منابع مشابه
Hardy-Littlewood Inequality for Quasiregular Maps on Carnot Groups
A natural generalization of analytic functions to n-dimensional Euclidean space are quasiregular mappings.(See [2] and [3].) An analogue of Theorem 1.1 for quasiregular mappings in John domains in Euclidean space appeared in [4]. Recently the analytical tools used in the proof of this result have been generalized to Carnot groups. We give an account of some of these advances and obtain an analo...
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The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Carathéodory distances and prove that this also fails f...
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Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between subRiemannian and Riemannian geometry is a strong one, one might conjecture that the subRiemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture is f...
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We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.
متن کاملHausdorff Measure of the Singular Set of Quasiregular Maps on Carnot Groups
Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Let ν stand for the homogeneous dimension of a Carnot group and let m be the index of the last vector space of the corresponding Lie algebra. We prove that the (ν −m− 1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimat...
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ژورنال
عنوان ژورنال: Annales Academiae Scientiarum Fennicae Mathematica
سال: 2015
ISSN: 1239-629X,1798-2383
DOI: 10.5186/aasfm.2015.4008