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Korevaar-Schoen spaces on Sierpiński carpets

Meng Yang

Abstract

We prove that certain $L^p$-regularity functional inequality holds on \emph{generalized Sierpiński carpets}. This gives an affirmative answer to an open question raised by Fabrice Baudoin. Our technique originates from an old idea of Alf Jonsson in 1996.

Korevaar-Schoen spaces on Sierpiński carpets

Abstract

We prove that certain -regularity functional inequality holds on \emph{generalized Sierpiński carpets}. This gives an affirmative answer to an open question raised by Fabrice Baudoin. Our technique originates from an old idea of Alf Jonsson in 1996.
Paper Structure (3 sections, 11 theorems, 72 equations, 1 figure)

This paper contains 3 sections, 11 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Result
  3. Proof

Key Result

Lemma 2.5

(Shi24) Assume that $p>\mathrm{dim}_{\mathrm{ARC}}(K,d)$. Let and Then $(\mathcal{F}_p,\lVert\cdot\rVert_{\mathcal{F}_p})$ is a reflexive separable Banach space which is continously embedded in the Hölder space Moreover, $\mathcal{F}_p$ is uniformly dense in $C(K)$.

Figures (1)

  • Figure 1: The standard Sierpiński carpet

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 2.7
  • Theorem 2.8
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 11 more