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

TL;DR

This work extends classical Hardy--Littlewood maximal theory to p.c.f. self-similar fractals by developing strong-type and weak-type inequalities for the Hardy--Littlewood maximal operator defined via basic cubes on $K$, with respect to the α-dimensional Hausdorff content $H^{oldsymbol{α}}_oldsymbol{μ}$ for all $0<oldsymbol{α}oldsymbol{≤} s$. It simultaneously establishes a Lebesgue differentiation theorem on fractals and characterizes Lebesgue--Choquet spaces $L^{p}(K,H^{oldsymbol{α}}_oldsymbol{μ})$ and Zygmund spaces $L ext{log}L(K,μ)$ through the maximal operator, including Wiener's $L ext{log}L$ inequality and its Stein-type converse in this non-doubling, nonlinear Choquet context. The proofs blend dyadic-analytic ideas with fractal geometry via basic cubes, and hinge on a new packing lemma to overcome Choquet’s nonlinearity, yielding explicit constants in the strong-type and weak-type bounds. The results apply to canonical fractals such as Cantor-type sets, the Sierpiński gasket, Vicsek set, and generalized Sierpiński carpets, providing a robust framework for harmonic analysis on fractals and paving the way for further function-space characterizations in non-doubling, non-Euclidean settings.

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 $μ$.

Figures (5)

• Figure 1: Three generations of the middle-forth Cantor set.
• Figure 2: Three generations of the generalized Sierpiński carpet.
• Figure 3: The unit interval can be regarded as a self-similar set.
• Figure 4: The first three approximations of the Sierpiński gasket.
• Figure 5: The first four approximations of the Vicsek set.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Proposition 1.6
• Proposition 1.7