K-regular Functions In Clifford Analysis

K-regular function is a kind of function class of good nature and a natural generalization of the regular function.Regular function is k-regular function,but k-regular function is not sure regular function.For example:when f(x)is regular function,xf(x)is not regular function, but it is 2-regular function.And there is a rule that k-1-regular function is k-regular function. On the other hand,k-regular function is not certainly k-1-regular function.Thus,there is some theoretical and applicable value to study it.In chapter 1,the preliminaries and some important lemmas are given.The lemmas we obtained in this part play a very important role in this paper.In chapter 2,we study some properties of k-regular functions,such as uniqueness theorem and the determination theorem of k-regular functions when k is even number and so on.The properties we discuss make us understand k-regular functions clearly.The series of k-regular functions and the connectivity of the domain which is the k-regular functions defined on are used to prove the uniqueness theorem.This tells us the uniqueness theorem of k-regular functions is on the base of connected open subset.And the method is applicable for the analytic functions with one complex variable and regular functions,biregular functions,hypermonogenic functions in real Clifford analysis.It becomes simple to judge k-regular functions by the determination theorem of k-regular functions when k is even number.In chapter 3,the Cauchy-Pompeiu formula of r times continuously differentiable functions defined over bounded domains in Rⁿ and with values in Clifford algebra C(V_n,0)is developed and higher order Cauchy integral formula,mean value theorem,Cauchy inequality of k-regular functions are studied.These theorems are essential theorems of k-regular functions. The higher order Cauchy integral formula is basal,because it is the integral representation of k-regular functions and the important tool to study k-regular functions.Otherwise,higher order Cauchy type integral is one of the main studying tools to research boundary value problems of k-regular functions.The difference between higher order Cauchy type integral and Cauchy type integral we notice is that there is some extra weak singularity items of higher order Cauchy type integral.The means we adopt in this paper is to put the higher order Cauchy type integral into two sections.One section is the Cauchy type integral which is familiar to us,and the other is the sum of the weak singularity items.Next,using the method of local generalized sphere coordinates transformation,we study Cauchy principle value,Plemelj formula and H(?)lder continuity of higher order Cauchy type integral for r times continuously differentiable functions defined over bounded domains in Rⁿ and with values in Clifford algebra C(V_n,0).Then we give the Privalov theorem of higher order Cauchy type integral.We mainly verify that the sum of the weak singularity items of higher order Cauchy type integral is continuous when we prove the Plemelj formula.The method we use is to summarize the ways of the point in the domain approaching the point on the boundary into two cases.One case is that the approaching way is not along the direction of the tangent plane,the other is that the approaching way is along the direction of the tangent plane.And the extension theorem of k-regular functions is obtained by higher order Cauchy integral formula and the Plemelj formula of the higher order Cauchy type integral.To prove the H(?)lder continuity of the boundary value and the Privalov theorem of the higher order Canchy type integral,the integral surface is divided into two parts,among which one has singularity and the other has no singularity.We prove the Privalov theorem from three cases.Firstly,the theorem can be proved by the H(?)lder continuity of the boundary value when the two points are both on the boundary of the domain;Then,the properties of the nearest distance point and some skills are used to prove the theorem when one point in the domain and the other on the boundary of the domain;Lastly,similar to the second ease,the theorem can be proved when the two points are both in the domain.In chapter 4,the Cauchy-Pompeiu formula of r times continuously differentiable functions defined over unbounded domains in Rⁿ and with values in Clifford algebra C(V_n,0)is introduced and higher order Cauchy type integral of k-regular functions,the Cauchy principle value and the Plemelj formula of higher order Cauchy type integral etc.are discussed.Inspired by the ideas that handling the Clifford regular functions over unbounded domains,and using the method of local generalized sphere coordinates transformation and the method applied in chapter 3,we proved the important results.