New Method of Smooth Extension of Local Maps on Linear Topological Spaces. Applications and Examples

TL;DR

This paper introduces blid maps, a new tool for extending local maps in Banach and non-smooth spaces, enabling solutions to problems like local linearization and cohomological equations.

Contribution

The paper proposes blid maps as a novel approach to local map extension, applicable to both smooth and non-smooth infinite-dimensional spaces.

Findings

Blid maps facilitate the extension of local maps without smooth bump functions.

They enable reconstruction of maps from derivatives at a point, akin to the Borel Lemma.

Application to local differentiable linearization of maps on Banach spaces.

Abstract

The question of extension of locally defined maps to the entire space arises in many problems of analysis (e.g., local linearization of functional equations). A known classical method of extension of smooth local maps on Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional spaces. We suggest a new approach to localization of Banach spaces with the help of locally identical maps, which we call blid maps. In addition to smooth spaces, blid maps also allow to extend local maps on non-smooth spaces (e.g., $C^q [0, 1]$, $q=0, 1, 2,...$). For the spaces possessing blid maps, we show how to reconstruct a map from its derivatives at a point (see the Borel Lemma). We also demonstrate how blid maps assist in finding global solutions of cohomological equations having linear transformation of the argument. We present application of blid maps to local differentiable linearization of maps on Banach spaces. We discuss differentiable localization for metric spaces (e.g., $C^{\infty}(\R)$), prove an extension result for locally defined maps and present examples of such extensions for the specific metric spaces. In conclusion, we formulate open problems.