Multipliers of Möbius invariant \(Q_{s}\) spaces
The space \(Q_s\), \(0\leq s<\infty\), consists of those analytic functions in the unit disc \(\mathbb D\) which satisfy \[ \sup_{a\in\mathbb D}\int_{\mathbb D}|f^{\prime}(z)|^2(1-|\varphi_a(z)|^2)^s\,dA(z)<\infty,\quad \text{where}\quad \varphi_a(z)=\frac{a-z}{1-\overline{a}z}. \] In the paper under review, the authors give a complete description of \(M(Q_s)\), the algebra of pointwise multipliers of \(Q_s\). They accomplish this by proving Xiao's conjecture: \(f\in M(Q_s)\) if and only if \(f\in H^{\infty}\) and \[ \sup_{a\in\mathbb D}\left( \log\frac{2}{1-|a|}\right)^2 \int_{\mathbb D}|f^{\prime}(z)|^2(1-|\varphi_a(z)|^2)^s\,dA(z)<\infty. \]
scientific article; zbMATH DE number 5533138