A REVISIT ON COMMUTATORS OF LINEAR AND BILINEAR FRACTIONAL INTEGRAL OPERATOR

摘要

Let I-alpha be the linear and I-alpha be the bilinear fractional integral operators. In the linear setting, it is known that the two-weight inequality holds for the first order commutators of I-alpha. But the method can't be used to obtain the two weighted norm inequality for the higher order commutators of I-alpha. In this paper, using some known results, we first give an alternative simple proof for the first order commutators of I-alpha. This new approach allows us to consider the higher order commutators. Then, by using the Cauchy integral theorem, we show that the two-weight inequality holds for the higher order commutators of I-alpha. In the bilinear setting, we present a dyadic proof for the characterization between BMO and the boundedness of [b, I-alpha]. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of [b, I-alpha].