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Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces

Riku Anttila, Sylvester Eriksson-Bique, Ryosuke Shimizu

Abstract

We construct self-similar $p$-energy forms as normalized limits of discretized $p$-energies on a rich class of Laakso-type fractal spaces. Collectively, we refer to them as IGS-fractals, where IGS stands for (edge-)iterated graph systems. We propose this framework as a rich source of "toy models" that can be consulted for tackling challenging questions that are not well understood on most other fractal spaces. Supporting this, our framework uncovers a novel analytic phenomenon, which we term as singularity of Sobolev spaces. This means that the associated Sobolev spaces $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ for distinct $p_1,p_2 \in (1,\infty)$ intersect only at constant functions. We provide the first example of a self-similar fractal on which this singularity phenomenon occurs for all pairs of distinct exponents. In particular, we show that the Laakso diamond space is one such example.

Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces

Abstract

We construct self-similar -energy forms as normalized limits of discretized -energies on a rich class of Laakso-type fractal spaces. Collectively, we refer to them as IGS-fractals, where IGS stands for (edge-)iterated graph systems. We propose this framework as a rich source of "toy models" that can be consulted for tackling challenging questions that are not well understood on most other fractal spaces. Supporting this, our framework uncovers a novel analytic phenomenon, which we term as singularity of Sobolev spaces. This means that the associated Sobolev spaces and for distinct intersect only at constant functions. We provide the first example of a self-similar fractal on which this singularity phenomenon occurs for all pairs of distinct exponents. In particular, we show that the Laakso diamond space is one such example.

Paper Structure

This paper contains 43 sections, 84 theorems, 312 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. General results
  4. Singularities of energy measures and Sobolev spaces
  5. Outline of the construction
  6. Organization of the paper
  7. Preliminary
  8. Metric spaces and measures
  9. Terminologies of graphs
  10. Discrete potential theory
  11. Construction of IGS-fractals
  12. Iterated graph systems
  13. Replacement rule
  14. First examples
  15. Geometric conditions
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $(X,d,\mu)$ be the Laakso diamond space and $\mathscr{F}_p$ the Sobolev space associated to the self-similar $p$-energy form $\mathscr{E}_p : L^p(X,\mu) \to [0,\infty]$ for $p \in (1,\infty)$. Then the following hold.

Figures (13)

  • Figure 1.1: The Laakso diamond space snowflake-embedded onto a self-similar set of $\mathbb{R}^2$. Lang and Plaut showed that Laakso diamond does not admit a biLipschitz embedding into any Euclidean space LangPlaut. See also CK_PI for a more general result and LaaksoLeeNaor for further related work.
  • Figure 1.2: Figure of the first iteration of the replacement that produces the Laakso diamond space. The blue vertices indicate the gluing rules.
  • Figure 1.2: $G_3$ in the construction of Laakso diamond.
  • Figure 3.4: Figure of the first replacement that produces the variant of a Laakso space discussed in Example \ref{['example: Laakso space']}. The colors of vertices indicate the gluing rules: Blue vertex is connected to the other blue vertex and orange to the other orange.
  • Figure 3.5: A figure of the IGS in Example \ref{['example: Counterexample']}.
  • ...and 8 more figures

Theorems & Definitions (201)

  • Theorem 1.1
  • Example 1.2: Laakso diamond
  • Definition 1.4
  • Theorem 1.5: Theorem \ref{['Thm: Existence of p-Energy']} and Corollary \ref{['cor: Self-similarity of domain']}
  • Corollary 1.7
  • Remark 1.8
  • Definition 1.11
  • Theorem 1.14: Propositions \ref{['prop: EM-Poincare']} and \ref{['prop: EM-Ucap']}
  • Theorem 1.15
  • Theorem 1.20
  • ...and 191 more