Construction of Sobolev spaces of fractional order with sub-riemannian vector fields

Sami Mustapha[1]; François Vigneron[2]

  • [1] Institut de Mathématiques de Jussieu 175, rue du Chevaleret 75013 Paris (France)
  • [2] Centre de Mathématiques Laurent Schwartz U.M.R. 7640 du C.N.R.S. École Polytechnique 91128 Palaiseau cedex (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 4, page 1023-1049
  • ISSN: 0373-0956

Abstract

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Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-riemannian distance.Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat-kernel associated to the sub-elliptic Laplacian.

How to cite

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Mustapha, Sami, and Vigneron, François. "Construction of Sobolev spaces of fractional order with sub-riemannian vector fields." Annales de l’institut Fourier 57.4 (2007): 1023-1049. <http://eudml.org/doc/10249>.

@article{Mustapha2007,
abstract = {Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-riemannian distance.Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat-kernel associated to the sub-elliptic Laplacian.},
affiliation = {Institut de Mathématiques de Jussieu 175, rue du Chevaleret 75013 Paris (France); Centre de Mathématiques Laurent Schwartz U.M.R. 7640 du C.N.R.S. École Polytechnique 91128 Palaiseau cedex (France)},
author = {Mustapha, Sami, Vigneron, François},
journal = {Annales de l’institut Fourier},
keywords = {functional space; Sobolev space; sub-riemannian distance; sub-elliptic Laplacian; Weyl-Hörmander calculus; sub-Riemannian distance},
language = {eng},
number = {4},
pages = {1023-1049},
publisher = {Association des Annales de l’institut Fourier},
title = {Construction of Sobolev spaces of fractional order with sub-riemannian vector fields},
url = {http://eudml.org/doc/10249},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Mustapha, Sami
AU - Vigneron, François
TI - Construction of Sobolev spaces of fractional order with sub-riemannian vector fields
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 4
SP - 1023
EP - 1049
AB - Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-riemannian distance.Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat-kernel associated to the sub-elliptic Laplacian.
LA - eng
KW - functional space; Sobolev space; sub-riemannian distance; sub-elliptic Laplacian; Weyl-Hörmander calculus; sub-Riemannian distance
UR - http://eudml.org/doc/10249
ER -

References

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  12. D. S. Jerison, A. Sánchez-Call, Subelliptic, Second Order Differential Operators, Complex Analysis III 1277 (1986), 46-77, Lect. Notes in Maths Zbl0634.35017
  13. D. S. Jerison, A. Sánchez-Calle, Estimates for the Heat Kernel for a Sum of Squares of Vector Fields, Indiana Univ. Math. J. 35 (1986), 835-854 Zbl0639.58026MR865430
  14. R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys & Monographs 91 (2002), A.M.S. Zbl1044.53022MR1867362
  15. F. Vigneron, The trace problem for Sobolev spaces over the Heisenberg group, École polytechnique (2006) Zbl1152.46023

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