$\mathbb{C}^n$ 中的加权正则性与多重细性
Weighted regularity and plurithinness in $\mathbb{C}^n$
- National University of Uzbekistan（乌兹别克斯坦国立大学）
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AI总结：
本文研究加权多重复极值函数的正则参数结构，给出与无权重格林函数相等的判据，构造两个加权正则性不保持的反例，并刻画局部多重正则性失效点，证明局部严格多重次调和性下正则性的保持。
AI中文摘要：
我们研究了具有固定权重和可变对数增长参数的加权多重复极值函数。我们证明了在某一点处的正则参数构成空集、仅包含最小接触参数或整个接触射线。对于连续权重，我们通过满足拟处处不等式的概率测度来表示右参数导数。这给出了与无权重格林函数相等的判据。我们构造了两个反例来说明加权正则性的不保持性，使用了光滑多重次调和权重以及在每一点都非多重极性和非多重细的多项式凸紧集。对于乘积例子，我们通过平面格林函数计算了整个例外多圆盘上的导数，并精确确定了等式成立的位置。第二个例子是全局多重正则的，但具有正的右导数。我们还利用可忽略子集和多重细拓扑刻画了局部多重正则性失效但多重细性不失效的点，并证明了在扩展权重的局部严格多重次调和性下正则性的保持性。
英文摘要：
We study weighted pluricomplex extremal functions with a fixed weight and a variable logarithmic growth parameter. We show that the regular parameters at a point form the empty set, the smallest contact parameter alone, or the whole contact ray. For continuous weights, we represent the right parameter derivative by probability measures satisfying quasi-everywhere inequalities. This gives a criterion for equality with the unweighted Green function. We construct two counterexamples to preservation of weighted regularity, using smooth plurisubharmonic weights and polynomially convex compact sets that are non-pluripolar and non-plurithin at every point. For the product example, we compute the derivative on the entire exceptional polydisc through planar Green functions and determine exactly where equality holds. The second example is globally pluriregular but has a positive right derivative. We also characterize the points where local pluriregularity fails without plurithinness, using negligible subsets and the plurifine topology, and prove preservation of regularity under local strict plurisubharmonicity of the extending weight.
