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

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Indian Institute of Science

1. 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 20, 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. 20 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 20]. 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

2. Feng, Han. Spherical h-Harmonic Analysis and Related Topics.

Degree: PhD, Department of Mathematical and Statistical Sciences, 2016, University of Alberta

This thesis contains the following three parts: \begin{description} \item[Part 1(Chapters 1-5):] Spherical h-harmonic analysis. \item[Part 2:] Reverse Hölder's inequality for spherical harmonics. \item[Part 3:] Multivariate Lagrange and Hermite approximation and pointwise limits of interpolants. \end{description} The main results of Part 1 are included in two journal papers, one long joint paper with Prof. F. Dai submitted to Adv. Math., and one single-authored paper to appear in Bull. Can. Math. Soc. Results of Part 2 are contained in a joint paper with Prof. F. Dai and Prof. S. Tikhonov to appear in Pro. AMS, and results of Part 3 are from a joint paper with Prof. M. Buhmann submitted to J.London Math. Soc. Part 1 consists of 5 chapters and is organized as follows. Chapter 1 is devoted to a brief description of some background information and main results for Part 1. Chapter 2 contains some preliminary materials on the Dunkl spherical h-harmonic analysis. After that in Chapter 3 the analogues of the classical Hardy-Littlewood-Sobolev (HLS) inequality for the spherical h-harmonics with respect to general reflection groups on the sphere is established. A critical index for the validity of the HLS inequality is obtained and is expressed explicitly involving in the multiplicity function and the structure of the reflection grouop, which allows us to compute the critical indexes for most known examples of reflection groups. One of the main difficulties in our proofs lies in the fact that an explicit formula for the Dunkl intertwining operator is unknown in the case of general reflection groups, and therefore, closed forms of the reproducing kernels for the spaces of spherical h-harmonics are not available. A novel feature in our argument is to apply weighted Christoffel functions to establish new sharp pointwise estimates of some highly localized kernel functions associated to the spherical h-harmonic expansions. In Chapter 4, we introduce Riesz transforms for the spherical h-harmonic expansions, which are motivated by a new elegant decomposition of the Dunkl-Laplace-Beltrami operator involving the tangent gradient and the difference operators. These Riesz transforms are shown to have properties similar to those of the classical Riesz means. In particular, the Lp boundedness of these operators is proved. % More importantly, the Lp-boundedness of the Riesz transforms is established. The proof of the main result in this chapter uses the Calderon-Zygmund decomposition, but the main difficulty is to establish some sharp kernel estimates related to the Riesz transforms. Finally, it is worthwhile to point out that the decomposition of the Dunkl-Laplace-Beltrami operator, discovered in this thesis, seems to be of independent interest. Indeed, as an application of this decomposition, in the last section of this chapter we establish the uncertainty principle with respect to the spherical h-harmonic expansions on the weighted spheres. Finally, we close this part by extending the results in preceding chapters to…

Subjects/Keywords: spherical harmonic analysis; Dunkl analysis; Hardy-Littlewood-Sobolev inequalities; Riesz transforms; Uncertainty Principle; Nikolskii type inequality

Dunkl operators, intertwining operator and angular derivatives . . . Spherical h-harmonic… …21 24 31 37 4. Riesz transforms… …A new decomposition of Dunkl-Laplace-Beltrami operator . . . . . . . . Riesz transform… …embedding theorem essentially by the re1 lationship between the Riesz transforms Rj = ∂j (… …i.e. the Riesz potentials). The HLS inequality and the Riesz transforms on Rd have been… 

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

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

APA (6th Edition):

Feng, H. (2016). Spherical h-Harmonic Analysis and Related Topics. (Doctoral Dissertation). University of Alberta. Retrieved from https://era.library.ualberta.ca/files/c5m60qs065

Chicago Manual of Style (16th Edition):

Feng, Han. “Spherical h-Harmonic Analysis and Related Topics.” 2016. Doctoral Dissertation, University of Alberta. Accessed October 20, 2020. https://era.library.ualberta.ca/files/c5m60qs065.

MLA Handbook (7th Edition):

Feng, Han. “Spherical h-Harmonic Analysis and Related Topics.” 2016. Web. 20 Oct 2020.

Vancouver:

Feng H. Spherical h-Harmonic Analysis and Related Topics. [Internet] [Doctoral dissertation]. University of Alberta; 2016. [cited 2020 Oct 20]. Available from: https://era.library.ualberta.ca/files/c5m60qs065.

Council of Science Editors:

Feng H. Spherical h-Harmonic Analysis and Related Topics. [Doctoral Dissertation]. University of Alberta; 2016. Available from: https://era.library.ualberta.ca/files/c5m60qs065

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