Solution operators for partial differential equations in weighted Gevrey spaces
The existence of continuous linear right inverses for concrete linear operators has recently been studied. These results are mainly based on the general splitting theorem of Vogt for exact sequences of power series spaces of infinite type, while the nonexistence of a continuous linear right inverse is often proved by the noncompatibility of certain linear topological invariants, which were developed in the recent study of the structure of nuclear (F)-spaces. The author considers partial differential equations in (weighted) spaces of ultradistributions of Roumieu type and obtains the following Theorem: Let W(x) satisfy certain three conditions and let P(D) be an \(r\times s\) system of partial differential operators with constant coefficients on \({\mathbb{R}}^ N\). Then, P(D) has tame linear right inverses \[ R_ -: Ker Q(D)\cap \Gamma^{\delta}(-W)^ r\to \Gamma^{\delta}(-W)^ s\text{ and } R_+: Ker Q(D)\cap (\Gamma^{\delta}(W)'_ b)^ r\to (\Gamma^{\delta}(W)'_ b)^ s, \] where Q(D) is the matrix of relations implied by P(D) and \(\Gamma^{\delta}(W)\) is a certain space of Gevrey functions with a weight W(x).
scientific article; zbMATH DE number 4161203