Hölder–Zygmund classes on smooth curves

  • Armin Rainer

    Universität Wien, Austria
Hölder–Zygmund classes on smooth curves cover
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We prove that a function in several variables is in the local Zygmund class Zm,1\mathcal{Z}^{m,1} if and only if its composite with every smooth curve is of class Zm,1\mathcal{Z}^{m,1}. This complements the well-known analogous result for local Hölder–Lipschitz classes Cm,α\mathcal{C}^{m,\alpha}, which we reprove along the way. We demonstrate that these results generalize to mappings between Banach spaces and use them to study the regularity of the superposition operator f ⁣:gfgf_*\colon g \mapsto f \circ g acting on the global Zygmund space Λm+1(Rd)\Lambda_{m+1}(\mathbb{R}^d). We prove that, for all integers m,k1m,k\ge 1, the map f ⁣:Λm+1(Rd)Λm+1(Rd)f_*\colon \Lambda_{m+1}(\mathbb{R}^d) \to \Lambda_{m+1}(\mathbb{R}^d) is of Lipschitz class Ck1,1\mathcal{C}^{k-1,1} if and only if fZm+k,1(R)f \in \mathcal{Z}^{m+k,1}(\mathbb{R}).

Cite this article

Armin Rainer, Hölder–Zygmund classes on smooth curves. Z. Anal. Anwend. 41 (2022), no. 1/2, pp. 189–209

DOI 10.4171/ZAA/1704