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Sufficient condition for boundedness of maximal operator on weighted generalized Orlicz spaces

Vertti Hietanen

TL;DR

The paper establishes a sufficiency condition for the boundedness of the Hardy-Littlewood maximal operator on weighted generalized Orlicz spaces $L^oldsymbol{ extphi}(oldsymbol{ extOmega},oldsymbol{ extomega})$. The result requires the weight to satisfy a Muckenhoupt condition $oldsymbol{ extomega} in A_p$ and the modular to obey the almost-increasing property $(aInc)_p$ in addition to standard $oldsymbol{ extphi}$-conditions, unifying unweighted and weighted theories. The core method treats the weight as a measure and derives a weighted Jensen-type key estimate, extending the classical $A_1$/$A_2$ framework to weighted variants $A1_w$, $A2_w$, which then yields the boundedness via the unit ball property. The findings apply to important special cases, including variable exponent spaces and double-phase functionals, under natural regularity like log-H"older continuity/decay and Nekvinda-type conditions, broadening the applicability of maximal operator bounds in Musielak-Orlicz settings.

Abstract

We prove that the Hardy-Littlewood maximal operator is bounded in the weighted generalized Orlicz space if the weight satisfies the classical Muckenhoupt condition $A_p$ and $t \to \frac{\varphi(x,t)}{t^p}$ is almost increasing in addition to the standard conditions.

Sufficient condition for boundedness of maximal operator on weighted generalized Orlicz spaces

TL;DR

The paper establishes a sufficiency condition for the boundedness of the Hardy-Littlewood maximal operator on weighted generalized Orlicz spaces . The result requires the weight to satisfy a Muckenhoupt condition and the modular to obey the almost-increasing property in addition to standard -conditions, unifying unweighted and weighted theories. The core method treats the weight as a measure and derives a weighted Jensen-type key estimate, extending the classical / framework to weighted variants , , which then yields the boundedness via the unit ball property. The findings apply to important special cases, including variable exponent spaces and double-phase functionals, under natural regularity like log-H"older continuity/decay and Nekvinda-type conditions, broadening the applicability of maximal operator bounds in Musielak-Orlicz settings.

Abstract

We prove that the Hardy-Littlewood maximal operator is bounded in the weighted generalized Orlicz space if the weight satisfies the classical Muckenhoupt condition and is almost increasing in addition to the standard conditions.
Paper Structure (5 sections, 16 theorems, 47 equations)

This paper contains 5 sections, 16 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of $A_p$ -weights
  4. Boundedness
  5. Special cases

Key Result

Lemma 2.4

Let $\varphi \in \Phi_\textnormal{w}(\Omega)$. Then $\varphi$ satisfies defn:A2w if and only if there exist $\varphi_{\infty}\in \Phi_\textnormal{w}$, $h\in L^1(\Omega,\omega)\cap L^{\infty}(\Omega)$ and $\beta \in (0,1]$ such that for almost every $x\in \Omega$.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: Og, Lemma 4.3.1
  • Definition 2.6
  • Lemma 2.7: Og, Lemma 3.2.3
  • Definition 3.1
  • Lemma 3.2: muck, Theorem 9
  • ...and 25 more