A decomposition lemma for elementary tensors

Abstract

We prove a decomposition lemma for elementary tensors and present an analogous result of a (DN)-(Ω) splitting theorem for the theory of tensoring an exact sequence of Fréchet spaces with a (DF)-space. This result is free of nuclearity and hilbertisability assumptions and can be applied in a natural way to vector-valued linear partial differential operators.

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

“…Hence, it has (Ω) if and only if it has (P Ω). By the explanations given in the introduction, the above polynomial P and the open set X therefore also give an example of a surjective hypoelliptic differential operator P (D) : C ∞ (X) → C ∞ (X) such that its kernel does not have property (Ω), hence also solving an open problem from [10,Section 3]. This should be compared with Vogt's classical result [11] that the kernel of an elliptic differential operator always has (Ω).…”