Bergman Space Theory Related To The Dunkl Operator

Bergman space is a mathematical branch of analytic function theory,functional analysis and operator theory,which has a rich knowledge system.The purpose of the present dissertation is to study the theory of the Bergman spaces with the Dunkl operator in the unit disc D(called ?-Bergman space).The Dunkl operator is a differential operator with a reflection term.The generalized analytic function defined by the Dunkl operator is called the ?-analytic function.It shows some excellent properties similar to classical analytical functions,but has completely different structures,which brings difficulties to the research of related problems.The main results of this paper include:1?The basic theory of the ?-Bergman spaces.The Bergman projection associated to the ?-Bergman spaces is introduced and its LP boundedness is proved;the exact pointwise estimation of function in ?-Bergman space is given;the completeness of ?-Bergman space and the density of ?-analytic polynomials in ?-Bergman space are proved;for 1<p<?,the dual spaces of ?-Hardy space and ?-Bergman space are determined;a characterization of ?-Bergman space is given by using Dunkl operator,and an interpolation theorem of?-Bergman space is proved.2?Various multiplier theorems on ?-Bergman space and ?-Hardy space.A power multiplier theorem from ? Hardy space to ?-Hardy space and a power multiplier theorem from ?-Bergman space to ?-Hardy space are established;and this part is devoted to various inequalities in ?-Bergman space:Hardy Littlewood type inequality and Hausdorff young type inequality;some necessary conditions and also some sufficient conditions on coefficient multipliers from ?-Bergman space to ?-Bergman space are given.3?The boundedness of operators acting on weighted ?-Bergman Spaces.Under appropriate conditions on weight functions,it is shown that the boundedness of linear operators from weighted ?-Bergman Spaces into general Banach spaces depends only upon the norm estimate of a single vector-valued ?-analytic function.At this time,a p-integral mean with parameters is introduced,and its exact estimation is given by using the properties of operator interpolation and harmonic control.Together with the estimation of a class of Bergman kernel functions,it is the key to prove the boundedness of the above operators;one application is to obtain a sufficient condition for the boundedness of multiplication operators from ?-Bergman space with power weight to LP space by Carleson type measure;a necessary and sufficient condition of multiplier theorem from general weighted ?-Bergman space to sequence space is given.