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

Let $(K,d)$ be a connected compact metric space and $p\in (1,\mathrm{\infty })$. 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 $p>{\dim }_{AR}(K,d)$, where ${\mathcal{W}}^p(K)$ is Kigami’s $(1,p)$-Sobolev space and ${\dim }_{AR}(K,d)$ is the Ahlfors regular dimension.