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You searched for subject:(Dunkl Laplacian). Showing records 1 – 2 of 2 total matches.

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1. Ayadi Ben Said, Fatma. Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique : Harmonic analysis and Schrödinger equation associated with the trigonometric Dunkl Laplacian.

Degree: Docteur es, Mathématiques, 2011, Orléans; Université de Tunis El Manar

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.

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.

Advisors/Committee Members: Anker, Jean-Philippe (thesis director), Sifi, Mohamed (thesis director).

Subjects/Keywords: Laplacien de Dunkl trigonométrique; Estimations de Strichartz; Trigonometric Dunkl Laplacian; Strichartz estiamtes

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APA (6th Edition):

Ayadi Ben Said, F. (2011). Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique : Harmonic analysis and Schrödinger equation associated with the trigonometric Dunkl Laplacian. (Doctoral Dissertation). Orléans; Université de Tunis El Manar. Retrieved from http://www.theses.fr/2011ORLE2060

Chicago Manual of Style (16th Edition):

Ayadi Ben Said, Fatma. “Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique : Harmonic analysis and Schrödinger equation associated with the trigonometric Dunkl Laplacian.” 2011. Doctoral Dissertation, Orléans; Université de Tunis El Manar. Accessed October 29, 2020. http://www.theses.fr/2011ORLE2060.

MLA Handbook (7th Edition):

Ayadi Ben Said, Fatma. “Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique : Harmonic analysis and Schrödinger equation associated with the trigonometric Dunkl Laplacian.” 2011. Web. 29 Oct 2020.

Vancouver:

Ayadi Ben Said F. Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique : Harmonic analysis and Schrödinger equation associated with the trigonometric Dunkl Laplacian. [Internet] [Doctoral dissertation]. Orléans; Université de Tunis El Manar; 2011. [cited 2020 Oct 29]. Available from: http://www.theses.fr/2011ORLE2060.

Council of Science Editors:

Ayadi Ben Said F. Analyse harmonique et équation de Schrödinger associées au laplacien de Dunkl trigonométrique : Harmonic analysis and Schrödinger equation associated with the trigonometric Dunkl Laplacian. [Doctoral Dissertation]. Orléans; Université de Tunis El Manar; 2011. Available from: http://www.theses.fr/2011ORLE2060


Indian Institute of Science

2. Boggarapu, Pradeep. Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.

Degree: PhD, Faculty of Science, 2018, Indian Institute of Science

This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the sub-Laplacian on Heisenberg group, special Hermite operator on Cd and Laplace-Beltrami operator on the group SU(2) are obtained using their Lpestimates and by making use of a lemma of Herz and Riviere along with an idea of Rubio de Francia. Applying these results to functions expanded in terms of spherical harmonics, we deduce certain vector valued inequalities for sequences of operators defined in terms of radial parts of the corresponding operators. In the second part, we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian ∆_κ on weighted LP-spaces. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted Lp-spaces, we only have partial results, but in the case of the heat semigroup generated by the standard Laplacian, a complete picture of the chaotic behavior is obtained on the spaces Lp ( Rd,〖 (φiρ (x ))〗2 dx) where φiρ the Euclidean spherical function is. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar. In the last part, we study mixed norm estimates for the Cesáro means associated with Dunkl-Hermite expansions on〖 R〗d. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator)〖 H〗_κ:=-Δ_κ+|x|2. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesáro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of Three Lines Lemma. Advisors/Committee Members: Thangavelu, Sundaram (advisor).

Subjects/Keywords: Dunkl Transforms; Dunkl Heat Semigroups; Differential Operators; Riesz Transforms; Dunkl-Laplacian; Dunkl Harmonic Oscillator; Heisenberg Group; Heat Semigroups Chaotic Behavior; Dunkl-Hermite Expansions; Dunkl-Hermite Operators; Dunkl Operators; Cesaro Means; Laguerre Function Expansions; Dunkl Heat Semigroup; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Boggarapu, P. (2018). Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups. (Doctoral Dissertation). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/2958

Chicago Manual of Style (16th Edition):

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Doctoral Dissertation, Indian Institute of Science. Accessed October 29, 2020. http://etd.iisc.ac.in/handle/2005/2958.

MLA Handbook (7th Edition):

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Web. 29 Oct 2020.

Vancouver:

Boggarapu P. Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups. [Internet] [Doctoral dissertation]. Indian Institute of Science; 2018. [cited 2020 Oct 29]. Available from: http://etd.iisc.ac.in/handle/2005/2958.

Council of Science Editors:

Boggarapu P. Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups. [Doctoral Dissertation]. Indian Institute of Science; 2018. Available from: http://etd.iisc.ac.in/handle/2005/2958

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