On Kigami's conjecture of the embedding 𝒲^p(K)⊂ C(K)

Let (K,d) be a connected compact metric space and p∈ (1, ∞). Under the assumption of and the conductive p-homogeneity, we show that 𝒲^p(K)⊂ C(K) holds if and only if p>dim_AR(K,d), where 𝒲^p(K) is Kigami's (1,p)-Sobolev space and dim_AR(K,d) is the Ahlfors regular dimension.