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A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type

Oleksiy Karlovych, Alina Shalukhina

Abstract

Let $(X,d,μ)$ be a space of homogeneous type and $p(\cdot):X\to[1,\infty]$ be a variable exponent. We show that if the measure $μ$ is Borel-semiregular and reverse doubling, then the condition ${\rm ess\,inf}_{x\in X}p(x)>1$ is necessary for the boundedness of the Hardy-Littlewood maximal operator $M$ on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$.

A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type

Abstract

Let be a space of homogeneous type and be a variable exponent. We show that if the measure is Borel-semiregular and reverse doubling, then the condition is necessary for the boundedness of the Hardy-Littlewood maximal operator on the variable Lebesgue space .
Paper Structure (4 sections, 3 theorems, 26 equations)

This paper contains 4 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction and the main result
  2. Preliminaries on spaces of homogeneous type
  3. Proof of the main result
  4. Final remark

Key Result

Theorem 1

Suppose $(X,d,\mu)$ is a space of homogeneous type which has the property that the measure $\mu$ is Borel-semiregular and reverse doubling. Given an exponent function $p(\cdot)\in\mathcal{P}(X)$, if the Hardy-Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof