\(Q_p\) spaces -- a survey (Q2738499)

For \(p\geq 0\), a function \(f\) holomorphic in the unit disk \(D\) is said to be a \(Q_p\) function, that is \(f\in Q_p\), if NEWLINE\[NEWLINE\|f\|^2_{Q_p} =\sup_{w \in D}\int_D \bigl|f'(z)\bigr|^2 \bigl(g(z,w) \bigr)^p dm(z)< \inftyNEWLINE\]NEWLINE where \(dm(z)\) denotes the element of area and NEWLINE\[NEWLINEg(z,w)=\log\left|{1-\overline wz\over w-z}\right |.NEWLINE\]NEWLINE The \(Q_p\) spaces and their analogues have been studied widely over the past ten years, and this paper gives a selected survey of known results. The \(Q_p\) spaces relate to many well known spaces, for example \(Q_0\) is the Dirichlet space, \(Q_1=\text{BMOA}\) and, for \(p>1\), \(Q_p\) is the Bloch space. However, for \(0\leq p<q\leq 1\), \(Q_p\subset Q_q\) but \(Q_p \neq Q_q\). The results surveyed fall into two basic types characterizations and applications. Several different characterizations of the \(Q_p\) spaces are given, including criteria involving Dirichlet space type norms, the area of the image of a function, Carleson-type measures, Cesàro means, Hadamard products, Taylor coefficients, boundary value behavior, Fefferman-Stein type decompositions, integrals with respect to hyperbolic area, and pseudoanalytic extensions. In the applications section, there are results about inner and outer functions in \(Q_q\), interpolation in \(Q_p\), a Corona theorem for \(Q_p\) spaces, some results on random power series and univalent functions, and some results connecting \(Q_p\) spaces with classical spaces such as Hardy spaces, Lipschitz spaces, and Besov spaces, among others. In addition, a number of open problems are stated. A very extensive bibliography of the subject is provided.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00035].