A Splitting Theorem for the Space of Smooth Functions

We will show using purely linear functional analytic methods that each exact complex0→F→(C∞(Ω))s0→p0(C∞(Ω))s1→p1(C∞(Ω))s2→p2…,wherepiare matrices of convolution operators (in particular, linear differential operators with constant coefficients), splits fromp1on (in the category of topological vector spaces). Moreover, we characterize when these complexes split completely. The obtained result covers all known particular cases as well as some more general cases independently of the analytic nature ofpi. The result is a consequence of studying splitting of short exact sequences0→F→jX→qG→0 (∗)whereF,X,Gare graded Fréchet spaces, i.e., Fréchet spaces equipped with fixed reduced spectra representing them andj,qare consistent with the graded structures. We characterize splitting of (∗) in case: (i)ForGare spacesC∞(Ω), whereΩis an open subset in Rn, or (ii)Gis the space of all sequences.