# Two-Weight Norm Inequality for the One-Sided Hardy-Littlewood Maximal Operators in Variable Lebesgue Spaces

Authors: Caiyin Niu; Zongguang Liu; Panwang Wang
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

The authors establish the two-weight norm inequalities for the one-sided Hardy-Littlewood maximal operators in variable Lebesgue spaces. As application, they obtain the two-weight norm inequalities of variable Riemann-Liouville operator and variable Weyl operator in variable Lebesgue spaces on bounded intervals.


---

## Body

## 1. Introduction and Main Results

The one-sided Hardy-Littlewood maximal operatorsM
+ and M
- are defined by(1)
M
+
f
x
=
sup
h
>
0
⁡
1
h
∫
x
x
+
h
f
t
d
t
,
M
-
f
x
=
sup
h
>
0
⁡
1
h
∫
x
-
h
x
f
t
d
t
.Sawyer [1] showed that M
+ is bounded from L
p
(
v
) to L
p
(
u
) if the pairs of nonnegative functions (
u
,
v
) satisfy Sawyer-type two-weight condition for the one-sided maximal operator. In [2] Martín-Reyes et al. generalized this result to M
g
+, where(2)
M
g
+
f
x
=
sup
h
>
0
⁡
1
∫
x
x
+
h
g
t
d
t
∫
x
x
+
h
f
t
g
t
d
twith g is positive locally integrable function. The similar results are also true for M
- and M
g
- (see [1–3]).LetE be a measurable set in R. Given a measurable function p
(
·
)
:
E
→
(
0
,
+
∞
), we denote(3)
p
E
-
=
ess
inf
x
∈
E
⁡
p
x
,
p
E
+
=
ess
sup
x
∈
E
⁡
p
x
.If E
=
R, we will simply note that p
-
=
p
R
- and p
+
=
p
R
+
.Definition 1.
Givenp
(
·
)
:
E
→
[
1
,
+
∞
), σ is a locally integrable function such that 0
<
σ
(
x
)
<
∞
a
.
e
.
x
∈
E. We say that p
(
·
)
∈
P
σ
+
(
E
) if there exists a constant C
>
0 such that for every (
x
,
x
+
h
)
⊂
E, with ∫
x
x
+
h
σ
(
t
)
d
t
⩽
1
/
2,(4)
p
x
-
p
x
,
x
+
h
-
⩽
C
-
ln
⁡
∫
x
x
+
h
σ
t
d
t
.Definition 2 (see [4]).
It is given thatp
(
·
)
:
E
→
[
1
,
+
∞
)
. We say that p
(
·
)
∈
L
H
∞
(
E
), if there exist constants p
∞
⩾
1 and C
∞
>
0 such that for all x
∈
E
(5)
p
x
-
p
∞
⩽
C
∞
ln
⁡
e
+
x
.Let1
⩽
p
E
-
⩽
p
E
+
<
∞
. The variable Lebesgue space L
p
(
·
)
(
E
) is the set of measurable functions f on E such that(6)
ρ
p
·
,
E
f
≔
∫
E
f
x
p
x
d
x
<
∞
.This is a Banach space (see [4–7]) with the norm(7)
f
L
p
·
E
=
inf
⁡
λ
>
0
:
ρ
p
·
,
E
f
λ
⩽
1
.If E
=
R, we will write f
L
p
(
·
)
(
E
) as f
p
(
·
). The variable Lebesgue space is the special case of the Musielak-Orlicz space (see [7, 8]). For the detail of L
p
(
·
) we refer to [4–7] and so on.Letw be a nonnegative locally integrable function on E. The weighted variable Lebesgue space L
