Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces

Abstract

Necessary and sufficient conditions are given for the fractional integral operator I α I_\alpha to be bounded from weighted strong and weak L p L^p spaces within the range p ≥ n / α p\geq n/\alpha into suitable weighted B M O BMO and Lipschitz spaces. We also characterize the weights for which I α I_\alpha can be extended to a bounded operator from weighted B M O BMO into a weighted Lipschitz space of order α \alpha . Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of I α I_\alpha between weighted Lipschitz spaces.