A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity
A domain \(\Omega\subset \mathbb C^n\) is called hyperconvex if there exists a weak plurisubharmonic barrier at every boundary point. Not every pseudoconvex domain is hyperconvex, e.g., the Hartogs triangle. The aim of the paper is to find a smoothness condition on the boundary of a domain, which is sufficient to conclude hyperconvexity from pseudoconvexity. This problem was considered by \textit{K. Diederich} and \textit{J. E. Fornaess} [Invent. Math. 39, 129--141 (1977; Zbl 0353.32025)], where they proved that any pseudoconvex \(\mathcal C^2\) domain is hyperconvex. Later \textit{N. Kerzman} and \textit{J.-P. Rosay} [Math. Ann. 257, 171--184 (1981; Zbl 0451.32012)] showed that the same is true for pseudoconvex \(\mathcal C^1\) domains. \textit{J.-P. Demailly} [Math. Z. 194, 519--564 (1987; Zbl 0595.32006)] proved that pseudoconvex Lipschitz domains are hyperconvex. Let \(\mathcal C^{0,\alpha}(\Omega)\) be the class of Hölder functions with exponent \(\alpha\) and let \[ \mathcal C^{0,1^-}(\Omega)=\bigcup_{\alpha\in (0, 1)}\mathcal C^{0,\alpha}(\Omega). \] A bounded open set \(\Omega\subset \mathbb R^n\) has \(\mathcal {C}^0\) boundary if there exist a finite set of balls \(\{B(x_j,r_j)\}_{j=1}^m\), \(x_j\in \partial \Omega\), \(r_j>0\), such that \(\partial \Omega\subset \bigcup_{j=1}^mB(x_j,r_j)\), and \(\varphi_j\in \mathcal C^0(\mathbb R^{n-1})\) such that \[ \Omega\cap B(x_j,4r_j)=\big\{y=(y',y_n)\in \mathbb R^n: y_n>\varphi_j(y')\big\}\cap B(x_j,4r_j), \] and \[ \partial\Omega\cap B(x_j,4r_j)=\big\{y=(y',y_n)\in \mathbb R^n: y_n=\varphi_j(y')\big\}\cap B(x_j,4r_j), \] in a coordinate system obtained by rotating and translating the standard basis. If \(\varphi_j\in C^{0,1^-}\) then \(\Omega\) is said to have \(C^{0,1^-}\) boundary. The main theorem of the paper is the following. If \(\Omega\) is a bounded pseudoconvex domain with \(C^{0,1^-}\) boundary, then \(\Omega\) is hyperconvex. Moreover it is proved that bounded Reinhardt domains with \(\mathcal C^0\) boundary are hyprerconvex.
scientific article; zbMATH DE number 6536143
scientific article