Calculus of Variations and Geometric Measure Theory
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A. Pinamonti - G. Speight

Universal Differentiability Sets in Carnot Groups of Arbitrarily High Step

created by pinamonti on 28 Nov 2017

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Submitted Paper

Inserted: 28 nov 2017
Last Updated: 28 nov 2017

Year: 2017

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.


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