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

B. Jöricke

Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 495-511

Résumé

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.