A sufficient condition of type (Ω) for tame splitting of short exact sequences of Fréchet spaces

Abstract

In a previous paper, the quotient spaces of (s) in the tame category of nuclear Frechet spaces have been characterized by property (ΩDZ) corresponding to the topological condition (Ω) of D. Vogt and M. J. Wagner. In addition, a splitting theorem has been proved which provides the existence of a tame linear right inverse of a tame linear map on the assumption that the kernel of the given map has property (ΩDZ) and that certain tameness conditions hold. In this paper it is proved that property (Ω) in standard form (i.e., the dual norms ‖ ‖n* are logarithmically convex) implies the tame splitting condition (ΩDZ) for any tamely nuclear Frechet space equipped with a grading defined by sermiscalar products. As an application, property (ΩDZ) is verified for the kernels of any hypoelliptic system of linear partial differential operators with constant coefficients on ℝN or on a bounded convex region in ℝN.