I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an answer to either.


First question
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Let $M$ be a real analytic manifold, and let $P$ be a linear partial differential operator on $M$ with real analytic coefficients. Let $f$ be a hyperfunction on $M$. Does the PDE $$Pu=f$$ have a hyperfunction solution $u$ on a neighborhood of every point of $M$?


Second question
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Is the sheaf of hyperfunctions on $M$ injective as a module over the sheaf $\mathcal{D}\_M$ of linear partial differential operators on $M$ with real analytic coefficients?



do hyperfunction solutions always exist?
The answer to both questions is Yes. It has been known (Grainger, Kohn, Stein, Proc. Nat. Academy USA, Vol. 72, No. 9, 3287-3289 (1975)) that when $f$ is Baire category 1, then the Lewy operator is locally solvable. This means that if the 'size'of the test space is reduced making its dual larger, then solvability occurs. In fact, this led Hormander to formulate his famous hypo-ellipticity condition in terms of the Lie brackets spanning the whole tangent space.


This has a nice interpretation in the hyperfunction setup, noting that the Dirac measure is a distributional limit of hyperfunctions.


