Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the symbol of $A$. I'm interested in such symbols $\sigma$ that for any function $u$ in the domain of definition of $A$ we have $\rho(\mathop{\mathrm{supp}} u, \mathop{\mathrm{supp}} Au) < \varepsilon$ for some fixed $\varepsilon > 0$, where $\rho$ is the Hausdorff distance. Are there some known results about such symbols, necessary and sufficient conditions on $\sigma$ to possess the desired property? Any references are very welcome.



Support-preserving pseudodifferential operators
A classical result due to Peetre (Math. Scand. 8, 1960) says that if for all $u$,
$$\text{supp}\ Au\subset \text{supp}u,$$
then $A$ is a differential operator. On the other hand if a Fourier multiplier $a(D)$ satisfies your property, then $\hat a$ must be compactly supported. It seems that the following condition is relevant: if $a(x,\xi)$ is the symbol of your pseudodifferential operator $a(x,D)$,
$$
\text{Fourier wrt $\xi$ of a(x,\xi)}={\widehat a}^{\_2}(x,y)
$$
is compactly supported in $y$.


