A version of scale calculus and the associated Fredholm theory

TL;DR

This paper develops a version of scale calculus tailored for nonlinear Fredholm maps between specific Fréchet spaces, enabling the application of the Nash-Moser inverse function theorem and extending classical Fredholm theory.

Contribution

It introduces a new framework of scale calculus that preserves key properties of Fredholm maps in a Fréchet space setting, including a constant rank theorem.

Findings

Extension of linear Fredholm maps to a scale calculus setting

Establishment of a constant rank theorem for nonlinear Fredholm maps

Application to examples like reparametrisation and elliptic PDEs

Abstract

This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending the notion of linear Fredholm maps from [HWZ14] and [Weh12] to a setting where the Nash-Moser inverse function theorem can be applied and which also encompasses the necessary examples such as the reparametrisation action and (nonlinear) elliptic partial differential operators.