Curvature dimension bounds on the deltoid model
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 1, pp. 65-90.

La courbe deltoide dans le plan est la frontière d’un domaine borné sur lequel il existe une famille de mesures de probabilité et des polynômes orthogonaux pour ces mesures qui sont aussi vecteurs propres d’opérateurs de diffusion. On peut donc considérer ces polynômes comme une extension des polynômes de Jacobi classiques. Ce domaine appartient à l’une des 11 familles de tels domaines bornés de 2 . Nous étudions les inégalités de courbure-dimension associés à ces opérateurs, en en déduisons diverses bornes sur les polyômes associés, ainsi que des inégalités de Sobolev relatives aux formes de Dirichlet correspondantes.

The deltoid curve in 2 is the boundary of a domain on which there exist probability measures and orthogonal polynomials for theses measures which are eigenvectors of diffusion operators. As such, those polynomials may be considered as a two dimensional extension of the classical Jacobi polynomials. This domain belongs to one of the 11 families of such bounded domains in 2 . We study the curvature-dimension inequalities associated to these operators, and deduce various bounds on the associated polynomials, together with Sobolev inequalities related to the associated Dirichlet forms.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1487
Bakry, Dominique 1 ; Zribi, Olfa 1

1 Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France.
@article{AFST_2016_6_25_1_65_0,
     author = {Bakry, Dominique and Zribi, Olfa},
     title = {Curvature dimension bounds on the deltoid model},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {65--90},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {1},
     year = {2016},
     doi = {10.5802/afst.1487},
     mrnumber = {3485291},
     zbl = {1371.60034},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1487/}
}
TY  - JOUR
AU  - Bakry, Dominique
AU  - Zribi, Olfa
TI  - Curvature dimension bounds on the deltoid model
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2016
SP  - 65
EP  - 90
VL  - 25
IS  - 1
PB  - Université Paul Sabatier, Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1487/
DO  - 10.5802/afst.1487
LA  - en
ID  - AFST_2016_6_25_1_65_0
ER  - 
%0 Journal Article
%A Bakry, Dominique
%A Zribi, Olfa
%T Curvature dimension bounds on the deltoid model
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2016
%P 65-90
%V 25
%N 1
%I Université Paul Sabatier, Toulouse
%U http://archive.numdam.org/articles/10.5802/afst.1487/
%R 10.5802/afst.1487
%G en
%F AFST_2016_6_25_1_65_0
Bakry, Dominique; Zribi, Olfa. Curvature dimension bounds on the deltoid model. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 1, pp. 65-90. doi : 10.5802/afst.1487. http://archive.numdam.org/articles/10.5802/afst.1487/

[1] Bakry (D.).— Remarques sur les semi-groupes de Jacobi, Hommage à P.A. Meyer et J. Neveu, vol. 236, Astérisque, p. 23-40 (1996). | Zbl

[2] Bakry (D.) and Émery (M.).— Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci. Paris Sér. I Math. 301, no. 8, p. 411-413 (1985). | Zbl

[3] Bakry (D.), Gentil (I.), and Ledoux (M.).— Analysis and Geometry of Markov Diffusion Operators, Grund. Math. Wiss., vol. 348, Springer, Berlin (2013).

[4] Bakry (D.), Orevkov (S.), and Zani (M.).— Orthogonal polynomials and diffusions operators, submitted, arXiv:1309.5632v2 (2013).

[5] L. Besse (A. L.).— Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin (2008), Reprint of the 1987 edition. | DOI | Zbl

[6] Bobkov (S. G.), Gentil (I.), and Ledoux (M.).— Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. (9) 80, no. 7, p. 669-696 (2001). | DOI | MR | Zbl

[7] Davies (E. B.).— Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge (1989). | Zbl

[8] Dunkl (C.).— Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311, no. 1, p. 167-183 (1989). | DOI | MR | Zbl

[9] Dunkl (C.) and Xu (Y.).— Orthogonal polynomials of several variables., Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge (2001). | DOI | Zbl

[10] Faraut (J.).— Analyse sur les groupes de Lie (2005).

[11] Gallot (S.), Hulin (D.), and Lafontaine (J.).— Riemannian geometry, third ed., Universitext, Springer-Verlag, Berlin (2004). | DOI | MR

[12] Heckman (G. J.).— Root systems and hypergeometric functions. II, Compositio Math. 64, no. 3, p. 353-373 (1987). | Zbl

[13] Heckman (G. J.) and Opdam (E. M.).— Root systems and hypergeometric functions. I, Compositio Math. 64, no. 3, p. 329-352 (1987). | Zbl

[14] Heckman (G. J.).— A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups (W Barker and P. Sally, eds.), vol. Progress in Math, 101, Birkhauser, p. 181-191 (1991). | DOI | Zbl

[15] Heckman (G. J.).— Dunkl operators, Séminaire Bourbaki 828, 1996-97, vol. Astérisque, SMF, p. 223-246 (1997). | Numdam | Zbl

[16] Helgason (S.).— Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI (2001), Corrected reprint of the 1978 original. | DOI | Zbl

[17] Koornwinder (T.).— Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. i., Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36, p. 48-58 (1974). | DOI | Zbl

[18] Koornwinder (T.).— Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. ii., Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36, p. 59-66 (1974). | DOI | Zbl

[19] Krall (H.L.) and Sheffer (I.M.).— Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. 76, p. 325-376 (1967). | DOI | MR | Zbl

[20] Macdonald (I. G.).— Symmetric functions and orthogonal polynomials., University Lecture Series, vol. 12, American Mathematical Society, Providence, RI (1998). | DOI | MR | Zbl

[21] Macdonald (I. G.).— Orthogonal polynomials associated with root systems., Séminaire Lotharingien de Combinatoire, vol. 45, Université Louis Pasteur, Strasbourg (2000). | Zbl

[22] Macdonald (I. G.).— Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge (2003). | DOI | Zbl

[23] Nicolaescu (L. I.).— Lectures on the geometry of manifolds, World Scientific Hackensack (2007). | DOI | Zbl

[24] Rösler (M.).— Generalized hermite polynomials and the heat equation for dunkl operators., Comm. in Math. Phys. 192, no. 3, p. 519-542 (1998). | DOI | MR | Zbl

[25] Rösler (M.).— Dunkl operators: theory and applications. Orthogonal polynomials and special functions (Leuven, 2002), Lecture Notes in Mathematics, vol. 1817, Springer, Berlin (2003). | DOI

[26] Saloff-Coste (L.).— On the convergence to equilibrium of brownian motion on compact simple lie groups, The Journal of Geometric Analysis 14, no. 4, p. 715-733 (2004). | DOI | MR | Zbl

[27] Varopoulos (N. Th.), Saloff-Coste (L.), and Coulhon (T.).— Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, (1992). | DOI

[28] Zribi (O.).— Orthogonal polynomials associated with the deltoid curve (2013). | DOI | MR

Cité par Sources :