Inner functions in \(W_{\alpha}\) as improving multipliers (Q781696)

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.