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Harmonic Analysis on Spherical hharmonics and Dunkl Transforms

 Author / Creator
 Ye, Wenrui

The thesis consists of two closely related parts: (i) Cesaro summability of the spherical hharmonic expansions on the unit sphere, and (ii) BochnerRiesz summability of the inverse Dunkl transforms on ddimensional 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 hharmonics at the critical index. This estimate allows us to improve several known results on spherical hharmonics, including the almost everywhere convergence of the Cesaro means at the critical index, the sufficient conditions in the Marcinkiewitcz multiplier theorem, and a FeffermanStein type inequality for the Cesaro operators. In particular, we obtain a new result on a.e. convergence of the Cesaro means of spherical hharmonics 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 BochnerRiesz mean of each L1function converges almost everywhere at the critical index. This result is surprising due to the celebrated counterexample of Kolmogorov on a.e. convergence of the Fourier partial sums of integrable functions in one variable, and the counterexample of E.M. Stein in several variables showing that a.e. convergence does not hold at the critical index even for H1functions. Next, we study the critical index for the a.e. convergence of the BochnerRiesz means in Lpspaces with p > 2. We obtain results that are in full analogy with the classical result of M. Christ on estimates of the maximal BochnerRiesz 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.

 Graduation date
 Fall 2016

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for noncommercial purposes. 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.