The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective
constant coefficient partial differential equation
In their deep paper on solvability of partial differential equations in the space of vector valued distributions [J. Funct. Anal. 230, No. 2, 329--381 (2006; Zbl 1094.46006)], \textit{J. Bonet} and \textit{P. Domański} were led to the following natural question. Let \(P(D)\) be a constant coefficient partial differential operator in d variables, which is surjective on the space of distributions \(D'(X)\) on some open set \(X\subset \mathbb{R}^d\). Is the augmented operator \(P^+(D_1,\dotsc,D_{d+1}):= P(D_1,\dotsc,D_d)\) necessarily surjective on \(D'(X\times \mathbb{R})\)?NEWLINENEWLINEIn the present interesting paper, this question is answered in the negative if \(d\geq 3\). In fact, a counterexample can be chosen such that \(P(D)\) is even hypoelliptic. This implies that the kernel of \(P(D)\) in \(C^\infty(X)\) does not have property \((\Omega)\) of Vogt providing the first example of this kind. The open set \(X\) is chosen as the complement of a suitable closed cone. The proof is then based on a characterization of the surjectivity of \(P^+(D)\) on \(D'(X\times \mathbb{R})\) by a variant of Hörmanders \(\sigma_P\)-function see [\textit{T. Kalmes}, J. Math. Anal. Appl. 386, No. 1, 125--134 (2012; Zbl 1248.35044)], where also positive solutions to the question of Bonet and Domanski [loc. cit.] are provided; in particular, the answer is always positive if \(d=2\)).
scientific article; zbMATH DE number 6048045
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