Hölder–Zygmund classes on smooth curves

Abstract

We prove that a function in several variables is in the local Zygmund class $\mathcal{Z}^{m,1}$ if and only if its composite with every smooth curve is of class $\mathcal{Z}^{m,1}$. This complements the well-known analogous result for local Hölder–Lipschitz classes $\mathcal{C}^{m,\alpha}$, 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 $f_*\colon g \mapsto f \circ g$ acting on the global Zygmund space $\Lambda_{m+1}(\mathbb{R}^d)$. We prove that, for all integers $m,k\ge 1$, the map $f_*\colon \Lambda_{m+1}(\mathbb{R}^d) \to \Lambda_{m+1}(\mathbb{R}^d)$ is of Lipschitz class $\mathcal{C}^{k-1,1}$ if and only if $f \in \mathcal{Z}^{m+k,1}(\mathbb{R})$.