Tohoku Mathematical Journal

A revisit on commutators of linear and bilinear fractional integral operator

Mingming Cao and Qingying Xue

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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

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

First available in Project Euclid: 21 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 47G10: Integral operators [See also 45P05]

commutator paraproduct Haar function dyadic analysis bilinear fractional integral operators


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.

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