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About boundary values in $ A(\Omega)$

Author: Piotr Kot
Journal: Trans. Amer. Math. Soc. 363 (2011), 4063-4079
MSC (2010): Primary 32A05, 32A40
Published electronically: March 21, 2011
MathSciNet review: 2792980
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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}\vert f(e^{2\pi it}z)\vert^{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 $ \vert g\vert<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}$, $ \vert(g+f_{2})(z)\vert\leq\max_{\vert\lambda\vert=1}h(\lambda z)$ for $ z\in\partial\Omega$ and

$\displaystyle \sigma\left(\left\{ z\in\partial\Omega:\left\Vert g+f_{1}\right\V... ... z)\vert \ne\max_{\vert\lambda\vert=1}h(\lambda z)\right\} \right)<\varepsilon.$

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-\vert g+f_{3}^{*}\vert\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.

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Piotr Kot
Affiliation: Instytut Matematyki, Politechnika Krakowska, ul. Warszawska 24, 31-155 Kraków, Poland

Keywords: Maximum modulus set, inner function.
Received by editor(s): December 16, 2007
Received by editor(s) in revised form: April 13, 2009
Published electronically: March 21, 2011
Article copyright: © Copyright 2011 American Mathematical Society