Regularity properties of energy densities on the Sierpinski gasket

TL;DR

This paper studies the regularity of energy densities related to harmonic functions on the Sierpinski gasket, revealing they are mostly discontinuous but H"older continuous along edges, advancing fractal analysis understanding.

Contribution

It provides new insights into the pointwise behavior of energy densities on fractals, showing their discontinuity almost everywhere and H"older continuity on edges.

Findings

Energy densities are almost everywhere discontinuous.

Energy densities are H"older continuous on each edge.

Advances understanding of fractal energy measure regularity.

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 H\"older continuous when restricted to each one-dimensional edge of the gasket.