A Revisit on Commutators of linear and bilinear Fractional Integral Operator

Let $I_α$ be the linear and $\mathcal{I}_α$ 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 $I_α$. But the method can't be used to obtain the two weighted norm inequality for the higher order commutators of $I_α$. In this paper, we first give an alternative proof for the first order commutators of $I_α$. This new approach allows us to consider the higher order commutators. This was done by showing that the commutator $[b,I_α]$ 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 $I_α$. In the bilinear setting, we present a dyadic proof for the characterization between $BMO$ and the boundedness of $[b,\mathcal{I}_α]$. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of $[b,\mathcal{I}_α]$.

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