# The Spaces of Functions of Two Variables of Boundedκ

Authors: A. Azócar; O. Mejía; N. Merentes; S. Rivas
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

The purpose of this paper is twofold. Firstly, we introduce the concept of boundedκ
Φ-variation in the sense of Schramm-Korenblum for real functions with domain in a rectangle of R
2. Secondly, we study some properties of these functions and we prove that the space generated by these functions has a structure of Banach algebra.


---

## Body

## 1. Introduction

During the last decades, several developments, extensions, and generalizations have been considered for the classical concept of the total variation of a function. It is well known that such extensions and generalization play significant role and find many applications in different areas of mathematics. In this paper we introduce a new definition of variation for functions defined on a nonempty rectangle subset of the plane, and we examine the algebra of functions of two variables of bounded generalized variation one obtains from this new definition. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The development of the theory of Fourier series in mathematical analysis began in the 19th century and it has been a source of new ideas for analysis during the last two centuries and is likely to be so in years to come. The first exactly proved result was published in Dirichlet’s paper in 1829. That theorem concerns the convergence of Fourier series of piecewise monotonic functions. According to Lakatos [1], functions of bounded variation were discovered by Jordan through a “critical reexamination” of Dirichlet’s famous flawed proof that arbitrary functions can be represented by Fourier series. Jordan [2] proved that if a continuous function has bounded variation, then its Fourier series converges uniformly on a closed bounded set [3]. Jordan gave the characterization of such functions as differences of increasing functions. It is well known that the space of functions of bounded variation on a compact interval [
a
,
b
]
⊂
R is a commutative Banach algebra with respect to pointwise multiplications [4–6]. Functions of bounded variation of one variable are of great interest and usefulness because of their valuable properties. Such properties, particularly with respect to additivity, decomposability into monotone functions, continuity, differentiability, measurability, integrability, and so on, have been much studied. It is largely to the possession of these properties that functions of bounded variation owe their important role in the study of rectifiable curves, Fourier series, Walsh-Fourier series, and other series, Stieltjes integrals, Henstock-Kurzweil integral, and other integrals, and the calculus of variations [7]. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis [8, 9]. Two well-known generalizations are the functions of bounded p-variation and the functions of bounded Φ-variation, due to N. Wiener and L. C. Young, respectively. In 1924 Wiener showed that the Fourier series of function in one variable of finite p-variation converges almost everywhere. In 1938 L. C. Young developed an integration theory with respect to functions of finite φ-variation and showed that the Fourier series of such functions converges everywhere. In 1972 Waterman [10] studied a class of bounded Λ-variation. Combining the notion of bounded Λ-variation with that of bounded Φ-variation, Leindler [11] introduced the class Λ
Φ
B
V of functions of bounded Λ
Φ-variation, and both classes of bounded Λ-variation and bounded Φ-variation are its special cases. In 1980 Shiba [12] introduced the class Λ
B
V
(
p
) expanding a fundamental concept of bounded Λ-variation formulated by Waterman.In 1986, S. K. Kim and J. Kim [13] introduced the notion of functions of bounded κ
Φ-variation on compact interval [
a
,
b
]
⊂
R which is a combination of concepts of bounded κ-variation and bounded ϕ-variation in the sense of Schramm [14]. In [15, 16] Castillo et al. introduce the notion of bounded κ-variation in the sense of Riesz-Korenblum, which is a combination of the notions of bounded p-variation in the sense of Riesz and bounded κ-variation in the sense of Korenblum. In the year 2014 Guerrero et al. [17] have introduced the space κ
B
V
(
I
a
b
,
R
) of the functions of two variables of bounded κ-variation in the sense of Hardy-Vitali-Korenblum and showed that the space κ
B
V
(
I
a
b
,
R
) is a Banach space.Soon after Jordan’s work, many mathematicians began to study notions of bounded variation for functions of several variables. There is no uniquely suitable way to extend the notion of variation to function of more than one variable. Proposers of definitions of bounded variation for functions of two variables have been actuated mainly by the desire to single out for attention a class of functions having properties analogous to some particular properties of a function of one variable of bounded variation. Clarkson and Adams [18] study six such generalizations, and Adams and Clarkson [19] mention two more. Two of these definitions are relevant to our purpose. Clarkson and Adams attribute the first to Vitali, Lebesgue, Fréchet, and De la Vallée Poussin and the second to Hardy [20] and Krause [21]. We will refer to them as Vitali variation and Hardy-Krause variation, respectively. Owen [22] provides a very helpful discussion of the concepts of Vitali and Hardy-Krause variation. Another useful reference is Hobson [23]. At the beginning of the past century Hardy [20] generalized the Jordan criterion to the double Fourier series and he proved that if a continuous function of two variables has bounded variation (in the sense of Hardy), then its Fourier series converges uniformly.Motivated by [13, 17] we introduce for functions of two variables the concept of bounded κ
Φ-variation in the sense of Schramm-Korenblum, which is a suitable combination of the notions of bounded Φ-variation in the sense of Schramm and bounded κ-variation in the sense of Korenblum for real functions defined on a rectangle of the plane. Our paper is structured as follows. Section 2 provides a review of the notion of Vitali and Hardy-Krause variations for multivariate functions. We recall some notions of variation and introduce for functions of two variables the definitions of bounded κ
Φ-variation in the sense of Schramm-Korenblum. In Section 3 we state and prove our main result: the linear space generated by the class of all bounded κ
Φ-variation functions is a Banach algebra.


## 2. Preliminaries, Background, and Notations

We begin with some general notation and definitions systematically used throughout the paper.As usual ifA and B are nonempty sets the symbol A
B denotes the family of functions f
:
B
→
A. We denote by S
n the set of all permutations σ of set {
1
,
…
,
n
}
n positive integer.LetI
=
[
a
,
b
]
⊂
R be any nonempty interval. For f
∈
R
I, we define f
∞
≔
sup
x
∈
B
⁡
f
(
x
), and if f
∞
<
∞ we say that f is bounded. We denote by P
(
I
) the class of all partitions of I. Let f
∈
R
[
a
,
b
] and ξ
=
t
i
i
=
1
k
∈
P
(
I
), and the following notations are used frequently:(1)
Δ
t
i
+
1
=
t
i
+
1
-
t
i
,
Δ
f
t
i
+
1
=
f
t
i
+
1
-
f
t
i
,
i
=
1
,
…
,
k
-
1
.We will establish the followinga
=
(
a
1
,
b
1
), b
=
(
a
2
,
b
2
)
∈
R
2, I
=
[
a
1
,
a
2
], J
=
[
b
1
,
b
2
], I
a
b
=
I
×
J, and k
1
,
k
2
≥
2 integer. If f
∈
R
I
a
b, ξ
=
t
i
i
=
1
k
1 and η
=
s
j
j
=
1
k
2 are partitions of the intervals I, J, respectively, denoted by (2)
Δ
10
f
t
i
+
1
,
s
j
+
1
≔
f
t
i
+
1
,
s
j
+
1
-
f
t
i
,
s
j
+
1
,
Δ
01
f
t
i
+
1
,
s
j
+
1
≔
f
t
i
+
1
,
s
j
+
1
-
f
t
i
+
1
,
s
j
,
Δ
11
f
t
i
+
1
,
s
j
+
1
≔
f
t
i
,
s
j
-
f
t
i
,
s
j
+
1
-
f
t
i
+
1
,
s
j
+
f
t
i
+
1
,
s
j
+
1
.Definition 1.
