Order, normalization, and extension of inverse Loewner chains

We consider extensions of inverse (i.e., decreasing) Loewner chains of order p ∈ [ 1 , ∞ ] , meaning the chains are not normalized at the origin but satisfy a local L p -continuity condition on their real parameter, as well as inverse Loewner chains normalized at the origin using an operator-valued function of their real parameter from the unit disk D in C or Euclidean unit ball B in C n to objects of the same type in higher dimensions. These notions of order and normalization, familiar in the study of (increasing) Loewner chains, have not been fully developed for inverse Loewner chains when n ≥ 2 . We remedy that here, including showing how both concepts interact with the inverse Loewner initial-value problem. A known link between order and normalization of Loewner chains (and also inverse Loewner chains) is strengthened here. Nonlinear resolvent families of holomorphic mappings f in B satisfying the range condition (equivalently, such that − f is an infinitesimal generator of a continuous semigroup of holomorphic self-mappings of B ) are well-known examples of inverse Loewner chains, and we consider their extensions to analogous objects in higher dimensions as well as those of infinitesimal generators.