Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators
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2025
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| author | Huang, Long Zhang, Yangzhi Zhuo, Ciqiang |
| author_facet | Huang, Long Zhang, Yangzhi Zhuo, Ciqiang |
| contents | 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 $δ\in(0,n]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_23839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators Huang, Long Zhang, Yangzhi Zhuo, Ciqiang Classical Analysis and ODEs Functional Analysis 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 $δ\in(0,n]$. |
| title | Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2509.23839 |