Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators

Abstract

Let b be a BMO function, \(0 1) and weighted endpoint estimates (p = 1) for the operator \(I^{+,k}_{\alpha,b}\) and for the pairs of weights of the type (w, \({\mathcal{M}_w}\)), where w is any weight and \(\mathcal{M}\) is a suitable one-sided maximal operator. We also prove that, for \(A^{+}_{\infty}\) weights, the operator \(I^{+,k}_{\alpha,b}\) is controlled in the Lp(w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators \(I^{+,k}_{\alpha,b}\). The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral \(I^{-,k}_{\alpha,b}\)