Average radial integrability spaces, tent spaces and integration operators

Abstract

We deal with a Carleson measure type problem for the tent spaces ATpq(α) in the unit disc of the complex plane. They consist of the analytic functions of the tent spaces Tpq(α) introduced by Coifman, Meyer and Stein. Well known spaces like the Bergman spaces arise as a special case of this family. Let s,t,p,q∈(0,∞) and α>0. We find necessary and sufficient conditions on a positive Borel measure μ of the unit disc in order to exist a positive constant C such that ∫T(∫Γ(ξ)|f(z)|tdμ(z))s/t|dξ|≤C‖f‖Ts,f∈ATpq(α), where Γ(ξ)=ΓM(ξ)={z∈D:|1−ξ¯z|<M(1−|z|2)}, M>1/2 and ξ is a boundary point of the unit disc. This problem was originally posed by D. Luecking. We apply our results to the study of the action of the integration operator Tg, also known as Pommerenke operator, between the average integrability spaces RM(p,q), for p,q∈[1,∞). These spaces have appeared recently in the work of the first author with M.D. Contreras and L. Rodríguez-Piazza. We also consider the action from an RM(p,q) to a Hardy space Hs, where p,q,s∈[1,∞).