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 CarnotCarathéodory (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 viceversa. 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2016
 arXiv:
 arXiv:1607.04681
 Bibcode:
 2016arXiv160704681P
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Functional Analysis;
 28A75;
 43A80;
 49Q15;
 53C17
 EPrint:
 23 pages