A decomposition theorem for BV functions

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November 2011, 10(6): 1549-1566. doi: 10.3934/cpaa.2011.10.1549

A decomposition theorem for $BV$ functions

Stefano Bianchini ^1, and Daniela Tonon ^1,

1.	SISSA, via Bonomea, 265, Trieste, 34136, Italy, Italy

Received May 2010 Revised March 2011 Published May 2011

Abstract
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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.

Keywords: Jordan decomposition., $BV$ functions, Monotone functions.

Mathematics Subject Classification: Primary: 26B30, 26B35; Secondary: 28A7.

Citation: Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549

References:

[1]	G. Alberti, S. Bianchini and G. Crippa, Invariants for weakly regular ODE flows,, to appear., ().
[2]	L. Ambrosio, V. Caselles, S. Masnou and J. M. Morel, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc. (JEMS), 3 (2001), 39-92. doi: 10.1007/PL00011302.
[3]	L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, 2000.
[4]	R. Engelking, "General Topology," PWN, Warsaw, 1977.
[5]	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.
[6]	J. J. Manfredi, Weakly monotone functions, J. Geom. Anal., 4 (1994), 393-402.

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References:

[1]	G. Alberti, S. Bianchini and G. Crippa, Invariants for weakly regular ODE flows,, to appear., ().
[2]	L. Ambrosio, V. Caselles, S. Masnou and J. M. Morel, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc. (JEMS), 3 (2001), 39-92. doi: 10.1007/PL00011302.
[3]	L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, 2000.
[4]	R. Engelking, "General Topology," PWN, Warsaw, 1977.
[5]	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.
[6]	J. J. Manfredi, Weakly monotone functions, J. Geom. Anal., 4 (1994), 393-402.

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