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The maximal operator on variable Lebesgue spaces: an ${\mathcal A}_{\infty}$-characterization

Andrei K. Lerner

Abstract

In this paper we obtain a new boundedness criterion for the maximal operator $M$ on variable exponent spaces $L^{p(\cdot)}$. It is formulated in terms of the variable exponent analogue of the well known weighted $A_{\infty}$ condition.

The maximal operator on variable Lebesgue spaces: an ${\mathcal A}_{\infty}$-characterization

Abstract

In this paper we obtain a new boundedness criterion for the maximal operator on variable exponent spaces . It is formulated in terms of the variable exponent analogue of the well known weighted condition.
Paper Structure (4 sections, 9 theorems, 72 equations)

This paper contains 4 sections, 9 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the ${\mathcal{A}}_{\infty}$ condition
  4. Proof of Theorem \ref{['boundml']}

Key Result

Theorem 1.1

Let $p_->1$ and $p_+<\infty$We use the standard notation $p_-:=\mathop{\mathrm{ess\,inf}}\limits p$ and $p_+:=\mathop{\mathrm{ess\,sup}}\limits p$.. Then $p(\cdot)\in {\mathcal{P}}$ if and only if $p(\cdot)\in {\mathcal{A}}$.

Theorems & Definitions (14)

  • Theorem 1.1: D05
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4: L27
  • Theorem 1.5
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 4 more