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Double Weighted Commutators Theorem for Pseudo-Differential Operators with Smooth Symbols

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Abstract

Let − (n + 1) < m ⩽ − (n + 1) (1 − ϱ) and let \({T_a} \in {\cal L}_{ϱ}^m,\delta \) be pseudo-differential operators with symbols a(x, ξ) ∈ ℝn × ℝn, where 0 < ϱ ⩽ 1,0 ⩽ δ < 1 and δ ⩽ ϱ. Let μ, λ be weights in Muckenhoupt classes Ap, ν = (μλ−1)1/p for some 1 < p < ∞. We establish a two-weight inequality for commutators generated by pseudo-differential operators Ta with weighted BMO functions b ∈ BMOν, namely, the commutator [b, Ta] is bounded from Lp(μ) into Lp(λ). Furthermore, the range of m can be extended to the whole m ⩽ − (n + 1)(1 − ϱ).

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Acknowledgement

The authors wish to express their thanks to the referee for his or her constructive suggestions and technological help.

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Yu-long Deng

  2. Institute of Computational Mathematics, School of Science, Hunan University of Science and Engineering, Yongzhou, 425199, China

    Yu-long Deng

  3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    Zhi-tian Chen & Shun-chao Long

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  1. Yu-long Deng
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  2. Zhi-tian Chen
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Correspondence to Yu-long Deng.

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Deng, Yl., Chen, Zt. & Long, Sc. Double Weighted Commutators Theorem for Pseudo-Differential Operators with Smooth Symbols. Czech Math J 71, 173–190 (2021). https://doi.org/10.21136/CMJ.2020.0246-19

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