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?