On The Representation And Convergence Of A Class Of Generalized Reharmonic Functions

The Dunkl operators are one-order operators with reflection terms.This thesis studies the non-tangential convergence and the analytic representations of the generalized biharmonic functions(the ?-biharmonic functions)on the upper-half plane and on the unit disc associated with the rank-one Dunkl operator.The main results of the thesis includes:The Green function(the ?-biharmonic Green function)on the upper-half plane associated with the -bilaplace operator is introduced,its various properties are proved,and for functions with compact support,the integral representation(the ?-biharmonic Green representation)is established by means of the -bilaplace operator and the ?-biharmonic Green function;the ?-biharmonic Poisson Kernels on the upper-half plane and on the unit disc are proved,and the non-tangential convergence of the ?-biharmonic Poisson integrals on the upper-half plane and on the unit disc are proved;and finally,the analytic representations of ?-biharmonic functions on an axis-symmetric,bounded,and simply connected domain are obtained,and the uniqueness on the unit disc of the Dirichlet problem of the?-biharmonic equation is proved,together with the representation of the solution.