The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective

Abstract

For $d\geq 3$ we give an example of a constant coefficient surjective differential operator $P(D):\mathscr{D}'(X)\rightarrow\mathscr{D}'(X)$ over some open subset $X\subset\R^d$ such that $P^+(D):\mathscr{D}'(X\times\R)\rightarrow\mathscr{D}'(X\times\R)$ is not surjective, where $P^+(x_1,...,x_{d+1}):=P(x_1,...,x_d)$. This answers in the negative a problem posed by Bonet and Doma\'nski in \cite[Problem 9.1]{Bonet}.