A decomposition theorem for BV functions

The Jordan decomposition 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. 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.

1. Introduction. 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 aim of this work is to give a generalization of this property to real valued BV functions of many variables.
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. Precisely the authors give the following definition of monotone function  where ⟨·, ·⟩ is the scalar product in R N .
Another possibility is to preserve the maximum principle: the supremum (infimum) of f in every set is assumed at the boundary. Taken Ω ⊂ R N , a Lebesgue monotone function is defined as a continuous function f : Ω → R, which satisfies the maximum and minimum principles in every subdomain. Manfredi, in [6], and Hajlasz and Malý, in [5], give a weaker formulations. Here, a weakly monotone function is defined as a function f : Ω → R in the Sobolev space W 1,p (Ω), which satisfies the weak maximum and the weak minimum principles in every subdomain. A natural generalization is given in the case f is in the Sobolev space W 1,p loc (Ω). In our case we choose to maintain the property that sub/super-level sets are connected. Differences and analogies from the case of functions of one variables arise.
On the one hand, it can be found an L 1 monotone function, which is not of bounded variation, that is a counterexample to the fact that monotonicity is a sufficient condition for being of bounded variation (Example 2).
On the other hand, it can be stated that a BV function is decomposable in a countable sum of monotone functions, similarly to the case of BV functions of one real variable.
The main result of the paper is the following.
This decomposition is in general not unique, see Remark 2.
The main tool for proving this theorem is a decomposition theorem for sets of finite perimeter, presented here in the form given in [2]. Theorem 1.6 (Decomposition Theorem for sets). Let E be a set with finite perimeter in R N . Then there exists a unique finite or countable family of pairwise disjoint indecomposable sets {E i } i∈I such that

Moreover, denoting with
} the essential interior of the set E, it holds The property stated in Theorem 1.2 (there is a disjoint partition {A i } i∈N of R N such that every derivative ∇f i of the decomposition is concentrated on A i ) is no longer preserved in the case of BV functions. Example 1 shows that, in general, this decomposition can generate monotone BV functions without mutually singular distributional derivatives.
Finally, we conclude the paper showing that there is no hope for a further generalization of this decomposition to vector valued BV functions, apart from the case of a function f : R → R m where the analysis is straightforward. We consider Lipschitz functions from R 2 to R 2 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. In fact, a necessary condition for the decomposability of a Lipschitz function, from R 2 to R 2 , is that some of its level sets must be of positive H 1 -measure. This is an additional property, which is clearly not shared by all the Lipschitz functions.
The paper is organized as follows.
In Section 2 we prove the main theorem and show that this decomposition can generate monotone BV functions without mutually singular distributional derivatives.
In Section 3 we give two counterexamples: the first to the fact that a monotone function is always a BV function, the second to a further extension of the main theorem to vector valued functions. We also give a proof of the fact that for Lipschitz functions Definition 1.1 and Definition 1.4 are equivalent.
2. The Decomposition Theorem for BV functions from R N to R. To generalize the Jordan decomposition property, let us concentrate on functions f : R N → R, which belong to BV (R N ). From now on N > 1.
Since we will consider functions of bounded variation, the Definition 1.4 of monotone function becomes the following: Indeed, we recall that, for BV functions, super-level sets and sub-level sets are of finite perimeter for L 1 -a.e. t ∈ R.
We now prove the main theorem of this paper.
Proof of Theorem 1.5. The proof will be given in several steps. Before entering into details, let us consider the following simple case. Let f = χ E with E ⊆ R N a decomposable set of finite perimeter such that R N \ E is indecomposable. Thanks to the Decomposition Theorem for sets, there exists a unique finite or countable family of pairwise disjoint indecomposable sets To see the properties of R N \ E i let us consider the following lemma. Proof. Letî ∈ I be fixed. Without loss of generality we can relabelî = 1. By contradiction, suppose R N \ E 1 be decomposable and let {F j } j∈J be the family of its indecomposable components given by the Decomposition Theorem for sets.
It holds where, we recall, (R N \ E) ∪ {E i } i∈I,i̸ =1 is a family of indecomposable and pairwise disjoint sets. From the maximal indecomposability of {F j } j∈J and {E i } i∈I , it follows that We relabelĵ = 1. Moreover, we can found two sub-families and Observe that where {E 1 , E i k k ∈ K} is precisely the family of indecomposable sets given by the Decomposition Theorem for sets. Therefore On the other hand From this lemma, for every i ∈ I, E i and R N \ E i are indecomposable. Therefore the functions χ Ei are BV (R N ) and monotone, so that the decomposition of χ E , Step 0. We can assume without loss of generality that f ≥ 0: in the general case one can decompose f + and f − separately.
Step 1. The sets E t := {f > t} are of finite perimeter for L 1 -a.e. t ∈ R + , thanks to the hypothesis that f is BV (R N ) and coarea formula. Therefore, the Decomposition Theorem for sets gives, for L 1 -a.e. t ∈ R + , pairwise disjoint indecomposable sets In particular, the property of maximal indecomposability yields a natural partial order relation between these sets: since . Taken a countable dense subset {t j } j∈J of R + , such that, for all j ∈ J, the sets E j := E tj are of finite perimeter, the countable family {E j i } j∈J,i∈It j can be equipped with the partial order relation . Therefore there exists at least one maximal countable ordered sequence (here we do not need the Axiom of Choice).
Let {E j i(j) } j∈J one of these maximal countable ordered sequences. Notice that, once one of these sequences is fixed, the index i is a function of j, by the uniqueness of the decomposition {E j i } i∈It j .

