Hölder-Zygmund classes on smooth curves
type: Item
instance of: scholarly article
title: Hölder-Zygmund classes on smooth curves
DOI: 10.4171/ZAA/1704
author: Armin Rainer
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2022-12-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/2203.04191
review text: 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 \(f \in \mathcal{Z}^{m+k,1} (\mathbb{R})\).
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 58B10
Mathematics Subject Classification ID: 58C25
zbMATH DE Number: 7626237
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7626237