Ensembles pics pour \(A^{\infty}(D)\) non globalement inclus dans une variété integrale
The authors consider the following situation: Let \(D\subset {\mathbb{C}}^ n\) be a bounded strictly pseudoconvex domain with \(C^{\infty}\)-smooth boundary and let \(A^{\infty}(D)\) be the algebra of functions holomorphic in D and \(C^{\infty}\)-smooth up to bD. A subset \(E\subset bD\) is said to be a peak set for \(A^{\infty}\) if there exists \(f\in A^{\infty}(D)\) such that \(f\equiv 1\) on E and \(| f| <1\) on \(\bar D\setminus E.\) Among other results, in previous papers the authors proved the following theorem: A closed subset \(E\subset bD\) is a peak set for \(A^{\infty}(D)\) if and only if E is locally contained in a submanifold N of bD such that for every \(p\in N\), \(T_ pN\subset T_ p^{{\mathbb{C}}}bD\). (Such an N can be called an integral submanifold for the contact structure on bD.) In this paper the authors construct examples of peak sets for \(A^{\infty}(D)\) which are not globally contained in an integral submanifold for the contact structure of bD.
integral submanifold for the contact structure
scientific article; zbMATH DE number 3916714