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arXiv · 1607.04681

Structure of Porous Sets in Carnot Groups

Andrea Pinamonti·Gareth Speight
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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.

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Andrea Pinamonti, Gareth Speight. 2016-07-15. Structure of Porous Sets in Carnot Groups. https://arxiv.org/abs/1607.04681

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