# Estimates ofL

Authors: Heping Wang; Zhaoyang Wu
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

Some sharp estimates of theL
p
1
≤
p
<
∞ modulus of continuity of classes of Λ
φ-bounded variation are obtained. As direct applications, we obtain estimates of order of Fourier coefficients of functions of Λ
φ-bounded variation, and we also characterize some sufficient and necessary conditions for the embedding relations H
p
ω
⊂
Λ
φ
B
V. Our results include the corresponding known results of the class Λ
B
V as a special case.


---

## Body

## 1. Introduction and Main Results

To generalize the notion of functions of bounded variation, Wiener [1] introduced the class B
V
β
(
β
>
1
) of functions of β-bounded variation. Young [2] introduced the notion of functions of φ-bounded variation, and Waterman [3] studied a class of Λ-bounded variations. Combining the notion of Λ-bounded variation with that of φ-bounded variation, Leindler [4] introduced the class Λ
φ
B
V of functions of Λ
φ-bounded variation, and both classes of Λ-bounded variation and φ-bounded variation are its special cases. Actually the class Λ
φ
B
V first appeared in Schramm and Waterman’s paper [5], and some restrictions are imposed on φ in their definition. Here we adopt Leindler’s definition.Definition 1.
Letφ
:
[
0
,
∞
)
→
ℝ be a nondecreasing function with φ
(
0
)
=
0, and let Λ
=
:
{
λ
k
} be a nondecreasing sequence of positive numbers such that ∑
k
=
1
∞
(
1
/
λ
k
)
=
+
∞. Let Γ be the set of all sequences of nonoverlapping subintervals [
a
k
,
b
k
] in [
a
,
b
]. If for any Δ
=
{
(
a
k
,
b
k
)
⊂
[
a
,
b
]
,
k
=
1,2
,
…
,
n
,
n
∈
Z
+
}
∈
Γ, a real valued function f
:
[
a
,
b
]
→
ℝ satisfies the condition
(1)
∑
k
=
1
n
‍
φ
(
|
f
(
b
k
)
-
f
(
a
k
)
|
)
λ
k
<
∞
,
then f is said to be of Λ
φ-bounded variation, and this fact is denoted by f
∈
Λ
φ
B
V. And the quantity
(2)
V
Λ
φ
(
f
;
[
a
,
b
]
)
:
=
sup
⁡
Δ
∈
Γ
{
∑
k
=
1
n
‍
φ
(
|
f
(
b
k
)
-
f
(
a
k
)
|
)
λ
k
}
is said to be Λ
φ-total variation of f.
In the special case whenφ
(
x
)
=
x
β
(
β
≥
1
), f is said to be of Λ
β-bounded variation, and we write f
∈
Λ
β
B
V and V
Λ
β
(
f
;
[
a
,
b
]
)
=
V
Λ
φ
(
f
;
[
a
,
b
]
), and if β
=
1, f is said to be of Λ-bounded variation, and we denote f
∈
Λ
B
V and V
Λ
(
f
;
[
a
,
b
]
)
=
V
Λ
1
(
f
;
[
a
,
b
]
).
In the caseΛ
=
{
1
}, we get the class of φ-bounded variation, and f is said to be of φ-bounded variation, and we denote V
φ
(
f
;
[
a
,
b
]
)
=
V
Λ
φ
(
f
;
[
a
,
b
]
). More specifically, when φ
(
x
)
=
x
β
(
β
≥
1
), we say that f is of β-bounded variation, and we denote f
∈
B
V
β and V
β
(
f
;
[
a
,
b
]
)
=
V
φ
(
f
;
[
a
,
b
]
). The class B
V
β is also called the Wiener class and B
V
1 is the well- known class of bounded variation B
V.
It is easily seen from the definition thatΛ
φ
B
V functions are bounded; that is, Λ
φ
B
V
[
a
,
b
]
⊆
B
[
a
,
b
], and the discontinuities of a Λ
φ
B
V function are simple and therefore at most denumerable, where B
[
a
,
b
] denotes the class of bounded real valued functions on [
a
,
b
].
LetΛ
=
:
{
λ
k
} be a nondecreasing sequence of positive numbers such that ∑
k
=
1
∞
(
1
/
λ
k
)
=
+
∞. If a continuous and nondecreasing function λ
(
s
) on [
0
,
∞
) such that λ
(
s
)
≡
λ
1, 0
≤
s
≤
1 and λ
(
k
)
=
λ
k, k
=
1,2
,
…, then we say that λ
(
s
) generates Λ. By the nondecreasing property of Λ, it is easily verified that if λ
(
s
) generates Λ, then
(3)
1
2
∫
0
n
‍
d
s
λ
(
s
)
≤
∑
k
=
1
n
‍
1
λ
k
≤
∫
0
n
‍
d
s
λ
(
s
)
,
n
≥
1
.
Letλ
(
x
) be a nonnegative real-valued function on [
0
,
∞
). If there exists a
>
0 such that λ
(
2
x
)
≤
a
λ
(
x
), x
∈
(
0
,
A
) for some positive constant A, then we say that λ
(
x
) satisfies the condition Δ
2. If λ
(
x
) generates Λ
=
{
λ
k
} and satisfies the condition Δ
2, we say that Λ satisfies the condition Δ
2. Obviously the condition Δ
2 here is a very weak restriction on λ
(
x
) and Λ.
Letω
(
t
) be a modulus of continuity, that is, a continuous and nondecreasing function on [
0
,
+
∞
) satisfying ω
(
0
)
=
0 and ω
(
t
1
+
t
2
)
≤
ω
(
t
1
)
+
ω
(
t
2
) for nonnegative t
1 and t
2. As usual, for 1
≤
p
≤
∞, denote by H
p
ω
≡
H
p
ω
(
t
) the class of functions for which ∥
f
∥
H
p
ω
:
=
sup
⁡
t
>
0
(
ω
(
f
,
t
)
p
/
ω
(
t
)
)
<
∞, where
(4)
ω
(
f
;
t
)
p
:
=
{
sup
⁡
0
≤
h
≤
t
{
∫
a
b
‍
|
f
(
x
+
h
)
-
f
(
x
)
|
p
d
x
}
1
/
p
,
1
≤
p
<
∞
,
sup
⁡
0
≤
h
≤
t
sup
⁡
x
∈
[
a
,
b
]
|
f
(
x
+
h
)
-
f
(
x
)
|
,
p
=
∞
is the L
p modulus of continuity of f. We write H
ω instead of H
∞
ω and H
p
α
(
0
<
α
≤
1
) instead of H
p
t
α
≡
Lip
(
α
,
p
), the Lipschitz class, for brevity.
Functions of classesB
V
β, φ
B
V, Λ
B
V, and Λ
φ
B
