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

. In this paper, we obtain a complete generalization of Rudin’s theorem for a holomorphic map

f : D → D ′ f\colon D\to D^{\prime}

between convex domains of finite type. In particular, given a point

ξ ∈ ∂ D \xi\in\partial D

with finite dilation, we show that the 𝐾-limit of 𝑓 at 𝜉 exists and is a point

η ∈ ∂ D ′ \eta\in\partial D^{\prime}

, and we obtain asymptotic estimates for all entries of the Jacobian matrix of the differential

d ⁢ f z df_{z}

in terms of the multitypes at the points 𝜉 and at 𝜂. We introduce a generalization of Bracci–Patrizio–Trapani’s pluricomplex Poisson kernel which, together with the dilation at 𝜉, gives a formula for the restricted 𝐾-limit of the normal component of the normal derivative

⟨ d ⁢ f z ⁢ ( n ξ ) , n η ⟩ \langle df_{z}(n_{\xi}),n_{\eta}\rangle

. 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

ξ ∈ ∂ D \xi\in\partial D

in terms of the multitype at 𝜉.