Geometric completeness of distribution spaces
The author considers \(\mathcal A\)--submodules \(\mathcal P\) of \({\mathcal A}^ k\), where \(k\in {\mathbb N}\) and \({\mathcal A}={\mathbb C}[\partial_1,\ldots,\partial_n]\) is the ring of linear partial differential operators with constant complex coefficients, and studies properties of the solution spaces \(\text{ker}_{\mathcal F}({\mathcal P})\) associated with \(\mathcal P\) and \(\mathcal F\), where \(\mathcal F\) is a finitely generated \(\mathcal A\)-module and, in particular, any \(\mathcal A\)-submodule of \({\mathcal D}'\) the space of distributions on \({\mathbb R}^ n\). The author also studies the notion of geometrically completeness for a given \({\mathcal F} \subset {\mathcal D}'\) and proves that if \(\mathcal F\) is \(\mathcal A\)-injective then it is geometrically complete and gives an example proving that the converse is not true.
scientific article; zbMATH DE number 1968884