V are considered in trigonometric Fourier series and some of them share good approximative properties (see [1–3, 6–11], etc.). What we mention here is the following theorem proved by Shiba [12], Schramm and Waterman [5], and Wang [13]:Theorem A.
(a) Iff
∈
Λ
β
B
V, 1
≤
β
<
2
r, 1
≤
r
<
∞, and
(5)
∑
n
=
1
∞
‍
(
∑
k
=
1
n
‍
(
1
/
λ
k
)
)
-
1
/
2
r
(
ω
β
+
(
2
-
β
)
s
(
f
;
π
/
n
)
)
1
-
β
/
2
r
n
1
/
2
<
∞
,
where 1
/
r
+
1
/
s
=
1, then the Fourier series of f converges absolutely.
(b) Ifφ is Δ
2, f
∈
Λ
φ
B
V, 1
≤
β
<
2
r, 1
≤
r
<
∞, and
(6)
∑
n
=
1
∞
‍
[
φ
-
1
(
(
∑
k
=
1
n
‍
(
1
λ
k
)
)
-
1
ω
β
+
(
2
-
β
)
s
2
r
-
β
(
f
;
π
n
)
)
]
1
/
2
r
n
1
/
2
<
∞
,
where 1
/
r
+
1
/
s
=
1, then the Fourier series of f converges absolutely.Embedding relations between various generalized bounded variation classes and the classH
p
ω (or the Lipschitz class H
p
α) are also investigated in recent years. It is well known that H
p
1
⊂
H
1
1
=
B
V
⊂
Λ
B
V. For f
∈
Λ
B
V, the estimates of L
p modulus of continuity of f had been given in [13] for p
=
1 and in [9, 14] for 1
<
p
<
∞. Furthermore, Goginava in [15] and Li and Wang in [9] proved that, for 1
≤
p
<
∞
,
0
<
α
,
δ
≤
1,
(7)
{
n
δ
}
B
V
⊂
H
p
α
iff
α
≤
min
⁡
{
1
p
,
1
-
δ
}
.
For more detailed results on this topic, we refer readers to [4, 9, 10, 14–25].In this paper, we obtain some sharp estimates ofL
p
(
1
≤
p
<
∞
) modulus of continuity of the classes Λ
φ
B
V in the case of that φ is convex. More specifically, our results include estimates of L
1 modulus of continuity of the classes Λ
φ
B
V, estimates of L
p
(
1
≤
p
<
∞
) modulus of continuity of the classes Λ
β
B
V, and specially estimates of L
p
(
1
≤
p
<
∞
) modulus of the classes {
(
n
+
1
)
α
ln
⁡
γ
(
n
+
1
)
}
β
B
V
(
0
≤
α
≤
1
,
β
≥
1
,
γ
∈
ℝ
). Our results extend and include the corresponding known results of the class Λ
B
V as a special case and are also sharp in most cases. As direct applications, we obtain estimates of order of Fourier coefficients of functions of Λ
φ-bounded variation, and we also characterize some sufficient and necessary conditions for the embedding relations H
p
ω
⊂
Λ
φ
B
V and H
p
ω
⊂
φ
B
V.Now we state our main results, and in what follows, without loss of generality, we always assume that[
a
,
b
]
=
[
0,2
π
], and functions in various generalized bounded variation classes are 2
π periodic.Theorem 2.
LetΛ
φ
B
V be the class of functions of Λ
φ-bounded variation, and let λ
(
s
) generate Λ, where φ is convex and φ
-
1 is the inverse function of φ. Then, for f
∈
Λ
φ
B
V,
(8)
ω
(
f
;
t
)
1
≤
4
π
φ
-
1
(
V
Λ
φ
(
f
;
[
0,2
π
]
)
∫
0
1
/
t
‍
(
d
s
/
λ
(
s
)
)
)
,
and this estimate is sharp in the sense of order provided that Λ satisfies the condition Δ
2.Corollary3 is a direct result of Theorem 2 by choosing φ
(
x
)
=
x and Λ
=
{
1
}, respectively.Corollary 3.
(a) LetΛ
B
V be the class of functions of Λ-bounded variation, and let λ
(
s
) generate Λ. Then, for f
∈
Λ
B
V,
(9)
ω
(
f
;
t
)
1
≤
4
π
V
Λ
(
f
;
[
0,2
π
]
)
∫
0
1
/
t
‍
(
d
s
/
λ
(
s
)
)
,
and this estimate is sharp in the sense of order provided that Λ satisfies the condition Δ
2.
(b) Letφ
B
V be the class of functions of φ-bounded variation, where φ is convex. Then, for f
∈
φ
B
V,
(10)
ω
(
f
;
t
)
1
≤
4
π
φ
-
1
(
t
V
φ
(
f
;
[
0,2
π
]
)
)
,
and this estimate is sharp in the sense of order.The first part of Corollary3(a) is due to Wang [13].For specialΛ’s, one hasCorollary 4.
Letf
∈
Λ
φ
B
V and φ be convex. Then(a)
forΛ
=
{
n
α
}
(
0
≤
α
<
1
),
(11)
ω
(
f
;
t
)
1
≤
c
φ
-
1
(
t
1
-
α
V
Λ
φ
(
f
;
[
0,2
π
]
)
)
;
(b)
forΛ
=
{
n
},
(12)
ω
(
f
;
t
)
1
≤
c
φ
-
1
(
V
Λ
φ
(
f
;
[
0,2
π
]
)
ln
⁡
(
1
/
t
)
)
;
(c)
forΛ
=
{
(
n
+
1
)
ln
⁡
(
n
+
1
)
},
(13)
ω
(
f
;
t
)
1
≤
c
φ
-
1
(
V
Λ
φ
(
f
;
[
0,2
π
]
)
ln
ln
(
1
/
t
)
)
;
(d)
forΛ
=
{
ln
⁡
γ
(
n
+
1
)
}
(
γ
>
0
),
(14)
ω
(
f
;
t
)
1
≤
c
φ
-
1
(
t
ln
⁡
-
γ
(
1
t
)
V
Λ
φ
(
f
;
[
0,2
π
]
)
)
;
(e)
forΛ
=
{
(
n
+
1
)
ln
⁡
γ
(
n
+
1
)
}
(
-
∞
<
γ
<
1
),
(15)
ω
(
f
;
t
)
1
≤
c
φ
-
1
(
V
Λ
φ
(
f
;
[
0,2
π
]
)
ln
⁡
1
-
γ
(
1
/
t
)
)
;
(f)
forΛ
=
{
(
n
+
1
)
α
ln
⁡
γ
(
n
+
1
)
}
(
0
<
α
<
1
,
γ
∈
ℝ
∖
{
0
}
),
(16)
ω
(
f
;
t
)
1
≤
c
φ
-
1
(
t
1
-
α
ln
⁡
-
γ
(
1
t
)
V
Λ
φ
(
f
;
[
0,2
π
]
)
)
.
The above estimates are sharp in the sense of order.Letf
(
x
) be an integrable function on [
0,2
π
] and let its Fourier coefficients be defined as follows:
(17)
a
n
(
f
)
=
1
π
∫
0
2
π
‍
f
(
t
)
cos
n
t
d
t
,
b
n
(
f
)
=
1
π
∫
0
2
π
‍
f
(
t
)
sin
n
t
d
t
,
c
n
(
f
)
=
a
n
(
f
)
+
i
b
n
(
f
)
.
We note that
(18)
|
c
n
(
f
)
|
≤
1
2
π
∫
0
2
π
‍
|
f
(
t
)
-
f
(
t
+
π
n
)
|
d
t
≤
c
ω
1
(
f
;
1
n
)
,
c
>
0
.
Theorem 2 implies the following estimates of Fourier coefficients.Corollary 5.
(a) LetΛ
φ
B
V be the class of functions of Λ
φ-bounded variation, and let λ
(
s
) generate Λ, where φ is convex and φ
-
1 is the inverse function of φ. Then, for f
∈
Λ
φ
B
V,
(19)
|
a
n
(
f
)
|




