Uniformly continuous superposition operators in the Banach space of Hölder functions

Let I,J⊂R be intervals. One of the main results says that if a superposition operator H generated by a two place h:I×J→R,H(φ)(x):=h(x,φ(x)), maps the set Lipα(I,J) of all Hölder functions φ:I→J into the Banach space Lipα(I,R) and is uniformly continuous with respect to the Lipα-norm, thenh(x,y)=a(x)y+b(x),x∈I,y∈J, for some a,b∈Lipα(I,R).