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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

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

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 October 20, 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. 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

URL: https://era.library.ualberta.ca/files/c5m60qs065

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 L^{p} boundedness of these operators is proved. %
More importantly, the L^{p}-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 Details Similar Records

❌

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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