Narrow Results

In this paper we survey some of the most important properties of Loewner chains and their transition mappings on the unit ball B of ℂⁿ with respect to an arbitrary norm. We begin with a discussion of the class of holomorphic mappings from B into ℂⁿ which plays the role of the Carathéodory class in higher dimensions. We consider the Lipschitz regularity in t of an arbitrary Loewner chain, and in s and t of the associated transition mapping v(z, s, t), improving known results. Various consequences are given. Next we consider the role of parametric representation in Loewner theory in higher dimensions. We show that the class S⁰(B) of mappings which have parametric representation on B has some properties in common with the class S of normalized univalent functions on the unit disc. For example, a mapping f in the set H(B) of holomorphic mappings from B into ℂⁿ has parametric representation if and only if there exists a Loewner chain f(z, t) such that the family {e^−tf(z, t)}_t≥0 is a normal family on B and f(z) = f(z, 0) for z ∈ B. Also, the set S⁰(B) is compact as a subset of H(B). Finally we consider some applications to the Roper-Suffridge extension operator.

In this paper we deduce some properties of Loewner chains and their transition mappings on the unit ball B of ℂⁿ with respect to an arbitrary norm. For this purpose, we deduce that any Loewner chain f(z,t) and its transition mapping v(z,s,t) satisfy some locally Lipschitz conditions in t locally uniformly with respect to z ∈ B. Also we deduce that a mapping f ∈ H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e^-tf(z, t)}_{t ≥ 0} is a normal family on B and f(z) = f(z,0) for z ∈ B. Finally we conclude that the set S⁰(B) of mappings which have parametric representation on B is compact.