# Space of Functions with Some Generalization of Bounded Variation in the Sense of de La Vallée Poussin

Authors: René Erlín Castillo; Humberto Rafeiro; Eduard Trousselot
Journal: Journal of Function Spaces
 (2015)

---

## 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.


---

## Body

## 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 f in the interval [
a
,
b
] by(1)
V
2
f
=
V
2
f
,
a
,
b
=
sup
Π
⁡
∑
j
=
1
n
-
1
f
t
j
+
1
-
f
t
j
t
j
+
1
-
t
j
-
f
t
j
-
f
t
j
-
1
t
j
-
t
j
-
1
,where the supremum is taken over all partitions Π
=
a
=
x
0
<
⋯
<
x
n
=
b of the interval [
a
,
b
]. The function f is said to be abounded second variation function in [
a
,
b
] if V
2
(
f
,
[
a
,
b
]
)
<
∞. The linear space of all such functions is denoted by B
V
2
[
a
,
b
]. The following characterization of B
V
2
[
a
,
b
] was given by de La Vallée Poussin in [7].Proposition 1.
A functionf is of bounded second variation if and only if f can be written as the difference of two convex functions.Since any convex functionf
:
[
a
,
b
]
→
R has right and left derivatives on each interior point of the interval [
a
,
b
] and moreover f is differentiable a.e. (see [8]), then from Proposition 1 we deduce that all bounded second variation functions in [
a
,
b
] have all the properties described above. The space B
V
2
[
a
,
b
] equipped with the norm(2)
f
B
V
2
a
,
b
=
f
a
+
f
′
a
+
V
2
f
,
a
,
bis a Banach space. These functions were generalized to bounded second variation functions with respect to a strictly increasing and continuous function α (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 useα as a strictly increasing continuous function, whenever not stated otherwise.


## 2. Preliminaries

We want to recall the so-calledPopoviciu variation (introduced in 1933 by Popoviciu in [10]) for a partition Π
=
a
=
x
1
<
x
2
<
⋯
<
x
m
=
b and a function f
:
[
a
,
b
]
→
R given by(3)
Va
r
k
,
1
⁡
f
,
Π
,
a
,
b
=
∑
j
=
1
m
-
k
+
1
f
x
j
,
…
,
x
j
+
k
-
1
-
f
x
j
-
1
,
…
,
x
j
+
k
-
2
,where f
[
·
,
…
,
·
] is defined recursively in the following way:(4)
f
x
0
≔
f
x
0
,
f
x
0
,
x
1
≔
f
x
1
-
f
x
0
x
1
-
x
0
f
x
0
,
x
1
,
x
2
≔
f
x
1
,
x
2
-
f
x
0
,
x
1
x
2
-
x
0
⋮
f
x
0
,
x
1
,
…
,
x
k
≔
f
x
1
,
x
2
,
…
,
x
k
-
f
x
0
,
x
1
,
…
,
x
k
-
1
x
k
-
x
0
.In the following we will consider a block partitionΠ of the interval [
a
,
b
] which we will callpartition of type II. It will be taken in the following way:(5)
Π
=
a
=
x
1,1
<
x
1,2
⩽
x
1,3
<
x
1,4
=
x
2,1
<
x
2,2
⩽
x
2,3
<
x
2,4
=
x
3,1
<
⋯
<
x
n
-
1,4
=
x
n
,
1
<
x
n
,
2
⩽
x
n
,
3
<
x
n
,
4
=
b
,in place of the regular partition Π which has been used in the study of functions with some variation. In what follows we call a regular partition Π
=
{
a
=
x
0
<
x
1
<
⋯
<
x
n
=
b
} apartition of type I.Definition 2.
Letf be a real-valued function defined on [
a
,
b
] and let Π be a type I partition of [
a
,
b
]. Let(6)
σ
2
,
α
f
,
Π
=
∑
j
=
1
n
-
2
f
α
x
j
,
x
j
+
1
-
f
α
x
j
+
1
,
x
j
+
2
,where(7)
f
α
a
,
b
=
f
b
-
f
a
α
b
-
α
a
,
V
2
,
α
f
,
a
,
b
=
V
2
,
α
=
sup
Π
⁡
σ
2
,
α
f
,
Π
,where the supremum is taken over all partitions of type I of [
