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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:C (X)C (X)P \colon C^\infty(X) \to C^\infty(X) a differential operator (self-adjoint say), there is an evident linear map

    DistribSolutions(P)(C (X)/im(P)) * DistribSolutions(P) \longrightarrow \left(C^\infty(X)/im(P)\right)^\ast

    from distributions uu for which Pu=0P u = 0 to linear duals on the cokernel of PP.

    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
    • CommentTime7 days ago

    Igor Khavkine has kindly provided a proof, now here.