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

Abstract

Let $\mathcal H_{\infty}^\delta$ denote the Hausdorff content of dimension $\delta\in(0,n]$ 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 $\delta$, denoted as $\mathcal A_{p,\delta}$, which enjoys the strict monotonicity on the dimension index $\delta$. Then we show that, for any $p\in(1,\infty)$ and $\delta\in(0,n]$, the weighted $L^p$-norm inequality holds true if and only if $w\in\mathcal A_{p,\delta}$, and the weighted weak $L^1$-norm inequality holds true if and only if $w\in\mathcal A_{1,\delta}$ 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 H\"older 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 $\mathcal A_{p,\delta}$. Finally, we also show that the maximal operator is bounded on the weak weighted Choquet-Lebesgue space $L_w^{p,\infty}(\mathbb R^n,{\mathcal H}_\infty^\delta)$ if and only if $w\in\mathcal A_{p,\delta}$ with $p\in(1,\infty)$ and $\delta\in(0,n]$.