a
,
b
]. The space V
2
,
α
(
f
) is called (
2
,
α
)
-variation in the sense of de La Vallée Poussin of type I. If V
2
,
α
(
f
)
<
∞ the function f is said to be of (
2
,
α
)-variation in the sense of de La Vallée Poussin of type I. The set of all such functions is denoted by B
V
2
,
α
[
a
,
b
].Definition 3.
Letf be a real-valued function defined on [
a
,
b
] and let Π be a type II partition of [
a
,
b
]. Let(8)
σ
2
,
α
f
,
Π
=
∑
j
=
1
n
f
α
x
j
,
1
,
x
j
,
2
-
f
α
x
j
,
3
,
x
j
,
4
,
V
2
,
α
f
,
a
,
b
=
V
2
,
α
f
=
sup
Π
⁡
σ
2
,
α
f
,
Π
,where the supremum is taken on all type II partition of [
a
,
b
]. The number V
2
,
α
(
f
,
[
a
,
b
]
) is called the (
2
,
α
)-variation in the sense of de La Vallée Poussin of type II in [
a
,
b
]. If V
2
,
α
(
f
,
[
a
,
b
]
)
<
∞, f is said to be a function of (
2
,
α
)-variation in the sense of de La Vallée Poussin of type II. The set of all such functions is denoted by B
V
2
,
α
[
a
,
b
].Definition 4.
A functionf
:
[
a
,
b
]
→
R is said to be an α-Lipschitz function if there exists a constant M
>
0 such that(9)
f
x
-
f
y
≤
M
α
x
-
α
yfor all x
,
y
∈
[
a
,
b
]. We define the space α
-
L
i
p
[
a
,
b
] as the space of all α-Lipschitz functions. This space is normable, via the norm(10)
f
α
-
L
i
p
≔
f
a
+
sup
x
≠
y
⁡
f
x
-
f
y
α
x
-
α
y
.We now introduce some concepts that will be of fundamental use (cf. [11]).Definition 5.
One said that a functionf is α-convex in [
a
,
b
] if, for a
⩽
x
⩽
z
⩽
y
⩽
b, one has(11)
f
z
⩽
α
z
-
α
x
α
y
-
α
x
f
y
+
α
y
-
α
z
α
y
-
α
x
f
x
.Definition 6.
A functionf
:
[
a
,
b
]
→
R is said to beabsolutely continuous with respect to
α if, for every ε
>
0, there exists some δ
>
0 such that if {
(
a
j
,
b
j
)
}
j
=
1
n are disjoint open subintervals of [
a
,
b
], then (12)
∑
j
=
1
n
α
b
j
-
α
a
j
<
δ
implies
∑
j
=
1
n
f
b
j
-
f
a
j
<
ɛ
.
All functions inα-A
C
[
a
,
b
] are bounded and form an algebra of functions under pointwise defined standard operations.Definition 7.
Supposef and α are real-valued functions defined on the same open interval I and let x
0
∈
I. One says that f
is
α
-differentiable at
x
0 if the following limit exists: (13)
lim
x
→
x
0
⁡
f
x
-
f
x
0
α
x
-
α
x
0
.If the limit exists we denote its value by f
α
′
(
x
0
), which we call the α
-derivative of
f
at
x
0.Byμ
α we denote the Lebesgue-Stieltjes measure induced by α. 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 α-A
C
[
a
,
b
] concept with the concept of α-derivative.Theorem 8.
Letf
∈
α-A
C
[
a
,
b
]; then f
α
′ exists and is finite μ
α a.e. on [
a
,
b
]. Moreover f
α
′ is integrable in the Lebesgue-Stieltjes sense and(14)
f
x
=
f
a
+
L
S
∫
a
x
f
α
′
t
d
α
t
,
x
∈
a
,
b
.Theorem 9.
Letp be an increasing function and α a continuous function in [
a
,
b
]. Then one has that(15)
x
⟼
S
x
=
L
S
∫
a
x
p
t
d
α
tis an α-convex function on [
a
,
b
].Lemma 10.
Letv be an α-convex function and a
⩽
λ
⩽
ξ
⩽
μ
⩽
b; then(16)
v
ξ
-
v
λ
α
ξ
-
α
λ
⩽
v
μ
-
v
λ
α
μ
-
α
λ
⩽
v
μ
-
v
ξ
α
μ
-
α
ξ
;that is, v
α
[
λ
,
ξ
]
⩽
v
α
[
λ
,
μ
]
⩽
v
α
[
ξ
,
μ
] with v
α
[
p
,
q
]
=
(
v
(
q
)
-
v
(
p
)
)
/
(
α
q
-
α
p
).


