Clifford分析中的若干问题研究

Clifford分析与多复变函数论都是源于单复变函数论，两者有很多类似之处.然而由于Clifford代数的不可交换性，两者又有一定的区别.该项目将研究两方面的内容： 1.利用多复变数几何函数论中已有的方法和技巧，结合微分几何、泛函分析、微分方程和代数等方面的相关知识,以拟置换为工具来克服Clifford代数的不可交换性带来的困难, 将几何函数论推广到实Clifford分析中，研究实Clifford分析中一些特殊函数的增长定理、掩盖定理和偏差定理. 2.复超正则函数是复Clifford分析中最重要的函数之一,但其Cauchy积分公式一直未能解决.借助于多复变函数论的方法和技巧,再结合代数矩阵的方法,我们将研究复Clifford分析中复超正则函数的Cauchy积分公式及其相关问题. 本项目的研究将进一步丰富和完善Clifford 分析理论,在理论上和实际中都有一定意义.

Clifford analysis and function theory of several complex variables all originated from function theory of one complex variable, and they are similar in many aspects. However,for the noncommutativity of Clifford algebra there are certain difference between them. This project will study the following two contents. 1. Making use of the methods and techniques in geometric function theory of several complex variables and combining knowledge on differential geometry, functional analysis, differential equation and algebra and using quasi-pertutations to overcome some difficulties brought by the noncommutativity of Clifford algebra, we extend geometric function theory to real Clifford analysis and we will study growth theorems, covering theorems and distortion theorems of some special functions in real Clifford analysis. 2. The complex hypermonogenic function is one of the most important functions in complex Clifford analysis, but its Cauchy integral formula has not been able to be obtained. By virtue of the methods and techniques in function theory of several complex variables and algebraic matrix approach we will study Cauchy integral formual of complex hypermonogenic functions and the related problems. Study on the project will further enrich Clifford analysis theory and make it perfect and the project is significant in both theory and practice.