Interpolation by holomorphic functions smooth to the boundary in the unit ball of \({\mathbb{C}}^ n\)
[Shortened version of the original review]. Let B denote the unit ball in \({\mathbb{C}}^ n\) and S its boundary; let \(A^{m,s}=H(D)\cap C^{m,s}(\bar D)\) and \(A^{\infty}=\cap_{m,s}A^{m,s}\). The paper concerns with interpolation by functions in \(A^{m,s}\) from closed sets \(E\subset S.\) For \(n>1\) and \(m=\infty\) there were given partial descriptions of this sets by Hakim, Sibony, Chaumat and Chollet. The aim of this work is to treat the intermediate case \(0<m+s<+\infty\). In this context it is natural to define, similarly as Whitney's definition in the real case, the space \(A^{m,s}(E)\) of non-isotropic holomorphic jets of class \(A^{m,s}\) on a closed set \(E\subset S\) in order to define E to be an \(A^{m,s}\)-interpolation set. With the definition of a K-set, given by \textit{J. Chaumat} and \textit{A. M. Chollet} in ''Ensembles de zéros et d'interpolation à la frontière de domaines strictement pseudoconvex'', Preprint (1984) the main results are: Theorem: A closed set E contained in a transverse curve \(\Gamma\) is an interpolation set for all the algebras \(A^{m,s}\), \(0<m+s<\infty\) if and only if it is a K-set. Theorem: Any closed set E contained in a complex-tangential submanifold of S is an \(A^{m,s}\)-interpolation set \(0<m+s<\infty.\) For \(m=\infty\) the authors also study the division property where E has the division property for \(A^{\infty}\) if, given arbitrary \(f_ 1,...,f_ r\) functions in \(C^{\infty}(\bar B)\) flat on E, there is a \(g\in A^{\infty}(\bar D)\) flat on E such that \(f_ i=h_ i\cdot g\). The main results for \(A^{\infty}\) are then: Every set with the division property for \(A^{\infty}\) is an \(A^{\infty}\)-interpolation set and every K-set has the division property.
algebra of holomorphic functions in the unit ball
scientific article; zbMATH DE number 3937533
scientific article