39 results on '"Jean-Philippe Anker"'
Search Results
2. Schrödinger equation on non-compact symmetric spaces
- Author
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Jean-Philippe Anker, Stefano Meda, Vittoria Pierfelice, Maria Vallarino, and Hong-Wei Zhang
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Mathematics - Analysis of PDEs ,Applied Mathematics ,Mathematics::Analysis of PDEs ,22E30, 35J10, 35P25, 43A85, 43A90 ,Analysis - Abstract
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schr\"odinger group to the wave propagator and an improved Hadamard parametrix., Comment: 19 pages, major revision with more details
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- 2023
3. Bottom of the $$L^2$$ spectrum of the Laplacian on locally symmetric spaces
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Jean-Philippe Anker, Hong-Wei Zhang, Institut Denis Poisson (IDP), Université d'Orléans (UO)-Université de Tours-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), and Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO)
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$L^2$ spectrum ,Poincaré series ,Critical exponent ,Mathematics - Spectral Theory ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Locally symmetric space ,Geometry and Topology ,[MATH]Mathematics [math] ,Spectral Theory (math.SP) ,Analysis of PDEs (math.AP) ,Heat kernel ,MSC 2010 : 22E40, 11N45, 35K08, 47A10, 58J50 ,[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] - Abstract
We estimate the bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincar\'e series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank., Comment: To appear in Geometriae Dedicata
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- 2022
4. Harmonic Functions, Conjugate Harmonic Functions and the Hardy Space $$H^1$$ H 1 in the Rational Dunkl Setting
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Agnieszka Hejna, Jacek Dziubański, and Jean-Philippe Anker
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Pure mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,020206 networking & telecommunications ,02 engineering and technology ,Hardy space ,01 natural sciences ,Square (algebra) ,Riesz transform ,symbols.namesake ,Harmonic function ,Fourier analysis ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Maximal function ,0101 mathematics ,Laplace operator ,Analysis ,Mathematics - Abstract
In this work we extend the theory of the classical Hardy space $$H^1$$ to the rational Dunkl setting. Specifically, let $$\Delta $$ be the Dunkl Laplacian on a Euclidean space $$\mathbb {R}^N$$ . On the half-space $$\mathbb {R}_+\times \mathbb {R}^N$$ , we consider systems of conjugate $$(\partial _t^2+\Delta _{\mathbf {x}})$$ -harmonic functions satisfying an appropriate uniform $$L^1$$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $$H^1_{\Delta }$$ , can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood–Paley square functions.
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- 2019
5. Comportement asymptotique des solutions de l'équation de la chaleur sur les espaces symétriques de type non-compact
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Jean-Philippe Anker, Effie Papageorgiou, Hong-Wei Zhang, Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Department of Mathematics, University of Crete, University of Crete [Heraklion] (UOC), Department of Mathematics [Gent/Ghent], Universiteit Gent = Ghent University [Belgium] (UGENT), and Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO)
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Asymptotic behaviour ,Mathematics - Analysis of PDEs ,Distinguished Laplacian ,Heat equation ,FOS: Mathematics ,Long-time convergence ,[MATH]Mathematics [math] ,Analysis ,Noncompact symmetric space ,Analysis of PDEs (math.AP) ,22E30, 35B40, 35K05, 58J35 - Abstract
This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-$K$-invariant $L^{1}$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-$K$-invariant case. These answer problems recently raised by J.L. V\'azquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-$K$-invariance condition and strong $L^{\infty}$ convergence., Comment: To appear in J. Funct. Anal
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- 2021
- Full Text
- View/download PDF
6. Relativity without light: A new proof of Ignatowski's theorem
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Jean-Philippe Anker, Francois Ziegler, Mathematical Sciences Department [Georgia Southern University], Georgia Southern University, and University System of Georgia (USG)-University System of Georgia (USG)
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[PHYS]Physics [physics] ,22E70, 83A05 ,special relativity ,010102 general mathematics ,General Physics and Astronomy ,FOS: Physical sciences ,Kinematics ,Mathematical Physics (math-ph) ,01 natural sciences ,Special relativity (alternative formulations) ,Lorentz group ,Theoretical physics ,Theory of relativity ,0103 physical sciences ,010307 mathematical physics ,Geometry and Topology ,[MATH]Mathematics [math] ,0101 mathematics ,applications of Lie groups ,Mathematical Physics ,Mathematics - Abstract
V. Ignatowski (1910) showed that assumptions about light are not necessary to obtain Lorentzian kinematics as one of only few possibilities. We give a much simplified proof of his result as formulated by V. Gorini (1971) for $n$+1-dimensional space-time., 9 pages. Accepted version, to appear in Journal of Geometry and Physics
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- 2020
7. An Introduction to Dunkl Theory and Its Analytic Aspects
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Jean-Philippe Anker, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Projet Régional Centre - Val de Loire MADACA (Marches Aléatoires et processus de Dunkl - Approches Combinatoires et Algébriques), G. Filipuk, Y. Haraoka, and S. Michalik
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Pure mathematics ,Dunkl theory ,Rank (linear algebra) ,010102 general mathematics ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,spherical Fourier analysis ,01 natural sciences ,special functions associated with root systems ,Algebra ,symbols.namesake ,Special functions ,Fourier analysis ,0103 physical sciences ,symbols ,MSC 2010: Primary 33C67 ,Secondary 05E05, 20F55, 22E30, 33C80, 33D67, 39A70, 42B10, 43A32, 43A90 ,010307 mathematical physics ,0101 mathematics ,Trigonometry ,Dunkl operator ,Mathematics - Abstract
International audience; Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was closely connected with certain classes of special functions in one variable. During the eighties, several attempts were made, mainly by the Dutch school, to extend these results in higher rank (i.e. in several variables), until the discovery of Dunkl operators in the rational case and Cherednik operators in the trigonometric case. Together with q-special functions introduced by Macdonald, this has led to a beautiful theory, developed by several authors, which encompasses in a unified way harmonic analysis on all Riemannian symmetric spaces and spherical functions thereon.In this series of lectures, delivered at the Summer School AAGADE 2015 (Analytic, Algebraic and Geometric Aspects of Differential Equations, Mathematical Research and Conference Center, Bedlewo, Poland, September 2015), we aimed at giving an updated overview of Dunkl theory, with an emphasis on its analytic aspects.
