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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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.
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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