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

[摘要] A model subspace $K_Theta$ of the Hardy space $H^2 = H^2(mathbb{C}_+)$ for the upper half plane $mathbb{C}_+$ is$H^2(mathbb{C}_+) ominus Theta H^2(mathbb{C}_+)$ where $Theta$is an inner function in $mathbb{C}_+$. A function $omega colon mathbb{R}mapsto[0,infty)$ is called an admissiblemajorant for $K_Theta$ if there exists an $f in K_Theta$, $fotequiv 0$, $|f(x)|leq omega(x)$ almost everywhere on$mathbb{R}$. For some (mainly meromorphic) $Theta$'s some partsof Adm $Theta$ (the set of all admissible majorants for$K_Theta$) are explicitly described. These descriptions depend onthe rate of growth of $arg Theta$ along $mathbb{R}$. This paperis about slowly growing arguments (slower than $x$). Our resultsexhibit the dependence of Adm $B$ on the geometry of the zeros ofthe Blaschke product $B$. A complete description of Adm $B$ isobtained for $B$'s with purely imaginary (``vertical'') zeros. We show that in this case a unique minimal admissible majorant exists.

[学科分类] 数学（综合）