# Universal differentiability sets and maximal directional derivatives in Carnot groups

## TL;DR

This paper proves the existence of a Hausdorff dimension one universal differentiability set in step 2 Carnot groups, where Lipschitz functions are Pansu differentiable, highlighting the link between maximal directional derivatives and differentiability.

## Contribution

It establishes the existence of universal differentiability sets in step 2 Carnot groups and clarifies the relationship between maximal directional derivatives and Pansu differentiability.

## Key findings

- Universal differentiability set exists in step 2 Carnot groups
- Maximal directional derivative implies Pansu differentiability in these groups
- The implication fails in the Engel group with step 3

## Abstract

We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set' N such that every real-valued Lipschitz map on G is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.