Admissible Majorants for Model Subspaces of H^2, Part II: Fast Winding of the Generating Inner Function

This paper is a continuation of [6]. We consider the model subspaces K(theta)=H^2 (circled minus)(theta) H^2 of the Hardy space H^2 generated by an inner function (theta) in the upper half plane. Our main object is the class of admissible majorants for K(theta), denoted by Adm (theta) and consisting of all functions (omega) defined on R such that there exists an f (not equal) 0, f (element of) K(theta) satisfying |f(x)| <= omega(x) almost everywhere on R. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any K(theta) generated by a meromorphic inner function. In contrast with [6], we consider the generating functions (theta) such that the unit vector theta(x) winds up fast as x grows from –(infinity) to (infinity). In particular, we consider (theta)=B where B is a Blaschke product with "horizontal" zeros, {\it i.e.}, almost uniformly distributed in a strip parallel to and separated from R. It is shown, among other things, that for any such B, any even (omega) decreasing on (0,infinity) with a finite logarithmic integral is in Adm B (unlike the "vertical" case treated in [6], thus generalizing (with a new proof) a classical result related to Adm exp(i(sigma) z), (sigma)>0. Some oscillating (omega)ʹs in Adm B are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm exp(i(sigma)z), (sigma)>0, and to de Brangesʹ space H(E).