Generalized Volterra type integral operators on large Bergman spaces
Let $ϕ$ be an analytic self-map of the open unit disk $\mathbb{D}$ and $g$ analytic in $\mathbb{D}$. We characterize boundedness and compactness of generalized Volterra type integral operators $$GI_{(ϕ,g)}f(z)= \int_{0}^{z}f'(ϕ(ξ))\,g(ξ)\, dξ$$ and $$GV_{ (ϕ, g)}f(z)= \int_{0}^{z} f(ϕ(ξ))\,g(ξ)\, dξ, $$ acting between large Bergman spaces $A^p_ω$ and $A^q_ω$ for $0<p,q\le \infty$. 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.
math.CV↗