On Peak-interpolation Manifolds for A(ω) for Convex Domains in C

Let Ω be a bounded, weakly convex domain in C, n ≥ 2, having real-analytic boundary. A(Ω) is the algebra of all functions holomorphic in Ω and continuous upto the boundary. A submanifold M ⊂ ∂Ω is said to be complex-tangential if Tp(M) lies in the maximal complex subspace of Tp(∂Ω) for each p ∈ M. We show that for real-analytic submanifolds M ⊂ ∂Ω, if M is complex-tangential, then every compact subset of M is a peak-interpolation set for A(Ω). 1. Statement of Main Result Let Ω be a bounded domain in C, and let A(Ω) be the algebra of functions holomorphic in Ω and continuous upto the boundary. Recall that a compact subset K ⊂ ∂Ω is called a peakinterpolation set for A(Ω) if given any f ∈ C(K), there exists a function F ∈ A(Ω) such that F |K = f and |F (ζ)| < supK |f | for every ζ ∈ Ω \K. We are interested in determining when a submanifold M ⊂ ∂Ω, M sufficiently smooth, is a peak-interpolation set for A(Ω). When Ω is a strictly pseudoconvex domain having C boundary, and M is of class C, the situation is very well understood; refer to the works of Henkin & Tumanov [2], Nagel [4], and Rudin [6]. In the strictly pseudoconvex setting, it turns out that what is crucial for M to be a peak-interpolation set for A(Ω) is its positioning in ∂Ω. In this situation, it suffices for M to be complex-tangential, i.e. for Tp(M ) ⊂ Hp(∂Ω) ∀p ∈ M . Here, and in what follows, for any submanifold M ⊆ ∂Ω, Tp(M) will denote the tangent space to M at the point p ∈ M, while Hp(∂Ω) will denote the maximal complex subspace of Tp(∂Ω). Very little is known, however, when Ω is weakly pseudoconvex. In view of a result by Nagel & Rudin [5], it is necessary for M to be complex-tangential. In this paper we show that when ∂Ω and M are real-analytic, it suffices for M to be complex-tangential. Our main result is as follows : Theorem 1.1. Let Ω be a bounded (weakly) convex domain in C having real-analytic boundary, and let M be a real-analytic submanifold of ∂Ω. If M is complex-tangential, then M (and thus, every compact subset of M) is a peak-interpolation set for A(Ω). 1991 Mathematics Subject Classification. Primary: 32A38, 32F32; Secondary: 32D99.