# Porosity and Differentiability of Lipschitz Maps from Stratified Groups to Banach Homogeneous Groups

## TL;DR

This paper investigates the differentiability and porosity properties of Lipschitz maps between stratified groups and Banach homogeneous groups, providing new proofs of differentiability almost everywhere and Pansu's Theorem.

## Contribution

It introduces new characterizations of differentiability and porosity for Lipschitz maps in this setting, offering alternative proofs of key theorems.

## Key findings

- Directional derivatives act as homogeneous homomorphisms at density points.
- Almost everywhere differentiability of Lipschitz maps is established.
- Provides an alternative proof of Pansu's Theorem.

## Abstract

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.