Extension and division in varieties with normal crossings (Q1348594)

In the paper under review, an extension problem of holomorphic functions from some analytic subvarieties in a bounded domain in \(\mathbb{C}^n\) and a division problem are dealt with. Let \(D\) be a bounded strictly pseudoconvex domain with smooth boundary in \(\mathbb{C}^n\) and \(f_j\) \((j = 1,\dots,p)\) holomorphic functions defined in \(D\). Define a holomorphic mapping \(f : D\to \mathbb{C}^p\) by \(f=(f_1,\dots,f_p)\). Denote by \(I\) the ideal generated by \(f_1,\dots,f_p\). Suppose that \(f^{-1}(0)\) is a complete intersection with normal crossings. The authors study an extension problem in \(L^\infty\)-norm for holomorphic functions defined on \(f^{-1}(0)\cap D\). Furthermore, they study a decomposition formula \(g = \sum^p_{j=1}f_jg_j\) for holomorphic functions \(g\in I\) in Lipschitz spaces. The division theorem in \(L^2\) due to \textit{H. Skoda} [Ann. Sci. Ec. Norm. Supér., IV. Ser. 5, 545-579 (1972; Zbl 0254.32017)] is well-known. In this paper, it is stressed that the classical theorems cannot be applied because \(f^{-1}(0)\) has singularities on the boundary of \(D\).