JournalsrmiVol. 35, No. 6pp. 1715–1744

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

  • Alexander Brudnyi

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

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Abstract

Let Cbk,ω(Rn)C_b^{k,\omega}(\mathbb R^n) be the Banach space of CkC^k functions on Rn\mathbb R^n bounded together with all derivatives of order k\le k and with derivatives of order kk having moduli of continuity O(ω)O(\omega) for some ωC(R+)\omega\in C(\mathbb R_+). Let Cbk,ω(S):=Cbk,ω(Rn)SC_b^{k,\omega}(S):=C_b^{k,\omega}(\mathbb R^n)|_S be the trace space to a closed subset SRnS\subset\mathbb R^n. The geometric predual Gbk,ω(S)G_b^{k,\omega}(S) of Cbk,ω(S)C_b^{k,\omega}(S) is the minimal closed subspace of the dual (Cbk,ω(Rn))(C_b^{k,\omega}(\mathbb R^n))^* containing evaluation functionals of points in SS. We study geometric properties of spaces Gbk,ω(S)G_b^{k,\omega}(S) and their relations to the classical Whitney problems on the characterization of trace spaces of CkC^k functions on Rn\mathbb R^n. In particular, we show that each Gbk,ω(S)G_b^{k,\omega}(S) is a complemented subspace of Gbk,ω(Rn)G_b^{k,\omega}(\mathbb R^n), describe the structure of bounded linear operators on Gbk,ω(Rn)G_b^{k,\omega}(\mathbb R^n), prove that Gbk,ω(S)G_b^{k,\omega}(S) has the bounded approximation property and that in some cases space Cbk,ω(S)C_b^{k,\omega}(S) is isomorphic to the second dual of its subspace consisting of restrictions to SS of C(Rn)C^\infty(\mathbb R^n) functions with compact supports.

Cite this article

Alexander Brudnyi, On properties of geometric preduals of Ck,ω{\mathbf C^{k,\omega}} spaces. Rev. Mat. Iberoam. 35 (2019), no. 6, pp. 1715–1744

DOI 10.4171/RMI/1099