Open Access
Spring and Summer 2017 Structure of porous sets in Carnot groups
Andrea Pinamonti, Gareth Speight
Illinois J. Math. 61(1-2): 127-150 (Spring and Summer 2017). DOI: 10.1215/ijm/1520046212


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.


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Andrea Pinamonti. Gareth Speight. "Structure of porous sets in Carnot groups." Illinois J. Math. 61 (1-2) 127 - 150, Spring and Summer 2017.


Received: 20 March 2017; Revised: 23 July 2017; Published: Spring and Summer 2017
First available in Project Euclid: 3 March 2018

zbMATH: 06864462
MathSciNet: MR3770839
Digital Object Identifier: 10.1215/ijm/1520046212

Primary: 28A75 , 43A80 , 49Q15 , 53C17

Rights: Copyright © 2017 University of Illinois at Urbana-Champaign

Vol.61 • No. 1-2 • Spring and Summer 2017
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