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

Havin, Victor ; Mashreghi, Javad

Canadian journal of mathematics, Tome 55 (2003) no. 6, pp. 1264-1301

Résumé

This paper is a continuation of $[6]$ . 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 $|f\left( x \right)|\,\le \,\omega \left( x \right)$ 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 that the unit vector $\Theta \left( x \right)$ winds up fast as $x$ grows from $-\infty \,\text{to}\,\infty $ . In particular, we consider $\Theta \,=\,B$ where $B$ is a Blaschke product with “horizontal” zeros, 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 $\left( 0,\,\infty\right)$ 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 $\exp \left( i\sigma z \right),\,\sigma \,>\,0$ . Some oscillating $\omega $ 's in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$ , and to de Branges’ space $H\left( E \right)$ .

Détail

DOI : 10.4153/CJM-2003-049-5

Mots-clés : 30D55, 47A15, Hardy space, inner function, shift operator, model subspace, Hilbert transform, admissible majorant

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