$C^{1,\omega}$ extension formulas for $1$-jets on Hilbert spaces

TL;DR

This paper characterizes when a 1-jet on a subset of a Hilbert space can be extended to a $C^{1, ext{omega}}$ function, providing explicit, dimension-independent extension formulas with optimal bounds and continuity properties.

Contribution

It introduces necessary and sufficient conditions for $C^{1, ext{omega}}$ extension of 1-jets in Hilbert spaces, with explicit formulas and dimension-independent nonlinear extension operators.

Findings

Extension operator norm does not depend on dimension.

Explicit formula for the extension function.

Extension depends continuously on initial data.

Abstract

We provide necessary and sufficient conditions for a $1$-jet $(f, G):E\rightarrow \mathbb{R} \times X$ to admit an extension $(F, \nabla F)$ for some $F\in C^{1, \omega}(X)$. Here $E$ stands for an arbitrary subset of a Hilbert space $X$ and $\omega$ is a modulus of continuity. As a corollary, in the particular case $X=\mathbb{R}^n$, we obtain an extension (nonlinear) operator whose norm does not depend on the dimension $n$. Furthermore, we construct extensions $(F, \nabla F)$ in such a way that: (1) the (nonlinear) operator $(f, G)\mapsto (F, \nabla F)$ is bounded with respect to a natural seminorm arising from the constants in the given condition for extension (and the bounds we obtain are almost sharp); (2) $F$ is given by an explicit formula; (3) $(F, \nabla F)$ depend continuously on the given data $(f, G)$; (4) if $f$ is bounded (resp. if $G$ is bounded) then so is $F$ (resp. $F$ is Lipschitz). We also provide similar results on superreflexive Banach spaces.