Some Properties Of Several Types Singular Integral Operators And Their Applications

Clifford algebra is named after W. K. Clifford who introduced geometric algebra by the combination of the high-dimensional geometry with algebra in 1878. It is an associative and incommutable algebra. Clifford Analysis is a branch of mathematic study which is to execute typical functional theory analysis on Clifford algebra An(R), such as the study of the properties of regular functions, hypermongenic functions and k-hypermongenic functions; the study of the properties of the Cauchy-type singular integral operators and the study of various boundary value problems. Clifford analysis is the natural extension of complex analysis. When n = 0, Clifford analysis is the real analysis; when n= 1, Clifford analysis is the complex analysis; when n= 2, Clifford analysis is the quaternionic analysis. Therefore as an active branch of mathematical study, it has significant theoretical and applied value in various fields of mathematical study.It is very important to study the properties of Cauchy-type integral in the typical functional theory analysis and it is one of the basic tools to solve various boundary value problems. Cauchy-type integral is a type of singular integral. It is widely used in the partial differential equational theories, the singular integral equational theories and the general functional theories, especially in the boundary value problems of the partial differential equations and the singular integral equations. It simplifies and stresses the process by using Cauchy-type integral. The transformation problem of Cauchy-type operators is crucial in the regularization and composition of the singular integral oper-ators. With the transformation formula, we can solve various boundary problems of the singular integral equation which has B-M core and is on the closed smooth manifold. As a result, the transformation problem is the core problem in the salvation of many problems. In complex analysis and multi-complex analysis, the nature and transfor-mation problem of Cauchy type integral operators are defined and solved thoroughly. It is widely used in elastic mechanics, fluid mechanics, hyper-dimensional singular integrals and integral equations. However, the nature and transformation problem of Cauchy-type integral operators have not been defined and solved although it is also very important and that is because Clifford algebra is incommutable. Consequently, it put us in great trouble in the composition and regularization of Cauchy type integral op-erators so that it has impeded the development of the boundary problems of the integral equations and the partial differential equations in Clifford analysis.In 1998 Huang Sha proved the P-B(Poincare-Bertrand) transformation formula of the Cauchy-type integrals in Clifford analysis. On the base of Huang Sha's works, this dissertation finds a new approach. It first gives a new definition of the the Cauchy-type integrals in Clifford analysis. Then it proves the transformation formula of the iter-ated integral in several fairly simple cases by using Cauchy singular integral operators' nature; then it proves a very important inequation about integrated element i.e. a in-equation with a differential element; and further it proves the P-B(Poincare-Bertrand) transformation formula about Clifford values functions with one or two variables by using that inequation and the aforementioned results.Besides, this dissertation also studies a high-order singular Teodorescu operator in the space Rn. By this kind of operator, we get the integral expression of the solution to the non-homogeneous Dirac equation so that many boundary value problems can be solved. It stresses on the boundedness, Holder continuity and the general differential of the high-order singular Teodorescu operator. Meanwhile it studies the perturbation stability concerning boundary surface of the integral domain and gives its error estima-tion. Ultimately, it gives an integral expression of a general Hn system in the space of Rn by using that operator.The dissertation consists of eight parts:1. Introduction. In this part, it recalls the history, academic significance and status que of Clifford analysis.2. Chapter 1. It discusses the transformation problems of the Cauchy-type singular integral operator and the general integral operator. Firstly, it proves the transformation formula of the two general integral on the Liapunov surface in Clifford analysis; then based on that it proves the transformation formula of the Cauchy-type singular integral operators and the general operators. In the aforementioned process, at the outset, it proves that the two iterated integrals are convergent according to the Cauchy principal values. The two iterated integral are separated into N1, N2and N1*, Ar2*, and first it proves N1= N1*, and then proves (?) N2 - (?) N2*= 0.3. Chapter 2. It studies the transformation problems of two Cauchy-type singular integral operators. Firstly, it separates the two iterated integrals into several Cauchy type singular integral operators and the sum of a function respectively to prove that the two iterated integrals are well defined. Then each of the aforementioned integrals are, respectively, separated into four parts. Part 1 are the integrals in the domain where the singular points are eliminated and the rest parts are integrals in the domain where the singular points exist. At the beginning it proves the values in the first part are equal and then the limit of the differences of the rest parts is zero.4. Chapter 3. It studies the transformation problem of a general integral operator with the Cauchy type singular integral operators whose singular point is the integral variable of a general integral. Firstly, it proves the nature of several relevant singular integral operators and then it proves that the two iterated integrals are well defined by using that nature. Next, it divides cleverly the integral domain into several parts. Naturally, the integral operators are grouped into two parts. One is with the singular integrals and the other is with non-singular operators. And it proves that the limit of the part with the singular operators is zero and the part without singularity are equal. Thus, it proves the transformation formula of a general integral with the Cauchy type integral whose singular point is the integral variable of a general integral.5. Chapter 4. It studies the transformation problem of two Cauchy type singular integral operators, among which the singular point of the second Cauchy type inte-gral is the integral variable of the first Cauchy singular integral. The conclusion of this problem is extremely different from those of the aforementioned chapters. Af-ter the transformation of the two operators, it has an extra function which agrees with the results in the multi-complex analysis. In the proving process, first it proves the existence of an inequation with differential elements. Then it proves that the iterated singular integrals under discussion are well defined. Meanwhile, by using that inequa-tion and eliminating the singular points, it proves the transformation formula, i.e., the P-B(Poincare-Bertrand) transformation formula of the Cauchy type singular integral operators in Clifford analysis.6. Chapter 5. It discusses the P-B(Poincare-Bertrand) transformation formula of the Cauchy singular integral operators in the two dimensional Clifford function by us-ing the results above. First, it defines the Cauchy singular integral operators in the two dimensional Clifford function and then discusses the convergence of the iterated singu-lar integrals. Next it separates the iterated singular integral operators into several parts. Then for each part it proves the correctness of the P-B(Poincare-Bertrand) transfor-mation formula of Cauchy singular integral operators in the two dimensional Clifford function by using the conclusion it has already got.7. Chapter 6. It studies the nature of a high-order singular Teodorescu operator in the space of Rn and this chapter consists of three sections:(1). First by using several inequations it proves the boundedness of this operator. Then it proves the Holder continuity of the operator in several special cases to sup-port that the Holder continuity of the operator in the whole space of Rn also exists. Meanwhile, it gets the general differential elements according to the definition.(2). By using several important inequations, it studies the perturbation stability concerning the boundary surface in the integral domain and it gives the error estimation too.(3). First by using variable replacement, it transforms a general Hn equation sys-tem into a general vector value Dirac equation in Clifford analysis. Then it gives the integral expression of the general Dirac equation by using high-order singular Teodor-escu operators so that it gets the integral expression of the generalâ…¡n equation system.8. Conclusions. The conclusions will be drawn and the problems get to be solved will be summarized in this part.