极值基、几何分离域及其应用
Extremal Bases, Geometrically Separated Domains and Applications
摘要：
本文在$\mathbb{C}^n$中引入了拟凸区域有限型边界点处切向量场极值基的概念。然后定义了在边界点上的几何分离域类，并给出了它们的复几何描述。例如，局部线性凸区域，具有局部可对角化Levi形式的区域，以及Levi形式在一点具有可比特征值的区域。此外，我们证明了这些域是局部化的。在此基础上，定义了“适应复下调和函数“的概念，并给出了它存在的充分条件。然后我们证明了Bergman ans Szegö投影的所有尖锐估计在这种情况下都是有效的。最后，我们将这些结果应用到例子中，得到全局和局部的尖锐估计，改进，例如，February man，Kohn和Machedon的Szegö投影的结果。
英文摘要：
We introduce the notion of extremal basis of tangent vector fields at a boundary point of finite type of a pseudo-convex domain in $\mathbb{C}^n$. Then we define the class of geometrically separated domains at a boundary point, and give a description of their complex geometry. Examples of such domains are given, for instance, by locally lineally convex domains, domains with locally diagonalizable Levi form, and domains for which the Levi form have comparable eigenvalues at a point. Moreover we show that these domains are localizable. Then we define the notion of "adapted pluri-subharmonic function" to these domains, and we give sufficient conditions for his existence. Then we show that all the sharp estimates for the Bergman ans Szegö projections are valid in this case. Finally we apply these results to the examples to get global and local sharp estimates, improving, for examlple, a result of Fefferman, Kohn and Machedon on the Szegö projection.
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