An Overview on Laakso Spaces

TL;DR

This paper explores Laakso spaces, special metric measure spaces with unique embedding properties, by expanding on Laakso's original work and providing new proofs and insights into their structure.

Contribution

It extends Laakso's original results by clarifying and proving previously unverified statements about Laakso spaces and their properties.

Findings

Laakso spaces are Ahlfors Q-regular and admit a weak (1,1)-Poincaré inequality.

They cannot be embedded into any Euclidean space.

The paper provides new proofs and expands on Laakso's original statements.

Abstract

Laakso's construction is a famous example of an Ahlfors $Q$-regular metric measure space admitting a weak $(1,1)$-Poincar\'{e} inequality that can not be embedded in $\mathbb{R}^n$ for any $n$. The construction is of particular interest because it works for any fixed dimension $Q>1$, even fractional ones. In this paper we will shed some light on Laakso's work by expanding some of his statements and proving results that were left unproved in the original paper.