End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

Yanping Chen,Wenyu Tao,Richard I Avery

doi:10.1155/2021/8867966

Abstract

Let L = − Δ + μ be the generalized Schrödinger operator on ℝ d , d ≥ 3 , where μ ≠ 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new BMO space associated to the generalized Schrödinger operator L , BM O θ , L , which is bigger than the BMO spaces related to the classical Schrödinger operators A = − Δ + V , with V a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to L in BM O θ , L also be proved.

Highlights

Consider the generalized Schrödinger operatorL = −Δ + μ, ð1Þ where μ is a nonnegative Radon measure on Rd, d ≥ 3
V ≥ 0 are in the reverse Hölder class, that is, there exists C = Cðd, VÞ > 0 such that
We will consider the follows Littlewood-Paley operators associated to the generality Schrödinger operator are bounded in BMOθ,L

Summary

Introduction

We shall be interested in a new BMO space associated to the generalized Schrödinger operator L. To give the definition of the new BMO space, we first recall the auxiliary function ρðx, μÞ (see [8]), ρðx, μÞ. We define the new BMO space, BMOθ,L , namely, BMOθ,L. After Wang [10] considered the g-function defined on BMO functions, more and more scholars pay attention to the end point estimate of the Littlewood-Paley operator [3, 8, 11,12,13]. We will consider the follows Littlewood-Paley operators associated to the generality Schrödinger operator are bounded in BMOθ,L. Throughout this paper, give a ball B, we denote by B∗ the ball with the same center and twice radius. c and C will denote positive constants that may not be the same in each occurrence

Preliminaries

Proof of Theorem 3