Let $\phi, \theta$ be two cutoff functions on a torus $\mathbb{T}$ in $\mathbb{R}^d$ such that their supports are disjoint. Let $P$ be a pseudodifferential operator on $\mathbb{T}$ and its symbol is in the Hormander class $S_{1,0}^{-2}$. Then it is known that $\phi P \theta$ is a smoothing operator. In particular, $\phi P \theta$ maps $H^{-m}(\mathbb{T})$ into $H^{m}(\mathbb{T})$ for $m \geq 0$.
Question:
"* Let $N>0$. There is some constant $C(N,m,d)>0$ depends on $N,m,d$ such that $\|\phi P \theta f\|_{H^{-m}(\mathbb{T})} \leq C(N,m,d)d(\phi, \theta)^{-2Nm}\|f\|_{H^{m}(\mathbb{T})}$ for any $f \in H^m$. Here $d(\phi, \theta)>0$ is the distance between supports of $\phi, \theta$*."
I wonder if the above estimate could be true, or at least a quantitative version depends on the distance $d(\phi, \theta)$.