On Properties of Geometric Preduals of ${\mathbf C^{k,\omega}}$ Spaces

TL;DR

This paper investigates the geometric properties of the preduals of certain function spaces of smooth functions with controlled derivatives, relating these properties to classical Whitney extension problems.

Contribution

It characterizes the geometric structure of the preduals of $C_b^{k, ext{omega}}$ spaces and connects these findings to Whitney's classical trace space problems.

Findings

Description of geometric properties of $G_b^{k, ext{omega}}(S)$

Relations established between preduals and Whitney extension problems

Insights into the structure of trace spaces for smooth functions

Abstract

Let $C_b^{k,\omega}(\mathbb R^n)$ be the Banach space of $C^k$ functions on $\mathbb R^n$ bounded together with all derivatives of order $\le k$ and with derivatives of order $k$ having moduli of continuity majorated by $c\cdot\omega$, $c\in\mathbb R_+$, for some $\omega\in C(\mathbb R_+)$. Let $C_b^{k,\omega}(S):=C_b^{k,\omega}(\mathbb R^n)|_S$ be the trace space to a closed subset $S\subset\mathbb R^n$. The geometric predual $G_b^{k,\omega}(S)$ of $C_b^{k,\omega}(S)$ is the minimal closed subspace of the dual $\bigl(C_b^{k,\omega}(\mathbb R^n)\bigr)^*$ containing evaluation functionals of points in $S$. We study geometric properties of spaces $G_b^{k,\omega}(S)$ and their relations to the classical Whitney problems on the characterization of trace spaces of $C^k$ functions on $\mathbb R^n$.