Hilbert-type operator induced by radial weight

Abstract

We consider the Hilbert-type operator defined byHω(f)(z)=∫01f(t)(1z∫0zBtω(u)du)ω(t)dt, where {Bζω}ζ∈D are the reproducing kernels of the Bergman space Aω2 induced by a radial weight ω in the unit disc D. We prove that Hω is bounded from H∞ to the Bloch space if and only if ω belongs to the class Dˆ, which consists of radial weights ω satisfying the doubling condition sup0⩽r<1⁡∫r1ω(s)ds∫1+r21ω(s)ds<∞. Further, we describe the weights ω∈Dˆ such that Hω is bounded on the Hardy space H1, and we show that for any ω∈Dˆ and p∈(1,∞), Hω:L[0,1)p→Hp is bounded if and only if the Muckenhoupt type conditionsup0<r<1⁡(1+∫0r1ωˆ(t)pdt)1p(∫r1ω(t)p′dt)1p′<∞, holds. Moreover, we address the analogous question about the action of Hω on weighted Bergman spaces Aνp.