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A decomposition theorem for $BV$ functions

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  • The Jordan decomposition states that a function $f: R\to R$ is of bounded variation if and only if it can be written as the difference of two monotone increasing functions.
    In this paper we generalize this property to real valued $BV$ functions of many variables, extending naturally the concept of monotone function. Our result is an extension of a result obtained by Alberti, Bianchini and Crippa.
    A counterexample is given which prevents further extensions.
    Mathematics Subject Classification: Primary: 26B30, 26B35; Secondary: 28A75.


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    L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, 2000.


    R. Engelking, "General Topology," PWN, Warsaw, 1977.


    P. Hajlasz and J. Malý, Approximation in Soblev spaces of nonlinear expressions involving the gradient, Ark. Mat., 40 (2002), 245-274.doi: 10.1007/BF02384536.


    J. J. Manfredi, Weakly monotone functions, J. Geom. Anal., 4 (1994), 393-402.

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