Hermitean Clifford分析是多复变的推广，也是古典的Clifford分析的复化。 2012年Sommen等建立了古典Clifford分析的四元数化理论，发现了由四元数twist Hermitean Dirac 算子族构造出的类似多复变d-bar算子的循环矩阵值算子具有平行的Cauchy积分理论， 推广了多复变中的Matinelli-Bochner积分公式，给出了相应的Hilbert变换理论。一个公开问题就是如何将这一理论从四元数化推广到一般情形，包括Clifford代数、八元数、 拟四元数等情形，这正是我们的研究内容。这一公开问题的解决将为微分流形理论提供类似于复结构和四元数结构的八元数结构和Clifford代数结构，为偏微分方程理论提供类似于多复变的d-bar算子， 从而为微分几何和复分析注入新的活力，也为Cliffod分析奠定自身的地位。

Hermitean Clifford analysis, as a generalization of the theory of several complex variables, is a complexification of the standard Clifford analysis. In 2012, Sommen and his collaborators established the theory of quaternionic Hermitean Clifford analysis, which is a quaternionification of the classical Clifford analysis. A circulant matrix valued operator was constructed from twist Hermitean Dirac operators. It has the Cauchy integral formula similar to the d-bar operator of the theory of several complex variables, which generalizes the Matinelli-Bochner integral formula . There holds the corresponding theory of Hilbert transforms. An open problem arises about how to extend the result from the complex case and the quaternion case to the general cases, e.g., the cases of Clifford algebra, octonion, quasi-quaternion. Solving this open problem is precisely the main objective of the project. It will provide for the theory of differential manifolds with the structure of octonion or Clifford algebra, similar to the classical complex structure or quaternion structure. It also provides for the theory of partial differential equations with new operators similar to d-bar operator so as to give new energy for the theory of differential geometry and analysis. As a result, it provides Clifford analysis an important role in mathematics.