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- 2017
8. The Hardy Space $$H^1$$ H 1 in the Rational Dunkl Setting
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Jean-Philippe Anker, Nabila Hamda, Jacek Dziubański, and Néjib Ben Salem
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Mathematics(all) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Hardy space ,01 natural sciences ,Multiplier (Fourier analysis) ,Computational Mathematics ,Atomic decomposition ,symbols.namesake ,Fourier transform ,symbols ,Maximal operator ,0101 mathematics ,Analysis ,Heat kernel ,Dunkl operator ,Mathematics - Abstract
This paper is perhaps the first attempt at a study of the Hardy space $$H^1$$ in the rational Dunkl setting. Following Uchiyama’s approach, we characterize $$H^1$$ atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for $$H^1$$ . These results are proved here in the one-dimensional case and in the product case.
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- 2014
9. An elementary proof of the positivity of the intertwining operator in one-dimensional trigonometric Dunkl theory
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Jean-Philippe Anker, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), and Projet Régional MADACA (Marches Aléatoires et processus de Dunkl - Approches Combinatoires et Algébriques, www.fdpoisson.fr/madaca)
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Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,Expression (computer science) ,01 natural sciences ,Proofs of trigonometric identities ,010101 applied mathematics ,Algebra ,Operator (computer programming) ,Simple (abstract algebra) ,Mathematics - Classical Analysis and ODEs ,MSC 33C67 ,Mathematics::Quantum Algebra ,Elementary proof ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Trigonometry ,Mathematics::Representation Theory ,Dunkl operator ,Mathematics - Abstract
This note is devoted to the intertwining operator in the one--dimensional trigonometric Dunkl setting. We obtain a simple integral expression of this operator and deduce its positivity., A para{\^i}tre dans Proc. Amer. Math. Soc
- Published
- 2016
10. Schrödinger Equations on Damek–Ricci Spaces
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Jean-Philippe Anker, Vittoria Pierfelice, Maria Vallarino, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Dipartimento di Matematica e Applicazioni [Milano], and Università degli Studi di Milano-Bicocca [Milano] (UNIMIB)
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spazi di Damek-Ricci ,Mathematics::Analysis of PDEs ,Schrödinger equation ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,Strichartz estimate ,symbols.namesake ,stime dispersive ,Mathematics - Analysis of PDEs ,Damek-Ricci spaces ,Operator (computer programming) ,equazione di Schrodinger ,0103 physical sciences ,Euclidean geometry ,dispersive estimate ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Mathematics ,varieta' non compatte ,Pointwise ,heat kernel estimate ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,35Q55, 43A85 ,22E30, 35J10, 35K08, 43A90, 58D25 ,Mathematics::Spectral Theory ,Kernel (algebra) ,Mathematics - Classical Analysis and ODEs ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Laplace operator ,Analysis ,Schrödinger's cat - Abstract
In this paper we consider the Laplace-Beltrami operator \Delta on Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(\tau \Delta), when \tau \in C* with Re(\tau) \geq 0. When \tau \in iR*, we obtain in particular pointwise estimates of the Schr\"odinger kernel associated with \Delta. We then prove Strichartz estimates for the Schr\"odinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schr\"odinger equation associated with a distinguished Laplacian on Damek-Ricci spaces, showing that in this case the standard dispersive estimate fails while suitable weighted Strichartz estimates hold.
- Published
- 2011
11. Three results in Dunkl analysis
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Jean-Philippe Anker, Mohamed Sifi, and Béchir Amri
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Pure mathematics ,Paley–Wiener theorem ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,01 natural sciences ,010101 applied mathematics ,Algebra ,Mathematics::Quantum Algebra ,Norm (mathematics) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics ,Dunkl operator - Abstract
In this article, we establish first a geometric Paley-Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the $L^p\to L^p$ norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.
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- 2010
12. The wave equation on Damek-Ricci spaces
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Vittoria Pierfelice, Maria Vallarino, Jean Philippe Anker, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), and Politecnico di Torino = Polytechnic of Turin (Polito)
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Electromagnetic wave equation ,Wave packet ,Mathematics::Analysis of PDEs ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,global well-posedness ,Strichartz estimate ,Strichartz estimates ,Damek-Ricci spaces ,Mathematics - Analysis of PDEs ,semilinear wave equation ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Acoustic wave equation ,dispersive estimate ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Mathematics ,Laplace's equation ,Partial differential equation ,Wave equation ,Damek-Ricci space ,Applied Mathematics ,Operator (physics) ,Mathematical analysis ,35L05, 43A85, 58J45 ,22E30, 35L71, 43A90, 47J35, 58D25 ,Nonlinear wave equation ,Mathematics - Classical Analysis and ODEs ,Mathematics::Differential Geometry ,Analysis of PDEs (math.AP) - Abstract
International audience; We study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on Damek-Ricci spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain global well-posedness results for the nonlinear wave equation.
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- 2015
13. Lie Theory : Lie Algebras and Representations
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Jean-Philippe Anker, Bent Orsted, Jean-Philippe Anker, and Bent Orsted
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- Topological groups, Lie groups, Algebra, Group theory, Harmonic analysis, Geometry, Number theory
- Abstract
Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, analysis, and mathematical physics. Three independent, self-contained volumes, under the general title'Lie Theory,'feature survey work and original results by well-established researchers in key areas of semisimple Lie theory. A wide spectrum of topics is treated, with emphasis on the interplay between representation theory and the geometry of adjoint orbits for Lie algebras over fields of possibly finite characteristic, as well as for infinite-dimensional Lie algebras. Also covered is unitary representation theory and branching laws for reductive subgroups, an active part of modern representation theory. Finally, there is a thorough discussion of compactifications of symmetric spaces, and harmonic analysis through a far-reaching generalization of Harish--Chandra's Plancherel formula for semisimple Lie groups. Ideal for graduate students and researchers,'Lie Theory'provides a broad, clearly focused examination of semisimple Lie groups and their integral importance to research in many branches of mathematics.
