Displaying similar documents to “Sobolev spaces of integer order on compact homogeneous manifolds and invariant differential operators”

Function spaces of Nikolskii type on compact manifold

Cristiana Bondioli (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Nikolskii spaces were defined by way of translations on R n and by way of coordinate maps on a differentiable manifold. In this paper we prove that, for functions with compact support in R n , we get an equivalent definition if we replace translations by all isometries of R n . This result seems to justify a definition of Nikolskii type function spaces on riemannian manifolds by means of a transitive group of isometries (provided that one exists). By approximation theorems, we prove that - for...

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,...

Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields

Daniele Morbidelli (2000)

Studia Mathematica

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We study the notion of fractional L p -differentiability of order s ∈ ( 0 , 1 ) along vector fields satisfying the Hörmander condition on ℝ n . We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different W s , p -norms are equivalent. We also prove a local embedding W 1 , p ⊂ W s , q , where q is a suitable exponent greater than p.

Metrics with homogeneous geodesics on flag manifolds

Dimitri V. Alekseevsky, Andreas Arvanitoyeorgos (2002)

Commentationes Mathematicae Universitatis Carolinae

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A geodesic of a homogeneous Riemannian manifold ( M = G / K , g ) is called homogeneous if it is an orbit of an one-parameter subgroup of G . In the case when M = G / H is a naturally reductive space, that is the G -invariant metric g is defined by some non degenerate biinvariant symmetric bilinear form B , all geodesics of M are homogeneous. We consider the case when M = G / K is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group G , and we give a simple necessary condition that M admits a non-naturally...

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...