Balls defined by nonsmooth vector fields and the Poincaré inequality
Annales de l'Institut Fourier, Volume 54 (2004) no. 2, pp. 431-452.

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.

DOI: 10.5802/aif.2024
Classification: 46E35
Keywords: vector fields, Carnot-Carathéodory distance, Poincaré inequality
Mot clés : champs de vecteurs, distance de Carnot-Carathéodory, inégalité de Poincaré
Montanari, Annamaria 1; Morbidelli, Daniele 

1 Università di Bologna, Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127 Bologna (Italie)
@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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {2},
     year = {2004},
     doi = {10.5802/aif.2024},
     zbl = {1069.46504},
     language = {en},
     url = {http://archive.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  - http://archive.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 http://archive.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, Volume 54 (2004) no. 2, pp. 431-452. doi : 10.5802/aif.2024. http://archive.numdam.org/articles/10.5802/aif.2024/

[1] G. Citti; E. Lanconelli; A. Montanari Smoothness of Lipschitz continuous graphs with non vanishing Levi curvature, Acta Math, Volume 188 (2002), pp. 87-128 | MR | Zbl

[2] G. Citti; A. Montanari Strong solutions for the Levi curvature equation, Adv. in Diff. Eq, Volume 5 (2000) no. 1-3, pp. 323-342 | MR | Zbl

[3] K. Deimling Nonlinear functional analysis, Springer-Verlag, Berlin, 1985 | MR | Zbl

[4] L. Evans; R. Gapiery Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992 | MR | Zbl

[5] C. Fefferman; D. H. Phong Subelliptic eigenvalue problems, Conference on Harmonic Analysis in honor of Antoni Zygmund (Wadsworth Mathematical Series) (1983), pp. 590-606 | MR | Zbl

[6] B. Franchi; E. Lanconelli 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 | MR | Zbl

[7] B. Franchi; E. Lanconelli 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 | EuDML | Numdam | MR | Zbl

[8] B. Franchi G. Lu; R. Wheeden A relationship between Poincaré type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices, Volume 1 (1996), pp. 1-14 | MR | Zbl

[9] B. Franchi; R. Serapioni; F. Serra Cassano 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 | MR | Zbl

[10] N. Garofalo; D. M. Nhieu Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math, Volume 49 (1996), pp. 1081-1144 | MR | Zbl

[11] N. Garofalo; D. M. Nhieu Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math, Volume 74 (1998), pp. 67-97 | MR | Zbl

[12] P. Hajlasz; P. Koskela Sobolev met Poincaré, Mem. Amer. Math. Soc, Volume 688 (2000) | MR | Zbl

[13] D. Jerison The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J, Volume 53 (1986), pp. 503-523 | MR | Zbl

[14] E. Lanconelli Stime subellittiche e metriche Riemanniane singolari, Seminario di Analisi Matematica, Università di Bologna (A.A 1982-83)

[15] E. Lanconelli; D. Morbidelli On the Poincaré inequality for vector fields, Ark. Mat, Volume 38 (2000), pp. 327-342 | MR | Zbl

[16] P. Maheux; L. Saloff-Coste Analyse sur le boules d'un opérateur sous-elliptique, Math. Ann, Volume 303 (1995), pp. 713-746 | MR | Zbl

[17] A. Montanari; D. Morbidelli Sobolev and Morrey estimates for non-smooth Vector Fields of step two, Z. Anal. Anwendungen, Volume 21 (2002) no. 1, pp. 135-157 | MR | Zbl

[18] D. Morbidelli Fractional Sobolev norms and structure of the Carnot--Carathéodory balls for Hörmander vector fields, Studia Math, Volume 139 (2000), pp. 213-244 | MR | Zbl

[19] A. Nagel; E. M. Stein; S. Wainger Balls and metrics defined by vector fields I: Basic properties, Acta Math, Volume 155 (1985), pp. 103-147 | MR | Zbl

[20] F. Rampazzo; H. J. Sussman 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] L. Saloff-Coste A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Res. Notices, Volume 2 (1992), pp. 27-38 | MR | Zbl

[22] N. Th. Varopoulos; L. Saloff-Coste; T. Coulhon Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992 | MR | Zbl

Cited by Sources: