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Inner functions in \(W_{\alpha}\) as improving multipliers. (English) Zbl 1451.30109

Let \(\mathbb{D}\) denote the open unit disk of the complex plane. For \(\alpha >-1\), the Dirichlet space \(\mathcal{D}_\alpha\) consists of all analytic functions \(f\) on the open unit disk for which \[\Vert f\Vert ^2_{\mathcal{D}_\alpha}=|f(0)|^2+\int_{\mathbb{D}} |f^\prime (z)|^2 dA_\alpha (z)<\infty,\] where \(dA_\alpha (z)=\pi^{-1} (\alpha +1)(1-|z|^2)^\alpha dxdy\). Similarly, the space \(\mathcal{W}_\alpha\) is defined as the space of all analytic functions \(f\) on \(\mathbb{D}\) such that \[\Vert f\Vert ^2_{\mathcal{W}_\alpha}=\sup_{\| g\|_{\mathcal{D}_\alpha \le 1}} \left (\int_{\mathbb{D}}|g(z)|^2 |f^\prime (z)|^2 dA_\alpha (z)\right )^{1/2}<\infty.\] These spaces are known to satisfy \(\mathcal{W}_\alpha\subseteq \mathcal{D}_\alpha\).
Let \(X\) and \(Y\) be two spaces of analytic functions on the unit disk such that \(X\subseteq Y\). An inner function \(\theta\) (a bounded analytic function on the unit disk whose boundary function has modulus \(1\) almost everywhere on the unit circle) is said to be \((X,Y)\)-improving if for each \(f\in X\) with \(\theta f\in Y\), we have \(\theta f\in X\). The main result of the paper under review states that for \(0<\alpha <1\), \(g\in \mathcal{D}_\alpha\), and an inner function \(\theta\), the following are equivalent:
1) \(\theta \in \mathcal{W}_\alpha\).
2) \(\theta\) is a Blaschke product with sequence of zeros \(\{z_j\}\) satisfying \[\sup_{\| g\|_{\mathcal{D}_\alpha \le 1}}\sum_{j=1}^\infty |g(z_j)|^2(1-|z_j|^2)^\alpha <\infty.\] 3) \(\theta\) is \((\mathcal{W}_\alpha, \mathrm{BMOA})\)-improving, where \(\mathrm{BMOA}\) is the space of analytic functions of bounded mean oscillation.
4) \(\theta\) is \((\mathcal{W}_\alpha, \mathcal{B})\)-improving, where \(\mathcal{B}\) is the Bloch space.
Reviewer: Ali Abkar (Qazvin)

MSC:

30H30 Bloch spaces
30H35 BMO-spaces
30H99 Spaces and algebras of analytic functions of one complex variable
30J05 Inner functions of one complex variable
30J10 Blaschke products
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