A functionκ
:
[
0,1
]
→
[
0,1
] is called distortion function if κ is continuous, increasing, and concave and satisfies κ
(
0
)
=
0, κ
(
1
)
=
1 and lim
t
→
0
+
⁡
κ
(
t
)
/
t
=
∞.The set of all distortion functions will be denoted byK. A distortion function is always subadditive (see [6, Section 2.5 page 170]) in the sense that (3)
κ
s
+
t
≤
κ
s
+
κ
t
0
≤
s
,
t
≤
1
.Important special cases for choices ofκ are κ
0
(
x
)
=
κ
(
1
-
log
⁡
x
) and κ
α
(
x
)
=
x
α, 0
<
α
<
1 (see [24], [25, Section 5], and [26]).Definition 2 (see [14]).
Aϕ
=
{
ϕ
n
}
n
≥
1 is a Φ-sequence if ϕ is a sequence of increasing convex functions, defined on R
+
=
[
0
,
∞
) such that ϕ
n
(
0
)
=
0, ϕ
n
+
1
(
t
)
≤
ϕ
n
(
t
) for all n and t, and ∑
n
ϕ
n
(
t
) diverges for t
>
0.Throughout the paper the double sequenceϕ
=
{
ϕ
i
,
j
}
i
,
j
≥
1 will be a Φ-sequence if for i or j fixed ϕ
=
{
ϕ
i
,
j
}
i
,
j
≥
1 is a Φ-sequence. It is worthy to recall that the initial works on double sequences can be found in [27, 28].In what follows we recall different notions of generalized bounded variation.The notion of variation was introduced by Jordan in 1881 in the one-dimensional case and Vitali and Hardy, which generalized the notion given by Jordan, in 1904–1906 (see [20, 29]). This generalization is for functions of two variables.Definition 3 (see [2]).
The functionf
∈
R
I is of bounded variation if(4)
V
f
;
a
,
b
=
V
f
≔
sup
P
I
⁡
∑
i
=
1
k
-
1
Δ
f
t
i
+
1
<
∞
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
). We denote by B
V
[
a
,
b
] the space of all functions of bounded variation and it is known that B
V
[
a
,
b
] is a Banach algebra with respect to the norm f
=
|
f
(
a
)
|
+
V
(
f
), f
∈
B
V
[
a
,
b
].Definition 4 (see [20, 29]).
Letf
∈
R
I
a
b and y
∈
J be fixed. The Jordan variation of the function f
(
·
,
y
)
∈
R
I is denoted by(5)
V
f
·
,
y
=
V
I
f
·
,
y
=
sup
P
I
⁡
∑
i
=
1
k
1
-
1
Δ
10
f
t
i
+
1
,
y
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
).
Forx
∈
I is fixed, the variation of Jordan of function f
(
x
,
·
)
∈
R
J is defined by(6)
V
f
x
,
·
=
V
J
f
x
,
·
=
sup
P
J
⁡
∑
j
=
1
k
2
-
1
Δ
01
f
x
,
s
j
+
1
,where the supremum is taken over all partitions s
j
j
=
1
k
2
∈
P
(
J
).
The variation off in the sense of Hardy-Vitali in the rectangle I
a
b is defined by(7)
V
I
a
b
f
≔
sup
P
I
,
P
J
⁡
∑
i
=
1
k
1
-
1
∑
j
=
1
k
2
-
1
Δ
11
f
t
i
+
1
,
s
j
+
1
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
) and s
j
j
=
1
k
2
∈
P
(
J
).
The total variation of the functionf is defined by(8)
T
V
f
=
T
V
f
,
I
a
b
≔
V
I
f
·
,
b
2
+
V
J
f
a
1
,
·
+
V
I
a
b
f
.
We denote byB
V
(
I
a
b
) the space of all functions having bounded total variation finite.The notion ofκ-variation was introduced by Korenblum in 1975 (see [24]) in one-dimensional case and Guerrero et al. in 2015 (see [17]) in two-dimension case.Definition 5 (see [24]).
Letκ
∈
K, and the function f
∈
R
[
a
,
b
] is of bounded κ-variation if(9)
κ
V
f
;
a
,
b
=
κ
V
f
≔
sup
P
a
,
b
⁡
∑
i
=
1
k
-
1
Δ
f
t
i
+
1
∑
i
=
1
k
-
1
κ
Δ
t
i
+
1
/
b
-
a
<
∞
,where the supremum is taken over all partitions t
i
i
=
1
k
∈
P
[
a
,
b
]. We denote by κ
B
V
[
a
,
b
] the space of functions of bounded κ-variation on [
a
,
b
].Definition 6 (see [17]).
Letκ
∈
K, f
∈
R
I
a
b, and y
∈
J be fixed, and the Jordan κ-variation of the function f
(
·
,
y
)
∈
R
I is denoted by(10)
κ
V
f
·
,
y
=
κ
V
I
f
·
,
y
=
sup
P
I
⁡
∑
i
=
1
k
1
-
1
Δ
10
f
t
i
+
1
,
y
∑
i
=
1
k
1
-
1
κ
Δ
t
i
+
1
/
a
2
-
a
1
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
).
Forx
∈
I  is fixed, the Jordan κ-variation of the function f
(
x
,
·
)
∈
R
J is defined by(11)
κ
V
f
x
,
·
=
κ
V
J
f
x
,
·
=
sup
P
J
⁡
∑
j
=
1
k
2
-
1
Δ
01
f
x
,
s
j
+
1
∑
j
=
1
k
2
-
1
κ
Δ
s
j
+
1
/
b
2
-
b
1
,where the supremum is taken over all partitions s
j
j
=
1
k
2
∈
P
(
J
).
The two-dimensional Hardy-Vitaliκ-variation of f in the rectangle I
a
b is defined by(12)
κ
V
I
a
b
f
≔
sup
P
I
,
P
J
⁡
∑
i
=
1
k
1
-
1
∑
j
=
1
k
2
-
1
Δ
11
f
t
i
+
1
,
s
j
+
1
∑
i
=
1
k
1
-
1
∑
j
=
1
k
2
-
1
κ
Δ
t
i
+
1
Δ
s
i
+
1
/
a
2
-
a
1
b
2
-
b
1
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
) and s
j
j
=
1
k
2
∈
P
(
J
).
The totalκ-variation of the function f is defined by(13)
κ
T
V
f
=
κ
T
V
f
,
I
a
b
≔
κ
V
I
f
·
,
b
2
+
κ
V
J
f
a
1
,
·
+
κ
V
I
a
b
f
.
