# MR2664252 Aziz, W.; Leiva, H.; Merentes, N.; Sánchez, J. L. Functions of two variables with bounded φ-variation in the sense of Riesz. J. Math. Appl. 32 (2010), 5–23. (Reviewer: Pasquale Vetro)

Risultato della ricerca: Review articlepeer review

## Abstract

The authors consider the space $BV_\varphi^R (I^b_a,\mathbb{R})$ offunctions $f:I^b_a =[a,b]\times [a,b]\subset \mathbb{R}^2 \to\mathbb{R}$ with a $\varphi$-bounded variation in the sense ofRiesz, where $\varphi: [0,+ \infty) \to [0,+ \infty)$ isnondecreasing and continuous with $\varphi(0)=0$ and $\varphi(t) \to+\infty$ as $t \to +\infty$. The authors show that $BV_\varphi^R(I^b_a,\mathbb{R})$ is a Banach algebra.Let $h: I^b_a \times \mathbb{R} \to \mathbb{R}$ and let $H:\mathbb{R}^{I^b_a} \to \mathbb{R}$ be the composition operatorassociated to $h$, that is the operator defined by $(Hf)(x)= h(x,f(x))$ for each $x \in I^b_a$.Then the authors consider the problem of characterizing thosefunctions $h$ such that the composition operator $H$ maps the space$BV_\varphi^R (I^b_a,\mathbb{R})$ to itself and is globallyLipschitzian. If we assume that $\varphi$ is also convex and suchthat $\limsup_{t \to +\infty}\frac{\varphi(t)}{t}= +\infty$, thenthe main result is that $H$ maps the space $BV_\varphi^R(I^b_a,\mathbb{R})$ to itself and is globally Lipschitzian if andonly if the function $h: I^b_a \times \mathbb{R} \to \mathbb{R}$ hasthe following representation $h(x,u)=h_0(x)+h_1(x)u$, for $(x,u) \inI^b_a \times \mathbb{R}$, with $h_0,h_1 \in BV_\varphi^R(I^b_a,\mathbb{R})$.
Lingua originale English 0 MATHEMATICAL REVIEWS 2011 Published - 2011

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