Weighted Spherical Harmonic Multiplier

Based on the theor of spherical harmonics for measures invariant under a finite reflection group developed by C. Diinkl recentl. the Littlewood-Pale^?theor is studied and an application is given to riiultipliers for ^plirioal li梙larllo(jlncs in IJ?h^: .Sr⁾.The paper is divided as follows: In Chapter Oiie I lie developrtieiit background of ;iitilti?pliers theory arid spherical h-harmonics theory are introduced. In Chapter T^vo the auxiliary functions g6(J. x?, S(f, x? and g^{f, x? related to spherical h-harmonics with respect t.o the weight function F=1 IxI?o > 0} are defined. We study the properties of those functionals and establish Littlewood-Palev theory. In Chapter Three the g⁽f. x?. S(J. x? and g(f. x? functionals are used to prove a iuiiultiphier tiieoreuiu for spherical u梙armonics with respect to the weight function fl .cJ (o > 0) on the unit sphere S in R^. 慖L proof of this theorem uses a number of the results which are proved in Chapter Two. In particular. we use the integral transformation formula of the intertwining operator V discovered by Y. Xu with respect to spherical h-harmonics. so that integral of several variables involved V turns to integral of a single variable. Finally using Parseval equality of ultraspherical series let us solve the problem in an easy way.Combined Fourier multiplier. Jacobi multiplier arid multipliers for classical spherical cxpanlsions, we study multipliers for spherical lu梙armonics for the first time. If all parameters o =0. then the multipliers for spherical h-harmonics reduce to multipliers for classical spherical expansions. But, since multipliers for spherical h-harmonics cani commute with tile action of the rotation group SO(n) on we meet a lot of difficulties in studying them. We use other method which involves complicated calculation. For the case tu², then the multipliers for spluerical h-hartnomiics reduce to multipliers for Jacobi expansions, where o o ?1/2.1 02?1/2. The convolution structure is used in studying the Littlewood桺alev theory and multipliers for Jacobi expansions, but there isn抰 corresponding convolution in higher dimensions.In this thesis, we obtain for spherical h-harmonics the coml)let.e analogue of the LittlewoodPaley theory to classical spherical expansions and Jacobi expansions. But in studying the bound conditions for multipliers of spherical h-harmonics, tile result in this paper is true only in the special case mint<^< o = 0. For general parameters a, > 0 we don抰 obtain an ideal result at present. This problem is well worth studying further.