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On Bi-dimensional Second Variation

100%

Ereu J. , Gimémez J. , Merentes N.

Commentationes Mathematicae

2012

tom 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].

Some Fine Properties of BV Functions on Wiener Spaces

100%

Ambrosio L. , Miranda Jr. M. , Pallara D.

2015

tom 3

nr 1

In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.

On second κ -variation

80%

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

Commentationes Mathematicae

2016

tom 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 second \(\kappa\)-variation

80%

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

Commentationes Mathematicae

2016

tom 56

nr 2

We present the notion of bounded second \(\kappa\)-variation for real functions defined on an interval \([a,b]\). We introduce the class \(\kappa BV^{2}([a,b])\) of all functions of bounded second \(\kappa\)-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

80%

Ereú J. , Giménez J. , Merentes N.

tom 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.

Functions of bounded variations on compact subsets of C

80%

Gimenez J. , Merentes N. , Vivas M.

Commentationes Mathematicae

2014

tom Vol 54, No. 1

3--19

In this paper we introduce the concept of bounded variation for functions defined on compact subsets of the complex plane C, based on the notion of variation along a curve as defined by Ashton and Doust; We describe in detail the space so generated and show that it can be equipped, in a natural way, with the structure of a Banach algebra. We also present a necessary condition for a composition operator Cφ to act between two such spaces.

Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

70%

Hakkarainen H. , Kinnunen J. , Lahti P. , Lehtelä P.

2016

tom 4

nr 1

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem involving the functional, boundary values can be presented as a penalty term.

Multiplication operators on the space of functions of bounded variation

70%

Astudillo-Villalba F. R. , Ramos-Fernández J. C.

Demonstratio Mathematica

2017

tom Vol. 50, nr 1

105--115

In this paper, we study the properties of the multiplication operator acting on the bounded variation space BV[0, 1]. In particular, we show the existence of non-null compact multiplication operators on BV[0, 1] and non-invertible Fredholm multiplication operators on BV[0, 1].