Extensions of vector-valued functions with preservation of derivatives

TL;DR

This paper develops an extension theorem for vector-valued functions with prescribed derivatives, ensuring the preservation of differentiability, continuity, and Lipschitz properties in Banach spaces, expanding the scope of previous extension results.

Contribution

It introduces a new extension theorem for vector-valued functions with pre-assigned derivatives, maintaining key smoothness and continuity properties.

Findings

Extension theorem applies to Banach and normed spaces.

Preserves differentiability, continuity, and Lipschitz properties.

Applicable to finite-dimensional domains for strict differentiability.

Abstract

Let $X$ and $Y$ be Banach or normed linear spaces and $F\subset X$ a closed set. We apply our recent extension theorem for vector-valued Baire one functions arXiv:1512.03717 to obtain an extension theorem for vector-valued functions $f\colon F\to Y$ with pre-assigned derivatives, with preservation of differentiability (at every point where the pre-assigned derivative is actually a derivative), preservation of continuity, preservation of (pointwise) Lipschitz property and (for finite dimensional domain $X$) preservation of strict differentiability and global (eventually local) Lipschitz continuity.