Smooth extension of functions on a certain class of non-separable Banach spaces

TL;DR

The paper proves that in certain Banach spaces, Lipschitz and $C^1$-smooth functions defined on subspaces or closed sets can be extended to the whole space while preserving smoothness and Lipschitz bounds.

Contribution

It establishes the existence of $C^1$-smooth and Lipschitz extensions of functions in Banach spaces bi-Lipschitz homeomorphic to subsets of $c_0( ext{Gamma})$ with smooth coordinate functions.

Findings

Extensions preserve Lipschitz constants within a fixed multiple.

Existence of $C^1$-smooth extensions for functions on subspaces.

Results apply to functions on closed subsets of the Banach space.

Abstract

Let us consider a Banach space $X$ with the property that every real-valued Lipschitz function $f$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (with $C$ depending only on the space $X$). This is the case for a Banach space $X$ bi-Lipschitz homeomorphic to a subset of $c_0(\Gamma)$, for some set $\Gamma$, such that the coordinate functions of the homeomorphism are $C^1$-smooth. Then, we prove that for every closed subspace $Y\subset X$ and every $C^1$-smooth (Lipschitz) function $f:Y\to\Real$, there is a $C^1$-smooth (Lipschitz, respectively) extension of $f$ to $X$. We also study $C^1$-smooth extensions of real-valued functions defined on closed subsets of $X$.