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

[摘要] This paper is a continuation of Part I [6]. We consider the modelsubspaces $K_Theta=H^2ominusTheta H^2$ of the Hardy space $H^2$generated by an inner function $Theta$ in the upper half plane. Ourmain object is the class of admissible majorants for $K_Theta$,denoted by Adm $Theta$ and consisting of all functions $omega$defined on $mathbb{R}$ such that there exists an $f e 0$, $f inK_Theta$ satisfying $|f(x)|leqomega(x)$ almost everywhere on$mathbb{R}$. 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 thatthe unit vector $Theta(x)$ winds up fast as $x$ grows from $-infty$to $infty$. In particular, we consider $Theta=B$ where $B$ is aBlaschke product with ``horizontal'' zeros, i.e., almostuniformly distributed in a strip parallel to and separated from $mathbb{R}$.It is shown, among other things, that for any such $B$, any even$omega$ decreasing on $(0,infty)$ with a finite logarithmic integralis in Adm $B$ (unlike the ``vertical'' case treated in [6]),thus generalizing (with a new proof) a classical result related toAdm $exp(isigma z)$, $sigma>0$. Some oscillating $omega$'s inAdm $B$ are also described. Our theme is related to theBeurling-Malliavin multiplier theorem devoted to Adm $exp(isigma z)$,$sigma>0$, and to de Branges' space $mathcal{H}(E)$.

[学科分类] 数学（综合）