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

Let b be a BMO function, 0 < alpha < 1 and I(alpha,b)(+,k) 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(alpha,b)(+,k) 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(infinity)(+) weights, the operator I(alpha,b)(+,k) 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(alpha,b)(+,k). The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral I(alpha,b)(+,k).