## 3. Equivalent Definitions

In this section we show thatB
V
(
2
,
α
)
[
a
,
b
]
=
B
V
(
2
,
α
)
[
a
,
b
]. In order to do that, we gather several previous results.Lemma 11.
Letf be defined on [
a
,
b
] and let Π
1
=
a
=
x
1
<
x
2
<
x
3
=
b be a partition of [
a
,
b
]. Then there exists λ and μ in [
0,1
] with λ
+
μ
=
1 such that(17)
f
α
x
1
,
x
3
=
λ
f
α
x
1
,
x
2
+
μ
f
α
x
2
,
x
3
.Proof.
Observe that(18)
f
α
x
1
,
x
3
=
α
x
3
-
α
x
2
α
x
3
-
α
x
1
f
x
3
-
f
x
2
α
x
3
-
α
x
2
+
α
x
2
-
α
x
1
α
x
3
-
α
x
1
f
x
2
-
f
x
1
α
x
2
-
α
x
1
=
λ
f
α
x
1
,
x
3
+
μ
f
α
x
2
,
x
3
,where λ and μ are straightforward from the equation.The following result shows that the numberσ
(
2
,
α
)
(
f
,
Π
) does not decrease if we add points in partition Π.Theorem 12.
Letf be defined on [
a
,
b
] and let Π
=
{
a
=
x
1
<
⋯
<
x
s
-
1
<
x
s
<
⋯
<
x
n
-
1
<
x
n
=
b
} be a partition of type I and y
∈
(
x
s
-
1
,
x
s
) with s
=
2
,
…
,
n. Let one consider(19)
Π
′
=
a
=
x
1
<
⋯
<
x
s
-
2
<
x
s
-
1
<
y
<
x
s
<
x
s
+
1
<
⋯
<
x
n
-
1
<
x
n
=
b
.Then(20)
σ
2
,
α
f
,
Π
⩽
σ
2
,
α
f
,
Π
′
.Proof.
Case  1 (
2
<
s
<
n
). Let us consider all the terms of σ
(
2
,
α
)
(
f
,
Π
) on which y play a role; we have to estimate the expression(21)
f
α
x
s
-
2
,
x
s
-
1
-
f
α
x
s
-
1
,
x
s
+
f
α
x
s
-
1
,
x
s
-
f
α
x
s
,
x
s
+
1
.Applying Lemma 11 in order to introduce the point y in (
x
s
-
1
,
x
s
), we obtain that there exists positive λ and μ with λ
+
μ
=
1 such that(22)
f
α
x
s
-
1
,
x
s
=
λ
f
α
x
s
-
1
,
y
+
μ
f
α
y
,
x
s
;replacing in (21) and using the fact that λ
+
μ
=
1 we have(23)
f
α
x
s
-
2
,
x
s
-
1
-
1
-
μ
f
α
x
s
-
1
,
y
-
μ
f
α
y
,
x
s
+
λ
f
α
x
s
-
1
,
y
+
1
-
λ
f
α
y
,
x
s
-
f
α
x
s
,
x
s
+
1
.From the triangle inequality in (23) we obtain(24)
⩽
f
α
x
s
-
2
,
x
s
-
1
-
f
α
x
s
-
1
,
y
+
f
α
x
s
-
1
,
y
-
f
α
y
,
x
s
+
f
α
y
,
x
s
-
f
α
x
s
,
x
s
+
1
.That is, the expression in (21) is less than or equal to (24). Since the remaining terms in σ
(
2
,
α
)
(
f
,
Π
) and σ
(
2
,
α
)
(
f
,
Π
′
) coincide, we get (20).
Case  2 (
s
=
2
). Let y
∈
(
x
1
,
x
2
); in this case we have a unique term of σ
(
2
,
α
)
(
f
,
Π
), where the interval (
x
1
,
x
2
) plays a role; applying Lemma 11 we obtain f
α
[
x
1
,
x
2
]
=
λ
f
α
[
x
1
,
y
]
+
(
1
-
λ
)
f
α
[
y
,
x
2
] with 0
<
λ
<
1. Hence(25)
f
α
x
1
,
x
2
-
f
α
x
2
,
x
3
=
λ
f
α
x
1
,
y
+
1
-
λ
f
α
y
,
x
2
-
f
α
x
2
,
x
3
⩽
λ
f
α
x
1
,
y
-
f
α
y
,
x
2
+
f
α
y
,
x
2
-
f
α
x
2
,
x
3
⩽
f
α
x
1
,
y
-
f
α
y
,
x
2
+
f
α
y
,
x
2
-
f
α
x
2
,
x
3
,and since the remaining terms of σ
(
2
,
α
)
(
f
,
Π
) and σ
(
2
,
α
)
(
f
,
Π
′
) coincide, we obtain (20).
Case  3 (
s
=
n
). This case is similar to Case 2.Corollary 13.
Letf be a function defined in [
a
,
b
] and let Π
′ be a refinement of partition Π. Then (26)
σ
2
,
α
f
,
Π
⩽
σ
2
,
α
f
,
Π
′
.Proof.
The proof follows from Theorem12 by induction.Theorem 14.
Letf be defined on [
a
,
b
]. Then(27)
V
2
,
α
f
⩽
V
2
,
α
f
⩽
7
V
2
,
α
f
.Therefore(28)
B
V
2
,
α
a
,
b
=
B
V
2
,
α
a
,
b
.Proof.
Consider the following.
Case  1. Let f
∈
B
V
(
2
,
α
)
[
a
,
b
] and Π be a type II partition of [
a
,
b
]. First we assume that x
j
,
2
<
x
j
,
3 for all j. On each σ
(
2
,
α
)
(
f
,
Π
) we may apply the triangle inequality for j
=
1
,
…
,
n to obtain (29)
f
α
x
j
,
1
