Plane harmonic measures live on sets of σ-finite length
Plane harmonic measures live on sets of σ-finite length
复制标题
平面谐波测量在 σ 有限长度组上实时进行
DOI：
10.1007/bf02559503
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
T. Wolff
中科院分区：
文献类型：
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作者：
T. Wolff
备注。(1)假设~对于Dirichlet问题是正则的，只是为了方便，实际上并不损失一般性，因为A.Ancona的结果[2]暗示，其补集具有正容量的任意域GT可以表示为一个L n\P，其中每个~，~对于Dirichlet问题是正则的，而P的容量为零。对每个~n有全调和测度的集合将有对~的全调和测度，并且由于有限长的集合的可数并显然对~n有-有限长定理1，所以~n‘s蕴含着对GT的相应陈述。(2)定理1改进了文献[5]的结果，即用“一维Hausdorff测度”代替了“Hausdorff维1”的“a-有限一维Hausdorff测度”。对于单连通区域，文[6，7]证明了定理1。(3)在某种意义上，我们得到了一个半显集F，即
Remarks. (1) The assumption that ~ be regular for the Dirichlet problem is made only for convenience and in fact is no loss of generality in view of A. Ancona's result [2] which implies that an arbitrary domain gt whose complement has positive capacity may be expressed as An l ' t n \P where each ~,~ is regular for the Dirichlet problem and P has zero capacity. A set with full harmonic measure for each ~n will have full harmonic measure for ~, and since a countable union of sets with a-finite length clearly has a-finite length Theorem 1 for the ~n'S implies the corresponding statement for gt. (2) Theorem 1 sharpens the result of [5] which says the same with "a-finite one-dimensional Hausdorff measure" replaced by "Hausdorff dimension one". For simply connected domains, Theorem 1 is proved in [6,7]. (3) In a sense we obtain a semiexplicit set F namely