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 $O(\omega)$ 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 $(C\_b^{k,\omega}(\mathbb R^n))^\*$ 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 $\mathbb R^n$. In particular, we show that each $G\_b^{k,\omega}(S)$ is a complemented subspace of $G\_b^{k,\omega}(\mathbb R^n)$, describe the structure of bounded linear operators on $G\_b^{k,\omega}(\mathbb R^n)$, prove that $G\_b^{k,\omega}(S)$ has the bounded approximation property and that in some cases space $C\_b^{k,\omega}(S)$ is isomorphic to the second dual of its subspace consisting of restrictions to $S$ of $C^\infty(\mathbb R^n)$ functions with compact supports.