Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators

TL;DR

This work extends Muckenhoupt weight theory to a capacitary setting based on the $\delta$-dimensional Hausdorff content. It introduces the capacitary Muckenhoupt class ${\mathcal A}_{p,\delta}$ and proves that the strong-type and weak-type weighted inequalities for the Hardy-Littlewood maximal operator associated with ${\mathcal H}_{\infty}^{\delta}$ are characterized precisely by membership in ${\mathcal A}_{p,\delta}$ and ${\mathcal A}_{1,\delta}$, respectively. A novel approach using sparse coverings and a weighted packing condition replaces classical linearity and Fubini-type tools, enabling a full analogue of the classical A_p theory, including a reverse Hölder inequality, self-improving property, and Jones' factorization within the capacitary framework. Applications include Jones' factorization in this setting and a capacitary criterion for the boundedness of the capacitary maximal operator on weak-type Choquet-Lebesgue spaces, thereby extending the theory to fractal and lower-dimensional contexts.

Abstract

Let $\mathcal H_{\infty}^δ$ denote the Hausdorff content of dimension $δ\in(0,n]$ defined on subsets of $\mathbb R^n$. The principal problem, considered in this paper, is to characterize the non-negative function $w$ for which the weighted $L^p$-norm inequality with $p\in(1,\infty)$ and the weighted weak $L^1$-norm inequality on Hardy-Littlewood maximal operators associated with Hausdorff contents hold true. To achieve this, we introduce a class of capacitary Muckenhoupt weights depending on the dimension $δ$, denoted as $\mathcal A_{p,δ}$, which enjoys the strict monotonicity on the dimension index $δ$. Then we show that, for any $p\in(1,\infty)$ and $δ\in(0,n]$, the weighted $L^p$-norm inequality holds true if and only if $w\in\mathcal A_{p,δ}$, and the weighted weak $L^1$-norm inequality holds true if and only if $w\in\mathcal A_{1,δ}$ by a new approach developed in this paper. As the second objective, applying this new approach, the seminal properties of classical Muckenhoupt $A_p$ weights, such as the reverse Hölder inequality [R. R. Coifman and C. Fefferman, Studia Math. 51 (1974), 241-250], the self-improving property [B. Muckenhoupt, Trans. Amer. Math. Soc. 165 (1972), 207-226], and Jones' factorization theorem [P. W. Jones, Ann. of Math. (2) 111 (1980), 511-530], are all established within the framework of capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$. Finally, we also show that the maximal operator is bounded on the weak weighted Choquet-Lebesgue space $L_w^{p,\infty}(\mathbb R^n,{\mathcal H}_\infty^δ)$ if and only if $w\in\mathcal A_{p,δ}$ with $p\in(1,\infty)$ and $δ\in(0,n]$.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Corollary 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10