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

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.

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