We denote byκ
B
V
(
I
a
b
) the space of all functions having bounded κ-variation total.The notion ofΦ-variation was introduced by Schramm in 1985 (see [14]) in one-dimensional case and Ereú et al. in 2010 (see [30]) in two-dimensional case.Definition 7 (see [14]).
Letϕ
=
{
ϕ
n
}
n
≥
1 be a Φ-sequence and σ
∈
S
k
-
1. The function f
∈
R
[
a
,
b
] is of bounded Φ-variation if(14)
V
ϕ
S
f
;
a
,
b
=
V
ϕ
S
f
≔
sup
P
a
,
b
⁡
∑
i
=
1
k
-
1
ϕ
i
f
Δ
t
σ
i
+
1
<
∞
,where the supremum is taken over all partitions t
i
i
=
1
k
∈
P
[
a
,
b
] and σ
∈
S
k
-
1. The class of functions with bounded Φ-variation is denoted by V
ϕ
S
[
a
,
b
] and the space generated by this class is denoted by B
V
ϕ
S
[
a
,
b
].Definition 8 (see [30]).
Letϕ
=
ϕ
i
,
j
i
,
j
≥
1 be a Φ-sequence and let f
∈
R
I
a
b and y
∈
J  be fixed, and the Φ-variation in the Schramm sense of the function f
(
·
,
y
)
∈
R
I is defined by(15)
V
ϕ
S
f
·
,
y
=
V
ϕ
,
I
S
f
·
,
y
=
sup
P
I
⁡
∑
i
=
1
k
1
-
1
ϕ
i
,
1
Δ
10
f
t
σ
1
i
+
1
,
y
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
) and σ
1
∈
S
k
1
-
1.
Forx
∈
I  is fixed, the Φ-variation, in the Schramm sense of the function f
(
x
,
·
)
∈
R
J, is defined by(16)
V
ϕ
S
f
x
,
·
=
V
ϕ
,
J
S
f
x
,
·
=
sup
P
J
⁡
∑
j
=
1
k
2
-
1
ϕ
1
,
j
Δ
01
f
x
,
s
σ
2
j
+
1
,where the supremum is taken over all partitions s
j
j
=
1
k
2
∈
P
(
J
) and σ
2
∈
S
k
2
-
1.
The bidimensional variation in the sense of Schramm of the functionf in the rectangle I
a
b is defined by(17)
V
ϕ
,
I
a
b
S
f
≔
sup
P
I
,
P
J
⁡
∑
i
=
1
k
1
-
1
∑
j
=
1
k
2
-
1
ϕ
i
,
j
Δ
11
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
) and s
j
j
=
1
k
2
∈
P
(
J
) of the intervals I, J, respectively, and σ
1
∈
S
k
1
+
1, σ
2
∈
S
k
2
+
1.
The totalΦ-variation of the function f is defined by(18)
T
V
ϕ
,
I
a
b
S
f
=
T
V
ϕ
S
f
,
I
a
b
≔
V
ϕ
,
I
S
f
·
,
b
2
+
V
ϕ
,
J
S
f
a
1
,
·
+
V
ϕ
,
I
a
b
S
f
.
The class of function with total boundedΦ-variation is denoted by V
ϕ
S
(
I
a
b
) and the space generated by this class is denoted by B
V
Φ
S
(
I
a
b
).In 1986 S. K. Kim and J. Kim (see [13]) combined the concepts of κ-variation and Φ-variation introduced by Korenblum and Schramm, respectively, to create the concept κ
Φ-variation in the Schramm-Korenblum sense.Definition 9 (see [13]).
Letϕ
=
{
ϕ
n
}
n
≥
1 be a Φ-sequence, κ
∈
K, and σ
∈
S
k
-
1. The function f
∈
R
[
a
,
b
] is of bounded κ
Φ-variation if (19)
κ
V
ϕ
f
;
a
,
b
≔
sup
P
a
,
b
⁡
∑
i
=
1
k
-
1
ϕ
i
f
Δ
t
σ
i
+
1
∑
i
=
1
k
-
1
κ
Δ
t
i
+
1
/
b
-
a
<
∞
,where the supremum is taken over all partitions t
i
i
=
1
k
∈
P
[
a
,
b
] and σ
∈
S
k
-
1. The vectorial space generated by this class of functions is denoted by κ
B
V
ϕ
[
a
,
b
].


## 3. Main Results

In this section we present the main result of this paper, we generalize the concept ofκ
Φ-variation in [
a
,
b
], presented by S. K. Kim and J. Kim in [13], to the two-dimensional total κ
Φ-variation in I
a
b in the sense of Schramm-Korenblum, and we prove that the space κ
B
V
ϕ
S
(
I
a
b
) is a Banach algebra.Definition 10.
Letϕ
=
ϕ
i
,
j
i
,
j
≥
1 be a Φ-sequence and let κ
∈
K, f
∈
R
I
a
b, and y
∈
J  be fixed. The variation of the function f in the sense of Schramm-Korenblum of the function f
(
·
,
y
)
∈
R
I is denoted by(20)
κ
V
ϕ
S
f
·
,
y
=
κ
V
ϕ
,
I
S
f
·
,
y
=
sup
P
I
⁡
∑
i
=
1
k
1
-
1
ϕ
i
,
1
Δ
10
f
t
σ
1
i
+
1
,
y
∑
i
=
1
k
1
-
1
κ
Δ
t
i
+
1
/
a
2
-
a
1
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
) and σ
1
∈
S
k
1
-
1.
Forx
∈
I  is fixed, the variation of Schramm-Korenblum of the function f
(
x
,
·
)
∈
R
J is defined by(21)
κ
V
ϕ
S
f
x
,
·
=
κ
V
ϕ
,
J
S
f
x
,
·
=
sup
P
J
⁡
∑
j
=
1
k
2
-
1
ϕ
1
,
j
Δ
01
f
x
,
s
σ
2
j
+
1
∑
j
=
1
k
2
-
1
κ
Δ
S
j
+
1
/
b
2
-
b
1
,where the supremum is taken over all partitions s
j
j
=
1
k
2
∈
P
(
J
) and σ
2
∈
S
k
2
-
1.
The two-dimensional variation in the sense of Schramm-Korenblum off in the rectangle I
a
b is defined by(22)
κ
V
ϕ
,
I
a
b
S
f
≔
sup
P
I
,
P
J
⁡
∑
i
=
1
k
1
-
1
∑
j
=
1
k
2
-
1
ϕ
i
,
j
Δ
11
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
∑
i
=
1
k
1
-
1
∑
j
=
1
k
2
-
1
κ
Δ
t
i
+
1
Δ
s
j
+
1
/
a
2
-
a
1
b
2
-
b
1
,where the supremum is taken over all partitions t
i
i
=
1
k
1
∈
P
(
I
) and s
j
j
=
1
k
2
∈
P
(
J
) of the intervals I, J, respectively, and σ
1
∈
S
k
1
-
1, σ
2
∈
S
k
2
-
1.
