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 \(\mathbb R^n\).