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

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