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First-order Sobolev spaces, self-similar energies and energy measures on the Sierpiński carpet

Mathav Murugan, Ryosuke Shimizu

TL;DR

This work constructs a canonical $(1,p)$-Sobolev framework on the planar Sierpiński carpet for all $p\in(1,\infty)$ by discretizing the space via a sequence of self-similar graphs and analyzing discrete $p$-energies. It establishes self-similarity, Lipschitz contractivity, and locality for the resulting $p$-energy ${\mathcal E}_p$ and introduces a corresponding family of $p$-energy measures ${\Gamma}_p\langle f\rangle$, connecting the energy theory to Korevaar–Schoen Besov–Lipschitz spaces and the Korevaar–Schoen framework. A key technical advance is the use of Loewner-type modulus bounds to derive discrete Poincaré inequalities and elliptic Harnack inequalities on graph approximations, which in turn yield regularity and the density of continuous functions in the Sobolev space. The paper also links the constructed Sobolev space to the Newton–Sobolev space in the ARC-attainment setting, showing that, if ARC is attained, the Sobolev space matches the Newton–Sobolev structure and the energy measure corresponds to an energy measure of an energy-harmonic function; these results illuminate the role of energy measures in the attainment problem for Ahlfors regular conformal dimension on self-similar spaces. Overall, the work provides a robust, self-similar, and analyzable Sobolev theory on a fractal planar carpet with potential implications for quasisymmetric uniformization and related geometric-analytic problems.

Abstract

We construct and investigate $(1, p)$-Sobolev space, $p$-energy, and the corresponding $p$-energy measures on the planar Sierpiński carpet for all $p \in (1, \infty)$. Our method is based on the idea of Kusuoka and Zhou [Probab. Theory Related Fields $\textbf{93}$ (1992), no. 2, 169--196], where Brownian motion (the case $p = 2$) on self-similar sets including the planar Sierpiński carpet were constructed. Similar to this earlier work, we use a sequence of discrete graph approximations and the corresponding discrete $p$-energies to define the Sobolev space and $p$-energies. However, we need a new approach to ensure that our $(1, p)$-Sobolev space has a dense set of continuous functions when $p$ is less than the Ahlfors regular conformal dimension. The new ingredients are the use of Loewner type estimates on combinatorial modulus to obtain Poincaré inequality and elliptic Harnack inequality on a sequence of approximating graphs. An important feature of our Sobolev space is the self-similarity of our $p$-energy, which allows us to define corresponding $p$-energy measures on the planar Sierpiński carpet. We show that our Sobolev space can also be viewed as a Korevaar-Schoen type space. We apply our results to the attainment problem for Ahlfors regular conformal dimension of the Sierpiński carpet. In particular, we show that if the Ahlfors regular conformal dimension, say $\dim_{\mathrm{ARC}}$, is attained, then any optimal measure which attains $\dim_{\mathrm{ARC}}$ should be comparable with the $\dim_{\mathrm{ARC}}$-energy measure of some function in our $(1, \dim_{\mathrm{ARC}})$-Sobolev space up to a multiplicative constant. In this case, we also prove that the Newton-Sobolev space corresponding to any optimal measure and metric can be identified as our self-similar $(1, \dim_{\mathrm{ARC}})$-Sobolev space.

First-order Sobolev spaces, self-similar energies and energy measures on the Sierpiński carpet

TL;DR

This work constructs a canonical -Sobolev framework on the planar Sierpiński carpet for all by discretizing the space via a sequence of self-similar graphs and analyzing discrete -energies. It establishes self-similarity, Lipschitz contractivity, and locality for the resulting -energy and introduces a corresponding family of -energy measures , connecting the energy theory to Korevaar–Schoen Besov–Lipschitz spaces and the Korevaar–Schoen framework. A key technical advance is the use of Loewner-type modulus bounds to derive discrete Poincaré inequalities and elliptic Harnack inequalities on graph approximations, which in turn yield regularity and the density of continuous functions in the Sobolev space. The paper also links the constructed Sobolev space to the Newton–Sobolev space in the ARC-attainment setting, showing that, if ARC is attained, the Sobolev space matches the Newton–Sobolev structure and the energy measure corresponds to an energy measure of an energy-harmonic function; these results illuminate the role of energy measures in the attainment problem for Ahlfors regular conformal dimension on self-similar spaces. Overall, the work provides a robust, self-similar, and analyzable Sobolev theory on a fractal planar carpet with potential implications for quasisymmetric uniformization and related geometric-analytic problems.

