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

Abstract.

Let b be a BMO function, $0 < \alpha < 1$ and $I^{+,k}_ {\alpha,b}$ 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 $I^{+,k}_{\alpha,b}$ 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 L ^p(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 $I^{-,k}_{\alpha,b}$