|
b
n
(
f
)
|




}
=
O
(
φ
-
1
(
1
∫
0
n
‍
(
d
s
/
λ
(
s
)
)
)
)
,
n
≥
1
.
(b) Letφ
B
V be the class of functions of φ-bounded variation, where φ is convex and φ
-
1 is the inverse function of φ. Then for f
∈
φ
B
V
(20)
|
a
n
(
f
)
|




|
b
n
(
f
)
|




}
=
O
(
φ
-
1
(
1
n
)
)
,
n
≥
1
.Corollary5(a) includes Wang’s result [13] for the class Λ
B
V as a special case of φ
(
x
)
=
x.Theorem 6.
LetΛ
β
B
V
(
β
≥
1
) be class of functions of Λ
β-bounded variation, and let λ
(
s
) generate Λ
=
{
λ
n
}. Set ϕ
(
z
)
=
∫
0
z
‍
(
d
s
/
λ
(
s
)
) and assume f
∈
Λ
β
B
V.
(a) For1
≤
p
≤
β,
(21)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
∫
0
1
/
t
‍
(
d
s
/
λ
(
s
)
)
)
1
/
β
,
and this estimate is sharp in the sense of order provided that Λ satisfies the condition Δ
2.
(b) For1
<
p
<
∞,
(22)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
1
/
β
(
t
∫
0
1
/
t
‍
d
z
ϕ
(
z
)
p
/
β
)
1
/
p
.
(c) If, for someβ
<
p
<
∞, ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
) is bounded on [
1
,
∞
), then
(23)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
1
/
β
t
1
/
p
.Unfortunately we cannot assert the sharpness of our estimates in (b) and (c) of Theorem6 for the case β
<
p
<
∞. Our next theorems concern some important special case of the class Λ
β
B
V and elaborate on the estimates in Theorem 6. Among the classes considered here are {
n
α
}
β
B
V
(
0
≤
α
≤
1
,
β
≥
1
) and {
(
n
+
1
)
α
ln
⁡
γ
(
n
+
1
)
}
β
B
V
(
0
≤
α
≤
1
,
β
≥
1
,
γ
∈
ℝ
).Theorem 7.
(a) Let{
n
α
}
β
B
V
(
0
≤
α
≤
1
,
β
≥
1
) be the class of functions of {
n
α
}
β-bounded variation. Then, for f
∈
{
n
α
}
β
B
V and 1
≤
p
<
∞, one has (i)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
min
⁡
{
(
1
-
α
)
/
β
,
1
/
p
}, 0
≤
α
<
1;
(ii)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
(
1
/
ln
⁡
t
-
1
)
1
/
β, α
=
1.
(b) LetB
V
β
(
β
≥
1
) be the class of β-bounded variation, that is, the Wiener class. Then, for f
∈
B
V
β and 1
≤
p
<
∞,
(24)
ω
(
f
;
t
)
p
≤
c
(
V
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
min
⁡
{
1
/
β
,
1
/
p
}
,
and both estimates in (a) and (b) are sharp in the sense of order.Li and Wang’s results in [9] are extended in the Theorems 6 and 7, which can be treated as the case β
=
1 of our theorems.Theorem 8.
Letf
∈
Λ
β
B
V
(
β
≥
1
). The following assertions are true.(a)
ForΛ
=
{
ln
⁡
γ
(
n
+
1
)
}
(
γ
>
0
),
(25)
ω
(
f
;
t
)
p
≤
{
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
(
t
ln
⁡
γ
(
1
t
)
)
1
/
β
,
1
≤
p
≤
β
,
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
1
/
p
,
β
<
p
<
∞
.
(b)
ForΛ
=
{
(
n
+
1
)
ln
⁡
(
n
+
1
)
},
(26)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
(
ln
⁡
⁡
ln
⁡
1
t
)
-
1
/
β
,
1
≤
p
<
∞
.
(c)
ForΛ
=
{
(
n
+
1
)
ln
⁡
γ
(
n
+
1
)
}
(
-
∞
<
γ
<
1
),
(27)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
(
ln
⁡
1
t
)
-
(
1
-
γ
)
/
β
,
sss
1
≤
p
<
∞
.
(d)
ForΛ
=
{
(
n
+
1
)
α
/
ln
⁡
γ
(
n
+
1
)
}
(
0
<
α
<
1
,
γ
>
0
),
(28)
ω
(
f
;
t
)
p
≤
{
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
(
1
-
α
)
/
β
(
ln
⁡
1
t
)
-
γ
/
β
,
1
≤
p
<
β
1
-
α
,
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
1
/
p
,
p
≥
β
1
-
α
.
(e)
ForΛ
=
{
(
n
+
1
)
α
ln
⁡
γ
(
n
+
1
)
}
(
0
<
α
<
1
,
γ
>
0
)
(29)
ω
(
f
;
t
)
p
≤
{
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
(
1
-
α
)
/
β
(
ln
⁡
1
t
)
γ
/
β
,
1
≤
p
<
β
1
-
α
,
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
1
/
p
,
p
>
β
1
-
α
.
And all above estimates are sharp in the sense of order.Remark 9.
In (e), forp
=
β
/
(
1
-
α
), we only have
(30)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
1
/
p
(
ln
⁡
1
t
)
1
/
p
+
γ
/
β
.
We do not know whether this estimate is sharp in the sense of order. However there exists f
n
∈
Λ
β
B
V such that
(31)
V
Λ
β
(
f
n
;
[
0,2
π
]
)
≍
1
,
n
⟶
+
∞
,
ω
(
f
n
;
π
n
)
p
≥
c
(
π
n
)
1
/
p
,
n
≥
10
.
This exception indicates that the estimates of the L
p modulus of continuity ω
(
f
;
t
)
p
(
β
<
p
<
+
∞
) of classes of Λ
β-bounded variation are complicated, and Theorem 6 cannot cover all cases of the class Λ
β
B
V
(
β
≥
1
).As direct applications of the above theorems, we characterize some sufficient and necessary conditions for the embedding relations between the generalized bounded variation classes and the classH
p
ω (or H
p
δ).Corollary 10.
On the embedding relations between the generalized bounded variation classes and the classH
p
ω or H
p
δ, the following assertions are true.(a)
LetΛ
φ
B
V be the class of functions of Λ
φ-bounded variation and let φ be convex; let λ
(
s
) generate Λ and Λ satisfy the condition Δ
2. Then
(32)
Λ
φ
B
V
⊂
H
1
ω
i
f
f
ω
(
t
)
=
O
(
φ
-
1
(
(
∫
0
1
/
t
‍
d
s
λ
(
s
)
)
-
1
)
)
,
sss
t
⟶
0
.
(b)
Letφ
B
V be the class of functions of φ-bounded variation and let φ be convex. Then
(33)
φ
B
V
⊂
H
1
ω
i
f
f
ω
(
t
)
=
O
(
φ
-
1
(
t
)
)
,
t
⟶
0
.
(c)
LetΛ
β
B
V be the class of functions of Λ
β-bounded variation and let λ
(
s
) generate Λ.
(
i
)
If, for1
≤
p
≤
β, Λ satisfies the condition Δ
2, then
(34)
Λ
β
B
V
⊂
H
p
ω
i
f
f
ω
(
t
)
=
O
(
(
∫
0
1
/
t
‍
d
s
λ
(
s
)
)
-
1
/
β
)
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
t
⟶
0
.
(
ii
)
If, forβ
<
p
<
∞, ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
) is bounded on [
1
,
∞
) and 0
<
δ
≤
1
/
p, then Λ
β
B
V
⊂
H
p
δ.
(d)
Let{
n
α
}
β
B
V
(
0
≤
α
≤
1
,
β
≥
1
) be the class of functions of {
n
α
}
β-bounded variation. Then, for 1
≤
p
<
∞,
(
i
)
{
n
α
}
β
B
V
⊂
H
p
δ
(
0
≤
α
<
1
) if and only if δ
≤
min
⁡
{
(
1
-
α
)
/
β
,
1
/
p
}.
(
ii
)
{
n
}
β
B
V
⊂
H
p
ω if and only if ω
(
t
)
=
O
(
(
ln
⁡
(
1
/
t
)
)
-
1
/
β
)
,
t
→
0.
(e)
LetB
V
β
(
β
≥
1
) be the class of functions of β-bounded variation, that is, Wiener class; then, for 1
≤
p
<
∞, B
V
β
⊂
H
p
δ if and only if δ
≤
min
⁡
{
1
/
β
,
1
/
p
}.This paper is organized as follows. In Section2, we first state three lemmas, and then by them we prove Theorem 2. Lemma 12 provides our proofs crucial upper estimates and Lemma 13 will be used repeatedly in proving the sharpness of our estimates. In Section 3, we prove Theorem 6, and the same estimate technique in [9] is partly used in our proof. Theorems 7 and 8 are proved in Section 4. For the case β
<
p
<
∞, the difficulty in our proofs of Theorems 7 and 8 is to prove the sharpness of our estimates and the key is to construct extreme functions by Lemma 13.


