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

Let $K\subset {\mathbb{R}}^d$ be a post-critically finite (p.c.f.) self-similar set with Hausdorff dimension $s$, and $\mu$ be a self-similar probability measure supported on $K$. Let $H_{\mu }^{\alpha }$, $0<\alpha \le s$, be the Hausdorff content on $K$, and $M_{\mathcal{D}}^{\mu }$ 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 $H_{\mu }^{\alpha }$ for all range $0<\alpha \le s$. As applications, the Lebesgue differentiation theorem on $K$ is proved. Moreover, via the Hardy–Littlewood maximal operator $M_{\mathcal{D}}^{\mu }$, we characterize the Lebesgue–Choquet space $L^p(K,H_{\mu }^{\alpha })$ and the Zygmund space $L\log L(K,\mu )$. To be exact, given $\alpha \mathrm{/}s<p\le \mathrm{\infty }$, we discover that $$f\in L^p(K,H_{\mu }^{\alpha })\text{ if and only if }{M}_{\mathcal{D}}^{\mu }f\in L^p(K,H_{\mu }^{\alpha })$$ and, for $f\in L^1(K,\mu )$ with $K$ satisfying the strong separation condition, $$M_{\mathcal{D}}^{\mu }f\in L^1(K,\mu )\text{ if and only if }f\in L\log L(K,\mu )\mathrm{.}$$ 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 $\mu$.