## Tohoku Mathematical Journal

### 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, 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 $[b,\mathcal{I}_{\alpha}]$. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of $[b,\mathcal{I}_{\alpha}]$.

#### Article information

Source
Tohoku Math. J. (2), Volume 71, Number 2 (2019), 303-318.

Dates
First available in Project Euclid: 21 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1561082600

Digital Object Identifier
doi:10.2748/tmj/1561082600

Mathematical Reviews number (MathSciNet)
MR3973253

Zentralblatt MATH identifier
07108041

#### Citation

Cao, Mingming; Xue, Qingying. A revisit on commutators of linear and bilinear fractional integral operator. Tohoku Math. J. (2) 71 (2019), no. 2, 303--318. doi:10.2748/tmj/1561082600. https://projecteuclid.org/euclid.tmj/1561082600

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