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On Bi-dimensional Second Variation

Ereu J., Gimémez J., Merentes N.

Commentationes Mathematicae

2012

Vol. 52, [Z] 1

39-59

In this paper we present the concept of bounded second variation of a real valued function defined on a rectangle in R2. 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 R. We introduce the class [formula] of all functions of bounded second variation on a rectangle [formula] 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 are in [formula].