Maximal Directional Derivatives in Laakso Space

TL;DR

This paper explores the relationship between maximal directional derivatives and differentiability in Laakso space, revealing unique behaviors that differ from Euclidean and Carnot group settings, with implications for understanding Lipschitz functions.

Contribution

It demonstrates that maximal directional derivatives imply differentiability only on a $\sigma$-porous set in Laakso space, contrasting with classical spaces.

Findings

Maximal directional derivatives imply differentiability only on a $\sigma$-porous set.

Distance functions are differentiable everywhere except on a $\sigma$-porous set.

Behavior in Laakso space differs from Euclidean and Carnot group cases.

Abstract

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a $\sigma$-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a $\sigma$-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces and Carnot groups.