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This question will just show my ignorance, please bear with me:
For $P \colon C^\infty(X) \to C^\infty(X)$ a differential operator (self-adjoint say), there is an evident linear map
$DistribSolutions(P) \longrightarrow \left(C^\infty(X)/im(P)\right)^\ast$from distributions $u$ for which $P u = 0$ to linear duals on the cokernel of $P$.
When is this surjective?
(I’d be happy to add various qualifiers if necessary, say compactly supported distributions, or whatever it takes.)
With help from Igor Khavkine, here is the sketch of an argument that this is indeed true at least for the case of “Green hyperbolic differential equations”: here
I should say that I created some stubby minimum at generalized solution
I have forwarded the question to MO here. I know it’s a straightforward check, but maybe somebody lends a hand.
Igor Khavkine has kindly provided a proof, now here.
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