Thin Sets at the Boundary
Thin Sets at the Boundary
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边界处的薄集
DOI：
10.1112/plms/s3-65.2.357
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发表时间：
1992
影响因子：
1.8
通讯作者：
Hiroaki Aikawa
中科院分区：
文献类型：
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作者：
Hiroaki Aikawa
在他的信[18]中，海曼提出了以下问题：如果u是单位圆盘中的负次调和函数，那么对于几乎所有的φ，u（z）都有极限，因为z非切向地趋向于e，并且在一个近薄集之外（[17，第7章]）。应该有一个全局的结果，即不在一个角度上，在一个合适的例外集之外。应该是什么呢？对于绿色电势可能更容易，其中极限总是为零。当然，这些问题也存在于更高的维度中。本文的目的是回答海曼的问题。我们将表明，有例外的集适合于描述（特别是切向）边界行为的超调和函数。因此，我们将证明上述非切向限制可以放宽（见推论1.1）。让我们首先回顾一下最小薄度的概念。这一概念可以在一个非常一般的框架中加以界定。设D是具有Martin边界的Martin空间.一个集合E D称为在极小边界点X极小薄，如果马丁核在X上的约化函数是一个势。Fatou-Naim-Doob定理描述了具有极小薄集的非负超调和函数（或等价的非正次调和函数）的边界行为。
In his letter [18], Hayman raised the following questions: If u is negative subharmonic in the unit disk, then for almost all φ, u(z) has a limit as z tends to e nontangentially and outside a near-thin set ([17, Chapter 7]). There ought to be a global result, i.e. not in an angle, outside a suitable exceptional set. What should it be? It might be easier for Green potentials, where the limit is always zero. Of course the questions also exist in higher dimensions. The purpose of this paper is to give an answer to Hayman’s questions. We shall show that there are exceptional sets suitable for the description of (especially tangential) boundary behavior of superharmonic functions. As a result we shall prove that the above nontangential restriction can be relaxed (see Corollary 1.1). Let us first recall the notion of minimal thinness. This notion can be defined in a very general framework. Let D be a Martin space with Martin boundary ∂MD. A set E ⊂ D is called minimally thin at a minimal boundary point X if the reduced function over E of the Martin kernel at X is a potential. The Fatou–Naim–Doob theorem describes the boundary behavior of nonnegative superharmonic functions (or equivalently nonpositive subharmonic functions) with minimally thin sets.