Weighted estimates for a bilinear fractional integral operator and its commutator: A union condition

Abstract

The main theme of this paper is to give sufficient conditions for the weighted boundedness of the bilinear fractional integral operator $\mathsf{BI}_\al$. The proposed condition involves the union of multilinear Muckenhoupt-type conditions. We have achieved new results in an unknown case and remarkably improved other known results by utilizing the hidden convolution nature inside the operator. We also study the effects of the general product commutators on the main operator and the weighted estimates for a related maximal operator that norm-wise dominates the main operator.