The Properties Of T-operator With Hyper-monogenic Kernel In Clifford Analysis

The basic properties of the T(Teodorescu) operator with hypermonogenic kernel in Clifford analysis are studied in this paper. The T-operator which we studied is a singular integral operator defined in bounded domain and it has been widely used in solving differential equations. In Complex analysis, many theories about T-operator are well development. But, in Clifford analysis, it hasn’t been studied about the properties of a T-operator with hypermonogenic kernel.In Clifford analysis, Hypermonogenic function is a class of function with good nature. Df+(n-1)(?)is obtain by modify the Dirac operator. The hypermonogenic function is a kind of function satisfied Mn-1f=0. At the same time, the T(Teodorescu) operator plays an important role in generalized analytic function, so it is very necessary to study the properties of the various types of T operator in high dimensional space.The first chapter, the preliminary knowledge and several important lemmas are given. The preliminary knowledge is the foundation and the lemma is the tool of the research.The second chapter, firstly, we give a definition about a T-operator with hypermono-genic kernel in Clifford analysis. The T-operator with hypermonogenic kernel defined as Ω(?)R+n=1is a bounded domain. Meanwhile, ωn+1is the area of unit hypersphere on Rn+1. Then we study the properties of the operator in Ω, including uniform boundedness, Holder continuity and γ times integrability.The third chapter, we reseaech the properties of the operator T on unbounded domain Rn+1-Ω, including uniform boundedness, Holder continuity and γ times integrability. Thus, we generalize the properties of the operator T to an unbounded domain.