Universal Differentiability Sets in Carnot Groups of Arbitrarily High\n Step

2017/11/28 by Andrea Pinamonti, Pinamonti, Andrea, Gareth Speight +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Arts and Humanities · #Dermatological and Skeletal Disorders #Geometric Analysis and Curvature Flows #French Historical and Cultural Studies

paper · pdf · doi:10.48550/arxiv.1711.11433

Abstract

We show that every model filiform group \𝔼_n contains a measure\nzero set N such that every Lipschitz map f colon \𝔼_n\→\n\ℝ is differentiable at some point of N. Model filiform groups are a\nclass of Carnot groups which can have arbitrarily high step. Essential to our\nwork is the question of whether existence of an (almost) maximal directional\nderivative Ef(x) in a Carnot group implies differentiability of a Lipschitz\nmap f at x. We show that such an implication is valid in model Filiform\ngroups except for a one-dimensional subspace of horizontal directions.\nConversely, we show that this implication fails for every horizontal direction\nin the free Carnot group of step three and rank two.\n

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