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

TL;DR

This paper proves that Kigami's Sobolev space embeds into continuous functions on a connected compact metric space if and only if the integrability parameter exceeds the Ahlfors regular dimension, under certain assumptions.

Contribution

It establishes a precise condition relating Sobolev space embedding to Ahlfors regular dimension for spaces satisfying specific assumptions.

Findings

Embedding holds if and only if p > AR dimension.

Under certain assumptions, the characterization is exact.

Connects Sobolev embedding with geometric dimension.

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.