We prove the Lorentz-Shimogaki and Boyd theorems for the spaces $\Lambda^{p}_{u}(w)$. As a consequence, we give the complete characterization of the strong boundedness of $H$ on these spaces in terms of some geometric conditions on the weights $u$ and $w$, whenever $p > 1$. For these values of $p$, we also give the complete solution of the weak-type boundedness of the Hardy-Littlewood operator on $\Lambda^{p}_{u}(w)$.