Weighted Bergman spaces induced by rapidly increasing weights (Q2925684)

Let \(H(\mathbb{D})\) denote the space of holomorphic functions on the unit disk \(\mathbb{D}\) of \(\mathbb{C}\). The principal objects of the monograph under review are sufficiently small weighted Bergman spaces \(A^p_\omega = A^p_\omega(\mathbb{D})\). The space \(A^p_\omega\) consists of those \(f\in H(\mathbb{D})\) for which NEWLINE\[NEWLINE \|f\|_{A^p_\omega}^p = \int_{\mathbb{D}} |f(z)|^p \omega(z)\, dA(z)<\infty, NEWLINE\]NEWLINE where \(\omega\) is assumed to be positive and integrable with respect to area measure \(dA\). Usually the authors also suppose that \(\omega\) is radial, i.e., \(\omega(z) = \omega(|z|)\) for a continuous function \(\omega: [0,1)\to (0, +\infty)\).NEWLINENEWLINEThe main class of radial weights under consideration is denoted by \(\mathcal{I}\). By definition, \(\omega\in \mathcal{I}\) if NEWLINE\[NEWLINE \lim_{r\to 1^-} \frac{\psi_\omega(r)}{1-r} = \infty, NEWLINE\]NEWLINE where the distortion function \(\psi_\omega\) is defined as NEWLINENEWLINE\[NEWLINE \psi_\omega(r) = \frac{\int_r^1 \omega(s)\, ds}{\omega(r)}. NEWLINE\]NEWLINE NEWLINENEWLINEThe elements of the class \(\mathcal I\) are called rapidly increasing. This term is somewhat misleading because, as indicated by the authors, such weights are not assumed to be increasing; moreover, \(\omega\in \mathcal{I}\) may admit a strong oscillatory behavior.NEWLINENEWLINEFor \(\omega\in\mathcal{I}\), the authors show that the spaces \(A^p_\omega\) are close neighbours of the Hardy spaces \(H^p = H^p(\mathbb{D})\). Also, many conventional methods used in the theory of the classical Bergman spaces \(A^p_\alpha\), \(\alpha>-1\), are not applicable to \(A^p_\omega\) with \(\omega \in \mathcal{I}\). For example, there is no characterization of \(p\)-Carleson measures for \(A^p_\omega\), \(\omega \in \mathcal{I}\), by a simple condition on pseudohyperbolic disks. Nevertheless, as shown in Chapter~2, the boundedness of the embedding \(A^p_\omega \subset L^q(\mu)\), \(\omega\in\mathcal{I}\), can be characterized by a geometric condition on the standard Carleson squares \(S(I)\), \(I\subset \partial\mathbb{D}\) is an arc, when \(0<p\leq q < \infty\).NEWLINENEWLINEA substantial part of the monograph is devoted to the integral operator NEWLINE\[NEWLINE T_g (f) (z) = \int_0^z f(\zeta) g^\prime(\zeta)\, d\zeta, \quad z\in\mathbb{D}, NEWLINE\]NEWLINE generated by a symbol \(g\in H(\mathbb{D})\). The study of the boundedness of \(T_g\) requires, among other things, a factorization of \(A^p_\omega\)-functions. So, in Chapter~3, the authors apply a probabilistic method of \textit{C.\,Horowitz} [Duke Math. J. 44, 201--213 (1977; Zbl 0362.30031)] under the assumption that \(\omega\in\mathcal{P}\). By definition, \(\omega\in\mathcal{P}\) if the polynomials are dense in \(A^p_\omega\) and \(\omega(z)\asymp \omega(\zeta)\) for all \(z\) in a pseudohyperbolic disk \(\Delta(\zeta, r)\).NEWLINENEWLINEIf \(\omega\in\mathcal{P}\), then each \(f\in A^p_\omega\) can be represented as a product \(f = f_1 f_2\), where \(f\in A^{p_1}_\omega\), \(f\in A^{p_2}_\omega\), \(\frac{1}{p_1}+ \frac{1}{p_2}= \frac{1}{p}\) and NEWLINE\[NEWLINE \|f_1\|^{p}_{A^{p_1}_\omega} \|f_2\|^{p}_{A^{p_2}_\omega} \leq C(p_1, p_2, \omega) \|f\|^{p}_{A^{p}_\omega}. NEWLINE\]NEWLINE NEWLINENEWLINEFor \(\omega\in \mathcal{I}\cap \mathcal{P}\) and \(0<q<p<\infty\) such results are used to describe those \(g\in H(\mathbb{D})\) for which \(T_g: A^p_\omega \to A^q_\omega\) is bounded. Also, the authors prove that each subset of an \(A^p_\omega\)-zero set is also an \(A^p_\omega\)-zero set. Moreover, they show that the \(A^p_\omega\)-zero sets depend on \(p\) when \(\omega\in \mathcal{I}\cap \mathcal{P}\).NEWLINENEWLINEIt is proved in Chapter~4 that \(T_g: A^p_\omega \to A^p_\omega\) is bounded if and only if \(g\) is in the space \(\mathcal{C}^1(\omega^*)\) which consists of those \(g\in H(\mathbb{D})\) for which NEWLINE\[NEWLINE \|g\|_{\mathcal{C}^1(\omega^*)}^2 = |g(0)|^2 + \sup_{I\subset \partial\mathbb{D}} \frac{\int_{S(I)} |g^\prime(z)|^2 \omega^*(z)\, dA(z)}{\omega(S(I))} <\infty, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \omega^*(z) = \int_{|z|}^1 \omega(s) \log\frac{s}{|z|} s\, ds, \quad z\in\mathbb{D}\setminus \{0\}. NEWLINE\]NEWLINE Note that the above \(\mathcal{C}^1(\omega^*)\)-norm is similar to the Carleson measure characterizations of the space \(\mathrm{BMOA}(\mathbb{D})\) and the Bloch space. Chapter~5 is devoted to the study of \(\mathcal{C}^1(\omega^*)\).NEWLINENEWLINEIn Chapter~6, the authors assume that \(\omega\in\mathcal{I}\) or \(\omega\) is regular, that is, \(\psi_\omega(r) \asymp 1-r\). They give a complete description of those \(g\in H(\mathbb{D})\) for which \(T_g\) belongs to the Schatten class \(\mathcal{S}_p(A^{2}_\omega)\).NEWLINENEWLINELinear differential equations with values in \(A^{p}_\omega\) or \(BN^{p}_\omega\), the Bergman-Nevanlinna class, are investigated in Chapter~7. The final Chapter~8 contains, in particular, several open problems related to special features of \(A^{p}_\omega\), \(\omega\in\mathcal{I}\).NEWLINENEWLINEIn general, the book under review successfully complements the monographs by \textit{H. Hedenmalm} et al. [Theory of Bergman spaces. New York, NY: Springer (2000; Zbl 0955.32003)], and by \textit{P. Duren} and \textit{A. Schuster} [Bergman spaces. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1059.30001)].