The total bidimensionalκ
Φ-variation in the sense of Schramm-Korenblum of the function f
∈
R
I
a
b is defined by(23)
κ
T
V
ϕ
,
I
a
b
S
f
=
κ
T
V
ϕ
S
f
,
I
a
b
≔
κ
V
ϕ
,
I
S
f
·
,
b
2
+
κ
V
ϕ
,
J
S
f
a
1
,
·
+
κ
V
ϕ
,
I
a
b
S
f
.The class of functionsf
∈
R
I
a
b with bounded κ
Φ-variation in the sense of Schramm-Korenblum is denoted by κ
V
ϕ
S
(
I
a
b
) and the space generated by this class is denoted by κ
B
V
ϕ
S
(
I
a
b
).Remark 11.
(
1
) If ϕ
i
,
j
(
t
)
=
t, t
≥
0, i
,
j
≥
1, then the κ
Φ-variation in the sense of Schramm-Korenblum coincides with the Hardy-Vitali-Korenblum variation studied by Guerrero et al. in [17].
(
2
) If ϕ
i
,
j
=
φ, i
,
j
≥
1, where φ is a Young function, then the Schramm-Korenblum κ
Φ-variation is a combination of the concepts of Wiener φ-variation and Korenblum-Hardy-Vitali bidimensional variation.
(
3
) For ϕ
i
,
j
(
t
)
=
t
/
λ
i
λ
j, t
≥
0, i
,
j
≥
1, where λ
i
i
≥
1 is a Λ-sequence in the sense of Waterman, our notion is a combination of the Waterman bidimensional variation introduced by Sahakyan and also studied by Sablim (see [31, 32]) with Korenblum variation [24].The following comprehensive type results give us interesting properties of the spaceκ
B
V
ϕ
S
(
I
a
b
).Theorem 12.
Letf
∈
R
I
a
b. Then (1)
κ
T
V
ϕ
S
(
f
,
I
a
b
)
≥
0 and κ
T
V
ϕ
S
(
f
)
=
0 if and only if f
=
c
t
t
e;
(2)
κ
V
ϕ
S
(
I
a
b
) is convex and symmetry set;
(3)
κ
B
V
ϕ
S
(
I
a
b
)
=
f
∈
R
I
a
b
:
∃
λ
>
0
,
κ
V
ϕ
S
(
λ
f
)
<
∞;
(4)
ifκ
1
,
κ
2
∈
K and κ
1
≤
κ
2, then κ
1
V
ϕ
S
(
I
a
b
)
⊂
κ
2
V
ϕ
S
(
I
a
b
). In particular κ
1
B
V
ϕ
S
(
I
a
b
)
⊂
κ
2
B
V
ϕ
S
(
I
a
b
);
(5)
B
V
(
I
a
b
)
⊂
V
ϕ
s
(
I
a
b
)
⊂
κ
V
ϕ
s
(
I
a
b
) and therefore B
V
(
I
a
b
)
⊂
B
V
ϕ
S
(
I
a
b
)
⊂
κ
B
V
ϕ
S
(
I
a
b
);
(6)
ifκ
T
V
ϕ
S
(
f
,
I
a
b
)
<
∞, then f is bounded and(24)
f
∞
≤
f
a
1
,
b
1
+
3
ϕ
1,1
-
1
4
κ
T
V
ϕ
,
I
a
b
S
f
.Proof.
(1) Suppose thatκ
T
V
ϕ
S
(
f
)
=
0; then κ
V
ϕ
,
I
S
(
f
)
=
κ
V
ϕ
,
J
S
(
f
)
=
0. In particular f
(
t
,
b
1
)
-
f
(
a
1
,
b
1
)
=
0 and f
(
t
,
s
)
-
f
(
t
,
b
1
)
=
0, (
s
,
t
)
∈
I
×
J; therefore(25)
f
t
,
s
=
f
t
,
s
-
f
t
,
b
1
+
f
t
,
b
1
=
f
a
1
,
b
1
.
The “only if” part is immediate.
Parts (2) and (4) follow from Definition10.
(3) is consequences of (2).
(5) Letf
∈
B
V
(
I
a
b
) and σ
∈
S
k
-
1. Consider a partition t
i
i
=
1
k
∈
P
(
I
) and let us establish the set(26)
A
≔
i
:
Δ
10
f
t
σ
i
+
1
,
b
2
≤
1
.
Since the functionsϕ
i
,
j, i
,
j
≥
1, are convex and V
ϕ
,
I
S
f
(
·
,
b
2
)
<
∞, then (27)
∑
i
=
1
k
-
1
ϕ
i
,
1
Δ
10
f
t
σ
i
+
1
,
b
2
=
∑
i
∈
A
ϕ
i
,
1
Δ
10
f
t
σ
i
+
1
,
b
2
+
∑
i
∉
A
ϕ
i
,
1
Δ
10
f
t
σ
i
+
1
,
b
2
≤
∑
i
∈
A
ϕ
1,1
1
Δ
10
f
t
σ
i
+
1
,
b
2
+
∑
i
∉
A
ϕ
i
,
1
Δ
10
f
t
σ
i
+
1
,
b
2
≤
ϕ
1,1
1
V
I
f
·
,
b
2
+
∑
i
∉
A
ϕ
i
,
1
Δ
10
f
t
σ
i
+
1
,
b
2
.
The last sum contains at mostV
ϕ
,
I
S
f
(
·
,
b
2
) terms, because otherwise there would exist at least V
ϕ
,
I
S
f
(
·
,
b
2
)
+
1 terms, so(28)
V
ϕ
,
I
S
f
·
,
b
2
≥
∑
i
∉
A
Δ
10
f
t
σ
i
+
1
,
b
2
≥
V
ϕ
,
I
S
f
·
,
b
2
+
1
,which is absurd. Therefore(29)
∑
i
∉
A
ϕ
i
,
1
Δ
10
f
t
σ
i
+
1
,
b
2
≤
∑
i
∉
A
ϕ
i
,
1
2
f
∞
≤
ϕ
1,1
4
f
∞
V
I
f
·
,
b
2
,so(30)
V
ϕ
S
f
·
,
b
1
≤
ϕ
1,1
1
+
ϕ
1,1
4
f
∞
V
ϕ
,
I
S
f
·
,
b
2
.
Similarly(31)
V
ϕ
S
f
a
1
,
·
≤
ϕ
1,1
1
+
ϕ
1,1
4
f
∞
V
ϕ
,
J
S
f
a
1
,
·
,
V
ϕ
,
I
a
b
S
f
≤
ϕ
1,1
1
+
ϕ
1,1
4
f
∞
V
I
a
b
f
.
Finally(32)
T
V
ϕ
,
I
a
b
S
f
≤
ϕ
1,1
1
+
ϕ
1,1
4
f
∞
V
I
a
b
f
,so the first inclusion B
V
(
I
a
b
)
⊂
V
ϕ
S
(
I
a
b
) holds.
The other inclusionV
ϕ
S
(
I
a
b
)
⊂
κ
V
ϕ
S
(
I
a
b
) is a consequence of the subadditive of the function κ.
