Weighted inequalities for some one-sided operators

We give a characterization of the pairs of weights $(u,v)$ such that the Weyl fractional integral operator maps $L^p(vdx)$ into weak $L^q(udx)$, $1<p\leq q<\infty$ or $p=1<q<\infty$. For the case $p<q$ we give necessary and sufficient conditions for the weak type of a maximal operator that includes as particular cases the Weyl fractional integral, the dual of the Hardy operator and the fractional one-sided maximal operator. As a consequence we give a new characterization of the pairs of weights for which the fractional one-sided maximal operator is bounded.