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arXiv · 2307.10449

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

Shiping Cao·Zhen-Qing Chen·Takashi Kumagai
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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.

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Shiping Cao, Zhen-Qing Chen, Takashi Kumagai. 2024-01-04. On Kigami's conjecture of the embedding $\mathcal{W}^p(K)\subset C(K)$. https://arxiv.org/abs/2307.10449

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