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Regularity properties of energy densities on the Sierpinski gasket

Masanori Hino, Kanji Inui, Kohei Nitta

TL;DR

This work investigates smoothness properties of energy densities $\frac{d\nu_h}{d\nu}$ for harmonic functions on the $N$-dimensional Sierpinski gasket with the Kusuoka measure $\nu$. By combining direct arguments, random-matrix product estimates, and invariant-cone/Hilbert-metric techniques, the authors show that, for nonconstant $h$, the density is almost surely discontinuous with respect to $\nu$ on a $\nu$-null set, while a natural edge-restricted derivative $\delta\nu_h/\delta\nu$ is Hölder continuous on each edge, and, for $N=2,3$, extends to every point on an edge with Hölder continuity. The results provide a refined edgewise regularity that complements the global irregularity of energy densities, offering a concrete step toward a first-order calculus on fractals. The methods yield new insights into gradient-like structures in measurable Riemannian settings and could inform broader analyses of fractal Dirichlet forms and their analytic/differential structure.

Abstract

We investigate the regularity properties of energy densities associated with harmonic functions on the Sierpinski gasket with respect to the Kusuoka measure. While energy measures themselves have been extensively studied in the framework of analysis on fractals, the fine pointwise behavior of their densities has remained less well understood. We prove that the density of the energy measure of a nonconstant harmonic function is almost everywhere discontinuous, whereas it is Hölder continuous when restricted to each one-dimensional edge of the gasket.

Regularity properties of energy densities on the Sierpinski gasket

TL;DR

This work investigates smoothness properties of energy densities for harmonic functions on the -dimensional Sierpinski gasket with the Kusuoka measure . By combining direct arguments, random-matrix product estimates, and invariant-cone/Hilbert-metric techniques, the authors show that, for nonconstant , the density is almost surely discontinuous with respect to on a -null set, while a natural edge-restricted derivative is Hölder continuous on each edge, and, for , extends to every point on an edge with Hölder continuity. The results provide a refined edgewise regularity that complements the global irregularity of energy densities, offering a concrete step toward a first-order calculus on fractals. The methods yield new insights into gradient-like structures in measurable Riemannian settings and could inform broader analyses of fractal Dirichlet forms and their analytic/differential structure.

Abstract

We investigate the regularity properties of energy densities associated with harmonic functions on the Sierpinski gasket with respect to the Kusuoka measure. While energy measures themselves have been extensively studied in the framework of analysis on fractals, the fine pointwise behavior of their densities has remained less well understood. We prove that the density of the energy measure of a nonconstant harmonic function is almost everywhere discontinuous, whereas it is Hölder continuous when restricted to each one-dimensional edge of the gasket.
Paper Structure (9 sections, 31 theorems, 136 equations, 2 figures)

This paper contains 9 sections, 31 theorems, 136 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Framework and statement of main theorems
  3. Preliminaries
  4. Behaviors of harmonic functions on edges
  5. Unimodality of harmonic functions on edges
  6. The locations of extremal points of a harmonic function along edges
  7. Proof of Theorem \ref{['th:main1']}
  8. Proof of Theorem \ref{['th:main2']}
  9. Proof of Theorem \ref{['th:main3']}

Key Result

Theorem 2.1

Let $h\in \mathcal{H}$ be a nonconstant function. Then, there exists a Borel subset $A$ of $K$ such that $\nu(K\setminus A)=0$ and an arbitrary $\nu$-modification of the Radon--Nikodym derivative $\mathrm{d}\nu_h/\mathrm{d} \nu$ is discontinuous at every point of $A$.

Figures (2)

  • Figure 1: The two-dimensional Sierpinski gasket.
  • Figure 2: Graphs of the Radon--Nikodym derivatives $\mathrm{d}\nu_h/\mathrm{d}\nu$ for harmonic functions $h$. Left figure: For $h$ with values $1,0,0$ at the vertices. Right figure: For $h$ with values $0,1,-1$ at the vertices.

Theorems & Definitions (63)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2: See HN06
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 53 more