On the length of chains in a metric space

TL;DR

This paper establishes an upper bound on the minimal length of epsilon-chains in metric spaces, generalizing previous results and enabling sharp heat kernel estimates without additional assumptions.

Contribution

It introduces a generalized upper bound on epsilon-chains in metric spaces, extending prior work and applying it to derive sharp heat kernel bounds under minimal conditions.

Findings

Derived a new upper bound on epsilon-chains in metric spaces.

Applied the bound to obtain sharp heat kernel estimates.

Showed the bound's sharpness using a snowflake transform.

Abstract

We obtain an upper bound on the minimal number of points in an $\epsilon$-chain joining two points in a metric space. This generalizes a bound due to Hambly and Kumagai (1999) for the case of resistance metric on certain self-similar fractals. As an application, we deduce a condition on $\epsilon$-chains introduced by Grigor'yan and Telcs (2012). This allows us to obtain sharp bounds on the heat kernel for spaces satisfying the parabolic Harnack inequality without assuming further conditions on the metric. A snowflake transform on the Euclidean space shows that our bound is sharp.