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- 2012
14. Comportement exact du noyau de la chaleur et de la fonction de Green sur les espaces symétriques non compacts
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Lizhen Ji and Jean-Philippe Anker
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Pure mathematics ,Symmetric space ,General Medicine ,Heat kernel ,Mathematics - Abstract
Resume Nous donnons un encadrement optimal du noyau de la chaleur h t ( x , y ) sur les espaces riemanniens symetriques non compacts G/K . lorsque t ≫ d( x.y ). Nous en deduisons un encadrement global de la fonction de Green sur G/K et une inegalite de type faible (1,1) pour l'operateur maximal associe a une diffusion particuliere sur les groupes d'Iwasawa AN .
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- 1998
15. The Shifted Wave Equation on Damek–Ricci Spaces and on Homogeneous Trees
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Pierre Martinot, Alberto G. Setti, Jean-Philippe Anker, Emmanuel Pedon, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Reims (LMR), Université de Reims Champagne-Ardenne (URCA)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Scienza e Alta Tecnologia, Universitá degli Studi dell’Insubria, HCM Network Fourier Analysis 1994-1997, TMR Network Harmonic Analysis 1998-2002, IHP Network HARP 2002-2006, PHC Galilée 25970QB VAMP 2011-2012, and M.A. Picardello
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Homogeneous tree ,Wave propagation ,Inverse ,wave propagation ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Huygens–Fresnel principle ,symbols.namesake ,Abel transform ,homogeneous tree ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Mathematics ,Huygens' principle ,35L05, 43A85 ,20F67, 22E30, 22E35, 33C80, 43A80, 58J45 ,Damek-Ricci space ,Hyperbolic space ,010102 general mathematics ,Mathematical analysis ,16. Peace & justice ,Wave equation ,010101 applied mathematics ,hyperbolic space ,Homogeneous ,symbols ,wave equation ,Mathematics::Differential Geometry - Abstract
International audience; We solve explicitly the shifted wave equation on Damek--Ricci spaces, using Asgeirsson's theorem and the inverse dual Abel transform. As an application, we investigate Huygens' principle. A similar analysis is carried out in the discrete setting of homogeneous trees.
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- 2013
16. Opdam's hypergeometric functions: product formula and convolution structure in dimension 1
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Fatma Ayadi, Jean-Philippe Anker, Mohamed Sifi, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Analyse Mathématique et Applications, Université de Tunis El Manar (UTM)-Ecole Préparatoire aux Etudes d'Ingénieurs de Tunis, and PHC Utique / CMCU 07G 1501 (aspects analytique et probabiliste de la théorie de Dunkl) et 10G 1503 (analyse et probabilités liées aux systèmes de racines)
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genetic structures ,General Mathematics ,Opdam-Cherednik transform ,MSC 2010: Primary 33C67, 43A62, 44A35 ,Secondary 33C45, 33C52, 43A15, 43A32 ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,Lambda ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,01 natural sciences ,Combinatorics ,product formula ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Dunkl-Cherednik operator ,Beta (velocity) ,Kunze-Stein phenomenon ,0101 mathematics ,Hypergeometric function ,Mathematics::Representation Theory ,Mathematics ,010102 general mathematics ,food and beverages ,Eigenfunction ,convolution product ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Mathematics - Classical Analysis and ODEs - Abstract
Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an integral in terms of $G_{\lambda}^{(\alpha,\beta)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. R\"osler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\lambda}^{(\alpha,\beta)}$., Comment: Adv. Pure Appl. Math. (2011) 27 pp
- Published
- 2012
17. The wave equation on hyperbolic spaces
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Maria Vallarino, Vittoria Pierfelice, Jean-Philippe Anker, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), and Politecnico di Torino = Polytechnic of Turin (Polito)
- Subjects
local well-posedness ,Mathematics::Analysis of PDEs ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,Hyperbolic space ,Strichartz estimate ,Mathematics - Analysis of PDEs ,semilinear wave equation ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,dispersive estimate ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,35L05, 43A85 ,22E30, 35L71, 43A90, 47J35, 58D25, 58J45 ,0101 mathematics ,Mathematics ,Applied Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,Wave equation ,Semilinear wave equation ,Dispersive estimate ,Local well-posedness ,Mathematics - Classical Analysis and ODEs ,Nonlinear wave equation ,010307 mathematical physics ,Analysis ,Analysis of PDEs (math.AP) - Abstract
International audience; In this paper, we study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on real hyperbolic spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain local well-posedness results for the nonlinear wave equation.
- Published
- 2010
- Full Text
- View/download PDF
18. Asymptotic finite propagation speed for heat diffusion on certain Riemannian manifolds
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Alberto G. Setti and Jean-Philippe Anker
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Pointwise ,010102 general mathematics ,Mathematical analysis ,Vector bundle ,Annulus (mathematics) ,Riemannian geometry ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Symmetric space ,symbols ,Heat equation ,0101 mathematics ,Laplace operator ,Analysis ,Heat kernel ,Mathematics - Abstract
Using pointwise upper bounds recently obtained by the first author, we first show that the heat kernel ht(x, y) on a Riemannian symmetric space of the noncompact type GK is asymptotically concentrated in an annulus centered at y and moving to infinity with finite speed 2 ¦ϱ¦, ϱ being as usual the half sum of all positive roots of GK. In the higher rank case we prove moreover that heat not only concentrates in an annulus but also along the (K-orbit) of the ϱ-axis. By applying wave equation techniques developed by M. E. Taylor, we partially extend the above results to Riemannian manifolds with exponential volume growth and L2 spectrum of the Laplacian bounded away from 0. Some extensions to the vector bundle case are also considered.
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- 1992
19. Besov-Type Spaces on Rdand Integrability for the Dunkl Transform
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Mohamed Sifi, Jean-Philippe Anker, Chokri Abdelkefi, Feriel Sassi, Analyse Mathématique et Applications, Université de Tunis El Manar (UTM)-Ecole Préparatoire aux Etudes d'Ingénieurs de Tunis, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), and Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO)
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Pure mathematics ,Mathematics::Classical Analysis and ODEs ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,Characterization (mathematics) ,Type (model theory) ,Besov-Dunkl spaces ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,Space (mathematics) ,01 natural sciences ,Convolution ,Mathematics::Quantum Algebra ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematical Physics ,Dunkl operator ,Mathematics ,Mathematics::Functional Analysis ,Dunkl operators ,lcsh:Mathematics ,010102 general mathematics ,Function (mathematics) ,Dunkl transform ,Dunkl translations ,lcsh:QA1-939 ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Algebra ,Mathematics - Classical Analysis and ODEs ,Schwartz space ,Interpolation space ,Geometry and Topology ,Dunkl convolution ,Analysis - Abstract
International audience; In this paper, we show the inclusion and the density of the Schwartz space in Besov-Dunkl spaces and we prove an interpolation formula for these spaces by the real method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable Besov-Dunkl space.
