On smoothing properties of the Bergman projection (Q1996521)

The author investigates smoothing properties of the Bergman and weighted Bergman projections in connection with the hyperconvexity index of pseudoconvex domains in \(\mathbb C^n\). Such a index was introduced by \textit{B.-Y. Chen} [Anal. PDE 10, No. 6, 1429--1454 (2017; Zbl 1368.32003)] and can also be use to study the integrability index of the Bergman kernel (Theorem 1.2). An example of a result contained in the paper is the following one. Corollary 1.1: Let \(\Omega\) be a bounded smooth pseudoconvex domain in \(\mathbb C^n\). For any \(k\in\mathbb Z^+\) and any number \(-\textstyle\frac{\alpha(\Omega)}{2}<s_1<s_2<\textstyle\frac{\alpha(\Omega)}{2}\), there is a positive constant \(C\) such that \[ \|P_\Omega(f)\|_{H^{s_2}(\Omega)}\leq C\|f\|_{H^{-k}(\Omega)} \] for all \(f\in H^{s_1}(\Omega\cap \overline{\mathcal{O}(\Omega)}\). Here \(P_\Omega\) is the Bergman projection of \(\Omega\), \(\alpha(\Omega)\) is the hyperconvexity index of the domain, \(H^{s_1}, H^{s_2}\) and \(H^{-k}\) are standard \(L^2\) Sobolev spaces, and \(\overline{\mathcal O(\Omega)}\) is the space of conjugate holomorphic functions.