AbstractThis paper is a continuation of [6]. We consider the model subspaces Kϴ = H2 ϴ ϴH2 of the Hardy space H2 generated by an inner function ϴ in the upper half plane. Our main object is the class of admissible majorants for Kϴ, denoted by Adm ϴ and consisting of all functions ω defined on ℝ such that there exists an f ≠ 0, f ∈ Kϴ satisfying |f(x)| ≤ ω(x) almost everywhere on ℝ. Firstly, using some simple Hilbert transformtechniques, we obtain a general multiplier theorem applicable to any Kϴ generated by a meromorphic inner function. In contrast with [6], we consider the generating functions ϴ such that the unit vector ϴ(x) winds up fast as x grows from –∞ to ∞. In particular, we consider ϴ = B where B is a Blaschke product with “horizontal” zeros, 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 ω decreasing on (0,∞) 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(iσz), σ > 0. Some oscillating ω's in Adm B are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm exp(iσz), σ > 0, and to de Branges’ space ℋ(E).
Subspaces of de Branges Spaces Generated by Majorants
