Characterization of inverse differential operators If I have a partial differential operator $p(D)$, where $p$ is a polynomial with constant coefficients and $D$ is the derivative in Euclidean space. Its inverse is easily described in Fourier space: $\mathcal{F}[p(D)^{-1}] = 1/p(k)$ with $k$ being the Fourier variable (of course up to specification of boundary conditions). Conversely, If I have a translation invariant integral operator with kernel $G(x-y)$, it is the inverse of a partial differential operator with constant coefficients whenever its Fourier transform is of the form $\mathcal{F}[G(x)] = 1/p(k)$ for some polynomial $p(k)$ with constant coefficients. I think this converse does a nice job of characterizing integral operators $G(x-y)$ that are inverse to a partial differential operator with constant coefficients.
Question: When translation invariance is dropped and one moves from Euclidean space to a general manifold, is it possible to characterize integral operators, say given by an integral kernel $G(x,y)$, which are inverse to some partial differential operator $p(x,D)$?
I have a feeling that the answer, if one exists, should have something to do with the wave-front set of $G(x,y)$. Unfortunately, my understanding of this subject is rather poor, so references to some general literature on this kind of problem are also appreciated.
 A: In some sense, much or all of microlocal analysis is precisely about these sorts of questions. As the previous answers indicate, different types of differential operators require rather different types of pseudodifferential or Fourier integral operators for their parametrices. Pseudodifferential operators are enough to study elliptic operators (or even more general operators in regions of phase space where they are micro-elliptic). There is a nice geometric definition of the structure of the Schwartz kernels of pseudodifferential operators, namely they are (classical) conormal distributions on the product space with respect to the diagonal.  For wave operators one requires the more complicated class of FIO's. Their Schwartz kernels are typically given as Lagrangian, or more generally, marked Lagrangian distributions. There is a nice coordinate invariant description of these too.  The question you ask (in one of your followups), namely to find simple conditions on G(x,y) that it be the retarded Green function of a hyperbolic differential operator, is pretty hopeless as stated. Certainly G would need to be a marked Lagrangian distribution, but to determine precisely which such distributions are actually the fundamental solutions of differential operators is not likely to have an easy answer. 
A: The problem is in fact a division problem: let $p(\xi)$ be a polynomial in $n$ variables. There exists a tempered distribution $T$ such that
$$
p(\xi) T(\xi)=T(\xi) p(\xi)=1.
$$
This is a result due to Hörmander and Lojasiewicz. The convolution by $\hat T(-x)$ provides then a fundamental solution for the operator $p(D)$.
Bazin.
A: It is straightforward to show that if a linear partial differential operator $P$ has a pseudodifferential right inverse (or even a pseudodifferential right parametrix) then it must be micro-hypoelliptic, i.e. $WF(Pu)=WF(u)$ for all $u\in\mathscr{D}'$, due to the fact that the wave front set of the Schwartz kernel of any pseudodifferential operator must be conormal to the diagonal. If $P$ has constant coefficients this amounts to saying that $P$ is hypoelliptic, i.e. $Pu\in C^\infty$ implies $u\in C^\infty$. For instance, the Laplace and heat operators are hypoelliptic, but the wave operator is not.
More generally, the existence of a right inverse to a linear partial differential operator $P$ entails that $P$ is locally solvable: for every compact subset $K$ and every $f$ in finite-codimensional subspace of $C^\infty$, there is $u\in\mathscr{D}'$ which satisfies $Pu=f$ in an open neighborhood of $K$. In that case, Hörmander has shown that the principal symbol $p$ of $P$ must satisfy the so-called condition $(\Psi)$: any compact subset $K$ has an open neighborhood $Y$ where there is no positively homogeneous, complex-valued $q\in C^\infty(T^*Y\smallsetminus 0)$ such that $\mathrm{Im}(pq)$ changes sign along any bicharacteristic strip $\gamma$ of $\mathrm{Re}(pq)$ over $Y$ such that $q\neq 0$ on $\gamma$. 
In general condition $(\Psi)$ does not entail that the right inverse will be a FIO (i.e. a linear operator whose Schwartz kernel is a Lagrangian distribution associated with an homogeneous canonical relation): if $p$ is real valued, this is indeed true as Deane Yang mentioned in his last comment to the OP's question (at least away from the diagonal, as Rafe Mazzeo remarked in his answer), but otherwise the bicharacteristic relation of $p$ may not be a manifold.
Finally, it should be remarked that all of the above applies to scalar $P$. For systems, the situation gets even more complicated, particularly when $P$ has characteristics of varying multiplicity.
