Weighted Bergman Spaces Induced by Rapidly Increasing Weights

This monograph is devoted to the study of the weighted Bergman space Aᵖ_ω of the unit disc D that is induced by a radial continuous weight ω satisfying limr→ 1⁻∫ᵣ¹ω(s)\,ds/ω(r)(1-r)=∞. Every such Aᵖ_ω lies between the Hardy space Hᵖ and every classical weighted Bergman space Aᵖ_α. Even if it is well known that Hᵖ is the limit of Aᵖ_α, as α→-1, in many respects, it is shown that Aᵖ_ω lies ``closer'' to Hᵖ than any Aᵖ_α, and that several finer function-theoretic properties of Aᵖ_α do not carry over to Aᵖ_ω.