Universal Differentiability Sets in Laakso Space

TL;DR

This paper demonstrates the existence of a measure zero universal differentiability set in Laakso space by constructing specific measures under which Lipschitz functions are differentiable almost everywhere, also supporting a Poincaré inequality.

Contribution

It introduces a family of mutually singular doubling measures on Laakso space that ensure differentiability of Lipschitz functions almost everywhere and support a Poincaré inequality, advancing understanding of differentiability in metric spaces.

Findings

Existence of a measure zero universal differentiability set in Laakso space.

Construction of mutually singular doubling measures supporting differentiability.

Measures support a Poincaré inequality in Laakso space.

Abstract

We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincar\'e inequality.