## 2. Proof of Theorem2

Before we start our proof of Theorem2, we prove three lemmas. Lemmas 11 and 12 will also be used in the proof of Theorem 6. Lemma 13 will be employed repeatedly in the proof of the sharpness of our estimates. Lemmas 12 and 13 are of independent interest for functions of Λ
φ-bounded variation.Lemma 11.
Letf
∈
L
p
[
a
,
b
]
(
p
≥
1
). Then
(35)
F
(
h
)
=
(
∫
a
b
‍
|
f
(
x
+
h
)
-
f
(
x
)
|
p
d
x
)
1
/
p
is continuous on [
0
,
∞
).Proof.
Using triangle inequality ofL
p norm, for any h
1, h
2
>
0, we have
(36)
|
F
(
h
1
)
-
F
(
h
2
)
|
≤
∥
f
(
·
+
h
1
)
-
f
(
·
+
h
2
)
∥
p
≤
ω
(
f
;
|
h
1
-
h
2
|
)
p
.
The right continuity of ω
(
f
;
t
)
p at t
=
0 implies the continuity of F
(
h
).Lemma 12.
LetΔ
=
{
[
a
k
,
b
k
]
:
1
≤
k
≤
n
,
n
∈
Z
+
}
∈
Γ be an arbitrary sequence of nonoverlapping intervals in [
a
,
b
], Λ
=
{
λ
k
}, φ convex, and φ
-
1 the function of φ. Then, for f
∈
Λ
φ
B
V,
(37)
∑
k
=
1
n
‍
|
f
(
b
k
)
-
f
(
a
k
)
|
≤
n
φ
-
1
(
V
Λ
φ
(
f
;
[
a
,
b
]
)
∑
k
=
1
n
‍
λ
k
-
1
)
.
Specifically, if f
∈
φ
B
V, then ∑
k
=
1
n
|
f
(
b
k
)
-
f
(
a
k
)
|
≤
n
φ
-
1
(
V
φ
(
f
;
[
a
,
b
]
)
/
n
).Proof.
By the definition ofΛ
φ-total variation and letting A
i
=
1
/
λ
i and B
k
=
φ
(
|
f
(
b
k
)
-
f
(
a
k
)
|
) in summation transform
(38)
(
∑
i
=
1
n
‍
A
i
)
(
∑
k
=
1
n
‍
B
k
)
=
∑
k
=
0
n
-
1
‍
{
∑
i
=
1
n
-
k
‍
A
i
B
i
+
k
+
∑
i
=
n
-
k
+
1
n
‍
A
i
B
i
+
k
-
n
}
,
we have
(39)
∑
k
=
1
n
‍
φ
(
|
f
(
b
k
)
-
f
(
a
k
)
|
)
≤
n
∑
k
=
1
n
‍
λ
k
-
1
V
Λ
φ
(
f
;
[
a
,
b
]
)
.
Note that ifφ is an increasing convex function on [
0
,
∞
), then φ
-
1 is increasing and concave on [
0
,
∞
). The concavity of φ
-
1 implies that φ
-
1
(
a
x
)
≥
a
φ
-
1
(
x
)
,
0
<
a
≤
1. Therefore, by Jensen’s inequality and the above inequality, we finally get
(40)
∑
k
=
1
n
‍
|
f
(
b
k
)
-
f
(
a
k
)
|
=
n
∑
k
=
1
n
‍
φ
-
1
(
φ
(
|
f
(
b
k
)
-
f
(
a
k
)
|
)
)
n
≤
n
φ
-
1
(
1
n
∑
k
=
1
n
‍
φ
(
|
f
(
b
k
)
-
f
(
a
k
)
|
)
)
≤
n
φ
-
1
(
V
Λ
φ
(
f
;
[
a
,
b
]
)
∑
k
=
1
n
‍
λ
k
-
1
)
.
This completes the proof of Lemma 12.Lemma 13.
Leta
n
=
{
a
k
}
k
=
1
n be a given set of nonnegative and nonincreasing numbers. Define a function f
(
x
) on [
0,2
π
] as
(41)
f
a
n
(
x
)
=
{
a
i
,
(
2
i
-
1
)
π
n
≤
x
<
2
i
π
n
,
i
=
1,2
,
…
,
n
,
0
,
other
points
of
[
0,2
π
]
,
and extend it to ℝ with period 2
π. Then(a)
f
∈
Λ
φ
B
V
[
0,2
π
] and V
Λ
φ
(
f
a
n
;
[
0
;
2
π
]
)
=
∑
i
=
1
n
‍
(
1
/
λ
2
i
-
1
+
1
/
λ
2
i
)
φ
(
a
i
);
(b)
ω
(
f
a
n
;
π
/
n
)
p
=
(
(
2
π
/
n
)
∑
k
=
1
n
a
k
p
)
1
/
p, p
≥
1;
(c)
∑
i
=
1
n
(
φ
(
a
i
)
/
λ
i
)
≤
V
Λ
φ
(
f
a
n
;
[
0
;
2
π
]
)
≤
2
∑
i
=
1
n
(
φ
(
a
i
)
/
λ
i
).Proof.
By the definition ofΛ
φ-total variation and the nonnegative and nonincreasing properties of a
n, direct computation proves (a). Since Λ
=
{
λ
k
} is nondecreasing, it is obvious that (a) implies (c). For (b), computation shows that
(42)
ω
(
f
a
n
;
π
n
)
p
p
=
∑
k
=
1
n
‍
∫
(
2
(
k
-
1
)
π
)
/
n
2
k
π
/
n
‍
|
f
a
n
(
x
+
π
n
)
-
f
a
n
(
x
)
|
p
d
x
=
2
π
n
∑
k
=
1
n
‍
a
k
p
.Now we prove Theorem2.Proof of Theorem2.
We writeV
Λ
φ
(
f
)
=
V
Λ
φ
(
f
;
[
0,2
π
]
) for simplicity. By Lemma 11, there exists an h
t
∈
(
0
,
t
] such that
(43)
ω
(
f
;
t
)
1
=
∫
0
2
π
‍
|
f
(
x
+
h
t
)
-
f
(
x
)
|
d
x
.
If we set N
=
[
2
π
/
h
t
] and consider the periodicity of f
(
x
), then
(44)
ω
(
f
;
t
)
1
=
1
N
∫
0
2
π
‍
F
t
(
x
)
d
x
,
where
(45)
F
t
(
x
)
=
∑
k
=
1
N
‍
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
.
From Jensen’s inequality and Lemma 12, for all x
∈
[
0,2
π
], the concavity of φ
-
1 implies
(46)
F
t
(
x
)
≤
N
φ
-
1
(
V
Λ
φ
(
f
)
∑
k
=
1
N
‍
λ
k
-
1
)
≤
2
N
φ
-
1
(
V
Λ
φ
(
f
)
∫
0
N
‍
(
d
s
/
λ
(
s
)
)
)
.
Substituting (46) into (44), we get
(47)
ω
(
f
;
t
)
1
≤
4
π
φ
-
1
(
V
Λ
φ
(
f
)
∫
0
N
‍
(
d
s
/
λ
(
s
)
)
)
.
Since the right of (47) is decreasing with respect to N and N
=
[
2
π
/
h
t
]
≥
[
2
π
/
t
]
≥
1
/
t, the estimate in Theorem 2 is obtained from (47) directly.
Now we show that our estimate is sharp in the sense of order under the assumption thatΛ satisfies the condition Δ
2.
We choose(48)
a
k
=
φ
-
1
(
1
2
π
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
)
,
k
=
1
,
…
,
n
,
and consider function f
a
n
(
x
) defined in Lemma 13.
We have(49)
∫
0
n
‍
(
d
s
/
λ
(
s
)
)
4
∫
0
n
‍
(
d
s
/
λ
(
s
/
π
)
)
=
∫
0
n
‍
(
d
s
/
λ
(
s
)
)
4
π
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
≤
V
Λ
φ
(
f
a
n
)
≤
∫
0
n
‍
(
d
s
/
λ
(
s
)
)
π
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
=
∫
0
n
‍
(
d
s
/
λ
(
s
)
)
∫
0
n
‍
(
d
s
/
λ
(
s
/
π
)
)
,
n
≥
10
,
(50)
ω
(
f
a
n
;
π
n
)
1
=
2
π
φ
-
1
(
1
2
π
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
)
,
n
≥
10
.
IfΛ satisfies the condition Δ
2, that is, there exists a
>
0 such that λ
(
2
x
)
≤
a
λ
(
x
), x
>
0, then
(51)
λ
(
s
π
)
≤
λ
(
s
)
=
λ
(
π
·
s
π
)
≤
λ
(
4
·
s
π
)
≤
a
2
λ
(
s
π
)
,
and this yields
(52)
∫
0
n
‍
d
s
λ
(
s
)
≤
∫
0
n
‍
d
s
λ
(
s
/
π
)
≤
a
2
∫
0
n
‍
d
s
λ
(
s
)
.
From (49) and (52), we have
(53)
0
<
1
4
a
2
≤
V
Λ
φ
(
f
a
n
)
≤
1
,
n
≥
10
.
On the other hand, (50), the concavity and the monotonicity of φ
-
1 imply that
(54)
ω
(
f
a
n
;
π
n
)
1
≥
φ
-
1
(
1
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
)
≥
φ
-
1
(
V
Λ
φ
(
f
a
n
)
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
)
,
n
≥
10
.
Obviously, (53) and (54) mean the sharpness of our estimate in Theorem 2.Proof of Corollary4.
Obviously theΛ’s in Corollary 4 satisfy the condition Δ
2. The proofs of (a), (b), (c), and (e) in Corollary 4 are obvious. In (d), we have
(55)
λ
(
s
)
=
ln
⁡
γ
(
s
+
1
)
,
s
≥
1
,
∫
0
z
‍
d
s
λ
(
s
)
≍
z
ln
⁡
γ
z
,
z
⟶
+
∞
,
and in (f), we have
(56)
λ
(
s
)
=
(
s
+
1
)
α
ln
⁡
γ
(
s
+
1
)
,
s
≥
1
,
∫
0
z
‍
d
s
λ
(
s
)
≍
z
1
-
α
ln
⁡
γ
z
,
z
⟶
+
∞
,
which complete the proofs of (d) and (f) of Corollary 4.


