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Indian Institute of Science

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

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

URL: http://etd.iisc.ac.in/handle/2005/2958

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 C^{d} and Laplace-Beltrami operator on the group SU(2) are obtained using their L^{pestimates} 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 L^{P-spaces}. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted L^{p-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 L^{p} ( R^{d},〖 (φ_{i}ρ (x ))〗^{2} dx) where φ_{i}ρ 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 (6^{th} 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 (16^{th} Edition):

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Doctoral Dissertation, Indian Institute of Science. Accessed November 24, 2020. http://etd.iisc.ac.in/handle/2005/2958.

MLA Handbook (7^{th} Edition):

Boggarapu, Pradeep. “Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups.” 2018. Web. 24 Nov 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 Nov 24]. 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