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On the smallness of mean oscillations on metric-measure spaces and applications

Dung Le

Abstract

It will be established that the mean oscillation of a function on a metric-measure space $X\times Y$ will be small if its mean oscillation on $X$ is small and some simple information on its (partial $Y$) upper-gradient is given. Applications to the regularity and global existence of bounded solutions to strongly coupled elliptic/parabolic systems on thin domains are also considered.

On the smallness of mean oscillations on metric-measure spaces and applications

Abstract

It will be established that the mean oscillation of a function on a metric-measure space will be small if its mean oscillation on is small and some simple information on its (partial ) upper-gradient is given. Applications to the regularity and global existence of bounded solutions to strongly coupled elliptic/parabolic systems on thin domains are also considered.
Paper Structure (5 sections, 8 theorems, 48 equations)

This paper contains 5 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. The smallness of mean oscillations on metric-measure spaces
  3. A theoretic functional result
  4. A variance of Theorem \ref{['KEYBMOthmmetric']}
  5. Euclidean spaces and applications

Key Result

Theorem 2.1

Let $X,Y$ be separable metric-measure spaces with $\sigma$-finite Borel measures $\mu,\lambda$ respectively. We assume that For some $\varepsilon>0$, let $B_R, B'_r$ be balls in $X,Y$ respectively and let $W$ be a integrable function on $B_R\times B'_r$. We also assume that $\mu(B_R\times \{y_*\})=\mu(B_R)$ for all $y_*\in Y$, and Then, we can assert that for the cube $\mathbf{B}_{R,r}=B_R\times

Theorems & Definitions (12)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1
  • Remark 3.2
  • ...and 2 more