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On second κ -variation

Ereú J. J., Lopez L. M., Merentes N. J.

Commentationes Mathematicae

2016

Vol 56, No. 2

209--224

We present the notion of bounded second κ-variation for real functions defined on an interval [a,b]. We introduce the class κBV2([a,b]) of all functions of bounded second κ-variation on [a,b]. We show several properties of this class and present a sufficient condition under which a composition operator acts between these spaces.

On bi-dimensional second μ-variation

Ereú J., Giménez J., Merentes N.

Demonstratio Mathematica

2014

Vol. 47, nr 4

910--932

In this paper, we present a generalization of the notion of bounded slope variation for functions defined on a rectangle Iba in R2. Given a strictly increasing function μ, defined in a closed real interval, we introduce the class BVμ,2 (Iba), of functions of bounded second μ-variation on Iba ; and show that this class can be equipped with a norm with respect to which it is a Banach space. We also deal with the important case of factorizable functions in BVμ,2 (Iba) and finally we exhibit a relation between this class and the one of double Riemann–Stieltjes integrals of functions of bi-dimensional bounded variation.

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].