On Kigami's conjecture of the embedding $\mathcal{W}^p(K)\subset C(K)$

TL;DR

The paper addresses Kigami's conjecture that the Sobolev-type space $\mathcal{W}^p(K)$ embeds into $C(K)$ on compact connected metric spaces under $p$-conductive homogeneity and Ki2 assumptions, characterizing the embedding in terms of the Ahlfors regular conformal dimension. Using a graph-energy framework with partitions, effective conductance, and neighbor disparity constants, it proves that the embedding holds exactly when $p>\operatorname{dim}_{AR}(K,d)$ by combining Hölder regularity for $p>\operatorname{dim}_{AR}$ with a constructed unbounded counterexample for $p\le\operatorname{dim}_{AR}$. The proof employs a Mosco-type convergence of energies and a weak-limit argument to produce an unbounded $\mathbb{L}^p$-function with uniformly bounded discrete energies, confirming the conjecture. The result extends Kigami's analytical framework to fractal settings such as Sierpiński carpets and informs potential theory, extensions, and the structure of $p$-energy spaces on infinitely ramified fractals.

Abstract

Let $(K,d)$ be a connected compact metric space and $p\in (1, \infty)$. Under the assumption of \cite[Assumption 2.15]{Ki2} and the conductive $p$-homogeneity, we show that $\mathcal{W}^p(K)\subset C(K)$ holds if and only if $p>\operatorname{dim}_{AR}(K,d)$, where $\mathcal{W}^p(K)$ is Kigami's $(1,p)$-Sobolev space and $\operatorname{dim}_{AR}(K,d)$ is the Ahlfors regular dimension.

Theorems & Definitions (16)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.6
• Theorem 2.7
• Definition 2.8
• Remark 2.9
• Definition 2.10