Hölder–Zygmund classes on smooth curves

We prove that a function in several variables is the local Zygmund class $\mathcal{Z}^{m,1}$ if and only its composite with every smooth curve of $\mathcal{Z}^{m,1}$. This complements well-known analogous result for Hölder–Lipschitz classes $\mathcal{C}^{m,\alpha}$, which we reprove along way. demonstrate these results generalize to mappings between Banach spaces use them study regularity superposition operator $f\_\colon g \mapsto f \circ g$ acting on global space $\Lambda\_{m+1}(\mathbb{R}^d)$. that, all integers $m,k\ge 1$, map \Lambda\_{m+1}(\mathbb{R}^d) \to \Lambda\_{m+1}(\mathbb{R}^d)$ Lipschitz $\mathcal{C}^{k-1,1}$ $f \in \mathcal{Z}^{m+k,1}(\mathbb{R})$.