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$β$-dimensional sharp maximal function and applications

You-Wei Benson Chen, Alejandro Claros

TL;DR

This work extends the classical sharp maximal function framework to a beta-dimensional setting defined via the Hausdorff content $\mathcal{H}^\beta_\infty$ by introducing the $\beta$-dimensional sharp maximal operator $\mathcal{M}^{\#}_\beta$. The authors establish a Fefferman–Stein inequality and a Muckenhoupt–Wheeden type inequality in this Choquet–Hausdorff context using good-$\lambda$ and packing techniques, and they connect these sharp maximal estimates to fractional and Riesz potentials. A key contribution is the pointwise comparison $\mathcal{M}^{\#}_\beta(I_\alpha f) \sim \mathcal{M}_\alpha f$, enabling Adams-type Morrey-type implications and enabling $L^p(\mathcal{H}^\beta_\infty)$ control. They develop dyadic and continuous good-$\lambda$ estimates for Riesz potentials with respect to Hausdorff content, yielding sharp bounds and exponential decay. Finally, the results are applied to PDEs, proving local exponential integrability of gradient estimates for $p$-Laplace type equations with measure data in capacitary terms, highlighting the practical impact on nonlinear PDE analysis.

Abstract

In this paper, we study $β$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _βf(x) := \sup_{Q} \inf_{c \in \mathbb{R}} χ_{Q}(x) \frac{1}{\ell(Q)^β} \int_Q |f-c| \; d \mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all cubes in $\mathbb{R}^d$ with sides pararell to the coordinate axes, $\ell(Q)$ is the length side of $Q$ and $\mathcal{H}^β_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{M}^{\#} _βf$ by giving a good lambda estimate for $β$-dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest.

$β$-dimensional sharp maximal function and applications

TL;DR

This work extends the classical sharp maximal function framework to a beta-dimensional setting defined via the Hausdorff content by introducing the -dimensional sharp maximal operator . The authors establish a Fefferman–Stein inequality and a Muckenhoupt–Wheeden type inequality in this Choquet–Hausdorff context using good- and packing techniques, and they connect these sharp maximal estimates to fractional and Riesz potentials. A key contribution is the pointwise comparison , enabling Adams-type Morrey-type implications and enabling control. They develop dyadic and continuous good- estimates for Riesz potentials with respect to Hausdorff content, yielding sharp bounds and exponential decay. Finally, the results are applied to PDEs, proving local exponential integrability of gradient estimates for -Laplace type equations with measure data in capacitary terms, highlighting the practical impact on nonlinear PDE analysis.

Abstract

In this paper, we study -dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _βf(x) := \sup_{Q} \inf_{c \in \mathbb{R}} χ_{Q}(x) \frac{1}{\ell(Q)^β} \int_Q |f-c| \; d \mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all cubes in with sides pararell to the coordinate axes, is the length side of and is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for by giving a good lambda estimate for -dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest.
Paper Structure (6 sections, 21 theorems, 156 equations)

This paper contains 6 sections, 21 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. good lambda estimates for the $\beta$-dimensional sharp maximal function and proof of Theorem \ref{['Maintheorem3']}
  4. pointwise estimate of fractional maximal function and sharp maximal function of $I_\alpha f$
  5. Good lambda estimate for Riesz potential with respect to Hausdorff content and applications
  6. An application to PDEs: exponential integrability of gradient estimates for $p$-Laplace type equation with measure data

Key Result

Theorem 1.2

Let $0 < \beta \leq d \in \mathbb{N}$, $0<p_0<\infty$ and $f\in L^{1}_{loc}(\mathcal{H}^\beta_\infty)$ such that Then there exists a constant $C = C(p,d,\beta)>0$ such that for all $p_0<p<\infty$, and also for all $p_0\le p<\infty$. In here, $L^{p,\infty}( \mathcal{H}^\beta_\infty)$ denotes the vector space of all the functions $g: \mathbb{R}^d \to \mathbb{R}$ satisfying

Theorems & Definitions (43)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4: Adams Adams1975
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 33 more