The relation between the solid modulus of continuity and the modulus of continuity along the Shilov boundary for analytic functions of several variables

Let $G\subset\mathbf C^n$ be a bounded doamin and let $\omega$ be a modulus of continuity. This article is devoted to the following problem: which closed sets $S$ with $S\subset\overline G$ possess the property that, for an arbitrary function $f$ belonging to the algebra $A(G)$ of all functions analytic in $G$ and continuous in $\overline G$, the relation
$$ \max_{z,\zeta\in S,|z-\zeta|\leqslant\delta}|f(z)-f(\zeta)|\leqslant\omega(\delta) $$
for all $\delta>0$ implies
$$ \max_{z,\zeta\in\overline G,|z-\zeta|\leqslant\delta}|f(z)-f(\zeta)|\leqslant C\omega(\delta) $$
for all $\delta>0$, where the constant $C$ depends only on $G$ and $S$.
The main result is a theorem which asserts that if $G$ is a regular Weil domain then $S$ can be taken to be the Shilov boundary.
Bibliography: 20 titles.