- Published
- 2009
20. The spherical Fourier transform of rapidly decreasing functions. A simple proof of a characterization due to Harish-Chandra, Helgason, Trombi, and Varadarajan
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Jean-Philippe Anker
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Pure mathematics ,Discrete-time Fourier transform ,Mathematical analysis ,Fourier inversion theorem ,Characterization (mathematics) ,Fractional Fourier transform ,Image (mathematics) ,symbols.namesake ,Discrete Fourier transform (general) ,Fourier transform ,Simple (abstract algebra) ,symbols ,Analysis ,Mathematics - Abstract
Harish-Chandra, S. Helgason, P. C. Trombi and V. S. Varadarajan have described the image of the L p Schwartz spaces (0 p ⩽ 2), under the spherical Fourier transform. We give a short proof of their results, based on the Paley-Wiener theorem.
- Published
- 1991
21. Nonlinear Schr\'odinger equation on real hyperbolic spaces
- Author
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Jean-Philippe Anker, Vittoria Pierfelice, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), and Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO)
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Mathematics::Analysis of PDEs ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,Schrödinger equation ,Strichartz estimates ,symbols.namesake ,wellposedness ,Mathematics - Analysis of PDEs ,0103 physical sciences ,dispersive estimates ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Initial value problem ,Gauge theory ,0101 mathematics ,Nonlinear Schrödinger equation ,Mathematical Physics ,Mathematical physics ,Physics ,Applied Mathematics ,Hyperbolic space ,scattering ,010102 general mathematics ,Invariant (physics) ,hyperbolic space ,Schrôdinger equation ,Mathematics - Classical Analysis and ODEs ,symbols ,010307 mathematical physics ,power-like nonlinearities ,Hyperbolic partial differential equation ,Analysis ,Linear equation - Abstract
We consider the Schr\"odinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we get strong well-posedness results for NLS. Specifically, for small intial data, we prove $L^2$ and $H^1$ global well-posedness for any subcritical nonlinearity (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity $F$. On the other hand, if $F$ is gauge invariant, $L^2$ charge is conserved and hence, as in the Euclidean case, it is possible to extend local $L^2$ solutions to global ones. The corresponding argument in $H^1$ requires the conservation of energy, which holds under the stronger condition that $F$ is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles & Staffilani, for small radial $L^2$ data and for large radial $H^1$ data. The second important application of our global Strichartz estimates is "scattering" for NLS both in $L^2$ and in $H^1$, with no radial or gauge invariance assumption. Notice that, in the Euclidean case, this is only possible for the critical power $\gamma=1+\frac4n$ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small $L^2$ solutions holds for all powers $1, Comment: Version 1 : 18 January 2008. Version 2 : 29 February 2008
- Published
- 2008
22. The infinite Brownian loop on a symmetric space
- Author
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Jean-Philippe Anker, Philippe Bougerol, Thierry Jeulin, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), First author partially supported by the European Commission (TMR Network Harmonic Analysis 1997-2002), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), and Anker, Jean-Philippe
- Subjects
quotient limit theorem ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,General Mathematics ,central limit theorem ,58G11 ,Zonal spherical function ,58G32 ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,010104 statistics & probability ,Mathematics::Probability ,43A85 ,Quantum mechanics ,spherical function ,Almost surely ,ground state ,0101 mathematics ,Ricci curvature ,43A85, 53C35, 58G32, 60J60 ,22E30, 43A90, 58G11, 60H30, 60F17 ,60J60 ,Mathematics ,Mathematical physics ,Riemannian manifold ,010102 general mathematics ,Polar decomposition ,relativized process ,Brownian bridge ,[MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA] ,53C35 ,Loop (topology) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,symmetric space ,43A90 ,60F17 ,heat kernel ,Spectral gap ,Weyl chamber ,60H30 ,22E30 - Abstract
The infinite Brownian loop $\{B_t^0,t\ge 0\}$ on a Riemannian manifold $\mathbb M$ is the limit in distribution of the Brownian bridge of length $T$ around a fixed origin $0$, when $T\to+\infty$. It has no spectral gap. When $\mathbb M$ has nonnegative Ricci curvature, $B^0$ is the Brownian motion itself. When $\mathbb M=G/K$ is a noncompact symmetric space, $B^0$ is the relativized $\Phi_0$--process of the Brownian motion, where $\Phi_0$ denotes the basic spherical function of Harish--Chandra, i.e. the $K$--invariant ground state of the Laplacian. In this case, we consider the polar decomposition $B_t^0=(K_t,X_t)$, where $K_t\in K/M$ and $X_t\in\overline{\mathfrak a_+}$, the positive Weyl chamber. Then, as $t\to+\infty$, $K_t$ converges and $d(0,X_t)/t\to 0$ almost surely. Moreover the processes $\{X_{tT}/\sqrt{T},t\ge 0\}$ converge in distribution, as $T\to+\infty$, to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that $d(0,X_{tT})/\sqrt{T}$ converges to a Bessel process of dimension $D=\operatorname{rank}\mathbb M+2j$, where $j$ denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on $\Phi_0$.
