BMO Functions Generated by    A   X       ℝ   n      Weights on Ball Banach Function Spaces

Ruimin Wu,Songbai Wang

doi:10.1155/2021/6626787

Abstract

Let X be a ball Banach function space on ℝ n . We introduce the class of weights A X ℝ n . Assuming that the Hardy-Littlewood maximal function M is bounded on X and X ′ , we obtain that BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A X ℝ n . As a consequence, we have BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A L p · ℝ n ℝ n , where L p · ℝ n is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space X ω for any ω ∈ A X ℝ n , then the commutator b , T is bounded on X with b ∈ BMO ℝ n .

Highlights

It is well known that there is a relation between A∞ðRnÞ weights and BMOðRnÞ, i.e., for any p ∈ ð1, ∞, BMOðRnÞ = È α ln W : α ≥ ∈ Ap ðRn É Þ: ð1ÞSee, for instance, [1] (p. 409)
The purpose of this note is to reveal the relation between BMOðRnÞ and AXðRnÞ weights over the ball Banach function space X
We begin with the definition of the ball Banach function space

Summary

Introduction

It is well known that there is a relation between A∞ðRnÞ weights and BMOðRnÞ, i.e., for any p ∈ ð1, ∞, BMOðRnÞ. The purpose of this note is to reveal the relation between BMOðRnÞ and AXðRnÞ weights over the ball Banach function space X. We begin with the definition of the ball Banach function space. For any ball Banach function space X, the associate space (Köthe dual) X ′ is defined by setting n n oo. Let X be ball Banach function spaces. Lebesgue space Lpð·ÞðRnÞ is defined to be the set of all measurable functions f on Rn such that Let pð·Þ be a globally log-Hölder continuous function satisfying 1 < p− ≤ p+ < ∞.

Proof of Theorem 5

Applications