Comptes Rendus
Harmonic analysis, Representation theory
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.

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.

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.

Published online:
DOI: 10.5802/crmath.188
Classification: 33C67, 43A90, 53C35

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Sharp {Estimates} of {Radial} {Dunkl} and {Heat} {Kernels} in the {Complex} {Case} $A_n$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {427--437},
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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.

[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),

[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), | 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) ( | 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

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