,
x
j
,
2
-
f
α
x
j
,
3
,
x
j
,
4
⩽
f
α
x
j
,
1
,
x
j
,
2
-
f
α
x
j
,
2
,
x
j
,
3
+
f
α
x
j
,
2
,
x
j
,
3
-
f
α
x
j
,
3
,
x
j
,
4
.Then(30)
σ
2
,
α
f
,
Π
⩽
∑
j
=
1
n
f
α
x
j
,
1
,
x
j
,
2
-
f
α
x
j
,
2
,
x
j
,
3
+
∑
j
=
1
n
f
α
x
j
,
2
,
x
j
,
3
-
f
α
x
j
,
3
,
x
j
,
4
.Adding all of those terms we have(31)
σ
2
,
α
f
,
Π
⩽
∑
j
=
1
n
-
1
f
α
x
j
,
1
,
x
j
,
2
-
f
α
x
j
,
2
,
x
j
,
3
+
f
α
x
n
,
1
,
x
n
,
2
-
f
α
x
n
,
2
,
x
n
,
3
+
∑
j
=
1
n
-
1
f
α
x
j
,
2
,
x
j
,
3
-
f
α
x
j
,
3
,
x
j
,
4
+
f
α
x
n
,
2
,
x
n
,
3
-
f
α
x
n
,
3
,
x
n
,
4
.This last sum is bounded by σ
(
2
,
α
)
(
f
,
Π
) if we consider Π as a type I partition; thus(32)
σ
2
,
α
f
,
Π
⩽
σ
2
,
α
f
,
Π
⩽
V
2
,
α
f
<
∞
.In the case x
j
,
2
=
x
j
,
3 for some j, we do not need to apply the triangle inequality, since |
f
α
[
x
j
,
1
,
x
j
,
2
]
-
f
α
[
x
j
,
3
,
x
j
,
4
]
|
=
|
f
α
[
x
j
,
1
,
x
j
,
2
]
-
f
α
[
x
j
,
2
,
x
j
,
4
]
|. σ
(
2
,
α
)
(
f
,
Π
) is bounded by(33)
σ
2
,
α
f
,
Π
⩽
Σ
′
+
Σ
,where Σ
′ is the summation with the terms x
j
,
2
=
x
j
,
3 and Σ is the summation where x
j
,
2
<
x
j
,
3.
Since (32) holds for all type II partitions, we conclude that V
(
2
,
α
)
(
f
)
⩽
V
(
2
,
α
)
(
f
)
<
∞, implying that f
∈
B
V
(
2
,
α
)
[
a
,
b
].
Case  2. Conversely, let f
∈
B
V
(
2
,
α
)
[
a
,
b
] and Π be a type I partition of [
a
,
b
]; that is, Π
=
a
=
x
1
<
x
2
<
⋯
<
x
n
=
b. Eventually adding one or two points we can suppose that n
=
3
p, since the addition of points in the partition does not decrease the sum σ
(
2
,
α
) (Theorem 12). Hence(34)
σ
2
,
α
f
,
Π
=
∑
j
=
1
3
p
-
2
f
α
x
j
,
x
j
+
1
-
f
α
x
j
+
1
,
x
j
+
2
=
∑
k
=
0
p
-
2
∑
j
=
3
k
+
1
3
k
+
3
f
α
x
j
,
x
j
+
1
-
f
α
x
j
+
1
,
x
j
+
2
+
f
α
x
3
p
-
2
,
x
3
p
-
1
-
f
α
x
3
p
-
1
,
x
3
p
,where we group the terms in threes.
On each interval(
x
j
,
x
j
+
1
) we introduce two points: x
j
<
x
j
,
1
<
x
j
,
2
<
x
j
+
1 for j
=
3
,
…
,
n
-
1. On each term from the sum we apply the triangle inequality (for the last term we give a different treatment) to obtain(35)
f
α
x
j
,
x
j
+
1
-
f
α
x
j
+
1
,
x
j
+
2
⩽
f
α
x
j
,
x
j
+
1
-
f
α
x
j
+
2,1
,
x
j
+
2,2
+
f
α
x
j
+
1
,
x
j
+
2
-
f
α
x
j
+
2,1
,
x
j
+
2,2for j
=
1
,
…
,
n
-
3. We introduce these terms in σ
(
2
,
α
)
(
f
,
Π
) to get(36)
σ
2
,
α
f
,
Π
=
∑
k
=
1
3
∑
j
=
0
p
-
2
f
α
x
3
j
+
k
,
x
3
j
+
k
+
1
-
f
α
x
3
j
+
k
+
2,1
,
x
3
j
+
k
+
2,2
+
∑
k
=
1
3
∑
j
=
0
p
-
2
f
α
x
3
j
+
k
+
1
,
x
3
j
+
k
+
2
-
f
α
x
3
j
+
k
+
2,1
,
x
3
j
+
k
+
2,2
+
f
α
x
n
-
2
,
x
n
-
1
-
f
α
x
n
-
1
,
x
n
.A partition of type II is obtained in each summation in parenthesis and therefore it is bounded by V
(
2
,
α
)
(
f
); accordingly(37)
σ
2
,
α
f
,
Π
⩽
6
V
2
,
α
f
+
f
α
x
n
-
2
,
x
n
-
1
-
f
α
x
n
-
1
,
x
n
.The term |
f
α
[
x
n
-
2
,
x
n
-
1
]
-
f
α
[
x
n
-
1
,
x
n
]
| is bounded by V
(
2
,
α
)
(
f
). Finally σ
(
2
,
α
)
(
f
)
⩽
7
V
(
2
,
α
)
(
f
)
<
∞. Since this holds for all partitions of type I we conclude that V
(
2
,
α
)
(
f
)
⩽
7
V
(
2
,
α
)
(
f
)
<
∞ and this f
∈
B
V
(
2
,
α
)
[
a
,
b
].Remark 15.
This result guarantees that theB
V
(
2
,
α
)
[
a
,
b
] functions satisfy all the properties verified by the B
V
(
2
,
α
)
[
a
,
b
] functions and reciprocally. In the coming demonstrations we choose the partition that will be more appropriate.


