Metrics on the Sierpinski carpet by weight functions

TL;DR

This paper constructs new metrics on the Sierpinski carpet using self-similar weight functions and proves sub-Gaussian heat kernel estimates, confirming a conjecture by Kigami.

Contribution

It introduces a novel class of metrics on the Sierpinski carpet and establishes heat kernel estimates, advancing understanding of diffusions on fractals.

Findings

Established two-sided sub-Gaussian heat kernel estimates

Constructed metrics via self-similar weight functions

Proved a conjecture by Kigami

Abstract

We construct certain metrics on the Sierpinski carpet via a class of self-similar weight functions. Using these metrics and by applying known results, we obtain the two-sided sub-Gaussian heat kernel estimates of time change of the standard diffusion on the Sierpinski carpet with respect to self-similar measures. This proves a conjecture by Kigami.