关于${\mathbf C^{k,ω}}$空间的几何性质
On Properties of Geometric Preduals of ${\mathbf C^{k,ω}}$ Spaces
摘要：
设$C_b^{k,ω}(\mathbb R^n)$是$C^k$函数在$\mathbb R^n$上的Banach空间，对于某些$ω\in C(\mathbb R_+)$，所有$\le k$阶导数和$k$阶导数的连续模都被$c\cdotω$，$c\in\mathbb R_+$控制.设$C_b^{k,ω}(S):=C_b^{k,ω}(\mathbb R^n)|_S$是到闭子集$S\subset\mathbb R^n$的迹空间。$C_b^{k,ω}(S)$的几何预对偶$G_b^{k,ω}(S)$是包含$S$中点的评价泛函的对偶$\bigl(C_b^{k,ω}(\mathbb R^n)\bigr)^*$的最小闭子空间。本文研究了空间$G_b^{k,ω}(S)$的几何性质及其与经典Whitney问题的关系，给出了$\mathbb R^n$上$C^k$函数的迹空间的刻划.
英文摘要：
Let $C_b^{k,ω}(\mathbb R^n)$ 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 $c\cdotω$, $c\in\mathbb R_+$, for some $ω\in C(\mathbb R_+)$. Let $C_b^{k,ω}(S):=C_b^{k,ω}(\mathbb R^n)|_S$ be the trace space to a closed subset $S\subset\mathbb R^n$. The geometric predual $G_b^{k,ω}(S)$ of $C_b^{k,ω}(S)$ is the minimal closed subspace of the dual $\bigl(C_b^{k,ω}(\mathbb R^n)\bigr)^*$ containing evaluation functionals of points in $S$. We study geometric properties of spaces $G_b^{k,ω}(S)$ and their relations to the classical Whitney problems on the characterization of trace spaces of $C^k$ functions on $\mathbb R^n$.
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