On the behavior of meromorphic functions at the ideal boundary of a Riemann surface
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- by J. L. Schiff PDF
- Proc. Amer. Math. Soc. 54 (1976), 130-132 Request permission
Abstract:
In a former work the author established an analog of a classical theorem of Painlevé in the context of an arbitrary resolutive compactification of a Riemann surface. In the same setting, a refinement of the argument used in the above yields an elementary proof of a theorem of Riesz-Luzin-Privaloff type: If a meromorphic function $f$ tends to zero at each point of a subset $E$ of the ideal boundary and $E$ has positive harmonic measure, then $f \equiv 0$ on $R$. The well-known inclusion relations ${U_{HB}} \subset {\mathcal {O}_{M{B^{\ast }}}}$ and ${U_{HD}} \sim {\mathcal {O}_{M{D^{\ast }}}}$, are then established from the point of view of the resolutivity of the Wiener and Royden compactification respectively.References
- Corneliu Constantinescu and Aurel Cornea, Über den idealen Rand und einige seiner Anwendungen bei der Klassifikation der Riemannschen Flächen, Nagoya Math. J. 13 (1958), 169–233 (German). MR 96791 —, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 32, Springer-Verlag, Berlin, 1963. MR 28 #3151.
- Zenjiro Kuramochi, On the ideal boundaries of abstract Riemann surfaces, Osaka Math. J. 10 (1958), 83–102. MR 96790
- L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. MR 0264064
- J. L. Schiff, Harmonic null sets and the Painlevé theorem, Proc. Amer. Math. Soc. 43 (1974), 171–172. MR 330447, DOI 10.1090/S0002-9939-1974-0330447-8
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 130-132
- DOI: https://doi.org/10.1090/S0002-9939-1976-0390209-4
- MathSciNet review: 0390209