Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

Yanqi Yang; Shuangping Tao; Guanghui Lu; Yanqi Yang; Shuangping Tao; Guanghui Lu

doi:10.3934/math.2022333

AIMS Mathematics

2022, Volume 7, Issue 4: 5971-5990. doi: 10.3934/math.2022333

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Research article

Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 26 October 2021 Revised: 12 December 2021 Accepted: 10 January 2022 Published: 14 January 2022
MSC : 42B20, 42B25, 47G30

In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.
- bilinear pseudo-differential operator,
- commutator,
- Lipschitz function,
- the product of variable exponent Lebesgue space
Citation: Yanqi Yang, Shuangping Tao, Guanghui Lu. Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators[J]. AIMS Mathematics, 2022, 7(4): 5971-5990. doi: 10.3934/math.2022333

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Abstract

In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.

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