Uniformly continuous superposition operators in the Banach space of Hِlder 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 Lip α ( I , J ) of all Hölder functions φ : I → J into the Banach space Lip α ( I , R ) and is uniformly continuous with respect to the Lip α -norm, then h ( x , y ) = a ( x ) y + b ( x ) , x ∈ I , y ∈ J , for some a , b ∈ Lip α ( I , R ) .