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Loewner chains in the unit disk. (English) Zbl 1209.30011

A family \((\varphi_{s,t})_{0\leq s\leq t<\infty}\) of holomorphic self-maps of the unit disk \(\mathbb D\) is an evolution family of order \(d\in[1,\infty]\) if \(\varphi_{s,s}=\text{id}_{\mathbb D}\), \(\varphi_{s,t}=\varphi_{u,t}\circ\varphi_{s,u}\) for all \(0\leq s\leq u\leq t<\infty\), and, for all \(z\in\mathbb D\) and \(T>0\), there exists a non-negative function \(k_{z,T}\in L^d\big([0,T],\mathbb R\big)\) such that
\[ \big|\varphi_{s,u}(z)-\varphi_{s,t}(z)\big|\leq\int_u^tk_{z,T}(\xi)d\xi, \qquad 0\leq s\leq u\leq t\leq T. \]
Correspondingly, a family \((f_t)_{0\leq t<\infty}\) of holomorphic maps of \(\mathbb D\) is a Loewner chain of order \(d\) if the \(f_t\) are univalent, \(f_s(\mathbb D)\subset f_t(\mathbb D)\) for all \(0\leq s<t<\infty\), and, for any compact set \(K\subset\mathbb D\) and all \(T>0\), there exists a non-negative function \(k_{K,T}\in L^d\big([0,T],\mathbb R\big)\) such that
\[ \big|f_s(z)-f_t(z)\big|\leq\int_s^tk_{K,T}(\xi)d\xi, \qquad z\in K,\; 0\leq s\leq t\leq T. \]
The main results of the paper concerning relations between Loewner chains and evolution families are stated in the following theorems.
Theorem 1.3: For any Loewner chain \((f_t)\) of order \(d\), let \(\varphi_{s,t}:=f_t^{-1}\circ f_s\), \(0\leq s\leq t\). Then \((\varphi_{s,t})\) is an evolution family of the same order \(d\). Conversely, for any evolution family \((\varphi_{s,t})\), there exists a Loewner chain \((f_t)\) of the same order \(d\) such that \(f_t\circ\varphi_{s,t}=f_s\), \(0\leq s\leq t\).
A Loewner chain \((f_t)\) is normalized if \(f_0(0)=0\) and \(f_0'(0)=1\).
Theorem 1.6: Let \((\varphi_{s,t})\) be an evolution family. Then there exists a unique normalized Loewner chain \((f_t)\) associated (in the sense of Theorem 1.3) with \((\varphi_{s,t})\) such that \(\bigcup_{t\geq0}f_t(\mathbb D)\) is either an Euclidean disk or the whole complex plane \(\mathbb C\). Moreover, the following statements are equivalent:
(i) the family \((f_t)\) is the only normalized Loewner chain associated with the evolution family \((\varphi_{s,t})\);
(ii) for all \(z\in\mathbb D\),
\[ \beta(z):=\lim_{t\to\infty}\frac{|\varphi'_{0,t}(z)|}{1-|\varphi_{0,t}(z)|^2}=0; \]
(iii) there exist at least one point \(z\in\mathbb D\) such that \(\beta(z)=0\);
(iv) \(\bigcup_{t\geq0}f_t(\mathbb D)=\mathbb C\).
Theorem 1.7: Suppose that under conditions of Theorem 1.6, \(\Omega:=\bigcup_{t\geq 0}f_t(\mathbb D)\neq\mathbb C\). Then \(\Omega=\{z: |z|<1/\beta(0)\}\), and the set \(\mathcal L[(\varphi_{s,t})]\) of all normalized Loewner chains \((g_t)\) associated with the evolution family \((\varphi_{s,t})\) is given by the formula
\[ \mathcal L\big[(\varphi_{s,t})\big]=\bigg\{(g_t)_{t\geq0}: g(z)=\frac{h\big(\beta(0)f_t(z)\big)}{\beta(0)},\quad h\in S\bigg\}. \]
Here \(S\) is the class of all univalent holomorphic functions \(h\) in \(\mathbb D\), \(h(0)=0\), \(h'(0)=1\).
Besides that, the authors find an analogue of the Loewner-Kufarev partial differential equation and show that there is a one-to-one correspondence between the concept of generalized Loewner chains and the generalized Berkson-Porta vector fields. They consider the special case of evolution families induced by semigroups of holomorphic functions in \(\mathbb D\) and show that the uniqueness of the Kœnigs function is a consequence of Theorems 1.3 and 1.6.

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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