The Julia-Wolff-Carath\'eodory theorem in convex finite type domains

Rudin's version of the classical Julia-Wolff-Carath\'eodory theorem is a cornerstone of holomorphic function theory in the unit ball of Cᵈ. 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 dfz 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 dfz(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.