Superposition operators and functions of bounded p-variation (vol 22, pg 455, 2006)

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 $BV_{p}^{1}({\mathbb{R}})$ into itself. Here $BV_{p}^{1}({\mathbb{R}})$, $1 \leq p < \infty$, 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 $B^{s}_{p,q}({\mathbb{R}}^{n})$ are discussed.