Interpolation from curves in pseudoconvex boundaries (Q2639998)

Let D be a bounded pseudoconvex domain with smooth boundary in \({\mathbb{C}}^ n\) and \(A^{\infty}(D)\) the smooth functions on \(\bar D\) which are holomorphic on D. A compact subset \(K\subset \partial D\) is called an interpolation set for \(A^{\infty}(D)\) if any smooth function on K can be extended to a function in \(A^{\infty}(D)\). If \(M\subset \partial D\) is a smooth submanifold then M is said to be complex- tangential if for each \(p\in M\) the tangent space T(M,p) is contained in the maximal complex subspace of T(\(\partial D,p).\) The main result of this paper is the following: Let D be a smoothly bounded pseudoconvex domain of finite type in \({\mathbb{C}}^ 2\), and \(M\subset \partial D\) a smooth complex-tangential curve. a) If \(\partial D\) is of constant type along M, then every compact subset of M is an interpolation set for \(A^{\infty}(D)\). b) If \(\partial D\) and M are real-analytic, then for each \(p\in M\) there exists a neighbourhood V of p such that every compact subset of \(M\cap V\) is an interpolation set for \(A^{\infty}(D\cap V)\).