w
p
(
·
)
(
E
) is the set of measurable functions f on E such that f
w
∈
L
p
(
·
)
(
E
) and f
L
w
p
(
·
)
(
E
)
=
f
w
L
p
(
·
)
(
E
). When p
(
·
)
=
p is a constant, L
w
p
(
·
)
(
E
) coincides with the classical weighted Lebesgue space L
p
(
w
p
)
(
E
).Edmunds et al. [9] investigated the boundedness of M
+ and M
- in the variable Lebesgue spaces L
p
(
·
). The two-weight weak type modular inequalities of M
+ and M
- in L
p
(
·
) were discussed in [10]. In [11], Kokilashvili et al. acquired the sufficient condition such that M
+ and M
- are bounded from L
w
p
(
·
)
(
R
+
) to L
v
p
(
·
)
(
R
+
), where p
(
·
) is constant on some interval (
a
,
+
∞
)
(
a
>
0
) and T
v
,
w is bounded in L
p
(
·
)
(
R
+
) with(8)
T
v
,
w
f
x
=
v
x
∫
x
+
∞
f
y
y
w
y
d
y
,
x
∈
R
+
.Throughout this paper,u and v are nonnegative locally integrable functions and C is a positive constant whose value may change from one occurrence to the next. For exponent function p
(
·
) with p
(
x
)
⩾
1, its conjugate exponent will be denoted by p
′
(
·
) with 1
/
p
(
x
)
+
1
/
p
′
(
x
)
=
1. For a Lebesgue measurable set E
⊂
R, χ
E will be its characteristic function.Definition 3.
It is given thatp
(
·
)
:
E
→
[
1
,
+
∞
) such that 1
⩽
p
E
-
⩽
p
E
+
<
∞. We say that (
u
,
v
)
∈
S
p
(
·
)
+
(
E
) if there exists a constant C
>
0 such that for every interval I
=
(
a
,
b
)
⊂
E, with ∫
-
∞
a
u
(
x
)
p
(
x
)
d
x
>
0,(9)
∫
I
M
+
χ
I
σ
x
p
x
u
x
p
x
d
x
⩽
C
∫
I
σ
x
d
x
<
+
∞
,where v
(
x
)
>
0
a
.
e
.
x
∈
E and σ
(
x
)
=
v
(
x
)
-
p
′
(
x
).Definition 4.
It is given thatp
(
·
)
:
E
→
[
1
,
+
∞
) such that 1
⩽
p
E
-
⩽
p
E
+
<
∞. We say that (
u
,
v
)
∈
S
p
(
·
)
′
+
(
E
) if there exist constants m
⩾
1
/
p
E
- and C
1
,
C
2
>
0 such that for every interval I
=
(
a
,
b
)
⊂
E, with ∫
-
∞
a
u
(
x
)
p
(
x
)
d
x
>
0,(10)
∫
I
M
+
χ
I
σ
x
p
x
u
x
p
x
e
+
x
m
p
E
-
d
x
⩽
C
1
∫
I
σ
x
e
+
x
m
p
E
-
d
x
⩽
C
2
,where v
(
x
)
>
0
a
.
e
.
x
∈
E and σ
(
x
)
=
v
(
x
)
-
p
′
x.S
p
(
·
)
+
(
E
) and S
p
(
·
)
′
+
(
E
) can be considered to be the generalization of Sawyer-type two-weight condition (see [1]) for the one-sided maximal operator in variable exponents case. If E
=
R we will simply write S
p
(
·
)
+
(
R
) as S
p
(
·
)
+ and S
p
(
·
)
′
+
(
R
) as S
p
(
·
)
′
+. We can define p
(
·
)
∈
P
σ
-
(
E
), (
u
,
v
)
∈
S
p
(
·
)
-
(
E
), and (
u
,
v
)
∈
S
p
(
·
)
′
-
(
E
) in similar ways.Our main results are the following theorems.Theorem 5.
LetD
=
(
-
∞
,
d
), d
<
+
∞, and p
(