(6) Let(
x
,
y
)
∈
I
a
b, and then(33)
f
x
,
y
-
f
a
1
,
b
2
≤
f
x
,
y
-
f
a
1
,
y
-
f
x
,
b
2
+
f
a
1
,
b
2
+
f
a
1
,
y
+
f
x
,
b
2
-
2
f
a
1
,
b
2
≤
f
x
,
y
-
f
a
1
,
y
-
f
x
,
b
2
+
f
a
1
,
b
2
+
f
a
1
,
y
-
f
a
1
,
b
2
+
f
x
,
b
2
-
f
a
1
,
b
2
.
On the other hand(34)
ϕ
1,1
f
x
,
b
2
-
f
a
1
,
b
2
κ
x
-
a
1
/
a
2
-
a
1
+
κ
a
2
-
x
/
a
2
-
a
1
≤
κ
V
ϕ
,
I
S
f
·
,
b
2
.
As0
≤
κ
(
t
)
≤
1, it follows that(35)
ϕ
1,1
f
x
,
b
2
-
f
a
1
,
b
2
≤
2
κ
V
ϕ
,
I
S
f
·
,
b
2
.
Therefore(36)
f
x
,
b
2
-
f
a
1
,
b
2
≤
ϕ
1,1
-
1
2
κ
V
ϕ
,
I
S
f
·
,
b
2
.
Similarly(37)
f
a
1
,
y
-
f
a
1
,
b
2
≤
ϕ
1,1
-
1
2
κ
V
ϕ
,
J
S
f
a
1
,
·
,
f
x
,
y
-
f
a
1
,
y
-
f
x
,
b
2
+
f
a
1
,
b
2
≤
ϕ
1,1
-
1
4
κ
V
ϕ
,
I
a
b
S
f
.
By inequalities (36) and (37) and the definition of total bidimensional κ
Φ-variation in the sense of Schramm-Korenblum, it follows that(38)
f
∞
≤
f
a
1
,
b
2
+
3
ϕ
1,1
-
1
4
κ
T
V
ϕ
,
I
a
b
S
f
.The following proposition allows us to calculate the total bidimensionalκ
Φ-variation in the sense of Schramm-Korenblum of a sum of monotone functions.Proposition 13.
Letϕ
=
ϕ
i
,
j
i
,
j
≥
1 be a Φ-sequence, κ
∈
K, I
a
b
=
I
×
J, and f
∈
R
I
a
b, such that f
(
t
,
s
)
=
g
(
t
)
+
h
(
s
), (
t
,
s
)
∈
I
×
J, where g
∈
R
I and h
∈
R
J are monotone functions. Then(a)
κ
V
ϕ
S
f
(
·
,
y
)
=
ϕ
1,1
g
(
a
2
)
-
g
(
a
1
), y
∈
J;
(b)
κ
V
ϕ
S
f
(
x
,
·
)
=
ϕ
1,1
h
(
b
2
)
-
h
(
b
1
), x
∈
J;
(c)
κ
V
ϕ
,
I
a
b
S
(
f
)
=
0;
(d)
κ
T
V
ϕ
,
I
a
b
S
(
f
)
=
ϕ
1,1
g
(
a
2
)
-
g
(
a
1
)
+
ϕ
1,1
h
(
b
2
)
-
h
(
b
1
)
.Proof.
Since the functionsϕ
i
,
j, i
,
j
≥
1, are convex then the function t
→
ϕ
i
,
j
(
t
)
/
t, t
>
0, i
,
j
≥
1, is increasing.
(a) Lety
∈
J and σ
∈
S
k
1
-
1 and let t
i
i
=
1
k
1 and s
j
j
=
1
k
2 be partitions of the intervals I, J, respectively. Then(39)
∑
i
=
1
k
1
-
1
ϕ
i
,
1
Δ
10
f
t
σ
i
+
1
,
y
∑
i
=
1
k
1
-
1
κ
Δ
t
i
+
1
/
a
2
-
a
1
=
∑
i
=
1
k
1
-
1
ϕ
1,1
Δ
g
t
σ
i
+
1
Δ
g
t
σ
i
+
1
/
Δ
g
t
σ
i
+
1
∑
i
=
1
k
1
-
1
κ
Δ
t
i
+
1
/
a
2
-
a
1
.
Sinceg is monotone and κ is subadditive, we get(40)
∑
i
=
1
k
1
-
1
ϕ
1,1
Δ
g
t
σ
i
+
1
∑
i
=
1
k
1
-
1
κ
Δ
t
i
+
1
/
a
2
-
a
1
≤
ϕ
1,1
g
a
2
-
g
a
1
g
a
2
-
g
a
1
∑
i
=
1
k
1
-
1
Δ
g
t
σ
i
+
1
=
ϕ
1,1
g
a
2
-
g
a
1
.
Sinceg is monotone and κ is subadditive then(41)
∑
i
=
1
k
1
-
1
ϕ
1,1
Δ
g
t
σ
i
+
1
=
ϕ
1,1
g
a
2
-
g
a
1
,
κ
1
≤
∑
i
=
1
k
1
-
1
κ
Δ
t
i
+
1
a
2
-
a
1
.
Therefore(42)
κ
V
ϕ
S
f
·
,
y
≤
ϕ
1,1
g
a
2
-
g
a
1
,
y
∈
J
.
Reciprocally(43)
ϕ
1,1
g
a
2
-
g
a
1
=
ϕ
1,1
f
a
2
,
y
-
f
a
1
,
y
≤
κ
V
ϕ
S
f
·
,
y
,
y
∈
J
.
Thus(44)
κ
V
ϕ
S
f
·
,
y
=
ϕ
1,1
g
a
2
-
g
a
1
,
y
∈
J
.
(b) With a similar reasoning as in part (a) we get(45)
κ
V
ϕ
S
f
x
,
·
=
ϕ
1,1
h
b
2
-
h
b
1
,
x
∈
I
.
(c) Ifσ
1
∈
S
k
1
-
1  σ
2
∈
S
k
2
-
1, then(46)
Δ
11
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
=
f
t
σ
1
i
,
s
σ
2
j
-
f
t
σ
1
i
,
s
σ
2
j
+
1
+
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
-
f
t
σ
1
i
+
1
,
s
σ
2
j
=
0
,
i
∈
1
,
…
,
k
1
-
1, j
∈
1
,
…
,
k
2
-
1. Therefore V
ϕ
,
I
a
b
S
(
f
)
=
0.
(d) Is a consequence of Definition10.Example 14.
Particular cases are obtained whenf
:
[
0,1
]
×
[
0,1
]
→
R is a function given by the following: (1)
f
(
t
,
s
)
=
-
5
t
+
9
s
2
+
7; then κ
T
V
ϕ
S
(
f
)
=
ϕ
1,1
(
5
)
+
ϕ
1,1
(
9
);
(2)
f
(
t
,
s
)
=
3
t
+
ln
⁡
(
t
+
1
)
-
5
s
2
3; then κ
T
V
ϕ
S
(
f
)
=
ϕ
1,1
(
3
ln
⁡
2
)
+
ϕ
1,1
(
5
).Below we present two lemmas that play an important role in what follows.Lemma 15.