## 3. Proof of Theorem6

In this section we prove Theorem6. The proof of Theorem 6(a) is based on Hölder’s inequality and Lemma 12. We use techniques used by Li and Wang [9] in the proofs of (b) and (c) of Theorem 6. Lemma 12 plays a crucial role in the whole proof of Theorem 6.Proof of Theorem6.
As in the proof of Theorem2, it is easily seen from Lemma 11 that, for 1
≤
p
<
∞, there exists an h
t
∈
(
0
,
t
] such that
(57)
ω
(
f
;
t
)
p
p
=
1
N
∫
0
2
π
‍
F
t
(
x
)
d
x
,
where
(58)
F
t
(
x
)
=
∑
k
=
1
N
‍
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
p
,
s
s
s
s
N
=
[
2
π
h
t
]
.
We first prove (a). We note thatr
=
β
/
p
>
1 for 1
≤
p
<
β. Let s
>
0 satisfy 1
/
r
+
1
/
s
=
1. By Hölder’s inequality and Lemma 12, we have
(59)
F
t
(
x
)
≤
(
∑
k
=
1
N
‍
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
β
)
1
/
r
(
∑
k
=
1
N
‍
1
s
)
1
/
s
≤
(
N
V
Λ
β
(
f
)
∑
k
=
1
N
‍
λ
k
-
1
)
1
/
r
N
1
/
s
=
N
(
V
Λ
β
(
f
)
∑
k
=
1
N
‍
λ
k
-
1
)
p
/
β
.
For p
=
β, Lemma 12 directly yields
(60)
F
t
(
x
)
≤
N
(
V
Λ
β
(
f
)
∑
k
=
1
N
‍
λ
k
-
1
)
.
Thus, for 1
≤
p
≤
β, we obtain
(61)
F
t
(
x
)
≤
N
(
V
Λ
β
(
f
)
∑
k
=
1
N
‍
λ
k
-
1
)
p
/
β
.
Substituting (61) into (57) and noting that N
=
[
2
π
/
h
t
]
≥
[
2
π
/
t
]
≥
1
/
t, we prove the desired estimate in Theorem 6(a).
IfΛ satisfies the condition Δ
2, then from the proof of Theorem 2, we know that
(62)
∫
0
n
‍
d
s
λ
(
s
)
≍
∫
0
n
/
π
‍
d
s
λ
(
s
)
,
n
⟶
∞
.
If we choose
(63)
a
k
=
ε
n
1
/
β
(
1
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
)
1
/
β
,
k
=
1,2
,
…
,
n
,
where
ε
n
=
∫
0
n
/
π
‍
(
d
s
/
λ
(
s
)
)
2
∫
0
n
‍
(
d
s
/
λ
(
s
)
)
≍
1
,
n
⟶
+
∞
,
and consider the functions f
a
n
(
x
) defined in Lemma 13, then we have
(64)
1
4
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(65)
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
ε
n
p
/
β
(
1
∫
0
π
/
n
‍
(
d
s
/
λ
(
s
)
)
)
p
/
β
≥
c
(
V
Λ
β
(
f
a
n
)
∫
0
π
/
n
‍
(
d
s
/
λ
(
s
)
)
)
p
/
β
,
n
≥
10
.Equations (64) and (65) show that the estimate in Theorem 6(a) is sharp.
Now we prove (b). Without loss of generality, we assume thatλ
1
=
1 and denote ψ
(
x
)
=
x
β
(
β
≥
1
) and ψ
-
1
(
x
)
=
x
1
/
β. From the definition of V
Λ
β
(
f
), we first have
(66)
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
≤
ψ
-
1
(
V
Λ
β
(
f
)
)
.
Denote by σ
m
(
m
≥
0
) the set of integers k
(
1
≤
k
≤
N
) for which
(67)
2
-
m
-
1
ψ
-
1
(
V
Λ
β
(
f
)
)
<
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
≤
2
-
m
ψ
-
1
(
V
Λ
β
(
f
)
)
.
Then ∑
m
=
0
∞
|
σ
m
|
=
N, and there are at most N nonempty σ
m, where |
σ
m
| denotes the number of the elements in σ
m. Obviously |
σ
m
|
≤
N
,
m
≥
0.
Hence we have(68)
F
t
(
x
)
=
∑
m
=
0
∞
‍
∑
k
∈
σ
m
‍
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
p
≤
(
ψ
-
1
(
V
Λ
β
(
f
)
)
)
p
∑
m
=
0
∞
‍
2
-
m
p
|
σ
m
|
=
(
V
Λ
β
(
f
)
)
p
/
β
{
∑
m
≤
M
‍
2
-
m
p
|
σ
m
|
+
∑
m
>
M
‍
2
-
m
p
|
σ
m
|
}
=
:
(
V
Λ
β
(
f
)
)
p
/
β
(
A
1
+
A
2
)
,
with M
>
0 to be determined.
From (67) and Lemma 12, we have, for |
σ
m
|
≠
0,
(69)
2
-
m
-
1
ψ
-
1
(
V
Λ
β
(
f
)
)
|
σ
m
|
≤
∑
k
∈
σ
m
‍
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
≤
|
σ
m
|
ψ
-
1
(
V
Λ
β
(
f
)
∑
i
=
1
|
σ
m
|
‍
λ
i
-
1
)
,
and thus
(70)
∑
i
=
1
|
σ
m
|
‍
1
λ
i
≤
V
Λ
β
(
f
)
ψ
(
2
-
m
-
1
ψ
-
1
(
V
Λ
β
(
f
)
)
)
=
2
β
(
m
+
1
)
.
Therefore,
(71)
ϕ
(
|
σ
m
|
)
=
∫
0
|
σ
m
|
‍
d
s
λ
(
s
)
≤
2
∑
i
=
1
|
σ
m
|
‍
1
λ
i
≤
2
β
(
m
+
1
)
+
1
.
If we set z
m
=
ϕ
-
1
(
2
β
(
m
+
1
)
+
1
), where ϕ
-
1 is the inverse function of ϕ, then
(72)
|
σ
m
|
≤
z
m
,
2
-
m
=
2
1
+
1
/
β
ϕ
(
z
m
)
1
/
β
.
From the monotonicity of λ
(
x
), it is easily verified that ϕ
(
2
z
)
≤
2
ϕ
(
z
). And from this we also have
(73)
ϕ
(
2
z
m
-
1
)
≤
2
ϕ
(
z
m
-
1
)
=
2
1
-
β
ϕ
(
z
m
)
≤
ϕ
(
z
m
)
,
m
≥
1
.
This yields
(74)
2
z
m
-
1
≤
z
m
,
z
m
≤
2
(
z
m
-
z
m
-
1
)
,
m
≥
1
.
Now we estimateA
1 and A
2 in (68).
By means of (72) and (74), we first have
(75)
A
1
≤
∑
m
≤
M
‍
2
(
2
+
1
/
β
)
p
z
m
ϕ
(
z
m
)
p
/
β
≤
2
(
2
+
1
/
β
)
p
+
1
∑
m
≤
M
‍
z
m
-
z
m
-
1
ϕ
(
z
m
)
p
/
β
≤
c
p
∫
0
z
M
‍
d
z
ϕ
(
z
)
p
/
β
.
If we choose M
=
[
log
⁡
2
ϕ
(
N
)
/
2
/
β
]
-
1, then z
M
≤
N
≤
z
M
+
1.
From (72) and the monotonicity of ϕ
(
z
), we have
(76)
A
2
≤
N
∑
m
>
M
‍
2
-
p
m
=
N
·
2
-
p
(
M
+
1
)
1
-
2
-
p
≤
2
(
1
+
1
/
β
)
p
+
1
N
ϕ
(
z
M
+
1
)
p
/
β
≤
c
p
N
ϕ
(
N
)
p
/
β
.
Inserting (75) and (76) into (68) and noting that ∫
0
N
‍
(
d
z
/
ϕ
(
z
)
p
/
β
)
≥
(
N
/
ϕ
(
N
)
p
/
β
), for x
∈
[
0,2
π
], we get
(77)
F
t
(
x
)
≤
c
3
(
V
Λ
β
(
f
)
)
p
/
β
[
∫
0
N
‍
d
z
ϕ
(
z
)
p
/
β
+
N
ϕ
(
N
)
p
/
β
]
≤
2
c
p
(
V
Λ
β
(
f
)
)
p
/
β
∫
0
N
‍
d
z
ϕ
(
z
)
p
/
β
.
Combining (77) and (57), we finally obtain
(78)
ω
(
f
;
t
)
p
p
≤
c
p
(
V
Λ
β
(
f
)
)
p
/
β
1
N
∫
0
N
‍
d
z
ϕ
(
z
)
p
/
β
.
Since N
=
[
2
π
/
h
t
]
≥
[
2
π
/
t
]
≥
1
/
t and the right of (78) is decreasing with respect to N, the proof of Theorem 6(b) is complete.
Finally we prove Theorem6(c). Notice that Λ is nondecreasing. It follows from (67) that
(79)
F
t
(
x
)
=
∑
m
=
0
∞
‍
∑
k
∈
σ
m
‍
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
p
-
β
×
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
β
≤
(
V
Λ
β
(
f
)
)
(
p
-
β
)
/
β
×
∑
m
=
0
∞
‍
2
-
m
(
p
-
β
)
∑
k
∈
σ
m
‍
|
f
(
x
+
k
h
t
)
-
f
(
x
+
(
k
-
1
)
h
t
)
|
β
≤
(
V
Λ
β
(
f
)
)
(
p
-
β
)
/
β
∑
m
=
0
∞
‍
2
-
m
(
p
-
β
)
λ
k
m
Ω
m
,
where
(80)
k
m
=
|
σ
0
|
+
|
σ
1
|
+
⋯
+
|
σ
m
|
,
Ω
m
=
∑
j
=
k
m
-
1
+
1
k
m
‍
|
f
(
x
+
σ
m
(
j
)
h
t
)
-
f
(
x
+
(
σ
m
(
j
)
-
1
)
h
t
)
|
β
λ
j
,
and σ
m
(
j
)
∈
σ
m
,
k
-
1
=
0. It is obvious that ∑
m
=
0
∞
Ω
m
≤
V
Λ
β
(
f
).
By (72) and (74), we know that
(81)
k
m
≤
∑
j
=
0
m
‍
z
j
≤
∑
j
=
1
m
‍
2
(
z
j
-
z
j
-
1
)
+
z
0
≤
2
z
m
.
Assume that ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
) is bounded on [
1
,
∞
). From (72) and the fact ϕ
(
2
z
m
)
≤
2
ϕ
(
z
m
), m
≥
0, we obtain
(82)
2
-
m
(
p
-
β
)
λ
k
m
≤
2
(
1
+
1
/
β
)
(
p
-
β
)
λ
(
2
z
m
)
ϕ
(
z
m
)
(
p
-
β
)
/
β
≤
c
ϕ
(
2
z
m
)
(
β
-
p
)
/
β
λ
(
2
z
m
)
≤
c
,
m
≥
0
.
Finally, (79) yields
(83)
F
t
(
x
)
≤
c
(
V
Λ
β
(
f
)
)
(
p
-
β
)
/
β
∑
m
=
0
∞
‍
Ω
m
≤
c
(
V
Λ
β
(
f
)
)
p
/
β
,
x
∈
[
0,2
π
]
.
Since N
≥
1
/
t, we prove Theorem 6(c) from (57).