STEFANO BIANCHINI AND DANIELA TONON
Step 2. Definef of the maximal countable ordered sequence and contains another Due to the maximal indecomposability property, one has that . • Next we show the other inclusion up to countably many values of t. Observe and this implies |{f = t}| > 0. This last condition can be satisfied only for a countable number of t ∈ R + . Therefore the set of t's such that E t i(t) does not coincide withẼ t has zero Lebesgue measure, i.e. for L 1 -a.e. t ∈ R + the setsẼ t coincide with E t i(t) up to L N -negligible sets. Since the property of being indecomposable is invariant up to L N -negligible sets, they are indecomposable.
In the following we will denote witht k , k ∈ K, the countable family of values such that Step 4. The functionf is BV (R N ) and has indecomposable super-level sets.
The indecomposability of the super-level sets off was proved in the previous step.
Using coarea formula, see for example Theorem 2.93 of [3], we get Thus the functionf is BV (R N ).
Step 5. Define the functionf : The aim of the following steps is to show that its total variation satisfies Denote with E t 1 the super-level sets used to generate the functionf : this can be done setting i(t) = 1 for L 1 -a.e. t ∈ R + .
It has been proved that, for L 1 -a.e. t ∈ R + , one has {f > t} = E t 1 , up to L N -negligible sets, therefore for such t's We would like to show that, for L 1 -a.e. t ∈ R + , for every i ∈ I t , i > 1, E t i is equal, up to L N -negligible sets, to one of the indecomposable componentsÊt i of {f >t }, wheret = t −t i for a certaint i . The index i int i refers to the fact that its value varies with the indecomposable component E t i , i ∈ I t , i > 1. We prove it in the following three steps.
Step 6. Let t be such that the set E t is of finite perimeter and {E t i } i∈It are its indecomposable components.
Let us prove that there exists a unique k ∈ K such that the set E t i , i ∈ I t , i > 1, is contained in H k , up to L N -negligible sets. The Therefore, from the definition off , for L N -a.e. x ∈ E t i one hasf (x) ≤ t.

