Peak-interpolating curves for \(A(\Omega)\) for finite-type domains in \(\mathbb C^2\) (Q859815)

Let \(\Omega \subset {\mathbb C}^2 \) be a bounded pseudoconvex domain having smooth boundary; let \(C \subset \partial \Omega \) be a smooth curve. \(A (\Omega)\) is the algebra of all functions holomorphic in \(\Omega\) and continuous up to the boundary. The author has used the following notations: 1) the domain \(\Omega\) is finite type if for each \(p \in \partial \Omega\) there is a positive integer \(N\) such that the maximum order of contact of the germ of a 1-dimensional complex variety through \(p\) with \(\partial \Omega\) at \(p\) is at most \(N;\) 2) \(M\) is complex-tangential if the real tangent space \(T_p (M)\) to \(M\) at the point \(p \in M\) is contained in the maximal complex subspace of \(T_p (\partial \Omega), p \in M;\) 3) a compact set \(K \subset \partial \Omega\) is a peak-interpolation set if for any continuous function \(f\) on \(K,\) \(f \not\equiv 0,\) there is \(F \in A (\Omega)\) such that \(F| _K = f\) and \(| F (\zeta)| < \sup_K | f|\), \(\zeta \in {\overline \Omega} \setminus K.\) The main result of this paper is: Theorem 1.1. i)Let \(\Omega\) be finite type. If \(C\) is complex-tangential, and \(\partial \Omega\) is a constant type along \(C, \) then each compact set of \(C\) is a peak-interpolation set for \(A (\Omega).\) ii) Let \(\Omega\) have real-analytic boundary and \(C \subset \partial \Omega\) be a real-analytic complex-tangential curve. Then each compact subset of \(C\) is a peak-interpolation set for \(A (\Omega).\)