- Published
- 2002
23. Heat kernel and Green function estimates on noncompact symmetric spaces II
- Author
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Jean-Philippe Anker, Lizhen Ji, Anker, Jean-Philippe, J.C. Taylor, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Department of Mathematics - University of Michigan, University of Michigan [Ann Arbor], University of Michigan System-University of Michigan System, and First author partially supported by the European Commission (HCM Network Fourier Analysis 1994-1997 and TMR Network Harmonic Analysis 1998-2002). Second author partially supported by the U.S.A. National Science Foundation (postdoctoral fellowship DMS 9407
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Harnack inequality ,010102 general mathematics ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,[MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA] ,noncompact) ,spherical functions ,Green function ,heat kernel ,0103 physical sciences ,22E30, 22E46, 31C12, 43A80, 43A85, 43A90, 58G11 ,010307 mathematical physics ,0101 mathematics ,symmetric spaces (Riemannian ,semisimple Lie groups - Abstract
In a recent joint work with the same title, we have obtained optimal upper and lower bounds for the heat kernel $h_t(x,y)$ on a noncompact symmetric space $G/K$, under the assumption that $d(x,y)=O(1+t)$. In this article, we reprove our main result in a simpler way, using Harnack's inequality and avoiding this way hard analysis along faces.
- Published
- 2001
24. Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces
- Author
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Jean-Philippe Anker
- Subjects
Power sum symmetric polynomial ,Triple system ,43A85 ,General Mathematics ,Mathematical analysis ,58G11 ,Elementary symmetric polynomial ,Ring of symmetric functions ,Laplace operator ,22E30 ,Symmetric closure ,Mathematics - Published
- 1992
25. Handling the Inverse Spherical Fourier Transform
- Author
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Jean-Philippe Anker
- Subjects
Physics ,symbols.namesake ,Pure mathematics ,Discrete Fourier transform (general) ,Fourier transform ,Fourier analysis ,Symmetric space ,Hartley transform ,symbols ,Inverse Laplace transform ,Mathematics::Differential Geometry ,Fractional Fourier transform ,Fourier transform on finite groups - Abstract
We use the standard notation and refer to [GV], [H] for more details. Let X = G/K be a Riemannian symmetric space of the noncompact type.
- Published
- 1991
26. Lie Theory : Unitary Representations and Compactifications of Symmetric Spaces
- Author
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Jean-Philippe Anker, Bent Orsted, Jean-Philippe Anker, and Bent Orsted
- Subjects
- Symmetric spaces, Lie groups
- Abstract
Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, a self-contained work by A. Borel, L. Ji and T. Kobayashi, focuses on two fundamental questions in the theory of semisimple Lie groups: the geometry of Riemannian symmetric spaces and their compactifications; and branching laws for unitary representations, i.e. restricting unitary representations to (typically, but not exclusively, symmetric) subgroups and decomposing the ensuing representations into irreducibles. Ji's introductory chapter motivates the subject of symmetric spaces and their compactifications with carefully selected examples and provides a good background for the second chapter, namely, the Borel–Ji authoritative treatment of various types of compactifications useful for studying symmetric and locally symmetric spaces. Kobayashi examines the important subject of branching laws. Knowledge of basic representation theory of Lie groups and familiarity with semisimple Lie groups and symmetric spaces is required of the reader.
- Published
- 2005
27. Lie Theory : Harmonic Analysis on Symmetric Spaces – General Plancherel Theorems
- Author
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Jean-Philippe Anker, Bent Orsted, Jean-Philippe Anker, and Bent Orsted
- Subjects
- Lie groups, Harmonic analysis, Linear topological spaces, Symmetric spaces
- Abstract
Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, analysis, and mathematical physics. Three independent, self-contained volumes, under the general title of Lie Theory, feature survey work and original results by well-established researchers in key areas of semisimple Lie theory. Harmonic Analysis on Symmetric Spaces – General Plancherel Theorems presents extensive surveys by E.P. van den Ban, H. Schlichtkrull, and P. Delorme of the spectacular progress over the past decade in deriving the Plancherel theorem on reductive symmetric spaces. Well suited for both graduate students and researchers in semisimple Lie theory and neighboring fields, possibly even mathematical cosmology, it provides a broad, clearly focused examination of semisimple Lie groups and their integral importance and applications to research in many branches of mathematics and physics. Knowledge of basic representation theory of Lie groups as well as familiarity with semisimple Lie groups, symmetric spaces, and parabolic subgroups is required.
- Published
- 2005
28. A Basic Inequality for Scattering Theory on Riemannian Symmetric Spaces of the Noncompact Type
- Author
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Jean-Philippe Anker
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Scattering theory ,Type (model theory) ,Mathematics ,media_common - Published
- 1991
29. L p Fourier Multipliers on Riemannian Symmetric Spaces of the Noncompact Type
- Author
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Jean-Philippe Anker
- Subjects
symbols.namesake ,Mathematics (miscellaneous) ,Fourier transform ,Mathematical analysis ,symbols ,Statistics, Probability and Uncertainty ,Type (model theory) ,Mathematics - Published
- 1990
30. Applications de lap-induction en analyse harmonique
- Author
-
Jean-Philippe Anker
- Subjects
General Mathematics ,Calculus ,Mathematics - Published
- 1983
31. A short proof of a classical covering lemma
- Author
-
Jean-Philippe Anker
- Subjects
Discrete mathematics ,symbols.namesake ,Argument ,Vitali covering lemma ,General Mathematics ,symbols ,Lebesgue's number lemma ,Locally compact space ,Locally compact group ,Zorn's lemma ,Mathematics ,Covering lemma - Abstract
We give a short proof of a general covering lemma for locally compact groups, using an elementary packing argument.
