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)

The authors consider the space $BV_\varphi^R (I^b_a,\mathbb{R})$ of functions $f:I^b_a =[a,b]\times [a,b]\subset \mathbb{R}^2 \to \mathbb{R}$ with a $\varphi$-bounded variation in the sense of Riesz, where $\varphi: [0,+ \infty) \to [0,+ \infty)$ is nondecreasing 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 operator associated 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 those functions $h$ such that the composition operator $H$ maps the space $BV_\varphi^R (I^b_a,\mathbb{R})$ to itself and is globally Lipschitzian. If we assume that $\varphi$ is also convex and such that $\limsup_{t \to +\infty}\frac{\varphi(t)}{t}= +\infty$, then the main result is that $H$ maps the space $BV_\varphi^R (I^b_a,\mathbb{R})$ to itself and is globally Lipschitzian if and only if the function $h: I^b_a \times \mathbb{R} \to \mathbb{R}$ has the following representation $h(x,u)=h_0(x)+h_1(x)u$, for $(x,u) \in I^b_a \times \mathbb{R}$, with $h_0,h_1 \in BV_\varphi^R (I^b_a,\mathbb{R})$.