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Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups

Electronic Theses of Indian Institute of Science

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Title Mixed Norm Estimates in Dunkl Setting and Chaotic Behaviour of Heat Semigroups
 
Creator Boggarapu, Pradeep
 
Subject 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
 
Description 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_iρ (x ))〗^2 dx) where φ_iρ 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.
 
Contributor Thangavelu, Sundaram
 
Date 2018-01-02T20:46:10Z
2018-01-02T20:46:10Z
2018-01-03
2014
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2958
http://etd.ncsi.iisc.ernet.in/abstracts/3820/G26701-AbS.pdf
 
Language en_US
 
Relation G26701