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.
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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 $y$.
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$\begingroup$ Thank you for the answer, could you tell me please where can I find the mentionned theorem about Fourier multipliers? $\endgroup$– AppliquéCommented Jan 16, 2014 at 20:33
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$\begingroup$ The Fourier multiplier $a(D)$ is the convolution with $\hat a(-x)$. For that operation to send compactly supported functions into compactly supported distribution, it is sufficient that $\hat a$ is compactly supported since $supp (u\ast v)\subset supp u+supp v$. It should also be necessary since $\hat a(\zeta) \hat u(\zeta)$ must be entire of exponential type when $\hat u$ is entire of exponential type. $\endgroup$– BazinCommented Jan 16, 2014 at 21:00