A Revisit on Commutators of linear and bilinear Fractional Integral Operator

Abstract

Let $I_{\alpha }$ be the linear and ${\mathcal{I}}_{\alpha }$ 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, we first give an alternative proof for the first order commutators of $I_{\alpha }$. This new approach allows us to consider the higher order commutators. This was done by showing that the commutator $[b,I_{\alpha }]$ can be represented as a finite linear combination of some paraproducts. 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 $[b,{\mathcal{I}}_{\alpha }]$. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of $[b,{\mathcal{I}}_{\alpha }]$.

17 pages