Abstract
We introduce a function space with some generalization of bounded variation in the sense of de la Vallée Poussin and study some of its properties, like embeddings and decompositions, among others.
1. Introduction
Two centuries ago, around 1880, Jordan (see [1]) introduced the notion of a function of bounded variation and established the relation between these functions and monotonic ones when he was studying convergence of Fourier series. Later on the concept of bounded variation was generalized in various directions by many mathematicians, such as L. Ambrosio, R. Caccioppoli, L. Cesari, E. Conway, G. Dal Maso, E. de Giorgi, S. Hudjaev, J. Musielak, O. Oleinik, W. Orlicz, F. Riesz, J. Smoller, L. Tonelli, A. Vol’pert, and N. Wiener, among many others. It is noteworthy to mention that many of these generalizations were motivated by problems in such areas as calculus of variations, convergence of Fourier series, geometric measure theory, and mathematical physics. For many applications of functions of bounded variation in mathematical physics see the monograph [2]. For a thorough exposition regarding bounded variation spaces and related topics, see the recent monograph [3].
For recent generalization of the concept of bounded variation, see [4–6].
In 1908 de La Vallée Poussin [7] generalized the concept of bounded variation functions. He defined the second bounded variation of a function in the interval bywhere the supremum is taken over all partitions of the interval . The function is said to be a bounded second variation function in if . The linear space of all such functions is denoted by . The following characterization of was given by de La Vallée Poussin in [7].
Proposition 1. A function is of bounded second variation if and only if can be written as the difference of two convex functions.
Since any convex function has right and left derivatives on each interior point of the interval and moreover is differentiable a.e. (see [8]), then from Proposition 1 we deduce that all bounded second variation functions in have all the properties described above. The space equipped with the normis a Banach space. These functions were generalized to bounded second variation functions with respect to a strictly increasing and continuous function (see [9]). We introduce this concept considering usual partition (see [9]) and block partitions (see [3]). Then we show that the two definitions are related and study some of its properties.
From now on, we will always use as a strictly increasing continuous function, whenever not stated otherwise.
2. Preliminaries
We want to recall the so-called Popoviciu variation (introduced in 1933 by Popoviciu in [10]) for a partition and a function given bywhere is defined recursively in the following way:
In the following we will consider a block partition of the interval which we will call partition of type II. It will be taken in the following way:in place of the regular partition which has been used in the study of functions with some variation. In what follows we call a regular partition a partition of type I.
Definition 2. Let be a real-valued function defined on and let be a type I partition of . Letwherewhere the supremum is taken over all partitions of type I of . The space is called -variation in the sense of de La Vallée Poussin of type I. If the function is said to be of -variation in the sense of de La Vallée Poussin of type I. The set of all such functions is denoted by .
Definition 3. Let be a real-valued function defined on and let be a type II partition of . Letwhere the supremum is taken on all type II partition of . The number is called the -variation in the sense of de La Vallée Poussin of type II in . If , is said to be a function of -variation in the sense of de La Vallée Poussin of type II. The set of all such functions is denoted by .
Definition 4. A function is said to be an -Lipschitz function if there exists a constant such thatfor all . We define the space as the space of all -Lipschitz functions. This space is normable, via the norm
We now introduce some concepts that will be of fundamental use (cf. [11]).
Definition 5. One said that a function is -convex in if, for , one has
Definition 6. A function is said to be absolutely continuous with respect to if, for every , there exists some such that if are disjoint open subintervals of , then All functions in - are bounded and form an algebra of functions under pointwise defined standard operations.
Definition 7. Suppose and are real-valued functions defined on the same open interval and let . One says that is -differentiable at if the following limit exists: If the limit exists we denote its value by , which we call the -derivative of at .
By we denote the Lebesgue-Stieltjes measure induced by . For a standard treatment regarding Lebesgue-Stieltjes measure and integral, see, for example, [12].
The following two results are proved in [11, 13, 14]. Theorem 8 connects the - concept with the concept of -derivative.
Theorem 8. Let -; then exists and is finite a.e. on . Moreover is integrable in the Lebesgue-Stieltjes sense and
Theorem 9. Let be an increasing function and a continuous function in . Then one has thatis an -convex function on .
Lemma 10. Let be an -convex function and ; thenthat is, with .
3. Equivalent Definitions
In this section we show that . In order to do that, we gather several previous results.
Lemma 11. Let be defined on and let be a partition of . Then there exists and in with such that
Proof. Observe thatwhere and are straightforward from the equation.
The following result shows that the number does not decrease if we add points in partition .
Theorem 12. Let be defined on and let be a partition of type and with . Let one considerThen
Proof.
Case 1 (). Let us consider all the terms of on which play a role; we have to estimate the expressionApplying Lemma 11 in order to introduce the point in , we obtain that there exists positive and with such thatreplacing in (21) and using the fact that we haveFrom the triangle inequality in (23) we obtainThat is, the expression in (21) is less than or equal to (24). Since the remaining terms in and coincide, we get (20).
Case 2 (). Let ; in this case we have a unique term of , where the interval plays a role; applying Lemma 11 we obtain with . Henceand since the remaining terms of and coincide, we obtain (20).
Case 3 (). This case is similar to Case 2.
Corollary 13. Let be a function defined in and let be a refinement of partition . Then
Proof. The proof follows from Theorem 12 by induction.
Theorem 14. Let be defined on . ThenTherefore
Proof. Consider the following.
Case 1. Let and be a type II partition of . First we assume that for all . On each we may apply the triangle inequality for to obtain ThenAdding all of those terms we haveThis last sum is bounded by if we consider as a type I partition; thusIn the case for some , we do not need to apply the triangle inequality, since . is bounded bywhere is the summation with the terms and is the summation where .
