Functional Calculus in Hölder-Zygmund Spaces

Abstract

In this paper we characterize those functions f f of the real line to itself such that the nonlinear superposition operator T f T_{f} defined by T f [ g ] := f ∘ g T_{f}[ g]:= f\circ g maps the Hölder-Zygmund space S s ( R n ) \mathcal {S}^s(\mathbf {R}^n) to itself, is continuous, and is r r times continuously differentiable. Our characterizations cover all cases in which s s is real and s > 0 s>0 , and seem to be novel when s > 0 s>0 is an integer.