Nontangential weighted limit of the infinitesimal Carathéodory metric in an \(h\)-extendible boundary point of a smooth bounded pseudoconvex domain in \(\mathbb{C}^n\)
Since \textit{I. Graham}'s research on weighted limits of the infinitesimal Carathéodory and Kobayashi metrics for strongly pseudoconvex domains [Trans. Am. Math. Soc. 207, 219-240 (1975; Zbl 0305.32011)], there have been a number of investigations on related problems. The author's main result is in the spirit of \textit{J. Yu} [Trans. Am. Math. Soc. 347, 587-614 (1995; Zbl 0814.32006)]. Let \(D\) be a smooth bounded pseudoconvex domain in \(\mathbb C^n\), \(\Delta\) be the unit disk in \(\mathbb C\) and \(H(D,\Delta)\) be the set of holomorphic mappings from \(D\) to \(\Delta\). The infinitesimal Carathéodory metric is defined by \[ C_D(z,X)=\sup\{|(Xf)(z)|:f\in H(D,\Delta)\}, \] where \(z\in D\), \(X\in \mathbb C^n\) and \((Xf)(z)\) is the derivative of \(f\) at \(z\) in the direction \(X\). Let \(z_0\in \partial D\) be an \(h\)-extensible boundary point, i.e., a point at which its \textit{D. Catlin}'s multitype [Ann. Math., II. Ser. 120, 529-586 (1984; Zbl 0583.32048)] coincides with its \textit{J. P. d'Angelo}'s type [ibid. 115, 615-637 (1982; Zbl 0488.32008)]. To \(D\) and \(z_0\) the author associates a model domain \(E\). Then for any nontangential cone \(\Lambda\) in \(D\) with vertex at \(z_0\), we have \[ \lim[C_D(z,X)/C_E(\widehat z_0, \widehat X)]=1\quad (z\to z_0, z\in \Lambda) \] uniformly in all \((1,0)\)-vector fields \(X\), where \(\widehat z_0\) is a fixed interior point of \(E\) and the vector \(\widehat X\) depends on \(X\) and \(\text{dist} (z,\partial D)\). The explicit formulas of \(C_D\) and \(C_E\) are constructed.
\(h\)-extensible boundary point, Carathéodory metric
scientific article; zbMATH DE number 1355014