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

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

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?


1 Answer 1


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.

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    $\begingroup$ Do you have a source for the hyperfunction interpretation? $\endgroup$ Oct 16, 2021 at 21:22
  • $\begingroup$ The key to the interpretation is the ''edge of the wedge'' theorem. Please start wth Walter Rudin's lectures here: bookstore.ams.org/cbms-6 $\endgroup$ Oct 17, 2021 at 7:09
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    $\begingroup$ Do you have a reference for the specific statement that the sheaf of hyperfunctions is injective over $\mathcal{D}_M$? $\endgroup$ Oct 17, 2021 at 17:52
  • $\begingroup$ Please see: (1) Chapter 1.1-1.6 of Henrik Schlichtkrull, "Hyperfunctions and Harmonic Analysis on Symmetric Spaces", Birkhauser Boston, Inc., 1984, and (2) Chapter 4 of Seiichiro Wakabayashi, "Classical Microlocal Analysis in the Space of Hyperfunctions", © Springer-Verlag Berlin Heidelberg 2000 $\endgroup$ Oct 23, 2021 at 9:30

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