Average radial integrability spaces, tent spaces and integration operators

We deal with a Carleson measure type problem for the tent spaces $AT_{p}^{q}(\alpha)$ in the unit disc of the complex plane. They consist of the analytic functions of the tent spaces $T_{p}^{q}(\alpha)$ introduced by Coifman, Meyer and Stein. Well known spaces like the Bergman spaces arise as a special case of this family. Let $s,t,p,q\in (0,\infty)$ and $\alpha >0\,.$ We find necessary and sufficient conditions on a positive Borel measure $\mu$ of the unit disc in order to exist a positive constant $C$ such that $$\int_{\mathbb{T}} \left(\int_{\Gamma (\xi)} |f(z)|^{t}\ d\mu(z)\right)^{s/t}\ |d\xi|\leq C \|f\|^s_{T_{p}^{q}(\alpha)} \,,\quad f\in AT_{p}^{q}(\alpha)\,,$$ where $\Gamma (\xi) = \Gamma_M (\xi)=\{ z\in \mathbb{D} : |1-\bar{\xi} z |< M (1-|z|^2)\},$ $M> 1/2$ and $\xi$ is a boundary point of the unit disk. This problem was originally posed by D. Luecking. We apply our results to the study of the action of the integration operator $T_g$, also known as Pommerenke operator, between the average integrability spaces $RM(p,q) ,$ for $p,q\in [1,\infty)$. These spaces have appeared recently in the work of the first author with M. D. Contreras and L. Rodr\'iguez-Piazza. We also consider the action from an $RM(p,q)$ to a Hardy space $H^s$, where $p,q,s \in [1,\infty)$.