[Boules définies par des champs de vecteurs non réguliers et l'inégalité de Poincaré]
We provide a structure theorem for Carnot-Carathéodory balls defined by a family of Lipschitz continuous vector fields. From this result a proof of Poincaré inequality follows.
On prouve un théorème de structure pour les boules de Carnot-Carathéodory définies par des champs de vecteurs lipschitziens. Une inégalité de Poincaré est aussi démontrée.
Keywords: vector fields, Carnot-Carathéodory distance, Poincaré inequality
Mots-clés : champs de vecteurs, distance de Carnot-Carathéodory, inégalité de Poincaré
Montanari, Annamaria 1 ; Morbidelli, Daniele
@article{AIF_2004__54_2_431_0,
author = {Montanari, Annamaria and Morbidelli, Daniele},
title = {Balls defined by nonsmooth vector fields and the {Poincar\'e} inequality},
journal = {Annales de l'Institut Fourier},
pages = {431--452},
year = {2004},
publisher = {Association des Annales de l'Institut Fourier},
volume = {54},
number = {2},
doi = {10.5802/aif.2024},
zbl = {1069.46504},
language = {en},
url = {https://www.numdam.org/articles/10.5802/aif.2024/}
}
TY - JOUR AU - Montanari, Annamaria AU - Morbidelli, Daniele TI - Balls defined by nonsmooth vector fields and the Poincaré inequality JO - Annales de l'Institut Fourier PY - 2004 SP - 431 EP - 452 VL - 54 IS - 2 PB - Association des Annales de l'Institut Fourier UR - https://www.numdam.org/articles/10.5802/aif.2024/ DO - 10.5802/aif.2024 LA - en ID - AIF_2004__54_2_431_0 ER -
%0 Journal Article %A Montanari, Annamaria %A Morbidelli, Daniele %T Balls defined by nonsmooth vector fields and the Poincaré inequality %J Annales de l'Institut Fourier %D 2004 %P 431-452 %V 54 %N 2 %I Association des Annales de l'Institut Fourier %U https://www.numdam.org/articles/10.5802/aif.2024/ %R 10.5802/aif.2024 %G en %F AIF_2004__54_2_431_0
Montanari, Annamaria; Morbidelli, Daniele. Balls defined by nonsmooth vector fields and the Poincaré inequality. Annales de l'Institut Fourier, Tome 54 (2004) no. 2, pp. 431-452. doi: 10.5802/aif.2024
[1] Smoothness of Lipschitz continuous graphs with non vanishing Levi curvature, Acta Math, Volume 188 (2002), pp. 87-128 | Zbl | MR
[2] Strong solutions for the Levi curvature equation, Adv. in Diff. Eq, Volume 5 (2000) no. 1-3, pp. 323-342 | MR | Zbl
[3] Nonlinear functional analysis, Springer-Verlag, Berlin, 1985 | Zbl | MR
[4] Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992 | Zbl | MR
[5] Subelliptic eigenvalue problems, Conference on Harmonic Analysis in honor of Antoni Zygmund (Wadsworth Mathematical Series) (1983), pp. 590-606 | Zbl | MR
[6] Une métrique associée à une classe d'opérateurs elliptiques dégénérés, Conference on Linear Partial and Pseudodifferential Operators (Rend. Sem. Mat. Univ. Politec. Torino), Volume (special issue) (1984), pp. 105-114 | Zbl | MR
[7] Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 4 (1983) no. 10, pp. 523-541 | Zbl | MR | Numdam | EuDML
[8] A relationship between Poincaré type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices, Volume 1 (1996), pp. 1-14 | Zbl | MR
[9] Approximation and Imbedding Theorems for Weighted Sobolev Spaces Associated with Lipschitz Continuous Vector Fields, Boll. Un. Mat. Ital. (7), Volume B11 (1997) no. 1, pp. 83-117 | Zbl | MR
[10] Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math, Volume 49 (1996), pp. 1081-1144 | Zbl | MR
[11] Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math, Volume 74 (1998), pp. 67-97 | Zbl | MR
[12] Sobolev met Poincaré, Mem. Amer. Math. Soc, Volume 688 (2000) | Zbl | MR
[13] The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J, Volume 53 (1986), pp. 503-523 | Zbl | MR
[14] Stime subellittiche e metriche Riemanniane singolari, Seminario di Analisi Matematica, Università di Bologna (A.A 1982-83)
[15] On the Poincaré inequality for vector fields, Ark. Mat, Volume 38 (2000), pp. 327-342 | Zbl | MR
[16] Analyse sur le boules d'un opérateur sous-elliptique, Math. Ann, Volume 303 (1995), pp. 713-746 | Zbl | MR
[17] Sobolev and Morrey estimates for non-smooth Vector Fields of step two, Z. Anal. Anwendungen, Volume 21 (2002) no. 1, pp. 135-157 | Zbl | MR
[18] Fractional Sobolev norms and structure of the Carnot--Carathéodory balls for Hörmander vector fields, Studia Math, Volume 139 (2000), pp. 213-244 | Zbl | MR
[19] Balls and metrics defined by vector fields I: Basic properties, Acta Math, Volume 155 (1985), pp. 103-147 | Zbl | MR
[20] Set--valued differential and a nonsmooth version of Chow's theorem, Proceedings of the 40th IEEE Conference on Decision and Control; Orlando, Florida (2001)
[21] A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Res. Notices, Volume 2 (1992), pp. 27-38 | Zbl | MR
[22] Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992 | Zbl | MR
Cité par Sources :
