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 $BV^1_p (\mathbb R)$ into itself. Here $BV^1_p (\mathbb R)$, $1 \le p < \infty$, 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 $B^s_{p,q}({\mathbb R}^n)$ are discussed.
Citation
Gérard Bourdaud . Massimo Lanza de Cristoforis . Winfried Sickel . "Superposition operators and functions of bounded $p$-variation." Rev. Mat. Iberoamericana 22 (2) 455 - 487, September, 2006.
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