On the extension of holomorphic functions across a real hypersurface
Let \(M\) be a real hypersurface in \(\mathbb{C}^ n\) and \(\Omega\) a side of \(M\). Let us say that \(M\) has the extension property \((\text{EXT},z)\) at a point \(z\in M\) if every holomorphic function, defined near \(z\) in \(\Omega\), extends (holomorphically) across \(M\) at \(z\), and that \(M\) has the extension property EXT if \((\text{EXT},z)\) holds for all \(z\in M\). A subset \(N\) of \(M\) is called a propagator if every holomorphic function in \(\Omega\) along \(N\) which extends across \(M\) at some point of \(N\) extends across \(M\) along the whole of \(N\). It is shown that, if \(\Omega\) is nowhere strictly pseudoconvex along \(M\) and if nevertheless \((\text{EXT},z)\) does not hold, some propagation occurs. To be precise, in case \(M\) is real-analytic, there exists a non- trivial propagator in \(M\); the structure of such propagators with respect to the Levi form is given. There is a classical characterization of EXT when \(n=2\), but the problem is still open for \(n\geq 3\), and there is no known characterization of \((\text{EXT},z)\) even for \(n=2\). By an example, it is shown that a characterization of EXT for \(n=3\) would contain a characterization of \((\text{EXT},z)\) for \(n=2\).
scientific article; zbMATH DE number 126400