A Revisit on Commutators of linear and bilinear Fractional Integral Operator

TL;DR

This paper revisits commutators of fractional integral operators, providing new proofs and extending two-weight inequalities to higher order commutators in both linear and bilinear settings.

Contribution

It introduces an alternative proof method for first order commutators and extends the results to higher order commutators, also analyzing bilinear operators with new techniques.

Findings

Two-weight inequality holds for higher order commutators of $I_{\alpha}$.

Characterization of $BMO$ via boundedness of commutators in bilinear setting.

Representation of commutators as finite linear combinations of paraproducts.

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 $[b,\mathcal{I}_{\alpha}]$.