## 4. Proofs of Theorems7 and 8

In this section, we prove Theorems7 and 8.Proof of Theorem7.
Theorem7(a) implies Theorem 7(b) by letting α
=
0. We only need to prove Theorem 7(a). Obviously Λ
=
{
n
α
}
(
0
≤
α
<
1
) satisfies the condition Δ
2 and we have
(84)
λ
(
z
)
=
z
α
,
z
≥
1
,
ϕ
(
z
)
=
∫
0
z
‍
d
s
λ
(
s
)
≍
z
1
-
α
,
z
≥
1
.
For 1
≤
p
≤
β, it follows from Theorem 6(a) that ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
t
(
1
-
α
)
/
β, and the order of this estimate is sharp.
Forβ
<
p
<
β
/
(
1
-
α
)
,
1
-
(
p
/
β
)
(
1
-
α
)
>
0, we have
(85)
∫
0
N
‍
d
z
ϕ
(
z
)
p
/
β
≍
∫
1
N
‍
d
z
z
p
(
1
-
α
)
/
β
≍
N
1
-
(
p
/
β
)
(
1
-
α
)
,
sss
N
⟶
∞
.
Substituting into Theorem 6(b), we get ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
t
(
1
-
α
)
/
β.
Forβ
/
(
1
-
α
)
≤
p
<
∞, α
+
(
1
-
α
)
(
β
-
p
)
/
β
≤
0, we have
(86)
ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
)
≤
c
1
z
α
+
(
1
-
α
)
(
β
-
p
)
/
β
≤
C
<
∞
,
z
≥
1
.
By Theorem 6(c), we get the estimate
(87)
ω
(
f
;
t
)
p
≤
(
V
Λ
β
(
f
)
)
1
/
β
t
1
/
p
.
IfΛ
=
{
n
}, then Λ satisfies the condition Δ
2, and
(88)
λ
(
z
)
=
z
,
z
≥
1
;
ϕ
(
z
)
=
∫
0
z
‍
d
s
λ
(
s
)
≍
ln
⁡
(
z
+
1
)
≍
ln
⁡
z
,
z
≥
1
.
For 1
≤
p
≤
β, (ii) follows from Theorem 6(a) and the order in the estimate (ii) is also sharp.
Forβ
<
p
<
∞, we have
(89)
1
N
∫
0
N
d
z
ϕ
(
z
)
p
/
β
≍
1
N
‍
∫
1
N
d
z
ln
⁡
⁡
(
z
+
1
)
p
/
β
≍
1
(
ln
⁡
z
)
p
/
β
‍
,
sss
N
⟶
∞
.
Substituting into Theorem 6(b), we get (ii).
Now we show the sharpness of the estimates in (i) and (ii) forβ
<
p
<
∞.
For this purpose, we considerf
a
n
(
x
) defined in Lemma 13.
In (i), forβ
<
p
<
β
/
(
1
-
α
), we choose
(90)
a
k
=
ε
n
1
/
β
(
π
n
)
(
1
-
α
)
/
β
,
k
=
1,2
,
…
,
n
,
where
ε
n
=
n
1
-
α
2
π
1
-
α
∑
k
=
1
n
‍
k
-
α
≍
1
,
n
⟶
∞
.
From Lemma 13, we know that
(91)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
ε
n
p
/
β
(
π
n
)
p
(
1
-
α
)
/
β
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
(
π
n
)
p
(
1
-
α
)
/
β
,
n
≥
10
.
Forβ
/
(
1
-
α
)
≤
p
<
∞, p
(
1
-
α
)
/
β
≥
1, we choose
(92)
a
k
=
ε
n
1
/
β
k
-
(
1
-
α
)
/
β
(
ln
⁡
⁡
(
k
+
1
)
)
-
(
1
+
σ
)
/
β
(
σ
>
0
)
,
sssss
k
=
1,2
,
…
,
n
,
where
(93)
ε
n
=
(
2
∑
k
=
1
n
k
-
1
(
ln
⁡
⁡
(
k
+
1
)
)
-
(
1
+
σ
)
‍
)
-
1
≍
1
,
n
⟶
+
∞
.
From Lemma 13, we have
(94)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
n
ε
n
p
/
β
∑
k
=
1
n
‍
k
-
(
1
-
α
)
p
/
β
(
ln
⁡
⁡
(
1
+
k
)
)
-
(
1
+
σ
)
p
/
β
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
π
n
,
n
≥
10
.
In (ii), forβ
<
p
<
∞, we choose
(95)
a
k
=
ε
n
1
/
β
(
n
π
)
1
/
β
,
k
=
1,2
,
…
,
n
,
where
(96)
ε
n
=
ln
⁡
(
n
/
π
)
2
∑
k
=
1
n
‍
k
-
1
≍
1
,
n
⟶
+
∞
.
From Lemma 13, we know that
(97)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
ε
n
p
/
β
(
ln
⁡
⁡
(
n
π
)
)
-
p
/
β
≥
c
(
V
Λ
β
(
f
a
n
)
)
-
p
/
β
(
ln
⁡
n
π
)
-
p
/
β
,
n
≥
10
.
Obviously the above functionsf
a
n chosen prove the sharpness of our estimates in Theorem 7 for β
<
p
<
∞.The proof of Theorem8 is similar to that of Theorem 7, but computations are more complicated.Proof of Theorem8.
Letλ
(
s
) generate Λ and ϕ
(
z
)
=
∫
0
z
‍
(
d
s
/
λ
(
s
)
). It is readily seen that Λ’s in (a), (b), (c), (d), and (e) satisfy the condition Δ
2.
In (a), we have(98)
λ
(
s
)
=
ln
⁡
γ
(
s
+
1
)
(
γ
>
0
)
,
s
≥
1
,
ϕ
(
z
)
≍
z
+
1
ln
⁡
γ
(
z
+
1
)
≍
z
ln
⁡
γ
z
,
z
≥
2
.
In (b), we have
(99)
λ
(
s
)
=
(
s
+
1
)
ln
⁡
⁡
(
s
+
1
)
,
s
≥
1
,
ϕ
(
z
)
≍
ln ln
(
z
+
1
)
≍
ln ln
z
,
z
≥
10
.
In (c), we have
(100)
λ
(
s
)
=
(
s
+
1
)
ln
⁡
γ
(
s
+
1
)
(
-
∞
<
γ
<
1
)
,
s
≥
1
,
ϕ
(
z
)
≍
(
ln
⁡
⁡
(
z
+
1
)
)
1
-
γ
≍
(
ln
⁡
z
)
1
-
γ
,
z
≥
2
.
In (d), we have
(101)
λ
(
s
)
=
(
s
+
1
)
α
ln
⁡
γ
(
s
+
1
)
(
0
<
α
<
1
,
γ
>
0
)
,
s
≥
1
,
ϕ
(
z
)
≍
(
z
+
1
)
1
-
α
(
ln
⁡
γ
(
z
+
1
)
)
≍
z
1
-
α
ln
⁡
γ
z
,
z
≥
2
.
In (e), we have
(102)
λ
(
s
)
=
(
s
+
1
)
α
ln
⁡
γ
(
s
+
1
)
(
0
<
α
<
1
,
γ
>
