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Displaying similar documents to “Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields”

Homogeneous Carnot groups related to sets of vector fields

Andrea Bonfiglioli (2004)

Bollettino dell'Unione Matematica Italiana

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In this paper, we are concerned with the following problem: given a set of smooth vector fields X 1 , … , X m on R N , we ask whether there exists a homogeneous Carnot group G = ( R N , ∘ , δ λ ) such that ∑ i X i 2 is a sub-Laplacian on G . We find necessary and sufficient conditions on the given vector fields in order to give a positive answer to the question. Moreover, we explicitly construct the group law i as above, providing direct proofs. Our main tool is a suitable version of the Campbell-Hausdorff formula. Finally, we...

Some relations among volume, intrinsic perimeter and one-dimensional restrictions of B V functions in Carnot groups

Francescopaolo Montefalcone (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let 𝔾 be a k -step Carnot group. The first aim of this paper is to show an interplay between volume and 𝔾 -perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for 𝔾 -regular submanifolds of codimension one. We then give some applications of this result: slicing of B V 𝔾 functions, integral geometric formulae for volume and 𝔾 -perimeter and, making use of a suitable notion of convexity, called, we state a Cauchy type formula for 𝔾 -convex sets. Finally,...