Beurling’s theorem
Beurling’s theorem
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伯林定理
DOI：
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发表时间：
1973
期刊：
影响因子：
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通讯作者：
Hiroshi Tanaka
中科院分区：
文献类型：
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作者：
Hiroshi Tanaka
Let R, R’ be hyperbolic Riemann surfaces and phi be an analytic mapping of R into R’. Let K_{0} be a closed disk in R and let R_{0}=R-K_{0} . Let acute{C} be the Kuramochi capacity on R_{0}cupDelta_{N} and Delta_{1} be the set of all minimal Kuramochi boundary points of R. For a metrizable compactification R^{prime*} of R’, we denote by mathscr{F}(phi) the set of all points in Delta_{1} at which phi has a fine limit in R^{prime*} . There are two typical extensions of Beurling’s theorem [1] to analytic mappings of a Riemann surface to another one, i . e. , Z. Kuramochi’s [5, 6, 7] and C. Constantinescu and A. Cornea’s theorems [3, 4] . The former result states that if phi is an almost finitely sheeted mapping and R^{prime*} is H. D. separative, then ilde{C}(Delta_{1}-mathscr{F}(phi))=0 . The latter one states that if phi is a Dirichlet mapping and R^{prime*} is a quotient space of the Royden compactification of R’, then overline{C}(Delta_{1}-^{Gamma j}(phi))=0 . The present author [9] proved that these two results are independent. In this paper we shall give an another extension of Beurling’s theorem such that it contains the above two results: If phi is a Dirichlet mapping and R^{prime*} is H. D. separative, then Beurling’s theorem is valid. Notation and terminology Let R be a hyperbolic Riemann surface. For a subset A of R, we denote by partial A and A^{i} the (relative) boundary and the interior of A respectively. We call a closed or open subset A of R is regular if partial A is nonempty and consists of at most a countable number of analytic arcs clustering nowhere in R. We fix a closed disk K_{0} in R once for all and let R_{0}=
Let R, R’ be hyperbolic Riemann surfaces and phi be an analytic mapping of R into R’. Let K_{0} be a closed disk in R and let R_{0}=R-K_{0} . Let acute{C} be the Kuramochi capacity on R_{0}cupDelta_{N} and Delta_{1} be the set of all minimal Kuramochi boundary points of R. For a metrizable compactification R^{prime*} of R’, we denote by mathscr{F}(phi) the set of all points in Delta_{1} at which phi has a fine limit in R^{prime*} . There are two typical extensions of Beurling’s theorem [1] to analytic mappings of a Riemann surface to another one, i . e. , Z. Kuramochi’s [5, 6, 7] and C. Constantinescu and A. Cornea’s theorems [3, 4] . The former result states that if phi is an almost finitely sheeted mapping and R^{prime*} is H. D. separative, then ilde{C}(Delta_{1}-mathscr{F}(phi))=0 . The latter one states that if phi is a Dirichlet mapping and R^{prime*} is a quotient space of the Royden compactification of R’, then overline{C}(Delta_{1}-^{Gamma j}(phi))=0 . The present author [9] proved that these two results are independent. In this paper we shall give an another extension of Beurling’s theorem such that it contains the above two results: If phi is a Dirichlet mapping and R^{prime*} is H. D. separative, then Beurling’s theorem is valid. Notation and terminology Let R be a hyperbolic Riemann surface. For a subset A of R, we denote by partial A and A^{i} the (relative) boundary and the interior of A respectively. We call a closed or open subset A of R is regular if partial A is nonempty and consists of at most a countable number of analytic arcs clustering nowhere in R. We fix a closed disk K_{0} in R once for all and let R_{0}=