BIHOLOMORPHIC MAPPINGS AND PARAMETRIC REPRESENTATION IN SEVERAL COMPLEX VARIABLES

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.