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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 9th 2017
• (edited Nov 9th 2017)

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.)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 11th 2017

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 11th 2017

I should say that I created some stubby minimum at generalized solution

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 12th 2017

I have forwarded the question to MO here. I know it’s a straightforward check, but maybe somebody lends a hand.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 13th 2017

Igor Khavkine has kindly provided a proof, now here.