On Kigami’s conjecture of the embedding $\mathcal {W}^p(K)\subset C(K)$
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- by Shiping Cao, Zhen-Qing Chen and Takashi Kumagai;
- Proc. Amer. Math. Soc. 152 (2024), 3393-3402
- DOI: https://doi.org/10.1090/proc/16779
- Published electronically: June 5, 2024
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Abstract:
Let $(K,d)$ be a connected compact metric space and $p\in (1, \infty )$. 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 $\mathcal {W}^p(K)\subset C(K)$ 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.References
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Bibliographic Information
- Shiping Cao
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 1228708
- ORCID: 0000-0002-5711-6632
- Email: spcao@uw.edu
- Zhen-Qing Chen
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 242576
- ORCID: 0000-0001-7037-4030
- Email: zqchen@uw.edu
- Takashi Kumagai
- Affiliation: Department of Mathematics, Waseda University, Tokyo 169-8555, Japan
- MR Author ID: 338696
- ORCID: 0000-0001-7515-1055
- Email: t-kumagai@waseda.jp
- Received by editor(s): July 19, 2023
- Received by editor(s) in revised form: November 30, 2023, January 2, 2024, and January 3, 2024
- Published electronically: June 5, 2024
- Additional Notes: The research of the first author was partially supported by a grant from the Simons Foundation Targeted Grant (917524) to the Pacific Institute for the Mathematical Sciences. The research of the second author was partially supported by a Simons Foundation fund. The research of the third author was supported by JSPS KAKENHI Grant Number 22H00099 and 23KK0050.
- Communicated by: Nageswari Shanmugalingam
- © Copyright 2024 by the authors
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3393-3402
- MSC (2020): Primary 31E05
- DOI: https://doi.org/10.1090/proc/16779
- MathSciNet review: 4767270