Hi,

I have a question which involves pdo. Let us consider a pseudodifferential operator $A:S(\mathbb{R}^d)\rightarrow S(\mathbb{R}^d) $ whose symbol $a(x,\xi)$ lives in the $S_{0,0}^0$ class : $$ \forall \alpha,\beta \in \mathbb{N}^d \qquad |\partial_x^{\alpha} \partial_\xi^{\beta} a(x,\xi) |\leq C(\alpha,\beta) $$ Let be $f$ and $g$ smooth real functions on $\mathbb{R}^d$ such that $$ \forall \alpha \in \mathbb{N}^d\qquad |\partial_x^\alpha f(x)|+ |\partial_x^\alpha g(x)|\leq C(\alpha) $$ If the supports of $f$ and $g$ are disjoint (one may assume that their distance is positive), is it true that for all $u\in S(\mathbb{R}^d)$ and $s',s\in \mathbb{R}$ one has $$ \left\vert \left\vert f A(gu)\right\vert\right\vert_{H^s} \leq C(s,s',f,g,A) \left\vert \left\vert u \right\vert\right\vert_{H^{s'}} $$ Notice that the case $s'\geq s$ is clear since $fAg \in \mbox{Op}S_{0,0}^0$.

Thanks