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 λ-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 λ-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 λ-Bergman spaces.The Bergman projection associated to the λ-Bergman spaces is introduced and its LP boundedness is proved;the exact pointwise estimation of function in λ-Bergman space is given;the completeness of λ-Bergman space and the density of λ-analytic polynomials in λ-Bergman space are proved;for 1<p<∞,the dual spaces of λ-Hardy space and λ-Bergman space are determined;a characterization of λ-Bergman space is given by using Dunkl operator,and an interpolation theorem ofλ-Bergman space is proved.2、Various multiplier theorems on λ-Bergman space and λ-Hardy space.A power multiplier theorem from λ Hardy space to λ-Hardy space and a power multiplier theorem from λ-Bergman space to λ-Hardy space are established;and this part is devoted to various inequalities in λ-Bergman space:Hardy Littlewood type inequality and Hausdorff young type inequality;some necessary conditions and also some sufficient conditions on coefficient multipliers from λ-Bergman space to λ-Bergman space are given.3、The boundedness of operators acting on weighted λ-Bergman Spaces.Under appropriate conditions on weight functions,it is shown that the boundedness of linear operators from weighted λ-Bergman Spaces into general Banach spaces depends only upon the norm estimate of a single vector-valued λ-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 λ-Bergman space with power weight to LP space by Carleson type measure;a necessary and sufficient condition of multiplier theorem from general weighted λ-Bergman space to sequence space is given.