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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Radial limit sets on the torus


Author: Laurence D. Hoffmann
Journal: Trans. Amer. Math. Soc. 171 (1972), 283-290
MSC: Primary 43A70; Secondary 32E25, 46J15
MathSciNet review: 0330934
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Abstract: Let $ {U^N}$ denote the unit polydisc and $ {T^N}$ the unit torus in the space of $ N$ complex variables. A subset $ A$ of $ {T^N}$ is called an (RL)-set (radial limit set) if to each positive continuous function $ \rho $ on $ {T^N}$, there corresponds a function $ f$ in $ {H^\infty }({U^N})$ such that the radial limit $ \vert f{\vert^ \ast }$ of the absolute value of $ f$ equals $ \rho $, a.e. on $ {T^N}$ and everywhere on $ A$. If $ N > 1$, the question of characterizing (RL)-sets is open, but two positive results are obtained. In particular, it is shown that $ {T^N}$ contains an (RL)-set which is homeomorphic to a cartesian product $ K \times {T^{N - 1}}$, where $ K$ is a Cantor set. Also, certain countable unions of ``parallel'' copies of $ {T^{N - 1}}$ are shown to be (RL)-sets in $ {T^N}$. In one variable, every subset of $ T$ is an (RL)-set; in fact, there is always a zero-free function $ f$ in $ {H^\infty }(U)$ with the required properties. It is shown, however, that there exist a circle $ A \subset {T^2}$ and a positive continuous function $ \rho $ on $ {T^2}$ to which correspond no zero-free $ f$ in $ {H^\infty }({U^2})$ with $ \vert f{\vert^ \ast } = \rho $ a.e. on $ {T^2}$ and everywhere on $ A$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0330934-5
PII: S 0002-9947(1972)0330934-5
Keywords: Polydisc, torus, Hardy space $ {H^\infty }({U^N})$, radial limits, radial limit sets, outer functions, inner functions
Article copyright: © Copyright 1972 American Mathematical Society



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