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Mathematics > Metric Geometry

arXiv:2207.02949 (math)
[Submitted on 6 Jul 2022 (v1), last revised 24 Sep 2022 (this version, v2)]

Title:Sobolev spaces and Poincaré inequalities on the Vicsek fractal

Authors:Fabrice Baudoin, Li Chen
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Abstract:In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete $p$-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for $p>1$. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain $L^p$-Poincaré inequalities for all values of $p \ge 1$.
Comments: V2: Accepted for publication in Ann. Fenn. Math
Subjects: Metric Geometry (math.MG); Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
Cite as: arXiv:2207.02949 [math.MG]
  (or arXiv:2207.02949v2 [math.MG] for this version)
  https://doi.org/10.48550/arXiv.2207.02949

Submission history

From: Fabrice Baudoin Dr [view email]
[v1] Wed, 6 Jul 2022 20:04:08 UTC (288 KB)
[v2] Sat, 24 Sep 2022 13:04:12 UTC (288 KB)
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