1 99 6 The Julia - Wolff - Carathéodory theorem in polydisks by Marco Abate

The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls in C, and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of main tool for the proof is a general version of Lindelöf’s principle valid for not necessarily bounded holomorphic functions. 0. Introduction The classical Fatou theorem says that a bounded holomorphic function f defined on the unit disk ∆ ⊂ C admits non-tangential limit at almost every point of ∂∆, but it does not say anything about the behavior of f(ζ) as ζ approaches a specific point σ of the boundary. Of course, to be able to say something in this case one needs some hypotheses on f . For instance, one can assume that, in a very weak sense, f(ζ) goes to the boundary of the image of f as ζ goes to σ. This leads to the classical Julia’s lemma: Theorem 0.1: (Julia [Ju1]) Let f : ∆ → ∆ be a bounded holomorphic function such that lim inf ζ→σ 1− |f(ζ)| 1− |ζ| = α < +∞ (0.1) for some σ ∈ ∂∆. Then f has non-tangential limit τ ∈ ∂∆ at σ, and furthermore |τ − f(ζ)| 1− |f(ζ)|2 ≤ α |σ − ζ| 1− |ζ|2 (0.2) for all ζ ∈ ∆. This statement has a very interesting geometrical interpretation. The horocycle E(σ,R) ⊂ ∆ of center σ ∈ ∂∆ and radius R > 0 is the set E(σ,R) = {