## 4.B
V
(
2
,
α
)
=
B
V
(
2
,
α
) as Linear Spaces

First of all we will prove some embedding between sets, to conclude thatB
V
(
2
,
α
)
[
a
,
b
]
⊂
B
[
a
,
b
], where B
[
a
,
b
] 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 formf
α
[
p
,
q
] are bounded.Lemma 16.
Iff
∈
B
V
(
2
,
α
)
[
a
,
b
]
=
B
V
(
2
,
α
)
[
a
,
b
], then there exists M
>
0 such that, for all x
1
,
x
2 with a
⩽
x
1
<
x
2
⩽
b, it holds that(38)
f
x
2
-
f
x
1
α
x
2
-
α
x
1
=
f
α
x
1
,
x
2
⩽
M
.Proof.
Fixa
1 and a
2 such that a
<
a
1
<
a
2
<
b and(39)
f
a
2
-
f
a
1
α
a
2
-
α
a
1
=
f
α
a
1
,
a
2
=
A
.
Case  1 (
x
2
<
b
). Let x
3 and x
4 be points such that m
a
x
x
2
,
a
2
<
x
3
<
x
4
⩽
b; then a
1
<
a
2
<
x
3
<
x
4 and x
1
<
x
2
<
x
3
<
x
4 are points in partitions of type II of [
a
,
b
]. Therefore |
f
α
[
a
1
,
a
2
]
-
f
α
[
x
3
,
x
4
]
|
⩽
V
(
2
,
α
)
(
f
) and |
f
α
[
x
1
,
x
2
]
-
f
α
[
x
3
,
x
4
]
|
⩽
V
(
2
,
α
)
(
f
). Using the triangle inequality we have(40)
f
α
x
1
,
x
2
⩽
f
α
x
1
,
x
2
-
f
α
x
3
,
x
4
+
f
α
x
3
,
x
4
-
f
α
a
1
,
a
2
+
f
α
a
1
,
a
2
⩽
2
V
2
,
α
f
+
A
=
M
1
.
Case  2 (
x
2
=
b
and
a
<
x
1
). Let x
3
,
x
4 be points such that a
⩽
x
3
<
x
4
<
m
i
n
a
1
,
x
1; then x
3
<
x
4
<
a
1
<
a
2 and x
3
<
x
4
<
x
1
<
x
2 are partitions of type II of [
a
,
b
]. Therefore |
f
α
[
x
3
,
x
4
]
-
f
α
[
a
1
,
a
2
]
|
⩽
V
(
2
,
α
)
(
f
) and |
f
α
[
x
3
,
x
4
]
-
f
α
[
x
1
,
x
2
]
|
⩽
V
(
2
,
α
)
(
f
). Similarly as in Case 1 we obtain |
f
α
[
x
1
,
x
2
]
|
⩽
M
1.
Case  3 (
x
1
=
a
and
x
2
=
b
). Consider(41)
f
α
x
1
,
x
2
=
f
α
a
,
b
=
f
b
-
f
a
α
b
-
α
a
=
M
2
.Taking M
=
m
a
x
M
1
,
M
2 we deduce that for all x
1
,
x
2 such that a
⩽
x
1
<
x
2
⩽
b we have |
f
α
[
x
1
,
x
2
]
|
⩽
M.This allow us to prove that all(
2
,
α
)-variation function in [
a
,
b
] is α-Lipschitz in [
a
,
b
].Theorem 17.
Iff is a (
2
,
α
)-variation function in [
a
,
b
], then f is an α-Lipschitz function.Proof.
Iff
∈
B
V
(
2
,
α
)
[
a
,
b
]
=
B
V
(
2
,
α
)
[
a
,
b
] from Lemma 11 there exists M
>
0 such that for all a
⩽
x
<
y
⩽
b we have that |
f
α
[
x
,
y
]
|
⩽
M holds. Then by the definition of f
α, we get |
f
(
y
)
-
f
(
x
)
|
⩽
M
|
α
(
y
)
-
α
(
x
)
|; that is, f is an α-Lipschitz function in [
a
,
b
].Corollary 18.
Iff
∈
B
V
(
2
,
α
)
[
a
,
b
]
=
B
V
(
2
,
α
)
[
a
,
b
], then one has that f is α-absolutely continuous in [
a
,
b
].Corollary 19.
Iff
∈
B
V
(
2
,
α
)
[
a
,
b
]
=
B
V
(
2
,
α
)
[
a
,
b
], then f
∈
B
V
[
a
,
b
].Corollary 20.
Iff
∈
B
V
(
2
,
α
)
[
a
,
b
]
=
B
V
(
2
,
α
)
[
a
,
b
], then f is (uniformly) continuous in [
a
,
b
] and thus bounded in [
a
,
b
].The above results allow us to obtain the following chain of embeddings:(42)
B
V
2
,
α
a
,
b
=
B
V
2
,
α
a
,
b
⊂
α
-
Lip
⁡
a
,
b
⊂
α
-
A
C
a
,
b
⊂
C
a
,
b
⊂
B
a
,
b
.Theorem 21.
The setB
V
(
2
,
α
)
[
a
,
b
] equipped with the sum of functions and the scalar product is a linear space.Proof.
Letf
,
g
∈
B
V
(
2
,
α
)
[
a
,
b
] and Π be a type I partition of [
a
,
b
]. Then by the triangle inequality, we obtain σ
(
2
,
α
)
(
f
+
g
,
Π
)
⩽
σ
(
2
,
α
)
(
f
,
Π
)
+
σ
(
2
,
α
)
(
g
,
Π
) and thus V
(
2
,
α
)
(
f
+
g
)
⩽
V
(
2
,
α
)
(
f
)
+
V
(
2
,
α
)
(
g
) and from this we conclude that f
+
g
∈
B
V
(
2
,
α
)
[
a
,
b
]. Moreover σ
(
2
,
α
)
(
λ
f
,
Π
)
=
|
λ
|
σ
(
2
,
α
)
(
f
), λ
∈
R, and V
(
2
,
α
)
(
λ
f
)
=
|
λ
|
V
(
2
,
α
)
(
f
) from which we obtain that if f
∈
B
V
(
2
,
α
)
[
a
,
b
], then λ
f
∈
B
V
(
2
,
α
)
[
a
,
b
], λ
∈
R.Corollary 22.
One has that(1)
(
B
V
(
2
,
α
)
[
a
,
b
]
,
+
,
·
) is a real linear space;
(2)
(
B
V
(
2
,
α
)
[
a
,
b
]
,
+
,
·
) is a real linear space.


