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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.