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Harmonic Analysis on Spherical h-harmonics and Dunkl Transforms Open Access


Other title
Dunkl transform
Fourier analysis
almost everywhere convergence
weak type estimates
spherical harmonics
harmonic analysis
Type of item
Degree grantor
University of Alberta
Author or creator
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 of Mathematical and Statistical Sciences
Date accepted
Graduation date
2016-06:Fall 2016
Doctor of Philosophy
Degree level
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.
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.
Citation for previous publication
Chapter 3 and Chapter 4 of this thesis has been published as F. Dai, S. Wang and W. Ye, Maximal Estimates for the Cesaro Means of Weighted Orthogonal Polynomial Expansions on the Unit Sphere, Journal of Functional Analysis, vol. 265, issue 10, 2357-2387.Chapter 6 of this thesis has been published as F. Dai and W. Ye, Almost Everywhere Convergence of the Bochner-Riesz Means with the Dunkl Transform, Journal of Approximation Theory, vol. 29, 129-155.

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