Revista Matemática Iberoamericana

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Volume 35, Issue 6, 2019, pp. 1715–1744
DOI: 10.4171/rmi/1099

Published online: 2019-07-30

On properties of geometric preduals of ${\mathbf C^{k,\omega}}$ spaces

Alexander Brudnyi[1]

(1) University of Calgary, Canada

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.

Keywords: Predual space, Whitney problems, finiteness principle, linear extension operator, approximation property, dual space, Jackson operator, weak∗ topology, weak Markov set

Brudnyi Alexander: On properties of geometric preduals of ${\mathbf C^{k,\omega}}$ spaces. Rev. Mat. Iberoam. 35 (2019), 1715-1744. doi: 10.4171/rmi/1099