  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 $\Lambda^p_u(w)$.
