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

TL;DR

This paper constructs an example showing that the augmented operator of a surjective constant coefficient differential operator may not be surjective, answering a previously open question in the theory of differential operators.

Contribution

It provides the first known example demonstrating that the augmented operator of a surjective differential operator with constant coefficients can fail to be surjective.

Findings

Constructed a specific example for d ≥ 3

Showed the augmented operator is not surjective

Answered an open problem in the field

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}.