Superposition operators and functions of bounded p-variation

Abstract

We characterize the set of all functions f of ℝ to itself such that the associated superposition operator Tf : g → f o g maps the class BVp1(ℝ) into itself. Here BVp1(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 BV p,qs(ℝn) are discussed.