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Weighted HardyLittlewoodSobolev Inequality on the Unit Sphere Open Access
Descriptions
 Other title
 Subject/Keyword

Spherical harmonic
unite sphere
HardyLittlewoodSobolev Inequality
 Type of item
 Thesis
 Degree grantor

University of Alberta
 Author or creator

Feng, Han
 Supervisor and department

Dai, Feng(Mathematical Science)
 Examining committee member and department

Safouhi, Hassan(Mathematical Science)
Lau, Tony(Mathematical Science)
Han, Bin(Mathematical Science)
 Department

Department of Mathematical and Statistical Sciences
 Specialization

Mathematics
 Date accepted

20130919T14:22:34Z
 Graduation date

201311
 Degree

Master of Science
 Degree level

Master's
 Abstract

One of the main aims in this thesis is to establish analogues of the classical HardyLittlewoodSobolev (HLS) inequality for weighted orthogonal polynomial expansions (WOPEs) on the unit sphere, the unit ball and the simplex. An optimal condition for which this inequality holds is obtained. Classical proofs of the optimality of this inequality on the usual Euclidean spaces rely on the dilation operators and do not seem applicable in our setting, where dilations are not available.
The crucial ingredients in our proofs in this thesis are a series of new sharp pointwise estimates for some important kernel functions that appear naturally in the WOPEs. These estimates are more difficult to establish, andwill be useful for some other problems in WOPEs.
The HLS inequality for the first order fractional integral operator has been playing important roles in many applications. The second part in this thesis proves an equivalent version of the first order HLS inequality, which involves the tangent gradient and the difference operators. This equivalent version has the advantages that it is much simpler and much easier to deal with in applications. While the main tool for the proof of this equivalent version is the CalderonZygmund decomposition, the details are much more involved.
Of particular importance in our proof is an elegant decomposition of the second order differentialdifference operator associated with the WOPEs, discovered in this thesis. It turns out that this decomposition is very useful in several other problems, such as the uncertainty principle of the WOPEs.
The main results of this thesis have many interesting applications in $N$widths, embedding of function spaces and approximation theory.
 Language

English
 DOI

doi:10.7939/R3796D
 Rights
 Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
 Citation for previous publication

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