Linear and multilinear fractional operators: weighted inequalities, sharp bounds, and other properties

Abstract

In this work we consider various fractional operators, including the classical fractional integral operators, related fractional maximal functions, multilinear fractional integral operators, and multisublinear fractional maximal functions. We characterize the weighted inequalities for the multilinear fractional operators, and examine more general two-weight inequalities giving sufficient conditions for their boundedness. For the classical fractional integral operator we obtain sharp bounds on the operator norm between weighted Lebesgue spaces in terms of the constant associated to the weight. We also introduce a more general fractional maximal operators, characterize their boundedness on weighted Lebsegue spaces, and obtain sharp bounds on the operator norms in terms of the weighted constants. Finally, we examine singular integral operators and fractional integral operators acting on mixed Lebesgue spaces with weights. We provide endpoint estimates for singular integrals and an off-diagonal extrapolation theorem.