About boundary values in $A(\Omega )$

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 $0$.