- Published
- 1989
32. Wave equation on symmetric and locally symmetric spaces of noncompact type
- Author
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Zhang, Hong-Wei, Université d'Orléans (UO), Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Université d'Orléans, Jean-Philippe Anker, Nicolas Burq, Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), and STAR, ABES
- Subjects
Propriété de dispersion ,Inégalité de Strichartz ,[PHYS.PHYS]Physics [physics]/Physics [physics] ,Espace symétrique ,Symmetric space ,Strichartz inequality ,Espace localement symétrique ,Spectrum of Laplacian ,Locally symmetric space ,Spectre de laplacian ,Wave equation ,Équation des ondes ,[PHYS.PHYS] Physics [physics]/Physics [physics] ,Dispersive property - Abstract
This thesis is devoted to the study of the wave equation on symmetric and locally symmetric spaces of noncompact type. One of our main results is to obtain pointwise kernel estimates for the wave equation on noncompact symmetric spaces of higher rank. They allow us to prove the dispersive property and to establish the Strichartz inequality for a large family of admissible pairs. We deduce global well-posedness results for the corresponding semilinear wave equation with low regularity initial data. In other words, we extend the results obtained on real hyperbolic spaces to noncompact symmetric spaces of general rank. The other part of our work concerns analysis on locally symmetric spaces. On the one hand, we study the wave and Klein-Gordon equations on certain locally symmetric spaces of rank one. On the other hand, we establish a characterization for the bottom of L2 spectrum of Laplacian on locally symmetric spaces of general rank., Cette thèse est consacrée à l’étude de l’équation des ondes sur les espaces symétriques et localement symétriques de type non compact. Un de nos principaux résultats est l’obtention des estimations ponctuelles du noyau pour l’équation des ondes sur les espaces symétriques non compacts de rang supérieur. Elles nous permettent de démontrer la propriété de dispersion et d’établir l’inégalité de Strichartz pour une grande famille de paires admissibles. Nous en déduisions que l’équation des ondes semi-linéaire correspondante est globalement bien posée pour les données initiales de régularité faible. Autrement dit, nous étendons les résultats obtenus sur les espaces hyperboliques réels aux espaces symétriques non compacts de rang général. L’autre partie de nos travaux concerne l'analyse sur les espaces localement symétriques. D’un côté, nous étudions les équations des ondes et de Klein-Gordon sur certains espaces localement symétriques de rang un. D’autre part, nous établissons une caractérisation pour le bas du spectre L2 du laplacien sur les espaces localement symétriques de rang général.
- Published
- 2020
33. Équation des ondes sur les espaces symétriques et localement symétriques de type non compact
- Author
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Zhang, Hong-Wei, Université d'Orléans (UO), Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Université d'Orléans, Jean-Philippe Anker, Nicolas Burq, Luc Hillairet [Président], Michael Cowling [Rapporteur], Nikolay Tzvetkov [Rapporteur], Manuela Valeria Banica, and Michael Ruzhansky
- Subjects
locally symmetric space ,dispersive property ,espace symétrique ,Strichartz inequality ,spectre de laplacien ,espace localement symétrique ,kernel esti- mate ,symmetric space ,propriété de dispersion ,spectrum of Laplacian ,inégalité de Strichartz ,Spectre de laplacian ,estimation du noyau ,wave equation ,Klein-Gordon equation ,équation de Klein-Gordon ,[MATH]Mathematics [math] ,équation des ondes - Abstract
This thesis is devoted to the study of the wave equation on symmetric and locally symmetric spaces of noncompact type. One of our main results is to obtain pointwise kernel estimates for the wave equation on noncompact symmetric spaces of higher rank. They allow us to prove the dispersive property and to establish the Strichartz inequality for a large family of admissible pairs. We deduce global well-posedness results for the corresponding semilinear wave equation with low regularity initial data. In other words, we extend the results obtained on real hyperbolic spaces to noncompact symmetric spaces of general rank. The other part of our work concerns the study on locally symmetric spaces. On the one hand, we study the wave and Klein-Gordon equations on certain locally symmetric spaces of rank one. On the other hand, we esta- blish a characterization for the bottom of L2 spectrum of Laplacian on locally symmetric spaces of general rank.; Cette thèse est consacrée à l’étude de l’équation des ondes sur les espaces symétriques et localement symétriques de type non compact. Un de nos principaux résultats est l’obtention des estimations ponctuelles du noyau pour l’équation des ondes sur les espaces symétriques non compacts de rang supérieur. Elles nous permettent de démontrer la propriété de dispersion et d’établir l’inégalité de Strichartz pour une grande famille des paires admissibles. Nous en déduisions que l’équation des ondes semi-linéaire correspondante est globalement bien posée pour les données initiales de régularité faible. Autrement dit, nous étendons les résultats obtenus sur les espaces hyperboliques réels aux espaces symétriques non compacts de rang général. L’autre partie de nos travaux concerne l’étude sur les espaces localement symétriques. D’un côté, nous étudions les équations des ondes et de Klein-Gordon sur certains espaces localement symétriques de rang un. D’autre part, nous établissons une caractérisation pour le bas du spectre L2 du laplacien sur les espaces localement symétriques de rang général.
- Published
- 2020
34. Equations d'évolution sur certains groupes hyperboliques
- Author
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Jamal Eddine, Alaa, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Université d'Orléans, Jean-Philippe Anker, and Vittoria Pierfelice
- Subjects
Estimations de Strichartz ,[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] ,Strichartz estiamtes ,Graphes symétriques ,Heat equation ,Equation de Schrödinger ,Wave equation ,Schrödinger equation ,Homogeneous tree ,Symmetric graph ,Equation de la chaleur ,Arbre homogène ,Equation des ondes - Abstract
This thesis focuses on the study of evolution equations on certain hyperbolic groups, in particular, we study the heat equation, the Schrödinger equation and the modified wave equation first on homogeneous trees then on symmetric graphs. In the homogeneous trees case, we show that under a gauge invariance condition, we have global existence of solutions of the Schrödinger equation and scattering for arbitrary data in the space of square integrable functions without any restriction on the degree of the nonlinearity, in contrast to the euclidean and hyperbolic space cases. We then generalize this result on symmetric graphs of degree (k − 1)(r − 1) under the condition k < r . One of our main results on symmetric graphs is the estimate of the heat kernel associated to the combinatorial laplacian. Finally, we establish an explicit expression of solutions of the modified wave equation on symmetric graphs.; Cette thèse porte sur l’étude d’équations d’évolution sur certains groupes hyperboliques, en particulier, nous étudions l’équation de la chaleur, l’équation de Schrödinger et l’équation des ondes modifiée, d’abord sur les arbres homogènes, ensuite sur des graphes symétriques. Sur les arbres homogènes, nous montrons que, sous une hypothèse d’invariance de jauge, on a existence globale des solutions de l’équation de Schrödinger ainsi qu’un phénomène de ’scattering’ pour des données arbitraires dans l’espace des fonctions de carré intégrable sans restriction sur le degré de la non-linéarité, contrairement au cas euclidien ou au cas hyperbolique. Nous généralisons ensuite ce résultat sur les graphes symétriques de degré (k − 1)(r − 1) sous la condition k < r. Un de nos principaux résultats sur les graphes symétriques est l’estimation du noyau de la chaleur associé au laplacien combinatoire. Pour finir, nous établissons une expression explicite des solutions de l’équation des ondes modifiée sur les graphes symétriques.
