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

Leandro Arosio; Matteo Fiacchi

doi:10.1515/crelle-2025-0041

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

Leandro Arosio and Matteo Fiacchi

Published/Copyright: July 22, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 827

Abstract

Rudin’s version of the classical Julia–Wolff–Carathéodory theorem is a cornerstone of holomorphic function theory in the unit ball of C d . In this paper, we obtain a complete generalization of Rudin’s theorem for a holomorphic map f : D → D ′ between convex domains of finite type. In particular, given a point ξ ∈ ∂ D with finite dilation, we show that the 𝐾-limit of 𝑓 at 𝜉 exists and is a point η ∈ ∂ D ′ , 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 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 ⟩ . 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 in terms of the multitype at 𝜉.

Funding source: Ministero dell’Università e della Ricerca

Award Identifier / Grant number: 2022AP8HZ9

Award Identifier / Grant number: E83C23000330006

Funding source: European Research Council

Award Identifier / Grant number: 101053085

Funding source: Javna Agencija za Raziskovalno Dejavnost RS

Award Identifier / Grant number: P1-0291

Funding statement: Leandro Arosio was partially supported by INdAM, by PRIN Real and Complex Manifolds: Geometry and Holomorphic Dynamics n. 2022AP8HZ9, and by the MUR Excellence Department Project MatMod@TOV CUP: E83C23000330006. Matteo Fiacchi was partially supported by the European Union (ERC Advanced grant HPDR, 101053085 to Franc Forstnerič), the research program P1-0291 from ARIS, Republic of Slovenia, and by the MUR Excellence Department Project MatMod@TOV CUP: E83C23000330006.

Acknowledgements

We want to thank the anonymous referee for the careful reading of the paper and for many useful comments.

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Received: 2024-08-02

Revised: 2025-05-22

Published Online: 2025-07-22

Published in Print: 2025-10-01

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