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1. CJM 2003 (vol 55 pp. 1231)
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function |
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function 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 {\it an admissible
majorant\/} for $K_\Theta$ if there exists an $f \in K_\Theta$, $f
\not\equiv 0$, $|f(x)|\leq \omega(x)$ almost everywhere on
$\mathbb{R}$. For some (mainly meromorphic) $\Theta$'s some parts
of $\Adm\Theta$ (the set of all admissible majorants for
$K_\Theta$) are explicitly described. These descriptions depend on
the rate of growth of $\arg \Theta$ along $\mathbb{R}$. This paper
is about slowly growing arguments (slower than $x$). Our results
exhibit the dependence of $\Adm B$ on the geometry of the zeros of
the Blaschke product $B$. A complete description of $\Adm B$ is
obtained for $B$'s with purely imaginary (``vertical'') zeros. We
show that in this case a unique minimal admissible majorant exists.
Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant Categories:30D55, 47A15 |
2. CJM 2003 (vol 55 pp. 1264)
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function |
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function This paper is a continuation of \cite{HM02I}. We consider the model
subspaces $K_\Theta=H^2\ominus\Theta H^2$ 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 $\mathbb{R}$ such that there exists an $f \ne 0$, $f \in
K_\Theta$ satisfying $|f(x)|\leq\omega(x)$ 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
\cite{HM02I}, we consider the generating functions $\Theta$ such that
the unit vector $\Theta(x)$ winds up fast as $x$ grows from $-\infty$
to $\infty$. 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 $\mathbb{R}$.
It is shown, among other things, that for any such $B$, any even
$\omega$ decreasing on $(0,\infty)$ with a finite logarithmic integral
is in $\Adm B$ (unlike the ``vertical'' case treated in \cite{HM02I}),
thus generalizing (with a new proof) a classical result related to
$\Adm\exp(i\sigma z)$, $\sigma>0$. Some oscillating $\omega$'s 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~Branges' space $\mathcal{H}(E)$.
Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant Categories:30D55, 47A15 |