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

Abstract

For d⩾3 we give an example of a constant coefficient surjective differential operator P(D):풟′(X)→풟′(X) over some open subset X⊂ℝ^d such that P⁺(D):풟′(X×ℝ)→풟′(X×ℝ) is not surjective, where P⁺(x₁, …, x_d+1)≔P(x₁, …, x_d). This answers Problem 9.1 posed by Bonet and Domański (‘Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences’, J. Funct. Anal. 230 (2006) 329–381) in the negative.