Tohoku Mathematical Journal
2019
June
SECOND SERIES VOL. 71, NO. 2

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

Title A REVISIT ON COMMUTATORS OF LINEAR AND BILINEAR FRACTIONAL INTEGRAL OPERATOR

Author Mingming Cao and Qingying Xue

(Received October 31, 2016, revised May 19, 2017)
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 $[b,\mathcal{I}_{\alpha}]$.

Mathematics Subject Classification. Primary 42B20; Secondary 47G10.
Key words and phrases. Commutator, paraproduct, Haar function, dyadic analysis, bilinear fractional integral operators.

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