·
)
:
D
→
[
1
,
+
∞
) such that 1
<
p
D
-
⩽
p
D
+
<
+
∞. If (
u
,
v
)
∈
(
S
p
(
·
)
+
(
D
)
∩
S
p
(
·
)
′
+
(
D
)
) and p
(
·
)
∈
(
P
σ
+
(
D
)
∩
L
H
∞
(
D
)
) with σ
(
x
)
=
v
(
x
)
-
p
′
(
x
), then (11)
M
+
f
u
L
p
·
D
⩽
C
f
v
L
p
·
D
.Theorem 6.
LetD
=
(
d
,
+
∞
), d
>
-
∞, and p
(
·
)
:
D
→
[
1
,
+
∞
) such that 1
<
p
D
-
⩽
p
D
+
<
+
∞. If (
u
,
v
)
∈
(
S
p
(
·
)
+
(
D
)
∩
S
p
(
·
)
′
+
(
D
)
) and p
(
·
)
∈
(
P
σ
+
(
D
)
∩
L
H
∞
(
D
)
) with σ
(
x
)
=
v
(
x
)
-
p
′
(
x
), then (12)
M
+
f
u
L
p
·
D
⩽
C
f
v
L
p
·
D
.Theorem 7.
LetD be a bounded interval and let p
(
·
)
:
D
→
[
1
,
+
∞
) such that 1
<
p
D
-
⩽
p
D
+
<
+
∞. If (
u
,
v
)
∈
S
p
(
·
)
+
(
D
) and p
(
·
)
∈
P
σ
+
(
D
) with σ
(
x
)
=
v
(
x
)
-
p
′
(
x
), then (13)
M
+
f
u
L
p
·
D
⩽
C
f
v
L
p
·
D
.Corollary 8.
It is given thatp
(
·
)
:
R
→
[
1
,
+
∞
) such that 1
<
p
-
⩽
p
+
<
+
∞. If there exists a constant c
>
0 such that (
u
,
v
)
∈
(
S
p
(
·
)
+
∩
S
p
(
·
)
′
+
(
R
∖
[
-
c
,
c
]
)
) and p
(
·
)
∈
(
P
σ
+
(
R
)
∩
L
H
∞
(
R
∖
[
-
c
,
c
]
)
) with σ
(
x
)
=
v
(
x
)
-
p
′
(
x
), then(14)
M
+
f
u
p
·
⩽
C
f
v
p
·
.Corollary 9.
It is given thatp
(
·
)
:
R
→
[
1
,
+
∞ such that 1
<
p
-
⩽
p
+
<
+
∞, (
u
,
v
)
∈
S
p
(
·
)
+, and p
(
·
)
∈
P
σ
+
(
R
) with σ
(
x
)
=
v
(
x
)
-
p
′
(
x
). If(a)
there exists a constantc
>
0 such that p
(
·
)
∈
L
H
∞
(
R
∖
[
-
c
,
c
]
) and p
(
x
)
⩾
p
∞ for x
∈
(
R
∖
[
-
c
,
c
]
),
(b)
there exist constantsm
⩾
1
/
p
- and C
>
0 such that for every interval I
=
(
a
,
b
)
⊂
(
R
∖
[
-
c
,
c
]
), with ∫
-
∞
a
u
(
x
)
p
(
x
)
d
x
>
0,(15)
∫
I
σ
x
e
+
x
m
p
-
d
x
⩽
C
.Then(16)
M
+
f
u
p
·
⩽
C
f
v
p
·
.Remark 10.
In Theorems5 and 6, the set S
p
·
+
D
∩
S
p
(
·
)
′
+
(
D
) is not empty. In fact, if v
(
x
)
⩾
1 and 0
⩽
u
(
x
)
⩽
v
(
x
)
1
/
1
-
p
(
x
), when x
∈
D, then (
u
,
v
)
∈
(
S
p
·
+
(
D
)
∩
S
p
(
·
)
′
+
(
D
)
).Remark 11.
If we change the conditionsS
p
(
·
)
+, S
p
(
·
)
′
+, P
σ
+ to S
p
(
·
)
-, S
p
(
·
)
′
-, and P
σ
-, respectively, in above theorems, we will obtain similar results of M
-.Remark 12.
If we takeu
(
x
)
=
1 and v
(
x
)
=
1 for a
.
e
.
x
∈
D whenever D is an open interval, then P
σ
+
(
D
)
=
P
1
+
(
D
)
=
P
+
(
D
) (see Definition 18 below) and the results of this paper coincide with those of [9].Remark 13.
The Sawyer-type condition was used earlier in [12] to characterize the two-weight boundedness of the classical Hardy-Littlewood maximal operator M. Corresponding results for variable Lebesgue spaces can be found in [13–15].


