An explicit extension formula of bounded holomorphic functions from analytic varieties to strictly convex domains
Let \(D\subset {\mathbb{C}}^ n\) denote a strictly convex domain and take \(\phi_ 1,...,\phi_ m\) holomorphic in a larger domain \(\Omega\supset \bar D\). Let V denote the set of common zeros of the \(\phi_ k\) and put \(M=V\cap D\), \(\partial M=V\cap \partial D\). Assume that V has no singular points on \(\partial M\) and that V meets \(\partial D\) transversally. Under these conditions the author defines a kernel K(\(\zeta\),z), \(\zeta\in \partial M\), \(z\in D\), through an explicit formula and proves that \(E: H^{\infty}(M)\to H^{\infty}(D)\) given by \[ Ef(z) = \int_{\partial M}f(\zeta) K(\zeta,z) \] is a bounded extension operator, \(Ef(z)=f(z)\), \(\forall z\in M.\) The paper is written in a clear style in spite of considerable technical details that are involved. It develops and carries further ideas of \textit{G. M. Khenkin} [Math. USSR Izv. 6(1972), 536-563 (1973; Zbl 0255.32008)] on this subject.
strictly convex domains in \({bbfC}^ n\)
scientific article; zbMATH DE number 3981529