On peak sets for Lip \(\alpha\) classes (Q1824157)

Let \(A^{\alpha}\) \((0<\alpha \leq 1)\) be the space of all holomorphic functions in the upper half plane which satisfy a Lipschitz condition of order \(\alpha\) there. Such functions extend continuously to the real axis. A compact subset E of \({\mathbb{R}}\) is called a peak set for \(A^{\alpha}\), if there exists a function \(f\in A^{\alpha}\) with \(f(z)=1\) on E but \(| f(z)| <1\) elsewhere. In the first part of the paper the authors study general properties of peak sets, they show e.g. that compact subsets or finite unions of peak sets are again peak sets. In the second part necessary as well as sufficient conditions (in terms of the lengths of the intervals complementary to E) are given, so that E is a peak set. A characterization is achieved only in the case of some Cantor type sets. The general problem to find necessary and sufficient conditions seems more difficult.