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 $\sigma$-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 $\sigma$-porous set and show preimages of open sets under the horizontal gradient are far from being porous.

23 pages

References in corpus (1)

• Differentiability inside sets with upper Minkowski dimension one