Estimates for the Szegö kernel on decoupled domains
A regular pseudoconvex domain \(\Omega \Subset \mathbb{C}^ n\) is said to be decoupled of finite type near \(\zeta \in \partial \Omega\) if there exists a holomorphic coordinate system \((z_ 1, \dots, z_ n)\) mapping \(\zeta\) onto 0 and a neighborhood \(U_ \zeta\) of \(\zeta\) onto a neighborhood \(U\) of 0 and smooth, subharmonic, but not harmonic, functions \(\{f_ k\}_{1 \leq k \leq n - 1}\) with \(f_ k(0) = 0\), vanishing to finite order at 0, such that \[ \left \{z \in U : z_ n + \overline z_ n + \sum_{1 \leq k \leq n - 1} f_ k(z_ k) < 0 \right\} \simeq \Omega \cap U_ \zeta. \] The paper studies properties of the Szegö projection operator on a domain which is decoupled of finite type near each its boundary points of nonstrict pseudoconvexity. Namely, some \(L^ p\), \(H^ p\) and Hölder estimates are obtained. It is done by means of so- called non-isotropic smoothing operator techniques (in fact, the Szegö operator is proved to be such an operator of order 0).
scientific article; zbMATH DE number 703350