STEFANO BIANCHINI AND DANIELA TONON
Again from the indecomposability of E t i and from the fact that E t i is contained in {f > t j } for all t j ≤ t, it follows that there exists a unique l ∈ I tj such that, In particular, we can order the sets Note that B t k could be empty for some t ∈ R + , k ∈ K.
Step 7. Lett > 0 such that the setÊt is of finite perimeter and {Êt i } i∈Ît are its indecomposable components, for L 1 -a.e. t ∈ R + . Let us prove that there exists a unique k ∈ K, such that the setÊt i is contained in H k , up to L N -negligible sets. Definet It follows that f |Êt For every t j in the countable dense sequence such thatt < t j <t +t there exists a uniqueī ∈ I tj such thatÊt . Due to the indecomposability ofÊt i , and, for the definition oft, the indexī must be greater than 1.
In particular, we can order the setsÊt i , i ∈Ît, asÊt (k,i) where Note thatBt k could be empty for somet ∈ R + , k ∈ K.
Step 8. In this step we prove that, for L 1 -a.e. t ∈ R + , k ∈ K fixed, Let us consider only the t's such that the set {f > t −t k } is of finite perimeter. For its indecomposability, E t (k,i) must be contained, up to L N -negligible sets, in The same argument, reversed, shows that, In an equivalent way, we can also say that, for L 1 -a.e.t ∈ R + , k ∈ K fixed, In the following we relabelÊt (k,i) and Et +t k (k,i) in order to havê Et (k,i) = Et +t k (k,i) (mod L N ).

The final steps consist in showing that
Step 10. Let {t k | k ∈ K} the countable set of values such that f −1 (t k ) > 0.
Step 6 shows that, for L 1 -a.e. t ∈ R + and for all i ∈ I t , i > 1, there exists a unique k ∈ K such thatf | E t i =t k . For every k ∈ K, let {E t (k,i) | i ∈ B t k } be the set of indecomposable components of E t such thatf | E t (k,i) =t k , i > 1.

STEFANO BIANCHINI AND DANIELA TONON
Observe that ∑ i∈B t k P (E t (k,i) ) are measurable functions of t, for all k ∈ K: indeed we have is integrable for all k ∈ K. Using this notation, we can write Step 7 it holdsÊt Step 11. Finally we have Since f has bounded variation we can iterate this process at most a countable number of times generating the family of functionsf l ∈ BV (R N ), such that everyone of them has indecomposable super-level sets, for L 1 -a.e. t ∈ R + .
Step 12. Letf :=f l be one of the functions generated in the previous steps.
If {f < t} is indecomposable for L 1 -a.e. t ∈ R + , thenf is already monotone. Otherwise we must again decomposef . If we succeed in decomposingf in a countable sum of monotone BV functions which preserves total variation we are done, since the decomposition of every function of a countable family in a countable family gives at the end a countable family as required.
In that case defineF t := {f < t} and let {F t i } i∈It be the family of indecomposable sets given by the Decomposition Theorem for sets for L 1 -a.e. t in R + . As for the super-level sets, we equip the family {F j i } i∈It j with the natural partial order relationF

As in the previous case, one has that
Recall that, for L 1 -a.e. t ∈ R + , {f < t} is decomposable and R N \ {f < t} indecomposable. Since {f < t} = ∪ i∈ItF t i and {f < t} =F t 1 up to L N -negligible sets, Lemma 2.2 implies that R N \ {f < t} is indecomposable, hence the super-level set {f > t} is indecomposable for L 1 -a.e. t ∈ R + . Thereforef is monotone as required.
Sincef has bounded variation we can iterate this process at most a countable number of times generating the family of monotone functions f i ∈ BV (R N ), which satisfies the theorem.

