In this monograph the authors summarize the major recent achievements in Möbius-invariant \(Q_K\) spaces, introduced by the first author and his collaborators at the beginning of this century. These spaces are a generalization of the \(Q_\alpha\) spaces, extensively treated in the books [J. Xiao, Holomorphic \(Q\) classes. Berlin: Springer (2001; Zbl 0983.30001); J. Xiao, Geometric \(Q_p\) functions. Basel: Birkhäuser (2006; Zbl 1104.30036)].
The Möbius group \(\mathcal{M}\) of the unit disk \(\mathbb{D}\subset \mathbb{C}\) consists of all one-to-one analytic functions \(\varphi\) that map \(\mathbb{D}\) onto itself. A vector space \(X\) of analytic functions on \(\mathbb{D}\) equipped with a complete seminorm \(\|\cdot\|_X\) is Möbius-invariant if \(\mathcal{M}\) acts on \(X\) isometrically, that is, \(\|f\|_X=\|f\circ \varphi\|_X\), for all \(f\in X\) and \(\varphi\in\mathcal{M}\).
A classical example of such spaces is \(H^\infty\) with the sup-norm. Other classical examples can be obtained using the Möbius-invariant operator \(D f(z)=(1-|z|^2)\,|f'(z)|\) and the invariant measure \(d\lambda(z)=(1-|z|^2)^{-2}dA(z)\): the Bloch space \(\mathcal{B}\) with the semi-norm \(\|f\|_\mathcal{B}=\|D f\|_{L^\infty(dA)}\); the diagonal Besov spaces \(B^p_{1/p}\), \(1< p< \infty\), with \(\|f\|_{B^p_{1/p}}=\|D f\|_{L^p(d\lambda)}\); the spaces \(Q_\alpha\), \(\alpha\geq 0\), with \( \|f\|_{Q_\alpha}=\sup_{\varphi\in\mathcal{M}} \|D f\|_{L^2((-\log |\varphi|)^\alpha d\lambda)}. \) Note that \(B^2_{1/2}\) is the Dirichlet space \(\mathcal{D}\). Moreover, if \(1< p< q\), then \(B^p_{1/p}\subsetneq B^q_{1/q}\subsetneq\mathrm{BMOA}\), and if \(0<\alpha<1<\beta\), then \(Q_0=\mathcal{D}\subsetneq Q_\alpha \subsetneq Q_1=\mathrm{BMOA}\subsetneq Q_\beta=\mathcal{B}\). Therefore, the scale \(Q_\alpha\), \(0<\alpha<1\) and the scale \(B^p_{1/p}\), \(p>2\), provide examples of Möbius-invariant spaces between \(\mathcal{D}\) and \(\mathrm{BMOA}\). One motivation for the introduction of the \(Q_K\) spaces was to obtain examples of such spaces between \(\mathrm{BMOA}\) and \(\mathcal{B}\).
Let \(K:[0,\infty)\to [0,\infty)\) be a nondecreasing function, not identically zero. The space \(Q_K\) consists of all holomorphic functions \(f\) on \(\mathbb{D}\) such that \[ \|f\|_{Q_K}=\sup_{\varphi\in\mathcal{M}} \|D f\|_{L^2(K(-\log |\varphi|) d\lambda)} =\sup_{\varphi\in\mathcal{M}} \|D(f\circ\varphi)\|_ {L^2(K(-\log|z|) d\lambda(z))}<\infty. \] The book consists of nine chapters. It begins with the main properties of general Möbius-invariant spaces and with some examples of such spaces, including the above-mentioned ones. In the second chapter the authors introduce the spaces \(Q_K\) and \(Q_{K,0}\) (the little \(Q_K\) space). A natural assumption on \(K\) is that \(K(-\log|z|)\in L^1(dA)\), because \(Q_K\) contains nonconstant functions if and only if this is satisfied. In this case \(Q_K\) contains all the polynomials. Moreover, \(Q_K=\mathcal{B}\) if and only if \(K(-\log|z|)\in L^1(d\lambda)\), and \(Q_K=\mathcal{D}\) if and only if \(K(0)>0\). Hence, when studying \(Q_K\) spaces it can be assumed that \(K(-\log|z|)\in L^1(dA)\setminus L^1(d\lambda)\), \(K(0)=0\) and \(K(t)>0\) for \(t>0\). Since \(Q_K\) only depends on the behaviour of \(K(t)\) for small \(t\), it is also possible to assume that \(K\) is constant for \(t>t_0\) and that it is right-continuous everywhere. An interesting result states that if \(K\) and \(K'\) satisfy the above natural conditions and the ratio \(K(t)/K'(t)\) tends to \(0\) as \(t\to 0\), then \( Q_{K'} \subsetneq Q_{K}\). This result allows to obtain spaces \(Q_K\) between \(\mathrm{BMOA}\) and \(\mathcal{B}\).
