Weighted inequalities for some one-sided operators

Abstract

We give a characterization of the pairs of weights (u, v) such that the Weyl fractional integral operator maps Lp(vdx) into weak Lq(udx), 1 < p ≤ q < ∞ or p = 1 < q < ∞. 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.