## 5. Jordan Type Decomposition

In this section we want to obtain a decomposition theorem in the spirit of Proposition1. Before doing that, we want to give the following lemma, which is of direct verification.Lemma 23.
Letf
∈
B
V
(
2
,
α
)
[
a
,
b
]. If a
<
c
<
b
(43)
V
2
,
α
f
,
a
,
c
+
V
2
,
α
f
,
c
,
b
⩽
V
2
,
α
f
,
a
,
b
.We have the following decomposition theorem for functions inB
V
(
2
,
α
).Theorem 24.
One has thatf
∈
B
V
(
2
,
α
)
[
a
,
b
] if and only if f
=
f
1
-
f
2, where f
1 and f
2 are α-convex functions.Proof.
Observe that the setE
=
x
∈
[
a
,
b
]
:
f
α
′
(
x
)
d
o
e
s
n
o
t
e
x
i
s
t has measure zero. For x
∈
[
a
,
b
]
∖
E let us define(44)
p
x
=
1
2
V
2
,
α
f
,
a
,
x
+
f
α
′
x
,
q
x
=
1
2
V
2
,
α
f
,
a
,
x
-
f
α
′
x
;then f
α
′
(
x
)
=
p
(
x
)
-
q
(
x
). Using Lemma 23 we observe that for y
<
x we have(45)
2
p
x
-
p
y
=
V
2
,
α
f
,
a
,
x
-
V
2
,
α
f
,
a
,
y
+
f
α
′
x
-
f
α
′
y
≥
V
2
,
α
f
,
y
,
x
-
f
α
′
x
-
f
α
′
y
≥
0and a similar calculation can be made regarding the function q and therefore p and q are increasing in [
a
,
b
]
∖
E. Since f
∈
α-A
C
[
a
,
b
], then f
α
′ is integrable in the sense of Lebesgue-Stieltjes and f
(
x
)
=
f
(
a
)
+
(
L
S
)
∫
a
x
‍
f
α
′
(
t
)
d
α
(
t
), for x
∈
[
a
,
b
]. In E we define p and q such that p and q are increasing and bounded in [
a
,
b
] and therefore L
S-integrable in [
a
,
b
] with respect to α; from Theorem 8 we obtain(46)
f
x
=
L
S
∫
a
x
f
α
′
t
d
α
t
+
f
a
=
L
S
∫
a
x
p
t
d
α
t
-
L
S
∫
a
x
q
t
d
α
t
.Since each integral is α-convex (see Theorem 9), the result follows.
Conversely, suppose thatf
=
f
1
-
f
2 where f
1 and f
2 are α-convex functions. We are going to prove that f
∈
B
V
(
2
,
α
)
[
a
,
b
]. To do so, it is enough to verify that all α-convex functions have second bounded α-variation.
Letv
:
[
a
,
b
]
→
R be α-convex; then by Lemma 10 we have v
α
+
′
(
a
)
⩽
v
a
[
x
,
y
]
⩽
v
α
-
′
(
b
). Let Π
=
a
=
x
1
<
x
2
<
⋯
<
x
n
=
b be a partition of type I of the interval [
a
,
b
]. Since v
α
[
·
,
·
] is increasing we obtain(47)
σ
2
,
α
v
,
Π
=
∑
j
=
1
n
-
2
v
α
x
j
,
x
j
+
1
-
v
α
x
j
+
1
,
x
j
+
2
=
∑
j
=
1
n
-
2
v
α
x
j
,
x
j
+
1
-
v
α
x
j
+
1
,
x
j
+
2
.This amount is part of a telescopic series; one more time by Lemma 10 we have(48)
σ
2
,
α
v
,
Π
=
v
α
b
,
x
n
-
1
-
v
α
a
,
x
2
⩽
v
α
-
′
b
-
v
α
+
′
a
<
∞
.Hence V
(
2
,
α
)
(
v
)
<
∞ and v
∈
B
V
(
2
,
α
)
[
a
,
b
]. Since f
1 and f
2 are α-convex, therefore f
=
f
1
-
f
2
∈
B
V
(
2
,
α
)
[
a
,
b
].From Theorem24 we deduce the following.Corollary 25.
Iff
∈
B
V
(
2
,
α
)
[
a
,
b
], then f can be written as(49)
f
x
=
L
S
∫
a
x
g
t
d
α
t
+
f
a
,where g is a bounded variation function in [
a
,
b
].As another consequence of Theorem24 and the α-convex property, we derive the existence of f
α
+
′
(
x
0
) and f
α
-
′
(
x
0
) in each point x
0
∈
(
a
,
b
) and f
α
+
′
(
a
), f
α
-
′
(
b
).


