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A new type of bmo space for non-doubling measures

Shining Li, Haijing Zhao, Baode Li

Abstract

Let $μ$ be a Radon measure on $\mathbb R^{d}$ which may be non-doubling and only satisfies $μ(Q(x,l))\le C_{0}l^{n}$} for all $x\in \mathbb R^{d}$, $l(Q)>0$, with some fixed constants $C_{0}>0$ and $n\in (0,d]$. We introduce a new type of $bmo(μ)$ space which looks bigger than the $rbmo(μ)$ space of Dachun Yang (JAMS,\,2005). And its four equivalent norms are established by constructing some special types of auxiliary doubling cubes. Then we further obtain that this new $rbmo(μ)$ space actually coincides with the $rbmo(μ)$ space of Dachun Yang.

A new type of bmo space for non-doubling measures

Abstract

Let be a Radon measure on which may be non-doubling and only satisfies } for all , , with some fixed constants and . We introduce a new type of space which looks bigger than the space of Dachun Yang (JAMS,\,2005). And its four equivalent norms are established by constructing some special types of auxiliary doubling cubes. Then we further obtain that this new space actually coincides with the space of Dachun Yang.

Paper Structure

This paper contains 4 sections, 13 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Some results for $(\alpha,\beta)$-doubling cubes
  3. Equivalence of four $rbmo(\mu)$-type spaces
  4. $rbmo^{1}_{\mathscr Q}(\mu)=rbmo(\mu) \subset RBMO(\mu)$

Key Result

Lemma 2.1

(6) Let $E$ be a bouned set in $\mathbb R^{d}$. If, for every $x\in E$, there exists a closed cube $Q_{x}$ centered at $x$, then it is possible to choose, from among the given cubes $\left\{Q_{x}\right\}_{x\in E}$, a subsequence $\left\{Q_{k}\right\}_{k}$ (possibly finite) such that

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • ...and 19 more