0
)
,
s
≥
2
,
ϕ
(
z
)
≍
(
z
+
1
)
1
-
α
ln
⁡
γ
(
z
+
1
)
≍
z
1
-
α
ln
⁡
γ
z
,
z
≥
2
.
From Theorem 6(a) we obtain sharp estimates in (a), (b), (c), (d), and (e) for 1
≤
p
≤
β.
Now we prove the estimates in Theorem8 for β
<
p
<
∞. We apply (b) and (c) in Theorem 6 for upper estimates. The technique used here to show the sharpness is the same as that of the proof of Theorem 7 and the key is to choose a
n to construct extreme functions f
a
n
(
x
) defined in Lemma 13.
In (a), forβ
<
p
<
∞, we have
(103)
ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
)
≍
(
ln
⁡
⁡
(
z
+
1
)
)
p
γ
/
β
(
z
+
1
)
(
p
-
β
)
/
β
<
+
∞
,
z
≥
1
.
From Theorem 6(b), we obtain
(104)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
t
1
/
p
.
If we choose
(105)
a
k
=
ε
n
1
/
β
(
k
+
1
)
-
1
/
β
(
ln
⁡
(
k
+
1
)
)
-
(
1
+
γ
+
σ
)
/
β
ssssss
(
σ
>
0
)
,
k
=
1,2
,
…
,
n
,
where
(106)
ε
n
=
(
2
∑
k
=
1
n
(
k
+
1
)
-
1
‍
(
ln
⁡
⁡
(
k
+
1
)
)
-
(
1
+
σ
)
)
-
1
≍
1
,
n
⟶
+
∞
,
then
(107)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(108)
ω
(
f
a
n
;
π
n
)
p
p
=
2
ε
n
p
/
β
∑
k
=
1
n
(
(
k
+
1
)
-
p
/
β
(
ln
⁡
⁡
(
k
+
1
)
)
-
(
p
(
1
+
σ
)
/
β
)
)
‍
·
π
n
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
·
π
n
,
n
≥
10
.Equations (107) and (108) imply the sharpness of (104).
In (b), we have(109)
1
N
∫
0
N
d
z
ϕ
(
z
)
p
/
β
‍
≍
1
N
∫
1
N
d
z
(
ln ln
(
z
+
1
)
)
p
/
β
‍
≍
1
(
ln ln
N
)
p
/
β
,
N
⟶
+
∞
.
Again from Theorem 6(b), we have
(110)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
(
ln ln
1
t
)
-
1
/
β
.
If we choose
(111)
a
k
=
ε
n
1
/
β
(
ln ln
n
π
)
-
1
/
β
,
k
=
1,2
,
…
,
n
,
where
(112)
ε
n
=
ln ln
(
n
/
π
)
2
∑
k
=
1
n
‍
[
(
k
+
1
)
(
ln
⁡
⁡
(
k
+
1
)
)
]
-
1
≍
1
,
n
⟶
+
∞
,
then
(113)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(114)
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
n
ε
n
p
/
β
∑
k
=
1
n
(
ln ln
n
π
)
-
p
/
β
‍
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
(
ln ln
n
π
)
-
p
/
β
,
n
≥
10
.Equations (113) and (114) imply the sharpness of (110).
In (c), forβ
<
p
<
∞, we have
(115)
1
N
∫
0
N
d
z
ϕ
(
z
)
p
/
β
‍
≍
1
N
∫
1
N
d
z
(
ln
⁡
⁡
(
z
+
1
)
)
p
(
1
-
γ
)
/
β
‍
≍
1
(
ln
⁡
N
)
p
(
1
-
γ
)
/
β
,
N
⟶
∞
.
From Theorem 6(b), we get
(116)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
(
ln
⁡
1
t
)
-
(
1
-
γ
)
/
β
.
If we choose
(117)
a
k
=
ε
n
1
/
β
(
ln
⁡
n
π
)
-
(
1
-
γ
)
/
β
,
k
=
1,2
,
…
,
n
,
where
(118)
ε
n
=
(
ln
⁡
⁡
(
n
/
π
)
)
1
-
γ
2
∑
k
=
1
n
(
k
+
1
)
-
1
(
ln
⁡
⁡
(
k
+
1
)
)
-
γ
‍
≍
1
,
n
⟶
+
∞
,
then
(119)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(120)
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
ε
n
p
/
β
(
ln
⁡
n
π
)
-
p
(
1
-
γ
)
/
β
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
(
ln
⁡
n
π
)
-
p
(
1
-
γ
)
/
β
,
n
≥
10
.Equations (119) and (120) imply the sharpness of (116).
In (d), forβ
<
p
<
β
/
(
1
-
α
), we have
(121)
1
N
∫
0
N
‍
d
z
ϕ
(
z
)
p
/
β
≍
1
N
∫
1
N
‍
d
z
(
z
+
1
)
p
(
1
-
α
)
/
β
(
ln
⁡
⁡
(
z
+
1
)
)
p
γ
/
β
≍
1
N
p
(
1
-
α
)
/
β
(
ln
⁡
N
)
p
γ
/
β
,
N
⟶
∞
.
From Theorem 6(b), we get
(122)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
t
(
1
-
α
)
/
β
(
ln
⁡
1
t
)
-
γ
/
β
.
If we choose
(123)
a
k
=
ε
n
1
/
β
(
π
n
)
(
1
-
α
)
/
β
(
ln
⁡
n
π
)
-
γ
/
β
,
k
=
1,2
,
…
,
n
,
where
(124)
ε
n
=
(
n
/
π
)
1
-
α
(
ln
⁡
⁡
(
n
/
π
)
)
γ
2
∑
k
=
1
n
(
k
+
1
)
-
α
ln
⁡
γ
⁡
(
k
+
1
)
‍
≍
1
,
n
⟶
+
∞
,
then
(125)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(126)
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
ε
n
p
/
β
(
π
n
)
p
(
1
-
γ
)
/
β
(
ln
⁡
n
π
)
-
p
γ
/
β
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
(
π
n
)
p
(
1
-
γ
)
/
β
(
ln
⁡
n
π
)
-
p
γ
/
β
,
n
≥
10
.Equations (125) and (126) imply the sharpness of (122).
Forβ
/
(
1
-
α
)
≤
p
<
∞, β
-
p
(
1
-
α
)
≤
0, we have
(127)
ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
)
≍
(
z
+
1
)
