近些年，由于M2-brane模型以及六维空间上ADS/CFT技术及自对偶芽理论的发展，物理学家对高维空间上的物理非常感兴趣，比如，六维时空上的超共形场理论，自对偶三形式的六维理论（约化后能够给出四维空间上的自对偶string方程）等。本项目主要研究六维空间上的椭圆版本的massless 场算子及其复形。首先利用基本的微分几何办法，找到六维空间上相应的Dolbeault复形及每个算子的精确表达式。其次给出六维空间上多个六维向量的massless场算子及其复形。进而研究六维空间上椭圆版本的massless场算子的加权L2估计，从而给出非齐次的massless场方程的典型解。发展六维及多六维变量空间上的函数论。

In recent years, due to the development of the M2-brane model, ADS/CFT technology on 6-dimensional space and self dual germ theory, physicists are very interested in 6-dimensional physics, for example, super conformal field theory over 6-dimensional space-time and the 6-dimensional theory of self-dual three-forms, the reduction of which gives us the self-dual string equations in 4-dimensional. we plan to investigate the elliptic version of massless field operator and its complex over 6-dimensional space. First of all, we will study the corresponding Dolbeault complex over 6-dimensional space and the explicit expression of each operator by using the basic differential geometric method. Second, we will study the elliptic version of massless field operator and its complex over several 6-dimensional vector space. Then we will study the weighted L2 estimation of the elliptic version of massless field operator over 6-dimensional space, and then give the canonical solution of the non homogeneous massless field equation. We can develop the function theory of 6-dimensional space of one and several 6-dimensional variables.