Universal Differentiability Sets in Carnot Groups of Arbitrarily High Step

Abstract

We show that every model filiform group ${\mathbb{E}}_n$ contains a measure zero set $N$ such that every Lipschitz map $f:{\mathbb{E}}_n\to \mathbb{R}$ is differentiable at some point of $N$. Model filiform groups are a class of Carnot groups which can have arbitrarily high step. Essential to our work is the question of whether existence of an (almost) maximal directional derivative $Ef(x)$ in a Carnot group implies differentiability of a Lipschitz map $f$ at $x$. We show that such an implication is valid in model Filiform groups except for a one-dimensional subspace of horizontal directions. Conversely, we show that this implication fails for every horizontal direction in the free Carnot group of step three and rank two.

42 pages. arXiv admin note: text overlap with arXiv:1505.07986