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

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

Czechoslovak Mathematical Journal, Tome 71 (2021) no. 1, pp. 173-190

Résumé

Let $-(n+1)$ 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 $0\rho \leq 1$, $0\leq \delta 1$ and $\delta \leq \rho $. Let $\mu $, $\lambda $ be weights in Muckenhoupt classes $A_{p}$, $\nu =(\mu \lambda ^{-1})^{1/p}$ for some $1$. 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 $m\leq -(n+1)(1-\rho )$.

Détail

DOI : 10.21136/CMJ.2020.0246-19

Classification : 35S05, 42B25, 47G30
Keywords: pseudo-differential operator; reverse Hölder inequality; $A_p$ weight; commutator