Weighted Bergman spaces induced by rapidly incresing weights

This monograph is devoted to the study of the weighted Bergman space $A_{\text{om}}^p$ of the unit disc $\text{D}$ that is induced by a radial continuous weight $\text{om}$ satisfying {equation}\label{absteq} \lim_{r\to 1^-}\frac{\int_r^1\om(s),ds}{\om®(1-r)}=\infty.\tag{\dag} {equation} Every such $A_{\text{om}}^p$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A_{\text{a}}^p$. Even if it is well known that $H^p$ is the limit of $A_{\text{a}}^p$, as $\text{a}\to -1$, in many respects, it is shown that $A_{\text{om}}^p$ lies “closer” to $H^p$ than any $A_{\text{a}}^p$, and that several finer function-theoretic properties of $A_{\text{a}}^p$ do not carry over to $A_{\text{om}}^p$. As to concrete objects to be studied, positive Borel measures $\mu$ on $\text{D}$ such that $A_{\text{om}}^p\subset L^q(\mu )$, $0<p\le q<\mathrm{\infty }$, are characterized in terms of a neat geometric condition involving Carleson squares. It is also proved that each $f\in A_{\text{om}}^p$ 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$\frac{1}{p_1}+ \frac{1}{p_2}=\frac{1}{p}$\mathrm{.}Becauseofthetrickynatureof$A^p_\om$ several new concepts are introduced. It gives raise to a some what new approach to the study of the integral operator $$T_g(f)(z)={\int }_0^{z}f(\zeta )g^{\mathrm{\prime }}(\zeta )d\zeta \mathrm{.}$$ This study reveals the fact that $T_g:A_{\text{om}}^p\to A_{\text{om}}^p$ 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 ${\text{SSS}}_p(A_{\text{om}}^2)$ are also described. Furthermore, techniques developed are applied to the study of the growth and the oscillation of analytic solutions of (linear) differential equations.