Invariant metric estimates for \(\overline\partial\) on some pseudoconvex domains
Let \(\Omega\subset {\mathbb C}^n\) be a smoothly bounded pseudo-convex domain of finite type. The main theorem of the paper under review asserts that, if \(n=2\), or if \(\Omega\) satisfies the further assumption of being strongly pseudo-convex, convex, or decoupled, then there exists a constant \(C\) such that, if \(\alpha\) is a \(\overline{\partial}\)-closed \((n,1)\)-form on \(\Omega\), then there exists an \((n,0)\)-form \(u\) solving the equation \(\overline{\partial}u=\alpha\), and which satisfies the estimate \[ \| u\| _I\leq C\| \alpha\| _I, \] where \(\| \;\| _I\) is the norm associated to any of the Carathéodory, Bergman or Kobayashi metrics on \(\Omega\).
scientific article; zbMATH DE number 2057379
scientific article