Multipliers of Möbius invariant Q s spaces

Abstract

The spaceQs, 0≤s < ∞, consists of those f which are analytic in the unit disc \(\mathbb {D}\) and such that $$\sup_{a \in {\mathbb{D}}} \int\limits_{{\mathbb{D}}} |f'(z)|^2 (1-|\varphi_a(z)|^{2})^{s}\,dA(z) < \infty \,,$$ where \(\varphi_a(z) = \frac{a-z}{1-\overline{a}z}\). We give a complete description of M(Qs), the algebra of pointwise multipliers of Qs by proving Xiao’s conjecture which says that f∈M(Qs) if and only if f∈H∞ and $$\sup_{a \in {\mathbb{D}}}\left({\rm log}\frac{2}{1-|a|}\right)^2 \int\limits_{{\mathbb{D}}}|f'(z)|^2\,(1-|\varphi_ a(z)|^2)^s\,dA(z) < \infty.$$