Quantitative Hardy--Littlewood maximal inequalities and Wiener--Stein theorem on p.c.f. fractals

Abstract

Let $K\subset \mathbb{R}^d$ be a post-critically finite (p.c.f.) self-similar set with Hausdorff dimension $s$, and $μ$ be a self-similar probability measure supported on $K$. Let $H^α_μ$, $0<α\le s$, be the Hausdorff content on $K$, and $M_{\mathcal{D}}^μ$ be the Hardy--Littlewood maximal operator defined on $K$ associated with its basic cubes $\mathcal{D}$. In this paper, we establish quantitative strong type and weak type Hardy--Littlewood maximal inequalities on fractal set $K$ with respect to $H^α_μ$ for all range $0<α\le s$. As applications, the Lebesgue differentiation theorem on $K$ is proved. Moreover, via the Hardy--Littlewood maximal operator $M_{\mathcal{D}}^μ$, we characterize the Lebesgue--Choquet space $L^p(K,H^α_μ)$ and the Zygmund space $L\log L(K,μ)$. To be exact, given $α/s< p\le \infty$, we discover that \[ \text{$f\in L^p(K,H^α_μ)$ if and only if $M_{\mathcal{D}}^μf\in L^p(K,H^α_μ)$}\] and, for $f\in L^1(K,μ)$ with $K$ satisfying the strong separation condition, \[\text{$M_{\mathcal{D}}^μf\in L^1(K,μ)$ if and only if $f\in L\log L(K,μ)$}.\] That is, Wiener's $L\log L$ inequality and its converse inequality due to Stein in 1969 are extended to fractal set $K$ with respect to $μ$.