Average radial integrability spaces of analytic functions

In this paper we introduce the family of spaces $RM(p,q)$, $1\leq p,q\leq +\infty$. They are spaces of holomorphic functions in the unit disc with average radial integrability. This family contains the classical Hardy spaces (when $p=\mathrm{\infty }$) and Bergman spaces (when $p=q$). We characterize the inclusion between $RM(p_1,q_1)$ and $RM(p_2,q_2)$ depending on the parameters. For $1<p,q<\mathrm{\infty }$, our main result provides a characterization of the dual spaces of $RM(p,q)$ by means of the boundedness of the Bergman projection. We show that $RM(p,q)$ is separable if and only if $q<+\mathrm{\infty }$. In fact, we provide a method to build isomorphic copies of ${\mathrm{\ell }}^{\mathrm{\infty }}$ in $RM(p,\mathrm{\infty })$.