Abstract

We construct and investigate -Sobolev space, -energy, and the corresponding -energy measures on the planar Sierpiński carpet for all . Our method is based on the idea of Kusuoka and Zhou [Probab. Theory Related Fields (1992), no. 2, 169--196], where Brownian motion (the case ) on self-similar sets including the planar Sierpiński carpet were constructed. Similar to this earlier work, we use a sequence of discrete graph approximations and the corresponding discrete -energies to define the Sobolev space and -energies. However, we need a new approach to ensure that our -Sobolev space has a dense set of continuous functions when is less than the Ahlfors regular conformal dimension. The new ingredients are the use of Loewner type estimates on combinatorial modulus to obtain Poincaré inequality and elliptic Harnack inequality on a sequence of approximating graphs. An important feature of our Sobolev space is the self-similarity of our -energy, which allows us to define corresponding -energy measures on the planar Sierpiński carpet. We show that our Sobolev space can also be viewed as a Korevaar-Schoen type space. We apply our results to the attainment problem for Ahlfors regular conformal dimension of the Sierpiński carpet. In particular, we show that if the Ahlfors regular conformal dimension, say , is attained, then any optimal measure which attains should be comparable with the -energy measure of some function in our -Sobolev space up to a multiplicative constant. In this case, we also prove that the Newton-Sobolev space corresponding to any optimal measure and metric can be identified as our self-similar -Sobolev space.
Paper Structure (45 sections, 123 theorems, 660 equations, 4 figures)

This paper contains 45 sections, 123 theorems, 660 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Overview for the rest of the paper.
  3. Preliminaries
  4. Basic facts and terminologies of graphs
  5. Combinatorial p-modulus of path families
  6. Discrete p-energy, p-Laplacian and associated capacity
  7. Volume growth conditions
  8. Loewner-type lower bounds for p-modulus
  9. Discrete (p,p)-Poincare inequality
  10. Equivalence with two-point estimates
  11. Two-point estimates are implied by Loewner bounds
  12. Discrete elliptic Harnack inequality
  13. EHI for discrete p-harmonic functions
  14. Hölder continuous cut-off functions with controlled energy
  15. Sobolev space via a sequence of discrete energies
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $p \in (1, \infty)$ and let $(K, d, m)$ be the Sierpiński carpet equipped with the Euclidean metric and the self-similar measure described above. Then there exists $\rho(p) \in (0,\infty)$ such that the normed linear space $({\mathcal{F}}_p, \left\lVert\,\cdot\,\right\rVert_{{\mathcal{F}}_p})$ d and satisfies the following properties. Furthermore, there exist $C \ge 1$ and $\mathcal{E}_p \co

Figures (4)

  • Figure 1.1: The planar Sierpiński carpet and its approximation graphs $\{ \mathbb{G}_{n} \}$. ($\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ are drawn in blue.)
  • Figure 5.1: The paths $\theta_{\textup{min}}$ and $\theta_{\textup{max}}$
  • Figure 10.1: The graphs $\mathbb{G}_{3}$ (left) and $G_{3}^{\#}$ (right)
  • Figure 10.2: The subset $\bigcup_{z \in \theta}H(z)$ drawn in red. Each big box corresponds to a copy of $\mathbb{G}_{k}$ in $\mathbb{G}_{k + m}$.

Theorems & Definitions (306)

  • Theorem 1.1: Construction of $(1,p)$-Sobolev space and $p$-energy
  • Remark
  • Theorem 1.2: Existence of $p$-energy measure
  • Remark
  • Definition 1.3
  • Theorem 1.4: Self-similar Sobolev space is a Korevaar--Schoen space
  • Theorem 1.5: Poincaré inequality and capacity upper bound
  • Proposition 1.6: Axiomatic description of the self-similar Sobolev space
  • Definition 1.7: Conformal gauge
  • Theorem 1.8
  • ...and 296 more