- Published
- 2013
35. Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique
- Author
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Ayadi Ben Said, Fatma, Mathématiques appliquées et physique mathématique d'Orléans (MAPMO), Université d'Orléans (UO), Université d'Orléans, Jean-Philippe Anker, and Mohamed Sifi
- Subjects
Estimations de Strichartz ,[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] ,Strichartz estiamtes ,Trigonometric Dunkl Laplacian ,Laplacien de Dunkl trigonométrique - Abstract
This thesis consists of three chapters. The first one is concerned with energy properties of the wave equation associated with the trigonometric Dunkl Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case. These results were published in [8]. The second chapter, in collaboration with J.Ph. Anker and M. Sifi [6], shows that Opdam’s functions in the rank one case satisfy a product formula. We then define and study a convolution structure related to Opdam’s functions. In particular, we prove that this convolution fulfills a Kunze-Stein type phenomena. The last chapter deals with dispersive and Strichartz estimates for the linear Schrödinger equation associated with the one dimensional trigonometric Dunkl Laplacian [7]. We establish sharp estimates for the heat kernel in complex time, and therefore for the Schrödinger kernel. We then use these estimates together with tools from chapter 2 to deduce dispersive and Strichartz inequalities for the linear Schrödinger equation and apply them to well–posedness in the nonlinear case.; Cette thèse est constituée de trois chapitres. Le premièr chapitre porte sur l’examen desconditions de validité du principe d’équipartition de l’énergie totale de la solution de l’équationdes ondes associée au laplacien de Dunkl trigonométrique. Enfin, nous établissons lecomportement asymptotique de l’équipartition dans le cas général. Les résultats de cettepartie ont fait l’objet de la publication [8]. Le deuxième chapitre, publié avec J.Ph. Ankeret M. Sifi [6], montre que les fonctions d’Opdam dans le cas de rang 1 satisfont à uneformule produit. Cela nous a permis de définir une structure de convolution du genre hypergroupe.En particulier, on montre que cette convolution satisfait l’analogue du phénomènede Kunze-Stein. Le dernier chapitre est consacrée à l’étude des propriétés dispersives et estimationsde Strichartz pour la solution de l’équation de Schrödinger associée au laplaciende Dunkl trigonométrique unidimensionnel [7]. Cette étude commence par des estimationsoptimales du noyau de la chaleur et de Schrödinger. À l’aide de ces résultats, ainsi que lesoutils d’analyse harmonique dévellopée dans le chapitre 2, on montre des éstimées de typeStrichartz qui permettent de trouver des conditions d’admissibilité pour des équations deSchrödinger semi-linéaires.
- Published
- 2011
36. Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique
- Author
-
Ayadi, Fatma, STAR, ABES, Ayadi, Fatma, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Université d'Orléans, and M. Jean-Philippe Anker et M. Mohamed Sifi(jean-philippe.anker@univ-orleans.fr et mohamed.sifi@fst.run.tn)
- Subjects
"équation des ondes" ," product formula" ," Strichartz estiamtes" ,Strichartz estiamtes ," Laplacien de Dunkl trigonométrique" ," heat equation" ,"estimations de Strichartz" ,[MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM] ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ," Schrödinger equation" ,[MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA] ,"équation de la chaleur" ,Estimations de Strichartz ," formule produit" ,Trigonometric Dunkl Laplacian ," wave equation" ,Laplacien de Dunkl trigonométrique ,"équation de Schrödinger" ,"Trigonometric Dunkl Laplacian" - Abstract
This thesis consists of three chapters. The first one is concerned with energy properties of the wave equation associated with the trigonometric Dunkl Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case. These results were published in [8]. The second chapter, in collaboration with J.Ph. Anker and M. Sifi [6], shows that Opdam’s functions in the rank one case satisfy a product formula. We then define and study a convolution structure related to Opdam’s functions. In particular, we prove that this convolution fulfills a Kunze-Stein type phenomena. The last chapter deals with dispersive and Strichartz estimates for the linear Schrödinger equation associated with the one dimensional trigonometric Dunkl Laplacian [7]. We establish sharp estimates for the heat kernel in complex time, and therefore for the Schrödinger kernel. We then use these estimates together with tools from chapter 2 to deduce dispersive and Strichartz inequalities for the linear Schrödinger equation and apply them to well–posedness in the nonlinear case., Cette thèse est constituée de trois chapitres. Le premièr chapitre porte sur l’examen desconditions de validité du principe d’équipartition de l’énergie totale de la solution de l’équationdes ondes associée au laplacien de Dunkl trigonométrique. Enfin, nous établissons lecomportement asymptotique de l’équipartition dans le cas général. Les résultats de cettepartie ont fait l’objet de la publication [8]. Le deuxième chapitre, publié avec J.Ph. Ankeret M. Sifi [6], montre que les fonctions d’Opdam dans le cas de rang 1 satisfont à uneformule produit. Cela nous a permis de définir une structure de convolution du genre hypergroupe.En particulier, on montre que cette convolution satisfait l’analogue du phénomènede Kunze-Stein. Le dernier chapitre est consacrée à l’étude des propriétés dispersives et estimationsde Strichartz pour la solution de l’équation de Schrödinger associée au laplaciende Dunkl trigonométrique unidimensionnel [7]. Cette étude commence par des estimationsoptimales du noyau de la chaleur et de Schrödinger. À l’aide de ces résultats, ainsi que lesoutils d’analyse harmonique dévellopée dans le chapitre 2, on montre des éstimées de typeStrichartz qui permettent de trouver des conditions d’admissibilité pour des équations deSchrödinger semi-linéaires.
