If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.


Is the reverse true? Namely that if some PDO $P$ with a symbol $p(x,\xi)$ from some reasonable symbol class has a property that $Pf$ has compact $x$-support for any Schwartz $f$, then does it imply that $p(x,\xi)$ has compact $x$-support?



Pseudo-differential operators with compactly supported symbols
Yes. Denote by $S\_K$, $K$ compact, the closed subspace of Schwartz space $S$ consisting of all $f$ such that the support of $Pf$ is contained in $K$. The assumption and the Baire category theorem imply that $S=S\_K$ for some $K$. It follows that the $x$-support of the Schwartz kernel $k(x,y)$ of $P$ is contained in $K$, hence also the $x$-support of the symbol $p(x,\xi)=\int e^{i\xi y} k(x,x-y)\,\mathrm dy$.


