The Properties Of Two Types Of High-order T-operator In High Dimensional Space In Clifford Analysis

The basic properties of two types of high-order T(Teodorescu)-operators are studied in Clifford analysis in this paper. T-operator is a singular integral operator defined in the region. As we know, T-operator plays a very important role in the generalized analytic functions and Vekua system. And Vekua system closely related to the plane elasticity, shell theory, aerodynamics and so on.Then it's necessary to study the properties of T-operators in various spaces.While T-operator is the generalized solution of the non-homogeneous Dirac equation. Therefore it plays a key role when we study the integral expression of non-homogeneous Dirac equation and many boundary value problems.In Complex analysis,many theories about T-operator are well development. But in Clifford analysis, T-operator, especially the natures of high-order singular T-operator,has no correspond-ing conclusions. The T-operator in high-dimensional space in Clifford analysis is the promotion of T-operator in complex plane. It has many of the same natures of the T-operator in complex plane. Meanwhile,the high-order singular T-operator in Clifford analysis also has many good properties and applications. The paper focuses on the basic natures of a generalized Teodorescu operator which has high-order singular in Clifford analysis.We obtain uniform boundedness, Holder continuity of the operator in Rn space. At the same time we also prove the 7 times inte-grability of the operator. In addition, many basic properties of a high-order singular T-operator defined in Lp,n(Ω) space are studied in this paper.And we obtain the uniform boundedness, Holder continuity of the operator in Rn space.We also get the nature of the operator at infinity. This paper is divided into three chapters.In chapter 1, we give the prior knowledge of this article which we needed.In chapter 2, we study the properties of a high-order singular T-operator on a bounded area. And we prove the uniform boundedness,Holder continuity andγtimes integrability of this operator in Rn space.Firstly, we introduce several lemmas and inequalities,which play different and important role in the process of the following proof.Then, we give a definition of T-operator in bounded domain.We prove its uniform boundedness and Holder continuity by Holder inequality and lemma 5 in Rn space. And we directly use Holder inequality and Lemma 5 to prove the boundedness of the operator. Then we use Holder inequality, lemma 2, Hile lemma,Hadamard lemma to prove the Holder continuity of the operator in the whole Rn space.Finally, we use Holder inequality to proved this high-order T-operator is 7 times integrable function about x.It mapped the function space LP(Ω) into function space Lγ(Ω).The main technique used here is to split the singular index into the form which we need.In this chapter,it's worth emphasizing that we introduced indicator marks in the proving process of the theorem. Those indicator marks transform the complex formulation into a simple form, which greatly facilitates the calculation.In chapter 3,we mainly study many basic properties of a high-order singular T-operator defined in Lp,n(Ω) space.We obtain the operator's uniform boundedness, Holder Continuity in the whole Rn space.We also get the natures of the operator at infinity. Firstly, we give two inequalities which is mainly used to prove the Holder continuity of T-operator.Then we prove two important inequalities. And we give a definition of high-order T-operator in Lp,n(Ω). At the same time,we use inequalities in lemma to prove uniform boundedness and Holder continuity By Theorem 6, we have proved the Holder continuity of the high-order singular T-operator of which indexα∈[1/2,1). By theorem 7 we have proved the Holder continuity of the high-order singular T-operator of which indexα∈[0,1/2). Note that the condition of the function f(x) in the two theorems are different. Finally, We study the nature of this operator at infinity, that is, when x→∞,|(Tf)(x)|→∞, and in the vicinity of infinity, (Tf)(x) and |x|n/p-n+α-1 are the same infinitesimal.T-operator plays a key role in the integral representation of generalized regular function. The normal T-operator has a very good application in Complex analysis, Quaternion analysis and Clifford analysis. This paper shows that, like ordinary T-operator, the two different types of high-order function we studied also has many good properties, such as the boundedness, Holder continuity andγtimes integrability. This make the theory of T-operator perfect. It also lays a theoretical foundation for the research of the high singularity equations of generalized regular functions. Like the normal T-operator, these two types of high-order T-operator can be applied in the integral representation of generalized regular function. The author of this paper is researching related work in this area.