在本研究计划中我们将考虑一类次椭圆算子，该类算子可以看为具有强退化性质的椭圆算子。我们所研究的这类算子来源于多复变理论中的一个中心课题，即拟凸域上的复Neumann边值问题。我们的研究目标是得到该类次椭圆算子的基本解与热核的表达式并研究它们的基本性质。这些性质包括基本解与热核的精确估计以及热核在短时间尺度下的渐近展开。这些结果将有助于解决一类典型拟凸域上的复Neumann边值问题，从而为这一问题的一般理论提供新的研究思路。

In this proposed project, we concentrate on a class of sub-elliptic operators, which could be regarded as a family of degenerate elliptic operators. The sub-elliptic operators considered here come from one of the major problems in the theory of several complex variables, namely the complex Neumann problem. Our objective is to derive explicit formulas for the fundamental solutions and the heat kernels of the sub-elliptic operators under consideration. We shall also study the properties of these fundamental solutions and heat kernels, which include their sharp bounds and the short-time asymptotic expansions of the heat kernels. These results will be beneficial for solving the related complex Neumann problem on a typical pseudo-convex domain and shed some light on the general theory of the complex Neumann problem.