Continuity Problem Of Riesz Potential Associated With The Rank-1 Dunkl Operator

Poisson's equation and its homogeneous form(Laplace's equation)are basic models of linear elliptic equations,and in the study of their boundary problems,the fundamental solution and the Green function play an essential role.The solutions of the Poisson equation can be explicitly expressed by means of the Riesz potential.The Dunkl operators are one-order differential operators with reflection-terms,and the corresponding Laplace operator contains one-order terms with variable coefficients and reflection-terms.This thesis introduces the fundamental solution and the Green function(?-Green function)of the generalized Laplace operator(?-Laplace operator)related to the Dunkl operator of rank 1,and establishes the Green representation of functions on the unit ball in terms of the ?-laplace operator and the ?-Green function,so that the explicit expression of solutions of the ?-Poisson equation is giving;this thesis studies the continuity of the generalized Riesz potential(?-Riesz potential)associated with the ?-Laplace operator and the spherical mean property,and proves a pointwise estimate of the?-Riesz potential and a decaying estimate of its spherical mean.