Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case $A_n$

Piotr Graczyk; Patrice Sawyer

doi:10.5802/crmath.188

Analyse harmonique, Théorie des représentations

Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case

A_{n}

Piotr Graczyk ¹ ; Patrice Sawyer ²

¹ LAREMA, UFR Sciences, Université d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01, France
² Department of Mathematics and Computer Science, Laurentian University, Sudbury, ON Canada P3E 2C6

Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 427-437.

Résumé
Abstract

Dans cet article, nous considérons le cas géométrique radial de Dunkl $k = 1$ correspondant aux espaces symétriques riemanniens plats dans le cas complexe et nous prouvons des estimations exactes pour le noyau de Dunkl à valeur positive et pour le noyau de chaleur radial.

In this article, we consider the radial Dunkl geometric case $k = 1$ corresponding to flat Riemannian symmetric spaces in the complex case and we prove exact estimates for the positive valued Dunkl kernel and for the radial heat kernel.

Reçu le : 2020-12-21
Révisé le : 2021-02-09
Accepté le : 2021-02-10
Publié le : 2021-05-27

Zbl

DOI : 10.5802/crmath.188

Classification : 33C67, 43A90, 53C35

Affiliations des auteurs :

Piotr Graczyk ¹ ; Patrice Sawyer ²

¹ LAREMA, UFR Sciences, Université d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01, France
² Department of Mathematics and Computer Science, Laurentian University, Sudbury, ON Canada P3E 2C6

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2021__359_4_427_0,
     author = {Piotr Graczyk and Patrice Sawyer},
     title = {Sharp {Estimates} of {Radial} {Dunkl} and {Heat} {Kernels} in the {Complex} {Case} $A_n$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {427--437},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {4},
     year = {2021},
     doi = {10.5802/crmath.188},
     zbl = {07362164},
     language = {en},
}

TY  - JOUR
AU  - Piotr Graczyk
AU  - Patrice Sawyer
TI  - Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case $A_n$
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 427
EP  - 437
VL  - 359
IS  - 4
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.188
LA  - en
ID  - CRMATH_2021__359_4_427_0
ER  -

%0 Journal Article
%A Piotr Graczyk
%A Patrice Sawyer
%T Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case $A_n$
%J Comptes Rendus. Mathématique
%D 2021
%P 427-437
%V 359
%N 4
%I Académie des sciences, Paris
%R 10.5802/crmath.188
%G en
%F CRMATH_2021__359_4_427_0

Piotr Graczyk; Patrice Sawyer. Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case $A_n$. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 427-437. doi : 10.5802/crmath.188. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.188/

Bibliographie
Cité par

[1] Jean-Philippe Anker; Jacek Dziubański; Agnieszka Hejna Harmonic Functions, Conjugate Harmonic Functions and the Hardy Space $H^{1}$ in the Rational Dunkl Setting, J. Fourier Anal. Appl., Volume 25 (2019) no. 5, pp. 2356-2418 | DOI | MR | Zbl

[2] Jean-Philippe Anker; Lizhen Ji Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal., Volume 9 (1999) no. 6, pp. 1035-1091 | DOI | MR | Zbl

[3] Edward B. Davies Heat kernels and spectral theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, 1989 | MR | Zbl

[4] Marcel De Jeu Paley–Wiener theorems for the Dunkl transform, Trans. Am. Math. Soc., Volume 358 (2006) no. 10, pp. 4225-4250 | DOI | MR | Zbl

[5] Léonard Gallardo; Marc Yor A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab., Volume 34 (2006) no. 4, pp. 1530-1549 | MR | Zbl

[6] Piotr Graczyk; Tomasz Luks; Patrice Sawyer Potential kernels for radial Dunkl Laplacians (2019) to appear in theCanadian Journal of Mathematics (2021), https://arxiv.org/abs/1910.03105

[7] Piotr Graczyk; Margit Rösler; Marc Yor Harmonic and Stochastic Analysis of Dunkl Processes, Travaux en Cours, 71, Hermann, 2008 | Zbl

[8] Piotr Graczyk; Patrice Sawyer Integral Kernels on Complex Symmetric Spaces and for the Dyson Brownian Motion (2020) to appear in the Mathematische Nachrichten (2021), https://arxiv.org/abs/2012.10946 | Zbl

[9] Sigurdur Helgason Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, 83, American Mathematical Society, 2000 (corrected reprint of the 1984 original edition) | Zbl

[10] Sigurdur Helgason Differential geometry and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, 2001 (reprint with corrections of the 1978 original edition) | MR | Zbl

[11] Sigurdur Helgason The Bounded Spherical Functions on the Cartan motion group (2015) (https://arxiv.org/abs/1503.07598) | Zbl

[12] E. K. Narayanan; Angela Pasquale; Sanjoy Pusti Asymptotics of Harish–Chandra expansions, bounded hypergeometric functions associated with root systems, and applications, Adv. Math., Volume 252 (2014), pp. 227-259 | DOI | MR | Zbl

[13] Margit Rösler Generalized Hermite polynomials and the heat equation for Dunkl operators, Commun. Math. Phys., Volume 192 (1998) no. 3, pp. 519-542 | MR | Zbl

[14] Margit Rösler Positivity of Dunkl’s intertwining operator, Duke Math. J., Volume 98 (1999) no. 3, pp. 445-463 | MR | Zbl

[15] Patrice Sawyer The Abel transform on symmetric spaces of noncompact type, Lie groups and symmetric spaces. In memory of F. I. Karpelevich (Translations of the American Mathematical Society-Series 2), Volume 210, American Mathematical Society, 2003, pp. 331-335 | MR | Zbl

[16] Patrice Sawyer A Laplace-type representation of the generalized spherical functions associated with the root systems of type $A$ , Mediterr. J. Math., Volume 14 (2017) no. 4, 147 | MR | Zbl

[17] Bruno Schapira Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel, Geom. Funct. Anal., Volume 18 (2008) no. 1, pp. 222-250 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique