Weighted Bergman spaces induced by rapidly incresing weights

This monograph is devoted to the study of the weighted Bergman space $A^p_\om$ of the unit disc $\D$ that is induced by a radial continuous weight $\om$ satisfying {equation}\label{absteq} \lim_{r\to 1^-}\frac{\int_r^1\om(s)\,ds}{\om(r)(1-r)}=\infty.\tag{\dag} {equation} Every such $A^p_\om$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\a$. Even if it is well known that $H^p$ is the limit of $A^p_\a$, as $\a\to-1$, in many respects, it is shown that $A^p_\om$ lies "closer" to $H^p$ than any $A^p_\a$, and that several finer function-theoretic properties of $A^p_\a$ do not carry over to $A^p_\om$. As to concrete objects to be studied, positive Borel measures $\mu$ on $\D$ such that $A^p_\om\subset L^q(\mu)$, $0<p\le q<\infty $, are characterized in terms of a neat geometric condition involving Carleson squares. It is also proved that each $f\in A^p_\om$ can be represented in the form $f=f_1\cdot f_2$, where $f_1\in A^{p_1}_\om$, $f_2\in A^{p_2}_\om$ and $\frac{1}{p_1}+ \frac{1}{p_2}=\frac{1}{p}$. Because of the tricky nature of $A^p_\om$ several new concepts are introduced. It gives raise to a some what new approach to the study of the integral operator $$ T_g(f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta. $$ This study reveals the fact that $T_g:A^p_\om\to A^p_\om$ is bounded if and only if $g$ belongs to a certain space of analytic functions that is not conformally invariant. The symbols $g$ for which $T_g$ belongs to the Schatten $p$-class $\SSS_p(A^2_\om)$ are also described. Furthermore, techniques developed are applied to the study of the growth and the oscillation of analytic solutions of (linear) differential equations.