Letϕ
=
ϕ
i
,
j
i
,
j
≥
1 be a Φ-sequence and k
∈
K. Then the operator κ
T
V
ϕ
S
:
κ
V
ϕ
S
(
I
a
b
)
→
[
0
,
∞
), defined by κ
T
V
ϕ
S
(
f
)
≔
κ
T
V
ϕ
,
I
b
a
S
(
f
), is convex and the set(47)
κ
A
ϕ
S
I
a
b
≔
f
∈
κ
B
V
ϕ
S
I
a
b
:
κ
T
V
ϕ
,
I
a
b
S
f
≤
1is convex, symmetric, and absorbent.Proof.
The proof is straightforward; see [33].The Minkowski functional associated withκ
A
ϕ
S
(
I
a
b
), κ
μ
ϕ
S
:
κ
B
V
ϕ
S
(
I
a
b
)
→
[
0
,
∞
), defined by(48)
κ
μ
ϕ
S
f
≔
inf
⁡
λ
>
0
:
κ
T
V
ϕ
,
I
a
b
S
f
λ
<
1
,
f
∈
κ
B
V
ϕ
S
I
a
b
,is a seminorm.Lemma 16.
Letf
∈
κ
B
V
ϕ
S
(
I
a
b
)
. Then there exist λ
>
0, such that κ
μ
ϕ
S
(
f
)
≤
λ if and only if κ
T
V
ϕ
,
I
a
b
S
f
/
λ
<
1.Proof.
Ifκ
μ
ϕ
S
(
f
)
<
λ, λ
>
0, then there exist 0
<
k
<
λ, such that κ
T
V
ϕ
,
I
a
b
S
f
/
k
≤
1
. Then(49)
κ
T
V
ϕ
,
I
a
b
S
f
λ
=
κ
T
V
ϕ
,
I
a
b
S
k
λ
f
k
≤
k
λ
κ
T
V
ϕ
,
I
a
b
S
f
k
≤
1
.
Ifκ
μ
ϕ
S
(
f
)
=
λ, λ
>
0, then κ
μ
ϕ
S
(
f
)
<
λ
n. Therefore there exists a sequence λ
n
n
≥
1, such that λ
n
↓
λ; then(50)
κ
T
V
ϕ
,
I
a
b
S
f
λ
n
<
1
.
Taking limit asn
→
∞ and the result follow. The other part is a consequence of the definition of κ
T
V
ϕ
,
I
a
b
S
(
·
) and the continuity of the functions ϕ
i
,
j, i
,
j
≥
1, and taking limit as n
→
∞ we get desired result.
The “only if” part is consequence of the definition ofκ
μ
ϕ
S.In the next theorem we will prove that the functionalκ
·
ϕ
s
:
κ
B
V
ϕ
S
(
I
a
b
)
→
[
0
,
∞
), defined by(51)
κ
f
ϕ
s
≔
f
a
1
,
b
2
+
κ
μ
ϕ
S
f
,is a norm.Theorem 17.
κ
B
V
ϕ
S
(
I
a
b
)
,
κ
·
ϕ
s is a normed space.Proof.
Asκ
·
ϕ
s is a seminorm we only need to show that if κ
f
ϕ
s
=
0 then f
≡
0.
Ifκ
μ
ϕ
S
(
f
)
=
0, then f
(
a
1
,
b
2
)
=
0 and κ
μ
ϕ
S
(
f
)
<
1
/
n, for all n
≥
1. Accordingly(52)
n
κ
T
V
ϕ
,
I
a
b
S
f
≤
κ
T
V
ϕ
,
I
a
b
S
n
f
<
1
,
n
≥
1
.
Henceκ
T
V
ϕ
,
I
a
b
S
(
f
)
=
0. By part (a) of Theorem 12, f
=
c
t
t
e
. As f
(
a
1
,
b
2
)
=
0, we obtain f
≡
0.We will use the next lemma at several places in this work.Lemma 18.
Lett
i
i
=
1
k
1
∈
P
(
I
), s
j
j
=
1
k
2
∈
P
(
J
), λ
>
0, and f
,
g
:
I
a
b
→
R be bounded functions. Then (a)
Δ
10
(
f
g
)
(
t
i
+
1
,
y
)
≤
g
∞
Δ
10
f
(
t
i
+
1
,
y
)
+
f
∞
Δ
10
g
(
t
i
+
1
,
y
), y
∈
J, i
=
1
,
…
,
k
1
-
1;
(b)
Δ
01
(
f
g
)
(
x
,
s
j
+
1
)
≤
g
∞
Δ
01
f
(
x
,
s
j
+
1
)
+
f
∞
Δ
01
g
(
x
,
s
j
+
1
), x
∈
I, j
=
1
,
…
,
k
2
-
1;
(c)
Δ
11
(
f
g
)
(
t
i
+
1
,
s
j
+
1
)
≤
g
∞
Δ
11
f
(
t
i
+
1
,
s
j
+
1
)
+
f
∞
Δ
11
g
(
t
i
+
1
,
s
j
+
1
), i
=
1
,
…
,
k
1
-
1, j
=
1
,
…
,
k
2
-
1
.Proof.
(a) Consider(53)
Δ
10
f
g
t
i
+
1
,
y
=
f
g
t
i
+
1
,
y
-
f
g
t
i
,
y
=
f
t
i
+
1
,
y
g
t
i
+
1
,
y
-
f
t
i
,
y
g
t
i
,
y
=
f
t
i
+
1
,
y
g
t
i
+
1
,
y
-
f
t
i
,
y
g
t
i
,
y
+
f
t
i
,
y
g
t
i
+
1
,
y
-
f
t
i
,
y
g
t
i
+
1
,
y
=
f
t
i
+
1
,
y
-
f
t
i
,
y
g
t
i
+
1
,
y
+
g
t
i
+
1
,
y
-
g
t
i
,
y
f
t
i
,
y
=
g
t
i
+
1
,
y
Δ
10
f
t
i
+
1
,
y
+
f
t
i
,
y
Δ
10
g
t
i
+
1
,
y
≤
g
∞
Δ
10
f
t
i
+
1
,
y
+
f
∞
Δ
10
g
t
i
+
1
,
y
.
(b) It follows by reasoning as in part (a).
