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The Julia-Carathéodory theorem on the bidisk revisited. (English) Zbl 1389.32017

The authors study a new type of regularity (called B+) of a point \(\tau\in\mathbb T^2\) (\(\mathbb T:=\partial\mathbb D\), \(\mathbb D:=\) the unit disk) for a holomorphic mapping \(\varphi:\mathbb D^2\longrightarrow\overline{\mathbb D}\). We say that \(\varphi\) has a B-point at \(\tau\) if there exists an \(\omega\in\mathbb T\) such that, for every non-tangential set \(S\subset\mathbb D^2\) (i.e., there exists a constant \(M>0\) such that \(\operatorname{dist}(z,\tau)\leq M\operatorname{dist}(z,\partial\mathbb D^2)\), \(z\in S\)), there exist \(\alpha, \varepsilon>0\) such that \(| \varphi(z)-\omega| \leq\alpha\| \tau-z\| \), \(z\in S\cap B(\tau,\varepsilon)\). We say that \(\varphi\) has a B+ point at \(\tau\) if the above constant \(\alpha\) can be chosen independently of the non-tangential set \(S\). We say that \(\varphi\) has a C-point at \(\tau\) if there exist \(\omega\in\mathbb T\) and \(\lambda\in\mathbb C^2\) such that for every non-tangential set \(S\) and every \(\beta>0\) there exists an \(\varepsilon>0\) such that \(| \varphi(z)-\omega-\lambda\cdot z| \leq\beta\| \tau-z\| \), \(z\in S\cap B(\tau,\varepsilon)\). The type B+ is between B and C. The authors give an alternative characterization of \(C\): if
\[ \limsup_{z\overset{\text{non-tan.}}\longrightarrow\tau}\frac{1-| \varphi(z)|}{1-\| z\|}<+\infty, \] then \(\varphi\) has a C-point at \(\tau\).
The main result of the paper states that the following conditions are equivalent:
(1) \(\varphi\) has a B+ point at \(\tau\);
(2) \(\varphi\) has a B+ point at \(\tau\) and for some constant \(\alpha\) we have \(| D\varphi(\tau)[h]| \leq\alpha\| h\| \) for all \(h\in\mathbb C^2\) such that \(\tau+th\in\mathbb D^2\), \(0<t\ll1\), where \(D\varphi(\tau)[h]:=\lim_{t\searrow 0}\frac{\varphi(\tau+th)-\omega}t\);
(3) there exist \(\omega\in\mathbb T\), \(A, B\geq0\), and a bounded holomorphic function \(g\) on the domain \(\mathbb C\setminus[-1,1]\) such that \(D\varphi(\tau)[h]=Ah_1+Bh_2+(h_2-h_1)g(\frac{h_1+h_2}{h_1-h_2})\).

MSC:

32A40 Boundary behavior of holomorphic functions of several complex variables
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