Generalized parametric representation and extension operators

Abstract

We consider conditions under which mappings of the open Euclidean unit ball of Cn with generalized parametric representation with respect to an appropriately defined locally integrable function A:[0,∞)→L(Cn), where L(Cn) is the algebra of linear operators from Cn into Cn, on an interval [s,∞), s≥0, are extended to mappings of the same type defined in a higher-dimensional ball (with respect to a related operator-valued function) using two classes of extension operators, namely perturbations of the Pfaltzgraff–Suffridge extension operator and modifications of the Roper–Suffridge extension operator introduced by the author. Extensions of Loewner chains of order p∈[1,∞], i.e., non-normalized Loewner chains that satisfy a locally uniform local Lp-continuity condition on their real parameter, using these operators have previously been considered by the author, but, unlike in the case of standard parametric representation, mappings with generalized parametric representation may not be as nicely compatible with Loewner chains. We will consider the ramifications of this. As part of our work, we will see that A-normalized Loewner chains and evolution families will have order p when ‖A(⋅)‖ is locally Lp. In addition, we will completely characterize generalized parametric representation in the open unit disk. This characterization involves families of bounded univalent functions and allows us to show that the modified Roper–Suffridge extension operators extend families of univalent functions bounded by some M>1 to biholomorphic mappings bounded by M under certain conditions as a consequence of our main extension results.