(c) Fori, j  are fixed, we obtain (54)
Δ
11
f
g
t
i
+
1
,
s
j
+
1
=
f
g
t
i
,
s
j
-
f
g
t
i
,
s
j
+
1
-
f
g
t
i
+
1
,
s
j
+
f
g
t
i
+
1
,
s
j
+
1
=
f
t
i
,
s
j
g
t
i
,
s
j
-
f
t
i
,
s
j
+
1
g
t
i
,
s
j
+
1
-
f
t
i
+
1
,
s
j
g
t
i
+
1
,
s
j
+
f
t
i
+
1
,
s
j
+
1
g
t
i
+
1
,
s
j
+
1
=
f
t
i
,
s
j
g
t
i
,
s
j
-
f
t
i
,
s
j
+
1
g
t
i
,
s
j
+
1
+
f
t
i
+
1
,
s
j
g
t
i
,
s
j
-
f
t
i
+
1
,
s
j
g
t
i
,
s
j
+
f
t
i
,
s
j
+
1
g
t
i
+
1
,
s
j
+
1
-
f
t
i
,
s
j
+
1
g
t
i
+
1
,
s
j
+
1
-
f
t
i
+
1
,
s
j
g
t
i
+
1
,
s
j
+
f
t
i
+
1
,
s
j
+
1
g
t
i
+
1
,
s
j
+
1
=
f
t
i
,
s
j
-
f
t
i
+
1
,
s
j
g
t
i
,
s
j
+
f
t
i
+
1
,
s
j
+
1
-
f
t
i
,
s
j
+
1
g
t
i
+
1
,
s
j
+
1
+
g
t
i
+
1
,
s
j
+
1
-
g
t
i
,
s
j
+
1
f
t
i
,
s
j
+
1
+
g
t
i
,
s
j
-
g
t
i
+
1
,
s
j
f
t
i
+
1
,
s
j
≤
f
t
i
,
s
j
-
f
t
i
+
1
,
s
j
g
∞
+
f
t
i
+
1
,
s
j
+
1
-
f
t
i
,
s
j
+
1
g
∞
+
g
t
i
+
1
,
s
j
+
1
-
g
t
i
,
s
j
+
1
f
∞
+
g
t
i
,
s
j
-
g
t
i
+
1
,
s
j
f
∞
=
Δ
11
f
t
i
+
1
,
s
j
+
1
g
∞
+
Δ
11
g
t
i
+
1
,
s
j
+
1
f
∞
.Theorem 19.
κ
B
V
ϕ
S
(
I
a
b
) is an algebra.Proof.
Letf
,
g
∈
κ
B
V
ϕ
S
(
I
a
b
) (f and g not identically zero) and ϕ
=
{
ϕ
i
,
j
}
i
,
j
≥
1, a Φ-sequence; then there exist α
>
0 and β
>
0 such that(55)
κ
T
V
ϕ
,
I
a
b
S
α
f
<
∞
,
κ
T
V
ϕ
,
I
a
b
S
β
g
<
∞
.
We set(56)
λ
≔
α
β
α
f
∞
+
β
g
∞
,
λ
1
≔
β
g
∞
α
f
∞
+
β
g
∞
,
λ
2
≔
α
f
∞
α
f
∞
+
β
g
∞
.
Now,x
∈
I, y
∈
J are fixed, {
t
i
}
i
=
1
k
1
∈
P
(
I
), {
s
j
}
j
=
1
k
2
∈
P
(
J
), σ
1
∈
S
k
1
-
1, and σ
2
∈
S
k
2
-
1. Then by Lemma 18 we have(57)
Δ
10
λ
f
g
t
σ
1
i
+
1
,
y
≤
λ
g
∞
Δ
10
f
t
σ
1
i
+
1
,
y
+
λ
f
∞
Δ
10
g
t
σ
1
i
+
1
,
y
=
β
g
∞
α
f
∞
+
β
g
∞
α
Δ
10
f
t
σ
1
i
+
1
,
y
+
α
f
∞
α
f
∞
+
β
g
∞
β
Δ
10
g
t
σ
1
i
+
1
,
y
;therefore(58)
Δ
10
λ
f
g
t
σ
1
i
+
1
,
y
≤
λ
1
α
Δ
10
f
t
σ
1
i
+
1
,
y
+
λ
2
β
Δ
10
g
t
σ
1
i
+
1
,
y
.
Similarly(59)
Δ
01
λ
f
g
x
,
s
σ
2
j
+
1
≤
λ
1
α
Δ
01
f
x
,
s
σ
2
j
+
1
+
λ
2
β
Δ
01
g
x
,
s
σ
2
j
+
1
,
(60)
Δ
11
λ
f
g
t
σ
1
i
+
1
,
s
σ
2
j
+
1
=
λ
Δ
11
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
g
∞
+
Δ
11
g
t
σ
1
i
+
1
,
s
σ
2
j
+
1
f
∞
=
β
g
∞
α
f
∞
+
β
g
∞
α
Δ
11
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
+
α
f
∞
α
f
∞
+
β
g
∞
β
Δ
11
g
t
σ
1
i
+
1
,
s
σ
2
j
+
1
;we get(61)
Δ
11
λ
f
g
t
σ
1
i
+
1
,
s
σ
2
j
+
1
≤
λ
1
α
Δ
11
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
+
λ
2
β
Δ
11
g
t
σ
1
i
+
1
,
s
σ
2
j
+
1
.
Next, combining (58), (59), and (61) with the fact that the functions ϕ
i
,
j, i
,
j
≥
1, are increasing and convex and λ
1
,
λ
2
>
0, λ
1
+
λ
2
=
1, we obtain(62)
κ
T
V
ϕ
,
I
a
b
λ
f
g
≤
λ
1
κ
V
ϕ
,
I
S
α
f
+
λ
2
κ
V
ϕ
,
I
S
β
g
+
λ
1
κ
V
ϕ
,
J
S
α
f
+
λ
2
κ
V
ϕ
,
J
S
β
g
+
λ
1
κ
V
ϕ
,
I
a
b
S
α
f
+
λ
2
κ
V
ϕ
,
I
a
b
S
β
g
=
λ
1
κ
T
V
ϕ
,
I
S
α
f
+
λ
2
κ
T
V
ϕ
,
I
S
β
g
<
∞
;therefore f
g
∈
κ
B
V
ϕ
S
(
I
a
b
).The main result of our paper now reads as follows.Theorem 20.
κ
B
V
ϕ
S
(
I
a
b
)
,
κ
·
ϕ
s is a Banach space.Proof.
Letf
n
n
≥
1 be a Cauchy sequence in κ
B
V
ϕ
S
(
I
a
b
); then, for each ε
>
0, there exists N
ε, such that if n
,
m
≥
N
ε, then(63)
f
n
a
1
,
b
2
-
f
m
a
1
,
b
2
<
ε
,and by Lemma 16 we get(64)
max
⁡
κ
V
ϕ
,
I
S
f
n
-
f
m
ε
·
,
b
2
,
κ
V
ϕ
,
J
S
f
n
-
f
m
ε
a
1
,
·
,
κ
V
ϕ
,
I
a
b
S
f
n
-
f
m
ε
<
ε
.
