An Overview on Laakso Spaces

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.