Superposition operators and functions of bounded p-variation

We characterize the set of all functions f of R to itself such that the associated superposition operator T_f: g → f ∘ g maps the class BV^1_p ( R) into itself. Here BV^1_p ( R) , 1 ≤ p < ∞ , denotes the set of primitives of functions of bounded p -variation, endowed with a suitable norm. It turns out that such an operator is always bounded and sublinear. Also, consequences for the boundedness of superposition operators defined on Besov spaces B^sp,q({ R}^n) are discussed.