On Bi-dimensional Second Variation

In this paper we present the concept of bounded second variation of a real valued function defined on a rectangle in ${\mathbb{R}}^2$. We use Hardy-Vitali type technics in the plane in order to extend the classical notion of function of bounded second variation on intervals of $\mathbb{R}$. We introduce the class $B{V}^2(I_a^b)$, of all functions of bounded second variation on a rectangle $I_a^b\subset {\mathbb{R}}^2$, and show that this class can be equipped with a norm with respect to which it is a Banach space. Finally, we present two results that show that integrals of functions of first bounded variation (on $I_a^b$) are in $B{V}^2(I_a^b)$.