Revista Matemática Iberoamericana

Superposition operators and functions of bounded $p$-variation

Gérard Bourdaud , Massimo Lanza de Cristoforis , and Winfried Sickel

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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.

Article information

Rev. Mat. Iberoamericana, Volume 22, Number 2 (2006), 455-487.

First available in Project Euclid: 26 October 2006

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Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]

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), no. 2, 455--487.

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