Structure of porous sets in Carnot groups
Copyright policy )Structure of porous sets in Carnot groups
We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $��$-porous with respect to the Carnot-Carath��odory (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 $��$-porous set and show preimages of open sets under the horizontal gradient are far from being porous.
23 pages
- University of Cincinnati United States
- University System of Ohio United States
- University of Trento Italy
Mathematics - Functional Analysis, 49Q15, 43A80, 53C17, Mathematics - Metric Geometry, 28A75, 43A80, 49Q15, 53C17, FOS: Mathematics, 28A75, Metric Geometry (math.MG), Functional Analysis (math.FA)
Mathematics - Functional Analysis, 49Q15, 43A80, 53C17, Mathematics - Metric Geometry, 28A75, 43A80, 49Q15, 53C17, FOS: Mathematics, 28A75, Metric Geometry (math.MG), Functional Analysis (math.FA)
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