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