Since (32) holds for all type II partitions, we conclude that , implying that .
Case 2. Conversely, let and be a type I partition of ; that is, . Eventually adding one or two points we can suppose that , since the addition of points in the partition does not decrease the sum (Theorem 12). Hencewhere we group the terms in threes.
On each interval we introduce two points: for . On each term from the sum we apply the triangle inequality (for the last term we give a different treatment) to obtainfor . We introduce these terms in to getA partition of type II is obtained in each summation in parenthesis and therefore it is bounded by ; accordinglyThe term is bounded by . Finally . Since this holds for all partitions of type I we conclude that and this .
Remark 15. This result guarantees that the functions satisfy all the properties verified by the functions and reciprocally. In the coming demonstrations we choose the partition that will be more appropriate.
4. as Linear Spaces
First of all we will prove some embedding between sets, to conclude that , where stands for the set of bounded functions, and therefore we may consider linear subspaces.
In the following lemma we will demonstrate that all terms of the form are bounded.
Lemma 16. If , then there exists such that, for all with , it holds that
Proof. Fix and such that andCase 1 (). Let and be points such that ; then and are points in partitions of type II of . Therefore and . Using the triangle inequality we haveCase 2 ( and ). Let be points such that ; then and are partitions of type II of . Therefore and . Similarly as in Case 1 we obtain .
Case 3 ( and ). ConsiderTaking we deduce that for all such that we have .
This allow us to prove that all -variation function in is -Lipschitz in .
Theorem 17. If is a -variation function in , then is an -Lipschitz function.
Proof. If from Lemma 11 there exists such that for all we have that holds. Then by the definition of , we get ; that is, is an -Lipschitz function in .
Corollary 18. If , then one has that is -absolutely continuous in .
Corollary 19. If , then .
Corollary 20. If , then is (uniformly) continuous in and thus bounded in .
The above results allow us to obtain the following chain of embeddings:
Theorem 21. The set equipped with the sum of functions and the scalar product is a linear space.
Proof. Let and be a type I partition of . Then by the triangle inequality, we obtain and thus and from this we conclude that . Moreover , , and from which we obtain that if , then , .
Corollary 22. One has that (1) is a real linear space;(2) is a real linear space.
5. Jordan Type Decomposition
In this section we want to obtain a decomposition theorem in the spirit of Proposition 1. Before doing that, we want to give the following lemma, which is of direct verification.
Lemma 23. Let . If
We have the following decomposition theorem for functions in .
Theorem 24. One has that if and only if , where and are -convex functions.
Proof. Observe that the set has measure zero. For let us definethen . Using Lemma 23 we observe that for we haveand a similar calculation can be made regarding the function and therefore and are increasing in . Since -, then is integrable in the sense of Lebesgue-Stieltjes and , for . In we define and such that and are increasing and bounded in and therefore -integrable in with respect to ; from Theorem 8 we obtainSince each integral is -convex (see Theorem 9), the result follows.
Conversely, suppose that where and are -convex functions. We are going to prove that . To do so, it is enough to verify that all -convex functions have second bounded -variation.
Let be -convex; then by Lemma 10 we have . Let be a partition of type I of the interval . Since is increasing we obtainThis amount is part of a telescopic series; one more time by Lemma 10 we haveHence and . Since and are -convex, therefore .
From Theorem 24 we deduce the following.
Corollary 25. If , then can be written aswhere is a bounded variation function in .
As another consequence of Theorem 24 and the -convex property, we derive the existence of and in each point and , .
6. as Banach Spaces
Corollary 22 shows us that is a linear space. Now we introduce a norm in this space.
Definition 26. Let . One defines asWhen there is no danger of confusion, one will denote .
Lemma 27. If and , then , where .
Proof. Note that if and only if for all partitions of . Consider the particular partition . Then from the fact that we deduceTakingthe result follows.
Theorem 28. The functional is a norm in the space .
Proof. Let . We haveLet , ; we get . Finally if and only if and and . Since , from Lemma 27 we have , . Moreover , but ; then . Thus , but ; then . Hence .
In this way we have shown that is a normed space.
Now, we are going to prove that this space is complete. In Theorem 17 we showed that . In what follows we prove that this inclusion is a normed space embedding to conclude that is complete.
Lemma 29. Let ; then
Proof. Let us consider ; then from Theorem 14 we obtainLetting we havethusfrom this we getThat is, .
Theorem 30. is a Banach space.
Proof. Let be a Cauchy sequence in . Given there exists such that if then . By the definition of the norm, we obtain , and . Invoking Lemma 29 it follows that and thus is a Cauchy sequence in -. In other wordswhich implies that and for . By the triangle inequality we have , for . This tells us that is a Cauchy sequence in . Since is complete we can define a function in the following way: given by , for . Next we need to prove that (I) and (II) converges to in -norm.
(I) Indeed, let be a partition of . Thenfor all and partition (the sequence is bounded since is a Cauchy sequence).
Now,for all partitions ; therefore ; thus . Also note that exists.
(II) Let be a partition of . For we obtainTaking limit as goes to infinity, we havewhich holds for any partition of ; then if . Since we have , for , if .
Next, for , we haveif . Taking limit as goes to zeroif . Since if , by virtue of . Finally, for we have , that is, .
7. as a Banach Algebra
As a result of the generalized Orlicz-Maligranda inequality proved in [15], we can show that the is a Banach algebra, as was done in [16]; namely, consider the following.
Theorem 31. The space equipped with the normis a Banach algebra and the norms and are equivalent.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referee for careful reading of the paper. The second named author was partially supported by Pontificia Universidad Javeriana under the research project Study of Boundedness Variation in the Sense of de La Vallée Poussin (ID PRY: 006780, ID PPTA: 006654).