Bounded functions in Möbius invariant Dirichlet spaces
For \(p>-1\) let \({\mathcal D}_p\) be the Dirichlet space of all analytic functions in the open unit disk \(\mathbb{D}\) for which \[ |f|^2_{{\mathcal D}_p} =\int_\mathbb{D} \bigl|f'(z) \bigr |^2 \bigl(1-|z|^2 \bigr)^p dm(z) <\infty, \] where \(dm\) is the two dimensional Lebesgue measure on \(\mathbb{D}\). The authors are concerned with the conformal invariant versions of these spaces, the so-called \(Q_p\) spaces, defined by \[ Q_p= \bigl\{f\in H(\mathbb{D}): |f|_{Q_p} =\sup_{w \in\mathbb{D}} |f\circ \varphi_w|_{{\mathcal D}_p} <\infty \bigr\}, \] where \(\varphi_w(z) =(w-z)/(1-\overline wz)\). A detailed study of these spaces was given in \textit{M. Essen} and \textit{J. Xiao} [J. Reine Angew. Math. 485, 173-195 (1997; Zbl 0866.30027)]. Moreover, let \[ Q_{p,0}= \bigl\{f\in Q_p: |f\circ \varphi_w |_{{\mathcal D}_p} \to 0\quad \text{for } |w|\to 1\bigr\}. \] We note that if \(-1<p<0\), then the space \(Q_p\) consists only of constant functions. If \(p=1\), then \(Q_1=\text{BMOA}\) and \(Q_{1,0} =\text{VMOA}\). If \(p>1\), then \(Q_p\) is the usual Bloch space of analytic functions in \(\mathbb{D}\) or which \(\sup_{z\in \mathbb{D}} (1-|z|^2) |f'(z) |<\infty\), whereas \(Q_{p,0}\) is the little Bloch space. Let us now review some of the remarkable results of this interesting paper, which is devoted to the study of the Banach algebras \(Q_p\cap H^\infty\) and \(Q_{p,0} \cap H^\infty\). It is shown for example that for \(p>0\) the algebras \(Q_p\cap H^\infty\) and \(Q_{p,0} \cap H^\infty\) have the corona property. (Note that for \(p\geq 1\), \(Q_p\cap H^\infty =H^\infty\), so this is Carleson's theorem, and that \(Q_{1,0} \cap H^\infty =QA\), so this gives Wolff's theorem). Especially interesting is the case of the algebra \(Q_{p,0} \cap H^\infty\), \(p>1\), of bounded functions in the little Bloch space. Here the corona theorem was known to be true, but a proof was never published. The authors also consider Fefferman-Stein decompositions of \(Q_p(\partial \mathbb{D})\) spaces defined on the circle. Finally, a complete and useful characterization of the interpolating sequences in the spaces \(Q_p\cap H^\infty\) and \(Q_{p,0} \cap H^\infty\) is given. Again here, of special interest, is the result that for the bounded analytic functions in the little Bloch space a sequence \((a_n)\) in \(\mathbb{D}\) is interpolating if and only if it is thin; that is if \(\lim_{n\to\infty} \prod_{k\neq n} |(a_k-a_n)/(1- \overline {a_k}a_n) |=1\).
scientific article; zbMATH DE number 1098589
scientific article