Universal Differentiability Sets in Carnot Groups of Arbitrarily High Step

TL;DR

This paper constructs measure zero sets in high-step Carnot groups where Lipschitz functions are differentiable, advancing understanding of differentiability in complex geometric structures.

Contribution

It demonstrates the existence of measure zero sets ensuring differentiability of Lipschitz functions in all model filiform Carnot groups of arbitrary step, and explores directional derivatives implications.

Findings

Existence of measure zero sets with differentiability properties in high-step Carnot groups.

Implication of maximal directional derivatives leading to differentiability holds in most directions in model filiform groups.

This implication fails in certain directions in free Carnot groups of step three, rank two.

Abstract

We show that every model filiform group $\mathbb{E}_{n}$ contains a measure zero set $N$ such that every Lipschitz map $f\colon \mathbb{E}_{n}\to \mathbb{R}$ 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.