Structure of Porous Sets in Carnot Groups

TL;DR

This paper investigates the structure of porous sets in Carnot groups, revealing the existence of measure-zero, non-σ-porous sets, and explores differences in porosity with respect to Carnot-Carathéodory and Euclidean distances, along with differentiability properties of Lipschitz functions.

Contribution

It constructs examples of porous and non-porous sets in Carnot groups and analyzes their properties, including differentiability and preimage porosity under Lipschitz functions.

Findings

Existence of measure-zero, non-σ-porous sets in Carnot groups.

Porosity differences between CC and Euclidean distances in the Heisenberg group.

Lipschitz functions that are nowhere Pansu differentiable on σ-porous sets.

Abstract

We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\sigma$-porous with respect to the Carnot-Carath\'eodory (CC) distance. In the first Heisenberg group we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups we then construct a Lipschitz function which is Pansu differentiable at no point of a given $\sigma$-porous set and show preimages of open sets under the horizontal gradient are far from being porous.