- Published
- 2011
37. Analytical and probabilistic study of Laplacians associated with root systems: hypergeometric Laplacian of Heckman--Opdam and combinatorial Laplacian on affine buildings
- Author
-
Schapira, Bruno, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Université d'Orléans, Jean-Philippe Anker et Philippe Bougerol(jean-philippe.anker@univ-orleans.fr et bougerol@ccr.jussieu.fr), Schapira, Bruno, Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
processusstochastiques ,immeubles affines ,Système de racines ,random walks ,[MATH] Mathematics [math] ,theory of Heckman--Opdam ,théorie deHeckman--Opdam ,affine buildings ,Poisson boundary ,théorèmes limites ,noyau de la chaleur ,heat kernel ,frontière de Poisson ,stochastic processes ,limit theorem ,marches aléatoires ,Root systems ,[MATH]Mathematics [math] - Abstract
In this thesis, we are interested inthe study of analytical and probabilistic aspects ofHeckman--Opdam and affine buildings of type $\tilde{A}_r$theories. We also study the Poisson boundary of rationaltriangular matrices.One of our main results, is to obtain new estimates of thehypergeometric functions of Heckman--Opdam. Our proofs arerelatively more elementary than in the particular case ofsymmetric spaces $G/K$. For instance for the proof of the basicestimates of spherical functions, obtained by Harish-Chandra orGangolli and Varadarajan, and for the recent estimate of theelementary spherical function $\phi_0$ by Anker, Bougerol andJeulin.Another main result is the estimate of the heat kernel associatedwith some combinatorial Laplacian on an affine building oftype $\tilde{A}_r$., Cette thèse porte sur une étudeanalytique et probabiliste des théories de Heckman--Opdam et desimmeubles affines de type $\tilde{A}_r$. On étudie aussi lafrontière de Poisson des matrices triangulaires inversiblesrationnelles.Un de nos principaux résultats est l'obtention de nouvellesestimations des fonctions hypergéométriques de Heckman--Opdam. Nospreuves sont relativement plus simples que dans le cas particulierdes espaces symétriques $G/K$. Par exemple pour les estimations debase des fonctions sphériques, obtenues par Harish-Chandra, ouGangolli et Varadarajan, ainsi que pour les estimations récentesde la fonction sphérique élémentaire $\phi_0$ par Anker, Bougerolet Jeulin.Un des autres principaux résultats est l'estimation du noyau de lachaleur associé à un certain laplacien combinatoire sur unimmeuble affine de type $\tilde{A}_r$.
- Published
- 2006
38. Estimations globales du noyau de la chaleur
- Author
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Ostellari, Patrick, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Université Henri Poincaré - Nancy 1, Jean-Philippe Anker, and Ostellari, Patrick
- Subjects
groupe de Lie réductif ,sous-groupes paraboliques ,sous-laplacien ,principe du maximum ,équation de la chaleur ,fonction de Green ,distance de Carnot-Carathéodory ,espace symétrique ,[MATH] Mathematics [math] ,métrique sous-riemannienne ,noyau de la chaleur ,laplacien ,[MATH]Mathematics [math] ,groupe de Lie semi-simple ,EDP parabolique - Abstract
This thesis deals with sharp heat kernel estimates in two related settings. We consider first noncompact Riemannian symmetric spaces X = G/K, and obtain in this case the same upper and lower bound for the heat kernel associated with the Laplace-Beltrami operator L. These bounds are global in space and time. We consider next the class of sub-Laplacians on a semisimple Lie group G which induce L on the associated symmetric space X = G/K. These sub-Laplacians share properties with L: they have the same L^2 spectral gap, the associated Carnot-Carathéodory distances are all comparable with the Riemannian metric on X and, most of all, their heat kernels are all comparable (for large time) with the heat kernel on X. This yields sharp heat kernel bounds and, consequently, optimal Green Green function estimates., Ce mémoire s'organise autour de deux cadres d'étude : d'une part, celui des espaces symétriques riemanniens non compacts X = G/K, pour lesquels nous prouvons un encadrement optimal et global en les variables d'espace et de temps, du noyau de la chaleur associé à l'opérateur de Laplace-Beltrami L ; d'autre part, dans le cas d'un groupe de Lie semi-simple G, nous montrons que tous les sous-laplaciens sur G qui induisent l'action de L sur X = G/K présentent des analogies avec L vis-à-vis de l'équation de la chaleur : le bas de leur spectre L^2 est le même, les distances de Carnot-Carathéodory associées sont comparables à la métrique riemannienne sur X et, surtout, les noyaux de la chaleur sont tous comparables (en temps grand) au noyau de la chaleur sur X. Nous en déduisons en particulier des encadrements très précis des noyaux de la chaleur dans ce cadre, ainsi que des fonctions de Green correspondantes.
- Published
- 2003
39. Harmonic analysis for differential forms on real hyperbolic spaces
- Author
-
Pedon, Emmanuel, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Université Henri Poincaré - Nancy 1, Jean-Philippe Anker, and UL, Thèses
- Subjects
Opérateurs différentiels ,Transformations (mathématiques) ,Espaces symétriques ,[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] ,Formes différentielles ,Laplacien ,Fonctions sphériques ,Variétés de ,[MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM] ,Riemann ,Équation de la chaleur ,Espaces hyperboliques ,Fibrés vectoriels - Abstract
Non disponible / Not available, Nous développons dans ce mémoire l'analyse harmonique L2 des p-formes différentielles (0 ? p ? n) sur l'espace hyperbolique réel Hn(R) ? SOe(n,1)/ SO(n). Les notions et résultats classiques de l'analyse harmonique des fonctions (i.e. des formes de degré zéro) sur Hn(R) sont ainsi généralisés. Les principaux outils employés sont la théorie des représentations des groupes de Lie semi-simples et la théorie des fonctions de Jacobi. Nous étudions notamment : la transformation de Poisson ; les fonctions sphériques (généralisées) ; la transformation de Fourier sphérique ; la transformation de Fourier ; la transformation d'Abel. Nous obtenons comme corollaires l'expression explicite du noyau de la chaleur et un nouveau calcul des invariants de Novikov-Shubin. Deux appendices sont consacrés à des résultats plus généraux : l'Appendice A décrit de manière élémentaire les séries discrètes intervenant dans la décomposition de l'espace des formes différentielles L2 sur un espace symétrique riemannien de type non compact général ; l'Appendice B introduit et développe la notion de « triplet de Gelfand», qui généralise à un cadre vectoriel la notion de paire de Gelfand, et permet l'étude des fonctions sphériques associées à un fibré homogène sur un espace symétrique riemannien de type non compact général.
- Published
- 1997
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