## 6.B
V
(
2
,
α
)
[
a
,
b
]
=
B
V
(
2
,
α
)
[
a
,
b
] as Banach Spaces

Corollary22 shows us that (
R
,
B
V
(
2
,
α
)
[
a
,
b
]
,
+
,
·
) is a linear space. Now we introduce a norm in this space.Definition 26.
Letf
∈
B
V
(
2
,
α
)
[
a
,
b
]. One defines ·
B
V
(
2
,
α
)
[
a
,
b
] as(50)
f
B
V
2
,
α
a
,
b
=
f
a
+
f
α
′
a
+
V
2
,
α
f
.When there is no danger of confusion, one will denote ∥
·
∥
B
V
(
2
,
α
)
[
a
,
b
]
=
:
·
(
2
,
α
).Lemma 27.
Iff
∈
B
V
(
2
,
α
)
[
a
,
b
] and V
(
2
,
α
)
(
f
)
=
0, then f
(
x
)
=
λ
α
(
x
)
+
μ, where λ
,
μ
∈
R.Proof.
Note thatV
(
2
,
α
)
(
f
)
=
0 if and only if σ
(
2
,
α
)
(
f
)
=
0 for all partitions Π of [
a
,
b
]. Consider the particular partition Π
0
=
a
<
x
<
b. Then from the fact that σ
(
2
,
α
)
(
f
,
Π
0
)
=
0 we deduce(51)
f
x
=
f
b
-
f
a
α
b
-
α
a
α
x
+
f
a
α
b
-
f
b
α
a
α
b
-
α
a
.Taking(52)
λ
=
f
b
-
f
a
α
b
-
α
a
,
μ
=
f
a
α
b
-
f
b
α
a
α
b
-
α
athe result follows.Theorem 28.
The functional·
B
V
(
2
,
α
)
[
a
,
b
]
=
·
(
2
,
α
) is a norm in the space B
V
(
2
,
α
)
[
a
,
b
].Proof.
Letf
,
g
∈
B
V
(
2
,
α
)
[
a
,
b
]. We have(53)
f
+
g
2
,
α
=
f
+
g
a
+
f
+
g
α
′
a
+
V
2
,
α
f
+
g
⩽
f
a
+
g
a
+
f
α
′
a
+
g
α
′
a
+
V
2
,
α
f
+
V
2
,
α
g
=
f
2
,
α
+
g
2
,
α
.Let λ
∈
R, f
∈
B
V
(
2
,
α
)
[
a
,
b
]; we get λ
f
(
2
,
α
)
=
λ
f
(
2
,
α
). Finally f
(
2
,
α
)
=
0 if and only if |
f
(
a
)
|
=
0 and |
f
α
′
(
a
)
|
=
0 and V
(
2
,
α
)
(
f
)
=
0. Since V
(
2
,
α
)
(
f
)
=
0, from Lemma 27 we have f
(
x
)
=
λ
α
(
x
)
+
μ, λ
,
μ
∈
R. Moreover f
α
′
(
x
)
=
λ, but f
α
′
(
a
)
=
0; then λ
=
0. Thus f
(
x
)
=
μ, but f
α
′
(
a
)
=
0; then μ
=
0. Hence f
=
0.
In this way we have shown that(
R
,
B
V
2
,
α
a
,
b
,
+
,
·
,
∥
·
∥
(
2
,
α
)
) is a normed space.Now, we are going to prove that this space is complete. In Theorem17 we showed that B
V
(
2
,
α
)
[
a
,
b
]
⊂
α
-
L
i
p
[
a
,
b
]. In what follows we prove that this inclusion is a normed space embedding to conclude that B
V
(
2
,
α
)
[
a
,
b
] is complete.Lemma 29.
Letf
∈
B
V
(
2
,
α
)
[
a
,
b
]; then(54)
f
α
-
L
i
p
a
,
b
⩽
f
B
V
2
,
α
a
,
b
.Proof.
Let us considera
<
a
+
h
<
s
<
t
<
b; then from Theorem 14 we obtain(55)
f
t
-
f
s
α
t
-
α
s
-
f
a
+
h
-
f
a
α
a
+
h
-
α
a
⩽
V
2
,
α
f
⩽
V
2
,
α
f
.Letting h
→
0 we have(56)
f
t
-
f
s
α
t
-
α
s
-
f
α
′
a
⩽
V
2
,
α
f
;thus(57)
f
t
-
f
s
α
t
-
α
s
⩽
f
α
′
a
+
V
2
,
α
f
,
(58)
f
a
+
f
t
-
f
s
α
t
-
α
s
⩽
f
a
+
f
α
′
a
+
V
2
,
α
f
;from this we get(59)
f
a
+
sup
s
≠
t
⁡
f
t
-
f
s
α
t
-
α
s
⩽
f
2
,
α
.That is, f
α
-
L
i
p
a
,
b
⩽
f
2
,
α.Theorem 30.
R
,
B
V
(
2
,
α
)
[
a
,
b
]
,
+
,
·
,
·
(
2
,
α
) is a Banach space.Proof.
Letf
p
p
∈
N be a Cauchy sequence in B
V
(
2
,
α
)
[
a
,
b
]. Given ε
>
0 there exists N
ɛ
∈
N such that if p
,
q
>
N
ɛ then f
p
-
f
q
(
2
,
α
)
<
ε. By the definition of the norm, we obtain |
f
p
(
a
)
-
f
q
(
a
)
|
<
ɛ, |
(
f
p
)
α
′
-
(
f
q
)
α
′
|
<
ε and V
(
2
,
α
)
(
f
p
-
f
q
)
<
ɛ. Invoking Lemma 29 it follows that f
p
-
f
q
α
-
L
i
p
[
a
,
b
]
⩽
f
p
-
f
q
(
2
,
α
)
<
ɛ and thus f
p
p
∈
N is a Cauchy sequence in α-L
i
p
[
a
,
b
]. In other words(60)
f
p
a
-
f
q
a
+
sup
x
≠
y
⁡
f
p
-
f
