An inverse function theorem for Frechet spaces satisfying a smoothing property and (DN)

Classical inverse function theorems of Nash-Moser type are proved for Frech et spaces that admit smoothing operators as introduced by NASH. In this not e an inverse function theorem is proved for Frechet spaces which only have to satisfy the condition (DN) of VOGT and the smoothing property (S-Omega)( t); for instance, any Frechet-Hilbert space which is an (Omega)-space in st andard form has property (S-Omega)(t) The main result of this paper general izes a theorem of LOJASIEWICZ and ZEHNDER. It can be applied to the space C -infinity(K) if the compact K subset of R-N is the closure of its interior and subanalytic; different from classical results the boundary of K may hav e singularities like cusps. The growth assumptions on the mappings are form ulated in terms of the weighted multiseminorms [](m,k) introduced in this p aper; nonlinear smooth partial differential operators on C-infinity(K) and their derivatives satisfy these formal assumptions.