黎曼流形上微分形式的■类

■-Classes of Differential Forms on Riemannian Manifolds

D.Franke等于2002年给出了黎曼流形上弱闭微分形式的■类定义,并利用这些类研究了A-调和张量和拟正则映射的一些性质.由于这些微分形式的WT类在几何函数论研究中有着重要作用,因此首先给出黎曼流形上一些新的微分形式类,称之为■和■类,然后利用D.Franke等人的思想方法给出A-调和张量与■类的关系,并利用Young不等式证明了■类与■类的等价关系,由这个等价关系推出A-调和张量的正则性性质.这些结果是经典结果的推广与发展,利用这些结果,可研究高维空间的几何函数论和映射问题.

In 2002, D. Franke, et al gave the definitions for some WT -classes of weakly closed differential forms on Riemannian manifolds, and studied some properties of A - harmonic tensors and quasi - regular mappings by using these classes. Due to the importance of WT classes to the study of the geometric function theory, the author in this paper first gives some new classes of differential forms, called WT1^- and WT2^- classes, and then discusses the relationship between A - harmonic tensors and WT1^- class by using the method by D. Franke, et al. , and the equivalence between WT2^-and WT2 classes by using Young＇s inequality, which implies the regularity property of A -harmonic tensors. The results can be regarded as generalizations and developments of the classical results, and can be used to study the geometric function theory and mapping problems in high dimensional spaces.