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Weighted Hardy-Littlewood-Sobolev Inequality on the Unit Sphere Open Access

Descriptions

Other title
Subject/Keyword
Spherical harmonic
unite sphere
Hardy-Littlewood-Sobolev 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
2013-09-19T14:22:34Z
Graduation date
2013-11
Degree
Master of Science
Degree level
Master's
Abstract
One of the main aims in this thesis is to establish analogues of the classical Hardy-Littlewood-Sobolev (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 Calderon-Zygmund decomposition, the details are much more involved. Of particular importance in our proof is an elegant decomposition of the second order differential-difference 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
R. Askey, and S. Wainger, On the behavior of special classes of ultraspherical expansions. I, J. Analyse Math. 15 (1965), 193-220.G. Brown and F. Dai., Approximation of smooth functions on compact two-point homogeneous spaces, J:F unc:Anal:220(2005), no, 2, 401- 423.C. Canuto and A. Quarteroni. Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comp. 38 (1982), 67-86R. Coifman, R L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math.Pures Appl. (9) 72(1993), 247-286.F.Dai, Mutivariate polynomial inequalities with respect to doubling weights and A1 weights, J:F unct:Anal:235(2006), no. 1, 137-170. MR2216443(2007f:41010)G. Brown, F. Dai, Approximation of smooth functions on compact twopoint homogeneous spaces, Journal of Functional Analysis 220 (2005), 401–423.F. Dai, Z. Ditzian and H. Huang. Equivalence of measures of smoothness in Lp(S^{d-1}). Studia Mathematica 196 (2011), 179-205.F. Dai, Y. Xu, Boundedness of projection operators and Cesaro means in weighted Lp space on the unit sphere, Transactions of the American Mathematical Society 361 (2009), 3189-3221.F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics , Springer, 2013.F. Dai,Y. Xu.Maximal function and multiplier theorem for weighted space on the unit sphere. J. Funct. Anal. 249(2), 477-504 (2007)F. Dai and Y. Xu Moduli of smoothness and approximation on the unit sphere and the unit ball. Advances in Mathematics 224(2010), no. 4, 1233-1310.F. Dai, Y. Xu. Cesaro Means of orthogonal expansions in several variables, Constr. Approx.29 (2009), no. 1, 129-155.F. Dai, and H. Wang, Positive cubature formulas and Marcinkiewicz– Zygmund inequalities on spherical caps, Constr. Approx.31 (2010), no. 1, 1-36.C.F. Dunkl. Reflection groups and orthogonal polynomials on the sphere. Math. Z. 197 (1988), 33-60.C.F. Dunkl. Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1989), 167-183.C.F. Dunkl. Operators commuting with Coxeter group actions on polynomials. In: Stanton, D. (ed.), Invariant Theory and Tableaux, Springer, 1990, pp. 107-117.C.F. Dunkl. Integral kernels with reflection group invariance. Canad. J. Math. 43 (1991), 1213-1227.C.F. Dunkl. Hankel transforms associated to finite reflection groups. In: Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications. Proceedings, Tampa 1991, Contemp. Math. 138 (1992), pp. 123-138.Charles F. Dunkl, Yuan Xu. Orthogonal Polynomials of Several Variables. Cambridge University Press, 2001.G.H. Hardy and J. E. Littlewood, Some properties of fractional integrals (1), Math. Zeitschr. 27 (1928), 565-606.M. Rosler, Positivey of Dunkl’s Intertwining Operator, DUKE MATHEMATICAL JOURNAL, VOL.98, NO.3.S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.) 4 (1938), 471-479. A. M. S. Transl. Ser. 2, 34 (1963), 39-68.E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, N. J.,1970.E.M. Stein, Elias. Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press.(1993)E. M. Stein and G.Weiss, Introduction to Fourier analysis on Euclidean spaces.G. Szegö, Orthogonal Polynomials, 4th edn. Am.Math. Soc. Colloq. Publ., vol. 23. AMS, Providence (1975)M. H. Taibleson and G. Weiss. The molecular characterization of certain Hardy spaces, Asterisque 77 (1980), 67-149.A. Torchinsky, Real-variable Methods in Harmonic Analysis.Dover Publications, INc. 2004.Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups. Proc. Am. Math. Soc. 125, 2963- 2973 (1997)Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres. Can. J. Math. 49, 175-192 (1997)Y. Xu, Uncertainty principles for weighted spheres, balls and simplexes. to appear.S. Wang, Generalized Ul’yanov type inequality on the sphere Sd 1. Acta Math. Sinica (Chin. Ser.) 54 (2011), no. 1, 115-124. (Chinese)K. Wang, L. Li. Harmonic Analysis and Approximation on the Unit Sphere. Science Press. Beijing. 2006.

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