Functional Calculus in Hoelder-Zygmund Spaces

In this paper we characterize those functions $f$ of the real line to itself, such that the nonlinear superposition operator $T_{f}$ defined by $T_{f}[ g]:= f\circ g$ maps the H\"older-Zygmund space $\HSn{s}$ to itself, is continuous, and is $r$ times continuously differentiable. Our characterizations cover all cases in which $s$ is real and $s>0$, and seem to be novel in case $s>0$ is integer