Peak sets in real-analytic convex boundaries

Abstract

Throughtout this paper we will denote by D a smoothly bounded domain in C with defining function r. We denote by A∞(D) the set of holomorphic functions in D which have C∞-extension to D. First, we give the necessary definitions and notations needed in this paper. A closed subset K ⊂ ∂D is a peak set for A∞(D) if there exists a function f ∈ A∞(D) so that f = 1 on K and |f | 0 on D \K. Such a function g is called a strong support function for K. We denote by Tp(M) the real tangent space to a smooth manifold M at the point p ∈ M. For a point p ∈ M , the complex tangent space of M at p denoted by T C p (M) is the maximal complex subspace of Tp(M), of complex dimension n− 1 if M = ∂D. A C∞-submanifold M ⊆ ∂D is integral at p ∈M if Tp(M) ⊆ T C p (∂D). M is an integral manifold if it is integral at each point p ∈M. A C∞-submanifold M ⊂ ∂D is totally real if T C p (M) = {0} for every p ∈M. We denote by w(∂D) the set of weakly pseudoconvex boundary points. For p ∈ ∂D, we let Np denote the null space in T C p (∂D) of the Levi form at