An Inverse Function Theorem for Fréchet Spaces Satisfying a Smoothing Property and (DN)

Abstract

AbstractClassical inverse function theorems of Nash‐Moser type are proved for Fréchet spaces that admit smoothing operators as introduced by Nash. In this note an inverse function theorem is proved for Fréchet spaces which only have to satisfy the condition (DN) of Vogt and the smoothing property (SΩ)t; for instance, any Fréchet‐Hilbert space which is an (Ω)‐space in standard form has property (SΩ)t. The main result of this paper generalizes a theorem of Lojasiewicz and Zehnder. It can be applied to the space C∞(K) if the compact K ⊂ ℝN is the closure of its interior and subanalytic; different from classical results the boundary of K may have singularities like cusps. The growth assumptions on the mappings are formulated in terms of the weighted multiseminorms [ ]m,k introduced in this paper; nonlinear smooth partial differential operators on C∞(K) and their derivatives satisfy these formal assumptions.