Locality of homogeneous pseudo-differential operator

Question

Let $P$ be a polynomial in several variables, and let $P(D)$ be the corresponding differential operator. Obviously, $P(D)$ is a local operator, in the sense that I need only to know the function $u$ in a neighborhood of a point $x$ in order to evaluate $P(D)(u)(x)$.

If $P$ is not a polynomial, we can still define $P(D)$ using the Fourier representation. Does this always produce a non-local operator? How can I check this, e.g. when $P$ is a rational fraction not reducible to a polynomial (if this matters)? (I browsed through Taylor's and Hörmander's books on pseudo-differential operators, but could not find this result. But I suppose that my question is very classical.)

Answer 1

A linear operator $P\colon C_c^\infty(X)\to \mathcal D'(X)$ with kernel $K\in\mathcal D'(X\times X)$ is local if and only if $\operatorname{supp}K\subseteq \Delta_X$, where $\Delta_X$ is the diagonal in $X\times X$. If $P$ happens to be a pseudodifferential operator, then its kernel $K$ is conormal with respect to $\Delta_X$. Having both properties the same time means that $P$ is a differential operator.

The original version of this theorem (in the language of sheaves) is due to Jaak Peetre (1959).