Remark 1. Notice that in
Step 10 we have also proved that

Remark 2.
In general the decomposition of f in BV monotone functions is not unique as the following example shows. The function f in Figure 1 can be decomposed either in the way shown in Figure  2  This property, which has been proved also for the decomposition of Lipschitz functions in Theorem 1.2, can be false in the general case. As shown in the example below, one can have monotone BV functions, whose distributional derivatives are concentrated on sets with non empty intersection.
where δ x is the Dirac measure, δ x (A) = 1 if x belongs to the set A, δ x (A) = 0 otherwise. Clearly these distributional derivatives are not mutually singular, since both have an atom in x = 3.
One can easily show that for any other monotone decomposition it is impossible to find two disjoint sets on which the distributional derivatives are concentrated.
3. Counterexamples. As we said in the Introduction, the definition of monotone function could be given even for a function which is only L 1 loc (R N ). In that case one has to require that this function must have super-level sets with finite perimeter, which is true L 1 -a.e. t ∈ R for the super-level sets of a BV function.
The Jordan decomposition states that monotonicity is a sufficient condition for a function of one variable to be of bounded variation. However, we cannot say that every monotone function f : R N → R defined as in Definition 1.4 is of bounded variation.
A counterexample is given below by a function, whose super-level sets are progressive configurations of the construction of a Koch snowflake.

Example 2.
The Koch snowflake is a curve generated iteratively from a unitary triangle T adding each time, on each edge, a smaller centered triangle with edges one third of the previous edge, see Figure 6. • the perimeter of the iterated curve is P ( • the area of the iterated curve is Denote with B the ball B = {x ∈ R 2 | ∥x∥ < R}, which contains the unitary triangle T centered in the origin: hence Clearly 0 ≤ f < 4, therefore f belongs to L 1 (B) and coarea formula can be used to obtain its variation. Let us note which are the super-level sets and their perimeter: Thus this function is monotone and computing its variation one has which implies that f does not belong to BV (B).
In the case of Lipschitz functions Definition 1.1 and Definition 1.4 are equivalent.
From its continuity, f must be greater than t or lower than t over the all ∂G. Let us fix f | ∂G < t. The compactness of {f = t} gives the existence of a δ > 0 such that f | ∂G ≤ t − δ. Thus, for all ε ∈ (0, δ),

STEFANO BIANCHINI AND DANIELA TONON
In addiction, defining L the Lipschitz constant of f , It follows that the open set {f > t − ϵ} can be decomposed into two open sets with positive distance, in particular it is decomposable.
In the case f | ∂G > t, one can similarly show that, for all ε in (0, δ), the set {f < t − ε} is decomposable. Therefore f is not monotone in the sense of Definition 1.4.
The Decomposition Theorem for real valued BV functions of R N is in some sense optimal. Considering BV functions from R 2 to R 2 one can find counterexamples to this theorem, i.e. BV functions which cannot be decomposed in sum of BV monotone functions preserving total variation.
The crucial point is that we require to our decomposition, besides being the sum of BV monotone functions, to preserve the the total variation, i.e.
For i = 1, ..., m, every f i is a BV function from R N to R so that Theorem 1.5 applies. Therefore, for every i = 1, ..., m, one has the decomposition in BV monotone Note that, if g : R N → R is a BV monotone function, the function However, this decomposition does not preserve the total variation of f and one can only say that We give now a counterexample in the case of Lipschitz function from R 2 to R 2 . In this situation we extend the Definition 1.1. We observe that if f : R N → R N Lipschitz is a monotone operator, then its level sets are closed convex. Hence the requirement to preserve the connectedness of the level sets is weaker than being a monotone operator.
Example 3. Let f : R 2 → R 2 be a Lipschitz function: in this particular case, by area formula it follows that f is monotone if and only if for L 1 -a.e. t ∈ R 2 f −1 (t) is a singleton.
Using Lipschitz continuity, it is simple to verify that if f 1 : R 2 → R 2 is a Lipschitz function such that |Df | = |Df 1 | + |D(f − f 1 )|, then either f = f 1 or there exists a set with positive length where f = f 1 is constant.
However, not all Lipschitz functions from R 2 to R 2 have this particular property. For example consider ) .
For this function the level sets {f = t} have zero length for every t ∈ R 2 . Thus any decomposition with the properties desired is impossible.  d(A, B) distance between the sets A and B