# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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## About boundary values in $A(\Omega )$HTML articles powered by AMS MathViewer

by Piotr Kot
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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