q
x
-
f
p
-
f
q
y
α
x
-
α
y
<
ɛ
,which implies that |
f
p
(
a
)
-
f
q
(
a
)
|
<
ɛ and (
(
f
p
-
f
q
)
(
x
)
-
(
f
p
-
f
q
)
(
y
)
)
/
(
α
x
-
α
y
)
<
ɛ for x
∈
[
a
,
b
]. By the triangle inequality we have |
f
p
(
x
)
-
f
q
(
x
)
|
<
ɛ
(
1
+
α
(
b
)
-
α
(
a
)
), for x
∈
[
a
,
b
]. This tells us that f
p
(
x
)
p
∈
N is a Cauchy sequence in R. Since R is complete we can define a function f in the following way: f
:
[
a
,
b
]
→
R given by f
(
x
)
=
l
i
m
p
→
∞
f
p
(
x
), for x
∈
[
a
,
b
]. Next we need to prove that (I) f
∈
B
V
(
2
,
α
)
[
a
,
b
] and (II) f
p
p
∈
N converges to f in ·
(
2
,
α
)-norm.
(I) Indeed, letΠ
=
a
=
x
1
<
x
2
<
⋯
<
x
n
=
b be a partition of [
a
,
b
]. Then(61)
∑
j
=
1
n
-
2
f
p
x
j
+
2
-
f
p
x
j
+
1
α
x
j
+
2
-
α
x
j
+
1
-
f
p
x
j
+
1
-
f
p
x
j
α
x
j
+
1
-
α
x
j
⩽
f
p
2
,
α
<
Mfor all p
∈
N and partition Π (the sequence f
p
(
2
,
α
)
p
∈
N is bounded since f
p
p
∈
N is a Cauchy sequence).
Now,(62)
lim
p
→
∞
⁡
∑
j
=
1
n
-
2
f
p
x
j
+
2
-
f
p
x
j
+
1
α
x
j
+
2
-
α
x
j
+
2
-
f
p
x
j
+
1
-
f
p
x
j
α
x
j
+
1
-
α
x
j
=
∑
j
=
1
n
-
2
f
x
j
+
2
-
f
x
j
+
1
α
x
j
+
2
-
α
x
j
+
2
-
f
x
j
+
1
-
f
x
j
α
x
j
+
1
-
α
x
j
⩽
Mfor all partitions Π; therefore V
(
2
,
α
)
(
f
)
⩽
M; thus f
∈
B
V
(
2
,
α
)
[
a
,
b
]. Also note that f
α
′
(
a
) exists.
(II) LetΠ
=
a
=
x
1
<
x
2
<
⋯
<
x
n
=
b be a partition of [
a
,
b
]. For p
,
q
>
N
ɛ we obtain(63)
∑
j
=
1
n
-
2
f
p
-
f
q
x
j
+
2
-
f
p
-
f
q
x
j
+
1
α
x
j
+
2
-
α
x
j
+
1
-
f
p
-
f
q
x
j
+
1
-
f
p
-
f
q
x
j
α
x
j
+
1
-
α
x
j
⩽
V
2
,
α
f
p
-
f
q
<
ɛ
.Taking limit as q goes to infinity, we have(64)
∑
j
=
1
n
-
2
f
p
-
f
x
j
+
2
-
f
p
-
f
x
j
+
1
α
x
j
+
2
-
α
x
j
+
2
-
f
p
-
f
x
j
+
1
-
f
p
-
f
x
j
α
x
j
+
1
-
α
x
j
⩽
ɛ
,which holds for any partition of [
a
,
b
]; then V
(
2
,
α
)
(
f
p
-
f
)
⩽
ɛ if p
>
N
ɛ. Since |
f
p
(
a
)
-
f
q
(
a
)
|
<
ɛ we have |
f
p
(
a
)
-
f
(
a
)
|
⩽
ɛ, for p
>
N
ɛ, if q
→
∞.
Next, fora
<
a
+
h
<
s
<
t
<
b, we have(65)
f
p
-
f
t
-
f
p
-
f
s
α
t
-
α
s
-
f
p
-
f
a
+
h
-
f
p
-
f
a
α
a
+
h
-
α
a
⩽
V
2
,
α
f
p
-
f
⩽
V
2
,
α
f
p
-
f
⩽
ɛif p
>
N
ɛ. Taking limit as h goes to zero(66)
f
p
-
f
t
-
f
p
-
f
s
α
t
-
α
s
-
f
p
α
′
a
+
f
α
′
a
⩽
ɛif p
>
N
ɛ. Since f
p
,
f
∈
B
V
2
,
α
a
,
b
⊂
α
-
L
i
p
[
a
,
b
]
(67)
f
p
α
′
a
-
f
α
′
a
⩽
ɛ
+
f
p
-
f
t
-
f
p
-
f
s
α
t
-
α
s
⩽
ɛ
+
f
p
-
f
α
-
L
i
p
a
,
b
⩽
2
ɛif p
>
N
ɛ, by virtue of f
p
-
f
α
-
L
i
p
[
a
,
b
]
=
l
i
m
q
→
∞
f
p
-
f
q
α
-
L
i
p
[
a
,
b
]
<
ɛ. Finally, for p
>
N
ɛ we have |
f
p
(
a
)
-
f
(
a
)
|
+
|
(
f
p
)
α
′
(
a
)
-
f
α
′
(
a
)
|
+
V
(
2
,
α
)
(
f
p
-
f
)
<
4
ɛ, that is, f
p
-
f
(
2
,
α
)
<
4
ɛ.


## 7.B
V
(
2
,
α
) as a Banach Algebra

As a result of the generalized Orlicz-Maligranda inequality proved in [15], we can show that the B
V
(
2
,
α
) is a Banach algebra, as was done in [16]; namely, consider the following.Theorem 31.
The spaceB
V
(
2
,
α
)
(
[
a
,
b
]
) equipped with the norm(68)
f
B
V
2
,
α
a
,
b
1
=
f
∞
+
2
α
b
-
α
a
f
B
V
2
,
α
a
,
b
,
f
∈
B
V
2
,
α
a
,
bis a Banach algebra and the norms ·
B
V
(
2
,
α
)
[
a
,
b
] and ·
B
V
(
2
,
α
)
1 are equivalent.



---

*Source: 605380-2015-06-04.xml*