Superposition operators and functions of bounded $p$-variation

Abstract

We characterize the set of all functions $f$ of $\mathbb{R}$ to itself such that the associated superposition operator $T_f:g\to f\circ g$ maps the class $B{V}_p^1(\mathbb{R})$ into itself. Here $B{V}_p^1(\mathbb{R})$, $1\le p<\infty$, 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 $B_{p,q}^s({\mathbb{R}}^n)$ are discussed.