The Properties Of A Class Of Singular Integral Operator With K-regular Kernel In Clifford Analysis

T operator is a singular integral operator defined in a region.It has some important applications in the theory of Vekua equations and the generalized analytic function theory.Over the years,the related theory of this operator has been developed in Clifford analysis.In this article,a T operator with k-regular kernel is defined and some properties of the operator are given.In Clifford analysis,k-regular function is a very important function class and it is the basic solution of k-order Dirac operator.The weighted k-regular function is a new function defined on the basis of the k-regular function.This function has a good application in solving high-order partial differential equations.Therefore,it is very necessary to study the T operator with weighted k-regular kernel in Clifford analysis.In this paper,a class of T operators with weighted k-regular kernel is defined in the Clifford analysis and we discuss some properties of the singular integral operators.Then we conclude that the integral operators with weighted k-regular kernel are uniformly boundedness in ?,H?lder continuity in ? and ?-integrability in ?.This article has the following conclusions:Based on the Cauchy integral formula,we define a class of T operators with weighted k-regular kernel on bounded regions in Clifford analysis.And we divide ? into the two regions and study the uniform boundedness of the singular integral operators in ?.Then we prove some important inequalities and use them to study H?lder continuity of the singular integral operator with weighted k-regular kernel in ?.Finally,we introduce a maker,and discuss the range of the y.Then we prove ?-integrability of the singular integral operator with weighted k-regular kernel in ?.