Weighted Endpoint Estimates Research Articles

WEIGHTED ESTIMATES FOR SINGULAR INTEGRAL OPERATORS WITH NONSMOOTH KERNELS AND APPLICATIONS

Let 𝒳 be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, two weighted estimates related to weights are established for singular integral operators with nonsmooth kernels via a new sharp maximal operator associated with a generalized approximation to the identity. As applications, the weighted Lp(𝒳) and weighted endpoint estimates with general weights are obtained for singular integral operators with nonsmooth kernels, their commutators with BMO (𝒳) functions, and associated maximal operators. Some applications to holomorphic functional calculi of elliptic operators and Schrodinger operators are also presented.

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

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 \(I^{-,k}_{\alpha,b}\)