Peak sets in real-analytic convex boundaries
For a domain \(D\Subset\mathbb{C}^n\) with smooth boundary, \(A^\infty (D)\) denotes the space of holomorphic in \(D\) and \({\mathcal C}^\infty\) on \(D\). The main result of the paper is the following. Theorem. Let \(D\Subset \mathbb{C}^n\) be a convex domain with real analytic boundary. Assume \(K\) is a peak set for \(A^\infty(D)\), and \(L\) is a compact subset of \(K\). Then \(L\) is a peak set for \(A^\infty (D)\). Recall that a closed subset \(K\subset \partial D\) is a peak set for \(A^\infty(D)\) if there exists a function \(f\in A^\infty (D)\) so that \(f\equiv 1\) on \(K\) and \(|f|<1\) on \(\overline D\setminus K\).
scientific article; zbMATH DE number 1344268