Hölder–Zygmund classes on smooth curves

Abstract

. We prove that a function in several variables is in the local Zyg- mund class Z m, 1 if and only if its composite with every smooth curve is of class Z m, 1 . This complements the well-known analogous result for local H¨older– Lipschitz classes C m,α 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 7→ f ◦ g acting on the Zygmund space Λ m +1 ( R d ). We prove that, for all integers m,k k, R ).