调和测度的一个极值分解问题
An extremal decomposition problem for harmonic measure
摘要：
设$E$是复$z$平面的闭单位圆盘$|z|\le 1$中的连续统，该平面将开圆盘$|z| < 1$分成$n\ge 2$两两非，相交单连通域$D_k,$，使得每个域$D_k$包含指定圆$|z| = ρ, 0 <ρ<1, k=1,...,n\,. $上的某个点$a_k$。结果表明，对于与$E$和点$a_k,$的选择无关的增函数$Ψ\,$，调和测度$$ Ψ^{-1}\[ \frac{1}{n} \sum_{k=1}^{k} Ψ(ω(a_k,E, D_k))] $$大于或等于调和测度$ω(ρ, E^*, D^*)\,,$，其中$E^* = \{z: z^n \in [-1,0] \}$和$D^* =\{z: |z|<1, |{\rm arg} z| < π/n\} \,.$这意味着例如R.W.巴纳德湖Cole和A. Yu. Solynin关于圆$|z| = ρ\,.$上任意点的量$\inf_{E} \max_{k=1,...,n} ω(a_k,E, D_k)\,$的较低估计这些作者在点均匀分布在圆$|z| = ρ\,.$上的特殊情况下陈述了这一假设
英文摘要：
Let $E$ be a continuum in the closed unit disk $|z|\le 1$ of the complex $z$-plane which divides the open disk $|z| < 1$ into $n\ge 2$ pairwise non-intersecting simply connected domains $D_k,$ such that each of the domains $D_k$ contains some point $a_k$ on a prescribed circle $|z| = ρ, 0 <ρ<1, k=1,...,n\,. $ It is shown that for some increasing function $Ψ\,$ independent of $E$ and the choice of the points $a_k,$ the mean value of the harmonic measures $$ Ψ^{-1}\[ \frac{1}{n} \sum_{k=1}^{k} Ψ(ω(a_k,E, D_k))] $$ is greater than or equal to the harmonic measure $ω(ρ, E^*, D^*)\,,$ where $E^* = \{z: z^n \in [-1,0] \}$ and $D^* =\{z: |z|<1, |{\rm arg} z| < π/n\} \,.$ This implies, for instance, a solution to a problem of R.W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity $\inf_{E} \max_{k=1,...,n} ω(a_k,E, D_k)\,$ for arbitrary points of the circle $|z| = ρ\,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $|z| = ρ\,.$
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