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Maximal operator on variable exponent spaces

Daviti Adamadze, Lars Diening, Tengiz Kopaliani

TL;DR

The paper addresses the boundedness of the Hardy-Littlewood maximal operator $M$ on variable exponent spaces $L^{p(\cdot)}$ under Nekvinda decay and Muckenhoupt-type conditions. It introduces a robust two-stage approach: first prove the boundedness for exponents with finite upper bound $p^+<\infty$ using a dyadic Calderón-Zygmund framework and local $A_\infty$-weights, then extend to unbounded exponents by approximating $p$ with a family $p_k$ that preserves $\mathcal{A}$ and $\mathcal{N}$ constants and passing to the limit via Fatou and lower semicontinuity. A novel exponent-approximation mechanism $\frac{1}{p_k(x)}=\frac{1}{k+1}+\frac{k-1}{k+1}\frac{1}{p(x)}$ ensures control of constants independent of $p^+$, enabling the final result for $p \in \mathcal{A}\cap\mathcal{N}$ with $p^->1$. The main contribution is extending known boundedness results to unbounded exponents and providing a concrete, transferable method for handling limits in variable-exponent harmonic analysis, with wide implications for PDEs and models with nonstandard growth.

Abstract

We explore the boundedness of the Hardy-Littlewood maximal operator $M$ on variable exponent spaces. Our findings demonstrate that the Muckenhoupt condition, in conjunction with Nekvinda's decay condition, implies the boundedness of $M$ even for unbounded exponents. This extends the results of Lerner, Cruz-Uribe and Fiorenza for bounded exponents. We also introduce a novel argument that allows approximate unbounded exponents by bounded ones while preserving the Muckenhoupt and Nekvinda conditions.

Maximal operator on variable exponent spaces

TL;DR

The paper addresses the boundedness of the Hardy-Littlewood maximal operator on variable exponent spaces under Nekvinda decay and Muckenhoupt-type conditions. It introduces a robust two-stage approach: first prove the boundedness for exponents with finite upper bound using a dyadic Calderón-Zygmund framework and local -weights, then extend to unbounded exponents by approximating with a family that preserves and constants and passing to the limit via Fatou and lower semicontinuity. A novel exponent-approximation mechanism ensures control of constants independent of , enabling the final result for with . The main contribution is extending known boundedness results to unbounded exponents and providing a concrete, transferable method for handling limits in variable-exponent harmonic analysis, with wide implications for PDEs and models with nonstandard growth.

Abstract

We explore the boundedness of the Hardy-Littlewood maximal operator on variable exponent spaces. Our findings demonstrate that the Muckenhoupt condition, in conjunction with Nekvinda's decay condition, implies the boundedness of even for unbounded exponents. This extends the results of Lerner, Cruz-Uribe and Fiorenza for bounded exponents. We also introduce a novel argument that allows approximate unbounded exponents by bounded ones while preserving the Muckenhoupt and Nekvinda conditions.

Paper Structure

This paper contains 7 sections, 12 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Variable exponent spaces
  3. Limits of functions and exponents
  4. Muckenhoupt condition for variable exponents
  5. Proof of the main result
  6. Using local A-infinity-weights
  7. Proof of the main theorem

Key Result

theorem 1

Let $p\in \mathcal{A}\cap\mathcal{N}$ with $p^->1$. Then there exists a constant $c>0$ depending on $(p')^+,\,[p]_{\mathcal{A}},\,[p]_{\mathcal{N}}$ and $n$ such that for any $f\in L^{p(\cdot)}(\mathbb{R}^{n})$

Theorems & Definitions (24)

  • theorem 1
  • remark 1
  • lemma 1: Norm conjugate formula
  • lemma 2
  • lemma 3: Fatou property
  • lemma 4: Lower semi-continuity
  • lemma 5: Approximation of semi-modular
  • remark 2
  • remark 3
  • definition 1: Muckenhoupt class
  • ...and 14 more