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Strong maximal function revisit on Heisenberg group

Chuhan Sun

TL;DR

This work studies the $L^p$-boundedness of a strong maximal operator on the Heisenberg group with respect to an absolutely continuous measure that satisfies a product $A_\infty$ condition. The authors adapt the Córdoba-Fefferman multi-parameter covering framework to the weighted Heisenberg setting, establishing a weighted weak-type bound and then obtaining $L^p$-boundedness through interpolation. The main result, Theorem A*, extends the classical unweighted theory (Christ) to a weighted, multi-parameter context on the Heisenberg group, and Section 3 provides a self-contained proof of the necessary covering lemma. Overall, the paper broadens the scope of maximal operator theory to weighted, multi-parameter analysis on nilpotent Lie groups, with potential applications in harmonic analysis on noncommutative spaces.

Abstract

We prove the $L^p$-boundedness of the strong maximal operator defined on a Heisenberg group w.r.t an absolutely continuous measure satisfying the product $A_\infty$-property.

Strong maximal function revisit on Heisenberg group

TL;DR

This work studies the -boundedness of a strong maximal operator on the Heisenberg group with respect to an absolutely continuous measure that satisfies a product condition. The authors adapt the Córdoba-Fefferman multi-parameter covering framework to the weighted Heisenberg setting, establishing a weighted weak-type bound and then obtaining -boundedness through interpolation. The main result, Theorem A*, extends the classical unweighted theory (Christ) to a weighted, multi-parameter context on the Heisenberg group, and Section 3 provides a self-contained proof of the necessary covering lemma. Overall, the paper broadens the scope of maximal operator theory to weighted, multi-parameter analysis on nilpotent Lie groups, with potential applications in harmonic analysis on noncommutative spaces.

Abstract

We prove the -boundedness of the strong maximal operator defined on a Heisenberg group w.r.t an absolutely continuous measure satisfying the product -property.
Paper Structure (3 sections, 1 theorem, 51 equations)

This paper contains 3 sections, 1 theorem, 51 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem A*
  3. Proof of the covering lemma

Key Result

Lemma 3.1

If ${\bf w}\in\bigotimes_{i=1}^k{\bf A}_\infty\left(\mathbb R^{2n+1}\right)$, then ${\bf w}$ satisfies the following: If ${\bf R}\subset\mathbb R^{2n+1}$ is any rectangle with its sides parallel to the axes and $E\subset {\bf R}$ is such that ${\bf vol}\left(E\right)>\frac{1}{2}{\bf vol}\left({\bf R

Theorems & Definitions (1)

  • Lemma 3.1