ON KIGAMI’S CONJECTURE OF THE EMBEDDING Wp(K) ⊂ C(K)

Shiping Cao; Zhen Qing Chen; Takashi Kumagai

doi:10.1090/proc/16779

ON KIGAMI’S CONJECTURE OF THE EMBEDDING W^p(K) ⊂ C(K)

Shiping Cao, Zhen Qing Chen, Takashi Kumagai

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

Let (K, d) be a connected compact metric space and p ∈ (1, ∞). Under the assumption of Kigami [Conductive homogeneity of compact metric spaces and construction of p-energy, Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] 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.

Original language	English
Pages (from-to)	3393-3402
Number of pages	10
Journal	Proceedings of the American Mathematical Society
Volume	152
Issue number	8
DOIs	https://doi.org/10.1090/proc/16779
Publication status	Published - 1 Aug 2024
Externally published	Yes

Keywords

conductance
Dirichlet form
energy measure
extension operator
harmonic function
Mosco convergence
Sierpiński carpets
weak convergence

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title = "ON KIGAMI{\textquoteright}S CONJECTURE OF THE EMBEDDING Wp(K) ⊂ C(K)",

abstract = "Let (K, d) be a connected compact metric space and p ∈ (1, ∞). Under the assumption of Kigami [Conductive homogeneity of compact metric spaces and construction of p-energy, Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that Wp(K) ⊂ C(K) holds if and only if p > dimAR(K, d), where Wp(K) is Kigami{\textquoteright}s (1, p)-Sobolev space and dimAR(K, d) is the Ahlfors regular dimension.",

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AU - Cao, Shiping

AU - Chen, Zhen Qing

AU - Kumagai, Takashi

PY - 2024/8/1

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N2 - Let (K, d) be a connected compact metric space and p ∈ (1, ∞). Under the assumption of Kigami [Conductive homogeneity of compact metric spaces and construction of p-energy, Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that Wp(K) ⊂ C(K) holds if and only if p > dimAR(K, d), where Wp(K) is Kigami’s (1, p)-Sobolev space and dimAR(K, d) is the Ahlfors regular dimension.

AB - Let (K, d) be a connected compact metric space and p ∈ (1, ∞). Under the assumption of Kigami [Conductive homogeneity of compact metric spaces and construction of p-energy, Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that Wp(K) ⊂ C(K) holds if and only if p > dimAR(K, d), where Wp(K) is Kigami’s (1, p)-Sobolev space and dimAR(K, d) is the Ahlfors regular dimension.

KW - conductance

KW - Dirichlet form

KW - energy measure

KW - extension operator

KW - harmonic function

KW - Mosco convergence

KW - Sierpiński carpets

KW - weak convergence

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JF - Proceedings of the American Mathematical Society

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ER -

ON KIGAMI’S CONJECTURE OF THE EMBEDDING W^p(K) ⊂ C(K)

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