β
-
p
(
1
-
α
)
/
β
(
ln
⁡
⁡
(
z
+
1
)
)
p
γ
/
β
<
+
∞
,
z
≥
1
.
From Theorem 6(c), we get
(128)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
t
1
/
p
.
If we choose
(129)
a
k
=
ε
n
1
/
β
(
k
+
1
)
-
(
1
-
α
+
σ
)
/
β
(
ln
⁡
(
k
+
1
)
)
-
γ
/
β
(
σ
>
0
)
,
k
=
1,2
,
…
,
n
,
where
(130)
ε
n
=
(
2
∑
k
=
1
n
‍
(
k
+
1
)
-
(
1
+
σ
)
)
-
1
≍
1
,
n
⟶
+
∞
,
then
(131)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(132)
ω
(
f
a
n
;
π
n
)
p
p
=
2
ε
n
p
/
β
∑
k
=
1
n
(
(
k
+
1
)
-
(
p
(
1
-
α
+
σ
)
/
β
)
(
ln
⁡
⁡
(
k
+
1
)
)
-
p
γ
/
β
)
‍
·
π
n
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
·
π
n
,
n
≥
10
.Equations (131) and (132) imply the sharpness of (128).
In (e), forβ
<
p
<
β
/
(
1
-
α
), we have
(133)
1
N
∫
0
N
d
z
ϕ
(
z
)
p
/
β
‍
≍
1
N
∫
1
N
(
ln
⁡
⁡
(
z
+
1
)
)
p
γ
/
β
(
z
+
1
)
p
(
1
-
α
)
/
β
‍
d
z
≍
(
ln
⁡
N
)
p
γ
/
β
N
p
(
1
-
α
)
/
β
,
N
⟶
∞
.
From Theorem 6(b), we get
(134)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
t
(
1
-
α
)
/
β
(
ln
⁡
1
t
)
γ
/
β
.
If we choose
(135)
a
k
=
ε
n
1
/
β
(
π
n
)
(
1
-
α
)
/
β
(
ln
⁡
n
π
)
γ
/
β
,
k
=
1,2
,
…
,
n
,
where
(136)
ε
n
=
(
n
/
π
)
1
-
α
(
ln
⁡
⁡
(
n
/
π
)
)
-
γ
2
∑
k
=
1
n
(
k
+
1
)
-
α
(
ln
⁡
⁡
(
k
+
1
)
)
-
γ
≍
1
n
⟶
+
∞
,
then
(137)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(138)
ω
(
f
a
n
;
π
n
)
p
p
=
2
π
ε
n
p
/
β
(
n
π
)
p
(
1
-
γ
)
/
β
(
ln
⁡
n
π
)
p
γ
/
β
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
(
n
π
)
p
(
1
-
γ
)
/
β
(
ln
⁡
n
π
)
p
γ
/
β
,
ss
n
≥
10
.Equations (137) and (138) imply the sharpness of (134).
Forβ
/
(
1
-
α
)
<
p
<
∞, β
-
p
(
1
-
α
)
<
0, we have
(139)
ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
)
≍
(
ln
⁡
⁡
(
z
+
1
)
)
p
γ
/
β
(
z
+
1
)
(
p
(
1
-
α
)
-
β
)
/
β
<
+
∞
,
z
≥
1
.
From Theorem 6(c), we get
(140)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
)
)
1
/
β
t
1
/
p
.
If we choose
(141)
a
k
=
ε
n
1
/
β
(
k
+
1
)
-
(
1
-
α
+
σ
)
/
β
(
ln
⁡
⁡
(
k
+
1
)
)
γ
/
β
(
σ
>
0
)
,
k
=
1,2
,
…
,
n
,
where
(142)
ε
n
=
(
2
∑
k
=
1
n
(
k
+
1
)
-
(
1
+
σ
)
‍
)
-
1
≍
1
,
n
⟶
+
∞
,
then
(143)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
(144)
ω
(
f
a
n
;
π
n
)
p
p
=
2
ε
n
p
/
β
∑
k
=
1
n
(
k
+
1
)
-
p
(
1
-
α
+
σ
)
/
β
(
ln
⁡
⁡
(
k
+
1
)
)
p
γ
/
β
‍
·
π
n
≥
c
(
V
Λ
β
(
f
a
n
)
)
p
/
β
·
π
n
,
n
≥
10
.Equations (143) and (144) imply the sharpness of (140).
But, forp
=
β
/
(
1
-
α
), we have
(145)
ϕ
(
z
)
(
β
-
p
)
/
β
λ
(
z
)
≍
(
ln
⁡
⁡
(
z
+
1
)
)
p
γ
/
β
⟶
+
∞
,
as
z
⟶
+
∞
,
1
N
∫
0
N
‍
d
z
ϕ
(
z
)
p
/
β
≍
1
N
∫
1
N
‍
(
ln
⁡
(
z
+
1
)
)
p
γ
/
β
z
+
1
d
z
≍
(
ln
⁡
N
)
1
+
p
γ
/
β
N
,
N
⟶
+
∞
.
Theorem 6(c) is not applicable for this case. From Theorem 6(b), we obtain
(146)
ω
(
f
;
t
)
p
≤
c
(
V
Λ
β
(
f
;
[
0,2
π
]
)
)
1
/
β
t
1
/
p
(
ln
⁡
1
t
)
1
/
p
+
γ
/
β
.
Unfortunately this estimate is not sharp in the sense of order. However, if we choose
(147)
a
k
=
ε
n
1
/
β
(
k
+
1
)
-
(
1
-
α
)
/
β
(
ln
⁡
⁡
(
k
+
1
)
)
-
(
1
-
γ
+
σ
)
/
β
(
σ
>
max
⁡
⁡
{
0
,
γ
-
α
}
)
,
k
=
1,2
,
…
,
n
,
where
(148)
ε
n
=
[
∑
k
=
1
n
(
k
+
1
)
-
1
(
ln
⁡
⁡
(
k
+
1
)
)
-
(
1
+
σ
)
‍
]
-
1
≍
1
,
n
⟶
+
∞
,
then we have
(149)
1
2
≤
V
Λ
β
(
f
a
n
)
≤
1
,
n
≥
10
,
ω
(
f
a
n
;
π
n
)
p
p
=
2
ε
n
p
/
β
∑
k
=
1
n
(
k
+
1
)
-
1
(
ln
⁡
⁡
(
k
+
1
)
)
-
(
1
-
γ
+
σ
)
/
(
1
-
α
)
‍
·
π
n
≥
c
·
π
n
,
n
≥
10
.
In other words, there exists f
n
∈
Λ
β
B
V such that
(150)
V
Λ
β
(
f
)
≍
1
,
n
⟶
+
∞
,
ω
(
f
n
;
π
n
)
p
≥
c
(
π
n
)
1
/
p
,
n
≥
10
.
This exception indicates that our methods used in this paper cannot cover all cases of estimates of L
p
(
1
≤
p
<
∞
) modulus of continuity of classes of functions of Λ
β-bounded variation.Proof of Corollary10.
Obviously Theorem2 implies (a) and (b). (c) follows from Theorem 6. Finally, (d) and (e) are obtained from Theorem 7 directly.



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*Source: 201801-2014-05-04.xml*