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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Riemann surfaces and bounded holomorphic functions
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by Walter Pranger PDF
Trans. Amer. Math. Soc. 259 (1980), 393-400 Request permission

Abstract:

The principal result of this article asserts the equivalence of the following four conditions on a hyperbolic Riemann surface X: (a) the following set $z| |f(z)| \leqslant {\text {sup}} |f|$ on K for every bounded holomorphic section f of $\xi$ is compact for every unitary vector bundle $\xi$ and every compact set K; (b) every unitary line bundle has nontrivial bounded holomorphic sections and the condition in (a) holds for $\xi = {i_d}$; (c) every unitary line bundle has nontrivial bounded holomorphic sections and X is regular for potential theory; (d) every unitary line bundle has nontrivial bounded holomorphic sections and X is its own B-envelope of holomorphy. If X is a subset of C, these are also equivalent to the following: (e) for every unitary line bundle $\xi$ the bounded holomorphic sections are dense in the holomorphic sections.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 259 (1980), 393-400
  • MSC: Primary 30F99; Secondary 14F05, 32L05
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0567086-7
  • MathSciNet review: 567086