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 $(C_b^{k,\omega }({\mathbb{R}}^n){)}^{\ast }$ 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.