Hölder–Zygmund classes on smooth curves
Armin RainerUniversität Wien, Austria
We prove that a function in several variables is in the local Zygmund class if and only if its composite with every smooth curve is of class . This complements the well-known analogous result for local Hölder–Lipschitz classes , 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 acting on the global Zygmund space . We prove that, for all integers , the map is of Lipschitz class if and only if .
Cite this article
Armin Rainer, Hölder–Zygmund classes on smooth curves. Z. Anal. Anwend. 41 (2022), no. 1/2, pp. 189–209DOI 10.4171/ZAA/1704