Ensembles de zéros et d'interpolation à la frontière de domaines strictement pseudoconvexes. (Zero sets and interpolation on the boundary of strictly pseudoconvex domains)
Let D be a bounded strongly pseudoconvex domain in \({\mathbb{C}}^ n\) with \(C^{\infty}\) boundary and \(\Gamma\) a totally real \(C^{\infty}\) curve in \(\partial D\). Given a compact subset K of \(\Gamma\) and \(\epsilon >0\) one denotes by \(N_{\epsilon}(K)\) the smallest number of balls in \(\partial D\) of radius \(\epsilon\) needed to cover K. (Here the balls are defined with respect to the usual pseudodistance on strongly pseudoconvex sets.) The authors obtain the following: Theorem. There is a function \(f\in A^{\infty}(D)\) such that \(\{\) \(z\in \bar D: f(z)=0\}= K\) if and only if K satisfies the Carleson condition \(\int^{1}_{0}N_{\epsilon}(K)d\epsilon <\infty\). Furthermore, K is the set of common zeros of all the derivatives of f. Theorem. K is an \(A^{\infty}\)-interpolation set if and only if there are constants c, c' such that for every \(r\in (0,1]\) and every ball \(B_ r\) in \(\partial D\) of radius r and center in a point of K one has \(\int^{r}_{0}N_{\epsilon}(K\cup B_ r)d\epsilon \leq cr \log (1/r)+c'r.\) For the case of totally real submanifolds of \(\partial D\) of dimension \(>1\) only sufficient conditions for the above type of theorems are proved.
scientific article; zbMATH DE number 3969054
scientific article