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

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