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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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About boundary values in $A(\Omega )$
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by Piotr Kot PDF
Trans. Amer. Math. Soc. 363 (2011), 4063-4079 Request permission

Abstract:

Assume that $\Omega \subset \mathbb C^{n}$ is a balanced bounded domain with a holomorphic support function (e.g. strictly pseudoconvex domain with $C^{2}$ boundary). We denote $\left \Vert f\right \Vert _{z}^{2}:=\int _{0}^{1}|f(e^{2\pi it}z)|^{2}dt$. Let $\varepsilon >0$ and $\sigma$ be a circular invariant Borel probability measure on $\partial \Omega$. If $g\in A(\Omega )$ and $h$ is a continuous function on $\partial \Omega$ with $|g|<h$ on $\partial \Omega$, then we construct nonconstant functions $f_{1},f_{2}\in A(\Omega )$ with $\left \Vert g+f_{1}\right \Vert _{z}\leq \left \Vert h\right \Vert _{z}$, $|(g+f_{2})(z)|\leq \max _{|\lambda |=1}h(\lambda z)$ for $z\in \partial \Omega$ and \[ \sigma \left (\left \{ z\in \partial \Omega :\left \Vert g+f_{1}\right \Vert _{z}\ne \left \Vert h\right \Vert _{z}\vee \max _{|\lambda |=1}|(g+f_{2})(\lambda z)| \ne \max _{|\lambda |=1}h(\lambda z)\right \} \right )<\varepsilon .\]

Additionally if $\Omega$ is a circular, bounded, strictly convex domain with $C^{2}$ boundary, then we give the construction of $f_{3}\in \mathbb O(\Omega )$, the holomorphic function with: $\left \Vert h-|g+f_{3}^{*}|\right \Vert _{z}=0$ for all $z\in \partial \Omega$, where $f^{*}$ denotes the radial limit of $f$. We also construct $f_{4}\in A(\Omega )$ with $\left \Vert g+f_{4}\right \Vert _{z}=\left \Vert h\right \Vert _{z}$ for $z\in \partial \Omega$.

In all cases we can make $f_{i}$ arbitrarily small on a given compact subset $F\subset \Omega$ and make it vanish to a given order at the point $0$.

References
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Additional Information
  • Piotr Kot
  • Affiliation: Instytut Matematyki, Politechnika Krakowska, ul. Warszawska 24, 31-155 Kraków, Poland
  • Email: pkot@pk.edu.pl
  • Received by editor(s): December 16, 2007
  • Received by editor(s) in revised form: April 13, 2009
  • Published electronically: March 21, 2011
  • © Copyright 2011 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 4063-4079
  • MSC (2010): Primary 32A05, 32A40
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05083-X
  • MathSciNet review: 2792980