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Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences. (English) Zbl 1094.46006
This paper extends the splitting theorem of D. Vogt and M. J. Wagner [Stud. Math. 67, 225–240 (1980; Zbl 0464.46010)] and applies this extension to problems in the theory of constant coefficient partial differential equations on spaces of vector-valued distributions. A (PLS)-space is the projective limit of a sequence of strong duals of Fréchet–Schwartz spaces. This class of spaces contains all Fréchet–Schwartz spaces as well as their duals, but also, e.g., the space of all real analytic functions on an arbitrary open set in \(\mathbb R^n\).
The main new concepts of the paper are the linear topological invariants (\(P\Omega\)) and (\(P\overline{\overline\Omega}\)), which extend the properties (\(\Omega\)) and (\(\overline{\overline\Omega}\)) of Vogt and Wagner to the class of (PLS)-spaces. It is shown that every short exact sequence of (PLS)-spaces splits, provided that the first element satisfies (\(P\Omega\)) and the last element is a nuclear Fréchet space with property (DN). Variants of this result with (\(P\Omega\)) replaced by (\(P\overline{\overline\Omega}\)) and (DN) replaced by (\(\underline{\text{DN}}\)) are also given. In the presence of a basis, the nuclearity condition can be weakened.
This work has many predecessors, which are listed in the extensive bibliography. Among them are investigations of spaces of real analytic functions, the first prominent result in this area coming from P. Domański and D. Vogt [Stud. Math. 142, 187–200 (2000; Zbl 0990.46015)], and homological methods in the theory of projective limit spaces as presented by J. Wengenroth [“Derived functors in functional analysis” (Lect. Notes Math. 1810, Springer, Berlin) (2003; Zbl 1031.46001)].

MSC:
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
35D05 Existence of generalized solutions of PDE (MSC2000)
35E20 General theory of PDEs and systems of PDEs with constant coefficients
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