λ-biharmonic Functions And Multipliers On The Bergman Spaces Associated With λ-analytic Functions

The purpose of the present dissertation is to study λ-biharmonic functions associated with the Dunkl operator in the unit disc D and the function multipliers in the associated Hardy spaces Hλp(D)(the λ-Hardy spaces)and Bergman spaces Aλp(D)(the λ-Bergman spaces).The former plays an essential role in the later.For 0<p<∞,the λ-Bergman space Aλp(D)consists of those functions in the weighted space Lλp(D):=LP(D;|y|2λdxdy)that are λ-analytic in D.The research contains four parts.In the first part,the Green function G(z,ζ)(the λ-Green function)associated with the λ-Laplacian △λ in D is introduced,and its various properties are obtained.As the main result of this part,the representation of Green-type associated with λ-Laplacian △λ(the λ-Green representation)for u∈C2(D)are established.In the second part,the Green function Γ(z,ζ)(the λ-bilaplacian Green function)associated with the λ-bilaplacian △2λ in D is introduced,and its various properties are proved.The main result of this part is to establish the representation of Green-type associated with λ-bilaplacian △λ2 for u∈C4(D)(the λ-bilaplacian Green representation).The aim of the third part is to study the contractive property and the expansive property of the function multipliers in the λ-Hardy spaces Hλp(D)and the λ-Bergman spaces Aλp(D).A subclass,denoted by of Lλ1(D)is defined in terms of some orthogonality and λ-subharmonicity,and several sufficient conditions for functions to be in the subclass Wλ(D)are obtained.The main results of this part are the following two conclusions:(i)the elements in the subclass Wλ(D)generate bounded multipliers from the λ-Hardy space Hλp(D)into Lλp(D)for<P<∞(p0=2λ/(2λ+1)),and also a contractive multipliers for p=2,or in the restricted sense for p0<p<2 and 2<p<∞;(ⅱ)the elements in the subclass Wλ(D)generate expansive multipliers from the λ-Bergman spaces Aλp(D)into Lλp(D)for p0<p<∞ in some senses.These conclusions are based upon a fundamental integral formula involving the Green function Γ(z,ζ)associated with the λ-bilaplacian Δλ2.In the fourth part,a Riesz-type representation formula for λ-superbiharmonic func-tions u is established.Roughly speaking,a real-valued function u defined on the unit disk D is called λ-superbiharmonic provided that u is locally integrable with respect to the mea-sure |y|2λ dxdy and the λ-bilaplacian △λ2u is a positive distribution on D.The sufficient and necessary conditions for λ-superbiharmonic functions to have the Riesz-type representation are given.