Revista Matemática Iberoamericana

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Volume 22, Issue 2, 2006, pp. 455–487
DOI: 10.4171/RMI/463

Published online: 2006-08-31

Superposition operators and functions of bounded p-variation

Gérard Bourdaud[1], Massimo Lanza de Cristoforis[2] and Winfried Sickel[3]

(1) Université Pierre et Marie Curie, Paris, France
(2) Università di Padova, Italy
(3) Friedrich-Schiller-Universität Jena, Germany

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.

Keywords: Functions of bounded p-variation, homogeneous and inhomogeneous Besov spaces, Peetre’s embedding theorem, boundedness of superposition operators

Bourdaud Gérard, Lanza de Cristoforis Massimo, Sickel Winfried: Superposition operators and functions of bounded p-variation. Rev. Mat. Iberoamericana 22 (2006), 455-487. doi: 10.4171/RMI/463