论文标题
在$ K $ -Cauchy-Fueter Complex的边界综合体上
On the boundary complex of the $k$-Cauchy-Fueter complex
论文作者
论文摘要
$ k $ -cauchy-fueter综合体，$ k = 0,1，\ ldots $，在Quaternionic Analysis中是多元变量理论中Dolbeault Complex的对应物。在本文中，我们在域的边界上明确构建了这些复合物的边界复合物，这是对应于复杂分析中的切向cauchy-riemann络合物的。它们仅是复杂分析以外的已知边界复合物，在功能理论中具有有趣的应用。作为一个应用程序，我们建立了$ k $ groumard函数的Hartogs-bochner扩展，即全体形函数的Quaternionic对应物。这些边界络合物在一种二次超曲面上具有非常简单的形式，该形式具有第二步的右型nilpotent Lie基团的结构。它们使我们能够介绍Quaternionic Monge-Ampère操作员，并为研究此类群体的多功能理论打开大门。我们还将抽象二元定理应用于边界复合物，以获得Malgrange的消失定理和Hartogs-Bochner扩展的概括，用于$ K $ -CF函数，这是CR函数的Quaternionic对应物，在这种组上。
The $k$-Cauchy-Fueter complex, $k=0,1,\ldots$, in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on boundaries of domains, corresponding to the tangential Cauchy-Riemann complex in complex analysis.They are only known boundary complexes outside of complex analysis that have interesting applications to the function theory. As an application, we establish the Hartogs-Bochner extension for $k$-regular functions, the quaternionic counterpart of holomorphic functions. These boundary complexes have a very simple form on a kind of quadratic hypersurfaces, which have the structure of right-type nilpotent Lie groups of step two. They allow us to introduce the quaternionic Monge-Ampère operator and opens the door to investigate pluripotential theory on such groups. We also apply abstract duality theorem to boundary complexes to obtain the generalization of Malgrange's vanishing theorem and Hartogs-Bochner extension for $k$-CF functions, the quaternionic counterpart of CR functions, on this kind of groups.
