A Smoothing Property for Fréchet Spaces

Abstract

A smoothing property (SΩ)tfor Fréchet spaces is introduced generalizing the classical concept of smoothing operators which are important in the proof of Nash–Moser inverse function theorems. For Fréchet–Hilbert spaces property (Ω) in standard form in the sense of D. Vogt is shown to be sufficient for (SΩ)t. For instance, the spaces E(K) of infinitely differentiable functions in the sense of Whitney have property (SΩ)tfor an arbitrary compactK⊂Rn; applications to extensions of Whitney functions with estimates are included. In a forthcoming paper, an inverse function theorem will be proved for Fréchet spaces with properties (SΩ)tand (DN); this applies to E(K) if the compactK=K⊂Rnis subanalytic.