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Construction of $p$-energy and associated energy measures on Sierpiński carpets

Ryosuke Shimizu

Abstract

We establish the existence of a scaling limit $\mathcal{E}_p$ of discrete $p$-energies on the graphs approximating generalized Sierpiński carpets for $p > \dim_{\text{ARC}}(\textsf{SC})$, where $\dim_{\text{ARC}}(\textsf{SC})$ is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space $\mathcal{F}_{p}$ defined as the collection of functions with finite $p$-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, $(\mathcal{E}_2, \mathcal{F}_2)$ recovers the canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and Zhou. We also provide $\mathcal{E}_{p}$-energy measures associated with the constructed $p$-energy and investigate its basic properties like self-similarity and chain rule.

Construction of $p$-energy and associated energy measures on Sierpiński carpets

Abstract

We establish the existence of a scaling limit of discrete -energies on the graphs approximating generalized Sierpiński carpets for , where is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space defined as the collection of functions with finite -energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, recovers the canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and Zhou. We also provide -energy measures associated with the constructed -energy and investigate its basic properties like self-similarity and chain rule.

Paper Structure

This paper contains 24 sections, 72 theorems, 441 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary and results
  3. Generalized Sierpiński carpets and graphical approximations
  4. $p$-energies and Poincaré constants on finite graphs
  5. Main results
  6. Estimates of Poincaré constants and conductances
  7. Basic estimates without (B$_{p}$) and (KM$_{p}$)
  8. Comparability under (B$_{p}$) and (KM$_{p}$)
  9. Checking (B$_{p}$) and (KM$_{p}$) for GSCs
  10. Proof of (B$_{p}$)
  11. Uniform Hölder estimate: $p > \dim_{\textup{ARC}}(K, d)$
  12. Proof of (KM$_{p}$): $p > \dim_{\textup{ARC}}(K, d)$
  13. The domain of $p$-energy
  14. Closability and regularity
  15. Separability
  16. ...and 9 more sections

Key Result

Proposition 2.6

For any $\omega = \omega_{1}\omega_{2}\cdots \in \Sigma$, the set $\bigcap_{m \ge 1}K_{[\omega]_{m}}$ contains only one point. If we define $\pi\colon \Sigma \to K$ by $\{ \pi(\omega) \} = \bigcap_{m \ge 1}K_{[\omega]_{m}}$, then $\pi$ is a continuous surjective map. Furthermore, it holds that $\pi

Figures (5)

  • Figure 1: Sierpiński carpet (left), two other generalized Sierpiński carpets and Menger sponge (right)
  • Figure 2: Graphical approximation $\{ G_{n} \}_{n \ge 1}$ of the SC (This figure draws $G_{1}$ and $G_{2}$ in blue)
  • Figure 3: The conductance $\mathcal{C}_{p}^{(n, M)}$ (the planar case)
  • Figure 4: Values of $f_{n, k}$
  • Figure 5: Modified Sierpiński carpet graph $\{ \mathbb{G}_{n} \}_{n \ge 1}$ (This figure draws $\mathbb{G}_{3}$ in the SC case)

Theorems & Definitions (152)

  • Conjecture 1.1
  • Definition 2.1: Generalized Sierpiński carpet, BBKT10*Subsection 2.2
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 142 more