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

Porosity and differentiability of Lipschitz maps from stratified groups to Banach homogeneous groups

created by pinamonti on 06 Jun 2017
modified by magnani on 26 Oct 2019


Published Paper

Inserted: 6 jun 2017
Last Updated: 26 oct 2019

Journal: Ann. Mat. Pura Appl.
Year: 2019
Links: journal


Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $\sigma$-porous set. At density points of $A$ we establish a pointwise characterization of differentiability in terms of directional derivatives. We use these new results to obtain an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property. As a consequence we also get an alternative proof of Pansu's Theorem.

Keywords: Carnot-Carathéodory distance, Carnot group, Rademacher's theorem, differentiability, Porous set, stratified group, Banach homogeneous group, Lipschitz map


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