rn)by localized Riesz transformsR^j; in addition, we give a characterization of weighted Hardy spacesHL1(ω)via Riesz transforms associated with Schrödinger operatorL, whereL=-Δ+Vis a Schrödinger operator onRn(n≥3) andVis a nonnegative function satisfying the reverse Hölder inequality.

Highlights

Let L = −Δ + V be a Schrodinger operator on Rn, n ≥ 3, where V ≢ 0 is a fixed nonnegative potential
We assume that V belongs to the reverse Holder class RHs(Rn) for some s ≥ n/2; that is, there exists C = C(s, V) > 0 such that
With regard to the Schrodinger operator L, we know that the operators derived from L behave “locally” quite similar to those corresponding to the Laplacian

Summary

Introduction

Bongioanni et al [3] introduced new classes of weights, related to Schrodinger operators L, that is, Aρp,∞(Rn) weight, which are in general larger than Muckenhoupt’s (see Section 2 for notions of Aρp,∞(Rn) weight). Since it is difficult to give the characterization by the same method in [5, 8], in this paper, we will establish the characterization by a new method, which will take advantage of weighted local Hardy spaces hρp(ω) and localized Riesz transforms. By this method, we can give the Riesz transforms characterization of hρp(ω) at the same time. We set N ≡ {1, 2, . . .} and Z+ ≡ N ∪ {0}

Preliminaries

Riesz Transform Characterization of Weighted Hardy Spaces