The Julia-Wolff-Carathéodory theorem in convex finite type domains

Abstract

Rudin's version of the classical Julia-Wolff-Carathéodory theorem is a cornerstone of holomorphic function theory in the unit ball of ${\mathbb{C}}^d$. In this paper we obtain a complete generalization of Rudin's theorem for a holomorphic map $f:D\to D^{\prime }$ between convex domains of finite type. In particular, given a point $\xi \in \partial D$ with finite dilation we show that the $K$-limit of $f$ at $\xi$ exists and is a point $\eta \in \partial D^{\prime }$, and we obtain asymptotic estimates for all entries of the Jacobian matrix of the differential $d{f}_z$ in terms of the multitypes at the points $\xi$ and at $\eta$. We introduce a generalization of Bracci-Patrizio-Trapani's pluricomplex Poisson kernel which, together with the dilation at $\xi$, gives a formula for the restricted $K$-limit of the normal component of the normal derivative $⟨d{f}_z(n_{\xi }),n_{\eta }⟩$. Our principal tools are methods from Gromov hyperbolicity theory, a scaling in the normal direction, and the strong asymptoticity of complex geodesics. To obtain our main result we prove a conjecture by Abate on the Kobayashi type of a vector $v$, proving that it is equal to the reciprocal of the line type of $v$, and we give new extrinsic characterizations of both $K$-convergence and restricted convergence to a point $\xi \in \partial D$ in terms of the multitype at $\xi$.

References in corpus (4)

• The horofunction boundary of a Gromov hyperbolic space
• Local and global notions of visibility with respect to Kobayashi distance, a comparison
• Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces
• Worm domains are not Gromov hyperbolic