# Harmonic Analysis on Spherical h-harmonics and Dunkl Transforms

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Dunkl transform, Bochner-Riesz, Fourier analysis, Cesaro, almost everywhere convergence, weak type estimates, spherical harmonics, harmonic analysis

Ye, Wenrui

Supervisor and department: Dai, Feng

Examining committee member and department: Dai, Feng Han, Bin Lau, Anthony T-M Safouhi, Hassan Yaskin, Vladyslav

Department: Department of Mathematical and Statistical Sciences

Specialization: Mathematics

Date accepted: 2016-09-30T09:49:53Z

Graduation date: 2016-06:Fall 2016

Degree: Doctor of Philosophy

Degree level: Doctoral

Abstract: The thesis consists of two closely related parts: i Cesaro summability of the spherical h-harmonic expansions on the unit sphere, and ii Bochner-Riesz summability of the inverse Dunkl transforms on d-dimensional real space, both being studied with respect to the weight that is invariant under an Abelian group {±1}^d in Dunkl analysis. In the first part, we prove a weak type estimate of the maximal Cesaro operator of the spherical h-harmonics at the critical index. This estimate allows us to improve several known results on spherical h-harmonics, including the almost everywhere convergence of the Cesaro means at the critical index, the sufficient conditions in the Marcinkiewitcz multiplier theorem, and a Fefferman-Stein type inequality for the Cesaro operators. In particular, we obtain a new result on a.e. convergence of the Cesaro means of spherical h-harmonics at the critical index, which is quite surprising as it is well known that the same result is not true for the ordinary spherical harmonics. We also establish similar results for weighted orthogonal polynomial expansions on the ball and the simplex. In the second part, we first prove that the Bochner-Riesz mean of each L1-function converges almost everywhere at the critical index. This result is surprising due to the celebrated counter-example of Kolmogorov on a.e. convergence of the Fourier partial sums of integrable functions in one variable, and the counter-example of E.M. Stein in several variables showing that a.e. convergence does not hold at the critical index even for H1-functions. Next, we study the critical index for the a.e. convergence of the Bochner-Riesz means in Lp-spaces with p > 2. We obtain results that are in full analogy with the classical result of M. Christ on estimates of the maximal Bochner-Riesz means of Fourier integrals and the classical result of A. Carbery, Jose L. Rubio De Francia and L. Vega on a.e. convergence of Fourier integrals. The proofs of these results for the Dunkl transforms are highly nontrivial since the underlying weighted space is not translation invariant. We need to establish several new results in Dunkl analysis, including: i local restriction theorem for the Dunkl transform which is significantly stronger than the global one, but more difficult to prove; ii the weighted Littlewood Paley inequality with Ap weights in the Dunkl noncommutative setting; iii sharp local pointwise estimates of several important kernel functions.

Language: English

DOI: doi:10.7939-R3DB7VZ2J

Rights: This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.

Author: ** Ye, Wenrui **

Source: https://era.library.ualberta.ca/

## Teaser

Harmonic Analysis on Spherical h-harmonics and
Dunkl Transforms
by
Wenrui Ye
A thesis submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Mathematics
Department of Mathematical and Statistical Sciences
University of Alberta
c
Wenrui
Ye, 2016
Abstract
The thesis consists of two closely related parts: (i) Cesàro summability of the spherical
h-harmonic expansions on the sphere Sd−1 , and (ii) Bochner-Riesz summability of the
inverse Dunkl transforms on Rd , both being studied with respect to the weight h2κ (x) :=
Qd
2κj
, which is invariant under the Abelian group Zd2 in Dunkl analysis.
j=1 |xj |
In the first part, we prove a weak type estimate of the maximal Cesàro operator
of the spherical h-harmonics at the critical index.

This estimate allows us to improve
several known results on spherical h-harmonics, including the almost everywhere (a.e.)
convergence of the Cesàro means at the critical index, the sufficient conditions in the
Marcinkiewitcz multiplier theorem, and a Fefferman-Stein type inequality for the Cesàro
operators.

In particular, we obtain a new result on a.e.

convergence of the Cesàro means
of spherical h-harmonics at the critical index, which is quite surprising as it is well known
that the same result is not true for the ordinary spherical harmonics.

We also establish
similar results for weighted orthogonal polynomial expansions on the ball and the simplex.
In the second part, we first prove that the Bochner-Riesz mean of each function in
L1 (Rd ; h2κ ) converges almost everywhere at the critical index.

This result is surprising
due to the celebrated counter-example of Kolmogorov on a.e.

convergence of the Fourier
partial sums of integrable functions in one variable, and the counter-example of E.M.
Stein in several variables showing that a.e.

convergence does not hold at the critical
index even for H 1 -functions.

Next, we study the critical index for the a.e.

convergence
of the Bochner-R...