Surjective partial differential operators on real analytic functions defined on a halfspace

Abstract

Let $P(D)$ be a partial differential operator with constant coefficients and let $A(ω)$ denote the real analytic functions defined on an open set $ω ⊂ Rⁿ$. Let H be an open halfspace. We show that $P(D)$ is surjective on $A(H)$ if and only if $P(D)$ is surjective on $A(Rⁿ)$ and $P(D)$ has a hyperfunction elementary solution which is real analytic on H.