Metrics and Uniform Harnack Inequality on the Strichartz Hexacarpet

TL;DR

This paper develops intrinsic metrics on the Strichartz hexacarpet, demonstrating they do not satisfy the chain condition, and establishes a uniform Harnack inequality on approximating graphs using these metrics.

Contribution

It introduces intrinsic metrics on the Strichartz hexacarpet and proves a uniform Harnack inequality with respect to these metrics, diverging from traditional graph metrics.

Findings

Intrinsic metrics do not satisfy the chain condition.

Uniform Harnack inequality holds on approximating graphs.

Intrinsic metrics provide new analytical tools for fractal analysis.

Abstract

We construct \emph{intrinsic} metrics on the Strichartz hexacarpet using weight functions and show that these metrics do \emph{not} satisfy the chain condition. We give uniform Harnack inequality on the approximating graphs of the Strichartz hexacarpet with respect to the intrinsic metrics instead of graph metrics.