多四元变量的k-Cauchy-Fueter算子及分析理论

Analysis of k-Cauchy-Fueter operator deals with analysis in the non-commutative realm, which generalizes theory of several complex variables into quaternionic analysis of several variables. It studies the k-Cauchy-Fueter operator, which involves elliptic version of the helicity k/2 massless field equations in physics. Recently, analysis of k-Cauchy-Fueter operator is in full development with an approach of algebraic geometry and complex geometry. A natural problem arises to find an approach of analysis which is not concerned with k-Cauchy-Fueter complex, algebraic geometry and complex geometry so that one can establish analytical theory of k-Cauchy-Fueter operator. The purpose of this project is to solve such a problem by providing an analytic approach. The new approach will apply analytic techniques from theory of several complex variables together with the technique dealing with non-commutativity in Clifford analysis. This innovative approach will enrich analysis of k-Cauchy-Fueter operator so that it will give new energy for the development of theory of equations of motion in electromagnetic fields.

k-Cauchy-Fueter分析理论是多复变函数论向多元四元数领域的推广，它将分析理论从交换领域推广到了非交换领域。k-Cauchy-Fueter算子所涉及的方程与理论物理密切相关，它是螺旋度为k/2的失重场方程的椭圆情形。近年来，k-Cauchy-Fueter算子理论得到迅猛的发展，但所采用的方法局限于代数几何和复几何的方法。一个自然的问题是：能否抛开k-Cauchy-Fueter复形，抛开代数几何和复几何，直接从分析的角度研究k-Cauchy-Fueter算子，消除原方法的局限。本项目针对这一问题，将利用多复变中纯分析的处理方式，结合Clifford分析中处理非交换分析理论的技巧，研究k-Cauchy-Fueter算子的分析理论。这是k-Cauchy-Fueter算子研究方法的革新，将丰富和深化k-Cauchy-Fueter分析理论，为其在电磁场运动方程理论中的应用注入新的活力。

k-Cauchy-Fueter分析理论是多复变函数论向多四元数分析的推广，它将分析理论从交换领域推广到了非交换领域。k-Cauchy-Fueter算子所涉及的方程与理论物理密切相关，它是螺旋度为k/2的失重场方程的椭圆情形。本项目抛开 k-Cauchy-Fueter 复形，采用纯分析的方法建立了非齐次k-Cauchy-Fueter方程解的具体表达式，研究了Bochner-Martinelli型积分的边界特性如给出了Plemelj公式和Poincare-Bertrand置换公式。该项目的成功实施，实现了多复变函数论向非交换领域推广的目标，丰富了k-Cauchy-Fueter分析理论，为其在电磁场运动方程理论中应用注入新的活力。