The Properties Of Singular Integral With Biregular Kernel In Cl_n+1,0（R）

As a theory of functions,Clifford analysis studies the properties of functions and related integral operators which are defined in real（or complex）n-dimensional Euclidean spaces and valued in noncommutative Clifford algebras.Cl_n+1,0（R）is a 2ⁿ⁺¹ dimensional real Clifford algebra.In recent years,people have begun to study the properties of various functions valued in Cl_n+1,0（R）,such as the Cauchy integral formula of the k-hypergenic function and the dual hypergenic function.But in this space,the properties of biregular function and their singular integrals remain to be studied.In this paper,we define biregular function in the real Clifford algebra Cl_n+1,0（R）and then construct biregular kernel,study the Cauchy integral formula for biregular functions in this space.Secondly,we discuss the existence and continuity of singular integrals with biregular kernel.Firstly,we introduce the basic structure and multiplication rules of real Clifford algebra Cl_n+1,0（R）,and define the biregular function in this space.we study the properties of biregular function,construct the proper biregular kernel and give the Cauchy integral formula of biregular function.Secondly,we study the convergence of the singular integralΦ（s,t）with biregular kernel in the principal value sense in Cl_n+1,0（R）,introduce the operator P1 and P2.By using the Holder continuity of the function φ（u,v）and the property of the biregular kernel function,we prove the convergence of the singular integralΦ（s,t）with biregular kernel in Cl_n+1,0（R）algebra in the principal value sense.Finally,we discuss the continuity of singular integrals Φ（s,t）with biregular kernel in Cl_n+1,0（R）algebra.We introduce the closest distance point,prove the continuity of χ（s,t）in four cases,and then prove the continuity of singular integrals Φ（s,t）with double regular kernel in algebra.The research of this paper lays a foundation for the boundary value problems of biregular functions on Cl_n+1,0（R）algebraic spaces.