Structure of porous sets in Carnot groups

Abstract

We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not σ-porous with respect to the Carnot-Carathéodory (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 σ-porous set and show preimages of open sets under the horizontal gradient are far from being porous.

Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

“…This is because doubling measures assign measure zero to porous sets, which is well known and follows from the fact that the Lebesgue differentiation theorem holds whenever the underlying measure is doubling. For an explicit proof one could follow the steps in [32], which do not depend on the Carnot group structure in that paper.…”

Maximal Directional Derivatives in Laakso Space

“…This is well known. For an explicit proof one could follow the steps in [23], which do not rely on the Carnot group structure in that paper.…”

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

“…In the Banach space setting, [21,22] give a version of Rademacher's theorem for Frechét differentiability of Lipschitz mappings on Banach spaces in which porous sets are negligible in a suitable sense. Other results have also been studied in stratified groups [20,29,31,32].…”