Average radial integrability spaces, tent spaces and integration operators

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