Czechoslovak Mathematical Journal, first online, pp. 1-18


Double weighted commutators theorem for pseudo-differential operators with smooth symbols

Yu-long Deng, Zhi-tian Chen, Shun-chao Long

Received May 31, 2019.   Published online September 16, 2020.

Abstract:  Let $-(n+1)<m\leq-(n+1)(1-\rho)$ and let $T_a\in\mathcal{L}^m_{\rho,\delta}$ be pseudo-differential operators with symbols $a(x,\xi)\in\mathbb{R}^n\times\mathbb{R}^n$, where $0<\rho\leq1$, $0\leq\delta<1$ and $\delta\leq\rho$. Let $\mu$, $\lambda$ be weights in Muckenhoupt classes $A_p$, $\nu=(\mu\lambda^{-1})^{1/p}$ for some $1<p<\infty$. We establish a two-weight inequality for commutators generated by pseudo-differential operators $T_a$ with weighted BMO functions $b\in{\rm BMO}_{\nu}$, namely, the commutator $[b,T_a]$ is bounded from $L^p(\mu)$ into $L^p(\lambda)$. Furthermore, the range of $m$ can be extended to the whole $m\leq-(n+1)(1-\rho)$.
Keywords:  pseudo-differential operator; reverse Hölder inequality; $A_p$ weight; commutator
Classification MSC:  47G30, 35S05, 42B25
DOI:  10.21136/CMJ.2020.0246-19

PDF available at:  Institute of Mathematics CAS

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Affiliations:   Yu-long Deng (corresponding author), School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, P. R. China, e-mail: yuldeng@163.com, Institute of Computational Mathematics, School of Science, Hunan University of Science and Engineering, Yongzhou 425199, P. R. China, e-mail: yuldeng@163.com; Zhi-tian Chen, Shun-chao Long, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, P. R. China, e-mail: kc-chan@foxmail.com, sclong@xtu.edu.cn


 
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