H\"older--Zygmund classes on smooth curves

TL;DR

This paper characterizes functions in Zygmund classes via their composition with smooth curves, extends these results to Banach space mappings, and analyzes the regularity of superposition operators on Zygmund spaces.

Contribution

It establishes a new characterization of Zygmund class functions through smooth curves and explores their properties in Banach spaces and superposition operators.

Findings

Functions in Zygmund classes are characterized by their composition with smooth curves.

Superposition operators are Lipschitz continuous if and only if the function belongs to a specific Zygmund class.

Results generalize classical H"older-Lipschitz class theorems to Zygmund classes.

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 $f \in \mathcal Z^{m+k,1}(\mathbb R)$.