Chapter 3 is devoted to studying the properties of the weights \(K\) satisfying one or more of the following properties: (1) the doubling condition \(K(2t)\leq CK(t)\), \(0< t< t_0\); (2) \(\varphi_K(s) =\sup_{0<t\leq 1} K(st)/K(t)\in L^1\bigl([0,1],s^{-1}\,ds\bigr)\); (3) \(\varphi_K(s)\in L^1\bigl([1,\infty),s^{-1-\sigma}\,ds\bigr)\), for some \(\sigma>0\). This study allows the authors to replace a weight \(K\) by a “better” weight \(K'\) such that \(Q_{K'}=Q_K\) and also estimate several integrals that will turn out to be useful in the forthcoming chapters.
In the next two chapters the authors give several characterizations of the \(Q_K\) spaces, with some of the above-mentioned constraints on \(K\). The characterizations of \(Q_K\) functions are given in terms of \(K\)-Carleson measures, of their inner factors and absolute values, of their fractional derivatives and of their boundary values, among others. Also interesting are the characterizations of the \(K\)-Carleson measures, that is, of the positive Borel measures \(\mu\) on \(\mathbb{D}\) such that \[ \|\mu\|_K=\sup_{I\subset\mathbb{T}}\int_{S(I)}K \left(\tfrac{1-|z|}{|I|}\right)d\mu(z)<\infty. \] Here \(S(I)\) denotes the so-called Carleson box based on the sub-arc \(I\subset \mathbb{T}\).
The remaining chapters are devoted to studying some classical aspects of function theory in the context of \(Q_K\) spaces.
Chapter 6 focuses on several problems concerning the Taylor series expansions of \(Q_K\) functions. This includes a characterization of lacunary series in \(Q_K\) and also \(Q_K\)-norm estimates of Hadamard products such as, for instance, \(\|f*g\|_{Q_K}\leq C\|f\|_{Q_K}\|g\|_{\mathcal{B}}\).
In Chapter 7 the authors prove an atomic decomposition for \(f\in Q_K\). Namely, \(f(z)=\sum_{k}\lambda_k\left(\frac{1-|z_k|^2} {1-z\overline{z_k}}\right)^b\), where \(\sum_k|\lambda_k|^2\delta_{z_k}\) is a \(K\)-Carleson measure. Further interesting results include a generalization of the classical Fefferman-Stein decomposition for \(\mathrm{BMOA}\) and an interpolation result for \(Q_K\).
The purpose of Chapter 8 is to generalize some of the geometric function theory of Teichmüller spaces to the context of \(Q_K\) spaces. It contains a characterization of the univalent functions on \(\mathbb{D}\) such that \(\log f'\in Q_K\).
Finally, Chapter 9 contains additional topics about \(Q_K\) spaces: preduals; the relationship between \(Q_K\) and Morrey-type spaces; the corona problem for the algebra \(H^\infty\cap Q_K\); the distance of \(f\in\mathcal{B}\) to \(Q_K\); a Korenblum-type inequality for \(Q_K\), which generalizes the well-known estimate \(\|f(r\cdot)\|_{\mathrm{BMOA}}\leq \|f\|_{\mathcal{B}} \sqrt{|\log(1-r^2)|}\).
This comprehensive monograph summarizes the main results about \(Q_K\) spaces obtained since the beginning of this century. It is reader-friendly, well written, self-contained and well structured. Each chapter contains a section with additional references and remarks which complement the results and also give a historical approach. The bibliography is up-to-date and very extensive. Therefore, this book will be useful to researchers interested in this field under active development. Moreover, each chapter ends with a list of exercises and it would be an excellent textbook for advanced students with a good background in real and complex analysis.