Abstract
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.
Highlights
One of the necessary and sufficient properties, which characterizes real valued BV functions of one variable, is the well-known Jordan decomposition: it states that a function f : R → R is of bounded variation if and only if it can be written as the difference of two monotone increasing functions
The starting point is a recent result presented in (1), which shows that a real Lipschitz function of many variables with compact support can be decomposed in sum of monotone functions
The authors give the following definition of monotone function
Summary
One of the necessary and sufficient properties, which characterizes real valued BV functions of one variable, is the well-known Jordan decomposition: it states that a function f : R → R is of bounded variation if and only if it can be written as the difference of two monotone increasing functions. There exists a finite or countable family of monotone BV (RN ) functions {fi}i∈I , such that f = fi and |Df | = |Dfi| This decomposition is in general not unique, see Remark 2.2. The Ei’s are maximal indecomposable sets, i.e. any indecomposable set F ⊆ E is contained, up to LN -negligible sets, in some set Ei. The property stated in Theorem 1 (there is a disjoint partition {Ai}i∈N of RN such that every derivative ∇fi of the decomposition is concentrated on Ai) is no longer preserved in the case of BV functions. We consider Lipschitz functions from R2 to R2 and the related definition of monotone function In this particular case, we construct a counterexample showing that the decomposition property is not true in general, see Example 3.2. We give a proof of the fact that for Lipschitz functions Definition 1 and Definition 3 are equivalent
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