Hölder-Zygmund classes on smooth curves
Summary: 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 \(f_* : g \mapsto f \circ g\) acting on the global Zygmund space \(\Lambda_{m+1} (\mathbb{R}^d)\). We prove that, for all integers \(m\), \(k \geq 1\), 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 \(f \in \mathcal{Z}^{m+k,1} (\mathbb{R})\).
scientific article; zbMATH DE number 7626237