  In this paper we characterize the boundedness, compactness, and weak
compactness of the integration operators \begin{align*} T_g (f)(z)=\int_{0}^{z}
f(w)g'(w)\ dw \end{align*} acting on the average radial integrability spaces
$RM(p,q)$. For these purposes, we develop different tools such as a description
of the bidual of $RM(p,0)$ and estimates of the norm of these spaces using the
derivative of the functions, a family of results that we call Littlewood-Paley
type inequalities.
