Weighted Bergman spaces induced by rapidly increasing weights

This monograph is devoted to the study of the weighted Bergman space $Apomega$ of the unit disc $mathbbD$ that is induced by a radial continuous weight $omega$ satisfying $limrto 1-fracintr1omega(s),dsomega(r)(1-r)=infty.$ Every such $Apomega$ lies between the Hardy space $Hp$ and every classical weighted Bergman space $Apalpha$. Even if it is well known that $Hp$ is the limit of $Apalpha$, as $alphato-1$, in many respects, it is shown that $Apomega$ lies closer'' to $Hp$ than any $Apalpha$, and that several finer function-theoretic properties of $Apalpha$ do not carry over to $Apomega$.