By the definition ofκ
V
ϕ
,
I
S
f
n
-
f
m
/
ε
(
·
,
b
2
) and considering a
1
<
x
≤
a
2 we get(65)
ϕ
1,1
f
n
-
f
m
x
,
b
2
-
f
n
-
f
m
a
1
,
b
2
/
ε
κ
x
-
a
1
/
a
2
-
a
1
+
κ
a
2
-
x
/
a
2
-
a
1
<
1
,
n
,
m
≥
N
ε
,
a
1
<
x
≤
a
2
.Due to 0
≤
κ
(
t
)
≤
1, t
∈
[
0,1
], and (63), we obtain(66)
f
n
-
f
m
x
,
b
2
<
ϕ
1,1
-
1
2
+
1
ε
,
n
,
m
≥
N
ε
,
x
∈
I
.
Similarly, we can consider the definition ofκ
V
ϕ
,
J
S
f
n
-
f
m
/
ε
(
a
1
,
·
)  and get(67)
f
n
-
f
m
a
1
,
y
<
ϕ
1,1
-
1
2
+
1
ε
,
n
,
m
≥
N
ε
,
y
∈
J
.
By the definition ofκ
V
ϕ
,
I
a
b
S
f
n
-
f
m
/
ε
(68)
ϕ
1,1
f
n
-
f
m
a
1
,
y
-
f
n
-
f
m
x
,
y
-
f
n
-
f
m
a
1
,
b
2
+
f
n
-
f
m
x
,
b
2
/
ε
κ
x
-
a
1
y
-
b
1
/
a
2
-
a
1
b
2
-
b
1
+
κ
x
-
a
1
b
2
-
y
/
a
2
-
a
1
b
2
-
b
1
+
κ
a
2
-
x
y
-
b
1
/
a
2
-
a
1
b
2
-
b
1
+
κ
a
2
-
x
b
2
-
y
/
a
2
-
a
1
b
2
-
b
1
<
1
,
n
,
m
≥
N
ε, (
t
,
s
)
∈
I
×
J, and a
1
<
x
≤
a
2. Consider(69)
f
n
-
f
m
x
,
y
<
3
ϕ
1,1
-
1
4
+
1
ε
,
n
,
m
>
N
ε
,
x
,
y
∈
I
×
J
.
Thereforef
n
n
≥
1 is a uniformly Cauchy sequence in I
×
J, so there exist f
∈
R
I
a
b, such that f
n
(
x
,
y
)
→
f
(
x
,
y
), as n
→
∞
. By the continuity of the functions ϕ
i
,
j, i
,
j
≥
1, and Definition 10 we conclude that κ
T
V
ϕ
i
,
j
,
I
a
b
S
(
f
)
<
∞, as n
→
∞.
With a similar argument(70)
lim
n
→
∞
⁡
lim
m
→
∞
⁡
κ
T
V
ϕ
i
,
j
,
I
a
b
S
f
n
-
f
m
=
lim
n
→
∞
⁡
κ
T
V
ϕ
i
,
j
,
I
a
b
S
f
n
-
f
=
0
.Theorem 21.
The spaceκ
B
V
ϕ
S
(
I
a
b
) equipped with the norm κ
·
ϕ
s is a Banach algebra.Proof.
Letf
,
g
∈
κ
B
V
ϕ
S
(
I
a
b
) such that they are not identically zero or constant functions and(71)
λ
≔
κ
μ
ϕ
s
f
g
∞
+
κ
μ
ϕ
s
g
f
∞
.
By Lemma16, κ
μ
ϕ
S
f
/
κ
f
ϕ
s
<
1. By Lemma 18
(72)
Δ
10
f
g
λ
t
σ
1
i
+
1
,
y
≤
g
∞
λ
Δ
10
f
t
σ
1
i
+
1
,
y
+
f
∞
λ
Δ
10
g
t
σ
1
i
+
1
,
y
=
κ
μ
ϕ
s
f
g
∞
λ
Δ
10
f
t
σ
1
i
+
1
,
y
κ
μ
ϕ
s
f
+
κ
μ
ϕ
s
g
f
∞
λ
Δ
10
g
t
σ
1
i
+
1
,
y
κ
μ
ϕ
s
g
.
Similarly(73)
Δ
01
f
g
λ
x
,
s
σ
2
j
+
1
≤
κ
μ
ϕ
s
f
g
∞
λ
Δ
01
f
x
,
s
σ
2
j
+
1
κ
μ
ϕ
s
f
+
κ
μ
ϕ
s
g
f
∞
λ
Δ
01
g
x
,
s
σ
2
j
+
1
κ
μ
ϕ
s
g
,
Δ
11
f
g
λ
t
σ
1
i
+
1
,
s
σ
2
j
+
1
≤
κ
μ
ϕ
s
f
g
∞
λ
Δ
11
f
t
σ
1
i
+
1
,
s
σ
2
j
+
1
κ
μ
ϕ
s
f
+
κ
μ
ϕ
s
g
f
∞
λ
Δ
11
g
t
σ
1
i
+
1
,
s
σ
2
j
+
1
κ
μ
ϕ
s
g
.
As(74)
κ
μ
ϕ
s
f
g
∞
λ
+
κ
μ
ϕ
s
g
f
∞
λ
=
1
.
By inequalities (72) and (73), Definition 1, the convexity of functions ϕ
i
,
j, i
,
j
≥
1, and the convexity of operator κ
T
V
ϕ
,
I
a
b
S we have(75)
κ
T
V
ϕ
,
I
a
b
S
f
g
λ
≤
κ
μ
ϕ
s
f
g
∞
λ
κ
T
V
ϕ
,
I
a
b
S
f
κ
μ
ϕ
S
f
+
κ
μ
ϕ
s
g
f
∞
λ
κ
T
V
ϕ
,
I
a
b
s
g
κ
μ
ϕ
s
g
.
We also have by Lemma16
(76)
κ
T
V
ϕ
,
I
a
b
S
f
κ
μ
ϕ
S
f
<
1
,
κ
T
V
ϕ
,
I
a
b
S
g
κ
μ
ϕ
S
g
<
1
.
Thusκ
T
V
ϕ
,
I
a
b
S
f
g
/
λ
<
1. Using Lemma 16 again(77)
κ
μ
ϕ
s
f
g
≤
λ
=
κ
μ
ϕ
s
f
g
∞
+
κ
μ
ϕ
s
g
f
∞
,
κ
f
g
ϕ
s
≤
2
f
a
1
,
b
2
g
a
1
,
b
2
+
κ
μ
ϕ
s
f
g
∞
+
κ
μ
ϕ
s
g
f
∞
≤
f
a
1
,
b
2
g
∞
+
g
a
1
,
b
2
f
∞
+
κ
μ
ϕ
s
f
g
∞
+
κ
μ
ϕ
s
g
f
∞
=
κ
f
ϕ
s
g
∞
+
κ
g
ϕ
s
f
∞
.
Clearly, all the conditions in Section4.1 of [4] are satisfied. Hence the conditions in Section 4.1 of [4] ensure that the space κ
B
V
ϕ
S
(
I
a
b
) is a Banach algebra equipped with the norm ·
∞
+
κ
·
ϕ
s and by part (
6
) of Theorem 12 and Lemma 18, this norm is equivalent to norm κ
·
ϕ
s. This completes the proof.



---

*Source: 727312-2015-07-07.xml*