부산대학교 도서관

This book gives a thorough account of a large family of Möbius invariant spaces of analytic functions, which have become known as $\Cal Q_K$ spaces. These spaces were introduced by the first author (Hasi Wulan) and his collaborators at the beginning of the 21st century, and in a short period of time, the theory has reached an impressive level of maturity. A monograph which summarizes the major achievements in the field is easily justified. This book provides an easy access point for the student meeting the subject for the first time and offers a compendium of approaches, methods and techniques for the researcher already acquainted with the subject. \par Each $\Cal Q_K$ space is induced by a non-decreasing (weight) function $K \colon [0,\infty) \to [0,\infty)$, and consists of those analytic functions $f$ in the unit disc $\Bbb D = \{z\in\Bbb C \: |z| < 1\}$ for which $$ \| f \|_{\Cal Q_K}^2 = \sup_{a\in\Bbb D} \int_{\Bbb D} |f'(z)|^2 \, K\big( g(z,a) \big) \, dm(z)< \infty. $$ Here, $g(z,a) = -\log |a-z|/|1-\overline{a} z|$ is Green's function of the unit disc with a logarithmic singularity at $a\in\Bbb D$, and $dm(z)$ is the Lebesgue area measure with $m(\Bbb D)=1$. This book concentrates on the case of the unit disc of the complex plane, and does not deal with generalizations to domains other than $\Bbb D$, or to higher dimensions. \par For different choices of $K$, many familiar spaces can be recovered. If $K(t) = t^p$ for a positive $p$, then the resulting space reduces to $Q_p$, whose study pre-dated the research of $\Cal Q_K$. In fact, much credit for the fast development of the subject is due to the previous research concerning $Q_p$ and $F(p,q,s)$ spaces. For more details on $Q_p$ spaces, see J. Xiao's monographs [{\it Holomorphic $Q$ classes}, Lecture Notes in Math., 1767, Springer, Berlin, 2001; MR1869752; {\it Geometric $Q_p$ functions}, Front. Math., Birkhäuser, Basel, 2006; MR2257688]. Recall that $Q_p$ is precisely the classical Bloch space for $p >1$, while $Q_{p_1} \subsetneq Q_{p_2} \subsetneq Q_1 = {\rm BMOA}$ for any $0