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Dominique Bakry; Olfa Zribi
Curvature dimension bounds on the deltoid model
Annales de la faculté des sciences de Toulouse Sér. 6, 25 no. 1 (2016), p. 65-90, doi: 10.5802/afst.1487
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[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). MR 1417973 | Zbl 0859.47026
[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). MR 808640 | Zbl 0579.60079
[3] Bakry (D.), Gentil (I.), and Ledoux (M.).— Analysis and Geometry of Markov Diffusion Operators, Grund. Math. Wiss., vol. 348, Springer, Berlin (2013). MR 3155209
[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. MR 2371700 | Zbl 1147.53001
[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). MR 1846020 | Zbl 1038.35020
[7] Davies (E. B.).— Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge (1989). MR 990239 | Zbl 0699.35006
[8] Dunkl (C.).— Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311, no. 1, p. 167-183 (1989). MR 951883 | Zbl 0652.33004
[9] Dunkl (C.) and Xu (Y.).— Orthogonal polynomials of several variables., Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge (2001). MR 3289583 | Zbl 0964.33001
[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). MR 2088027 | Zbl 0636.53001
[12] Heckman (G. J.).— Root systems and hypergeometric functions. II, Compositio Math. 64, no. 3, p. 353-373 (1987). Numdam | MR 918417 | Zbl 0656.17007
[13] Heckman (G. J.) and Opdam (E. M.).— Root systems and hypergeometric functions. I, Compositio Math. 64, no. 3, p. 329-352 (1987). Numdam | MR 918416 | Zbl 0656.17006
[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). MR 1168482 | Zbl 0749.33005
[15] Heckman (G. J.).— Dunkl operators, Séminaire Bourbaki 828, 1996-97, vol. Astérisque, SMF, p. 223-246 (1997). Numdam | MR 1627113 | Zbl 0916.33012
[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. MR 1834454 | Zbl 0993.53002
[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). MR 340673 | Zbl 0263.33011
[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). MR 340674 | Zbl 0263.33011
[19] Krall (H.L.) and Sheffer (I.M.).— Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. 76, p. 325-376 (1967). MR 228920 | Zbl 0186.38602
[20] Macdonald (I. G.).— Symmetric functions and orthogonal polynomials., University Lecture Series, vol. 12, American Mathematical Society, Providence, RI (1998). MR 1488699 | Zbl 0887.05053
[21] Macdonald (I. G.).— Orthogonal polynomials associated with root systems., Séminaire Lotharingien de Combinatoire, vol. 45, Université Louis Pasteur, Strasbourg (2000). MR 1817334 | Zbl 1032.33010
[22] Macdonald (I. G.).— Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge (2003). MR 1976581 | Zbl 1024.33001
[23] Nicolaescu (L. I.).— Lectures on the geometry of manifolds, World Scientific Hackensack (2007). MR 1435504 | Zbl 1155.53001
[24] Rösler (M.).— Generalized hermite polynomials and the heat equation for dunkl operators., Comm. in Math. Phys. 192, no. 3, p. 519-542 (1998). MR 1620515 | Zbl 0908.33005
[25] Rösler (M.).— Dunkl operators: theory and applications. Orthogonal polynomials and special functions (Leuven, 2002), Lecture Notes in Mathematics, vol. 1817, Springer, Berlin (2003). MR 2022853 | Zbl 1029.43001
[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). MR 2111426 | Zbl 1059.43006
[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). MR 1218884 | Zbl 0813.22003
[28] Zribi (O.).— Orthogonal polynomials associated with the deltoid curve (2013).
Dominique Bakry; Olfa Zribi
Curvature dimension bounds on the deltoid model
Annales de la faculté des sciences de Toulouse Sér. 6, 25 no. 1 (2016), p. 65-90, doi: 10.5802/afst.1487
Article PDF
Résumé - Abstract
The deltoid curve in $\mathbb{R}^2$ 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 $\mathbb{R}^2$. 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.
Bibliography
[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). MR 808640 | Zbl 0579.60079
[3] Bakry (D.), Gentil (I.), and Ledoux (M.).— Analysis and Geometry of Markov Diffusion Operators, Grund. Math. Wiss., vol. 348, Springer, Berlin (2013). MR 3155209
[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. MR 2371700 | Zbl 1147.53001
[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). MR 1846020 | Zbl 1038.35020
[7] Davies (E. B.).— Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge (1989). MR 990239 | Zbl 0699.35006
[8] Dunkl (C.).— Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311, no. 1, p. 167-183 (1989). MR 951883 | Zbl 0652.33004
[9] Dunkl (C.) and Xu (Y.).— Orthogonal polynomials of several variables., Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge (2001). MR 3289583 | Zbl 0964.33001
[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). MR 2088027 | Zbl 0636.53001
[12] Heckman (G. J.).— Root systems and hypergeometric functions. II, Compositio Math. 64, no. 3, p. 353-373 (1987). Numdam | MR 918417 | Zbl 0656.17007
[13] Heckman (G. J.) and Opdam (E. M.).— Root systems and hypergeometric functions. I, Compositio Math. 64, no. 3, p. 329-352 (1987). Numdam | MR 918416 | Zbl 0656.17006
[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). MR 1168482 | Zbl 0749.33005
[15] Heckman (G. J.).— Dunkl operators, Séminaire Bourbaki 828, 1996-97, vol. Astérisque, SMF, p. 223-246 (1997). Numdam | MR 1627113 | Zbl 0916.33012
[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. MR 1834454 | Zbl 0993.53002
[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). MR 340673 | Zbl 0263.33011
[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). MR 340674 | Zbl 0263.33011
[19] Krall (H.L.) and Sheffer (I.M.).— Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. 76, p. 325-376 (1967). MR 228920 | Zbl 0186.38602
[20] Macdonald (I. G.).— Symmetric functions and orthogonal polynomials., University Lecture Series, vol. 12, American Mathematical Society, Providence, RI (1998). MR 1488699 | Zbl 0887.05053
[21] Macdonald (I. G.).— Orthogonal polynomials associated with root systems., Séminaire Lotharingien de Combinatoire, vol. 45, Université Louis Pasteur, Strasbourg (2000). MR 1817334 | Zbl 1032.33010
[22] Macdonald (I. G.).— Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge (2003). MR 1976581 | Zbl 1024.33001
[23] Nicolaescu (L. I.).— Lectures on the geometry of manifolds, World Scientific Hackensack (2007). MR 1435504 | Zbl 1155.53001
[24] Rösler (M.).— Generalized hermite polynomials and the heat equation for dunkl operators., Comm. in Math. Phys. 192, no. 3, p. 519-542 (1998). MR 1620515 | Zbl 0908.33005
[25] Rösler (M.).— Dunkl operators: theory and applications. Orthogonal polynomials and special functions (Leuven, 2002), Lecture Notes in Mathematics, vol. 1817, Springer, Berlin (2003). MR 2022853 | Zbl 1029.43001
[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). MR 2111426 | Zbl 1059.43006
[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). MR 1218884 | Zbl 0813.22003
[28] Zribi (O.).— Orthogonal polynomials associated with the deltoid curve (2013).
