H\"older--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 H\"older--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 $f_* : g \mapsto f \circ g$ acting on the global Zygmund space $\Lambda_{m+1}(\mathbb R^d)$. We prove that, for all integers $m,k\ge 1$, the map $f_* : \Lambda_{m+1}(\mathbb R^d) \to \Lambda_{m+1}(\mathbb R^d)$ is of Lipschitz class $\mathcal C^{k-1,1}$ if and only if $f \in \mathcal Z^{m+k,1}(\mathbb R)$.