Generalized Volterra type integral operators on large Bergman spaces

Abstract

Let ϕ be an analytic self-map of the open unit disk D and g analytic in D. We characterize boundedness and compactness of generalized Volterra type integral operatorsGI(ϕ,g)f(z)=∫0zf′(ϕ(ξ))g(ξ)dξ andGV(ϕ,g)f(z)=∫0zf(ϕ(ξ))g(ξ)dξ, acting between large Bergman spaces Aωp and Aωq for 0<p,q≤∞. To prove our characterizations, which involve Berezin type integral transforms, we use the Littlewood-Paley formula of Constantin and Peláez and establish corresponding embedding theorems, which are also of independent interest. When ϕ(z)=z, our results for GV(ϕ,g) complement the descriptions of Pau and Peláez.