Sharp Two weight inequalities for commutators of Riemann-Liouville and Weyl fractional integral operators

Abstract

Let b be a BMO function, 0 < α < 1 and I+,k α,b the commutator of order k for the Weyl fractional integral. In this paper we prove weighted strong type (p, p) inequalities (p > 1) and weighted endpoint estimates (p = 1) for the operator I+,k α,b and for the pairs of weights of the type (w, Mw), where w is any weight and M is a suitable one-sided maximal operator. We also prove that, for A+∞ weights, the operator I +,kα,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 +,kα,b. The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral I -,kα,b.