Functional Calculus in Hölder-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 Hölder-Zygmund space ${\mathcal C}^{s}({\mathbf R}^{n})$ 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 when $s>0$ is an integer.