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

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 $\mathbb R^n$.