Argument of outer functions on the real line

Abstract

A complete description of the modulus of an outer function on the real line is well known. Indeed, this characterization is considered as one of the classical results of the theory of Hardy spaces. However, a satisfactory characterization of the argument of an outer function on the real line is not available yet. In this paper, we define some classes of real functions which can serve as the argument of an outer function. In particular, for any $0 < p \leq \infty$, an increasing bi-Lipschitz function is the argument of an outer function in $H^p(\mathbb{R})$.