## 2. Proof of the Main Results

In order to establish our main results, we will need following lemmas.Lemma 14 (see [4]).
It is given thatE and p
(
·
)
:
E
→
[
1
,
+
∞
) such that p
E
+
<
∞. (a)
Iff
L
p
(
·
)
(
E
)
⩽
1, then ρ
p
·
,
E
f
1
/
p
E
-
⩽
f
L
p
(
·
)
(
E
)
⩽
ρ
p
(
·
)
,
E
f
1
/
p
E
+.
(b)
Iff
L
p
(
·
)
(
E
)
>
1, then ρ
p
·
,
E
f
1
/
p
E
+
⩽
f
L
p
(
·
)
(
E
)
⩽
ρ
p
(
·
)
,
E
f
1
/
p
E
-.In particular, if f
L
p
(
·
)
(
E
)
⩽
1, then ρ
p
(
·
)
,
E
(
f
)
⩽
1
.Lemma 15 (see [4]).
Ifp
E
+
<
∞, the set of bounded functions with compact support is dense in L
p
(
·
)
(
E
).Lemma 16 (see [1]).
Supposeg
⩾
0 is integrable function with compact support on R. If I
=
(
a
,
b
) is a component interval of the open set {
x
∈
R
:
M
+
g
(
x
)
>
λ
} with λ
>
0, then (17)
1
b
-
x
∫
x
b
g
t
d
t
⩾
λ
f
o
r
a
⩽
x
<
b
.Lemma 17 (see [16]).
It is given that a setG
⊆
R and two exponents s
(
·
) and r
(
·
) such that (18)
s
y
-
r
y
⩽
C
0
ln
⁡
e
+
y
.Then for every t
⩾
1 there exists a constant C
=
C
(
t
,
C
0
) such that for all functions f with f
y
⩽
1,(19)
∫
G
f
y
s
y
d
μ
y
⩽
C
∫
G
f
y
r
y
d
μ
y
+
∫
G
1
e
+
y
t
r
G
-
d
μ
y
,where μ is a given nonnegative measure.Proof of Theorem5.
By the homogeneity, Lemma15, the Fatou lemma for variable Lebesgue spaces [4] and Lemma 14, we only need to prove(20)
∫
-
∞
d
M
+
f
x
p
x
u
x
p
x
d
x
⩽
Cfor the nonnegative bounded function f with compact support and f
v
L
p
(
·
)
(
D
)
=
1. Let N
>
d be a positive integer and for k
∈
Z, define (21)
Ω
k
,
N
=
x
∈
R
:
M
+
f
x
>
2
k
∩
-
N
,
d
.Obviously each Ω
k
,
N is a bounded open set. Let I
j
k
=
(
a
j
k
,
b
j
k
) be the component intervals of Ω
k
,
N, where j is an integer. Applying Lemma 16 to a fixed I
j
k, we have(22)
1
b
j
k
-
x
∫
x
b
j
k
f
t
d
t
⩾
2
k
for
x
∈
a
j
k
,
b
j
k
.Let E
j
k
=
I
j
k
∖
Ω
k
+
1
,
N, then the sets E
j
k are pairwise disjoint and for every k
(23)
⋃
j
E
j
k
=
x
∈
R
:
2
k
<
M
+
f
x
⩽
2
k
+
1
∩
-
N
,
d
.Therefore(24)
∫
-
N
d
M
+
f
x
p
x
u
x
p
x
d
x
=
∑
k
,
j
∫
E
j
k
M
+
f
x
p
x
u
x
p
x
d
x
⩽
∑
k
,
j
∫
E
j
k
2
k
+
1
p
x
u
x
p
x
d
x
⩽
2
p
D
+
∑
k
,
j
∫
E
j
k
1
b
j
k
-
x
∫
x
b
j
k
f
t
d
t
p
x
u
x
p
x
d
x
=
2
p
D
+
∑
k
,
j
∫
-
N
d
A
j
k
f
σ
-
1
x
p
x
φ
j
k
x
d
x
, where(25)
A
j
k
g
x
=
1
∫
x
b
j
k
σ
t
d
t
∫
x
b
j
k
g
t
σ
t
d
t
χ
E
j
k
x
,
φ
j
k
x
=
1
b
j
k
-
x
∫
x
b
j
k
σ
t
d
t
p
x
u
x
p
x
χ
E
j
k
x
.We show that for every j and k the inequality(26)
A
j
k
g
x
p
x
⩽
C
A
j
k
g
p
·
/
p
D
-
x
p
D
-
+
A
j
k
R
x
p
D
-
+
R
x
p
D
-is valid, where g
=
f
σ
-
1, R
(
x
)
=
C
/
e
+
x
m (m
⩾
1
/
p
D
- is the same as that in (10)), and C is independent of j and k.
Letq
(
x
)
=
p
(
x
)
/
p
D
-, then q
(
·
)
∈
(
P
σ
+
(
D
)
∩
L
H
∞
(
D
)
). By f
v
L
p
(
·
)
(
D
)
=
1 and Lemma 14 we get(27)
∫
D
g
x
p
x
σ
x
d
x
=
∫
D
f
x
p
x
v
x
p
x
d
x
⩽
1
. Let g
(
x
)
=
g
1
(
x
)
+
g
2
(
x
), where(28)
g
1
x
=
g
x
χ
x
∈
D
:
g
x
>
1
x
,
g
2
x
=
g
x
χ
x
∈
D
:
g
x
⩽
1
x
,then(29)
A
j
k
g
x
p
x
⩽
2
p
D
+
-
1
A
j
k
g
1
x
p
x
+
A
j
k
g
2
x
p
x
.Since g
1
(
t
)
⩾
1, by (27) we can get(30)
∫
x
b
j
k
g
1
t
σ
t
d
t
⩽
∫
D
g
1
t
p
t
σ
t
d
t
⩽
1for x
∈
E
j
k. Then(31)
A
j
k
g
1
x
q
x
⩽
∫
x
b
j
k
σ
t
d
t
-
q
x
∫
x
b
j
k
g
1
t
σ
t
d
t
q
x
,
b
j
k
-
=
∫
x
b
j
k
σ
t
d
t
-
q
x
-
q
x
,
b
j
k
-
1
∫
x
b
j
k
σ
t
d
t
∫
x
b
j
k
g
1
t
σ
t
d
t
q
x
,
b
j
k
-
.If ∫
x
b
j
k
σ
(
t
)
d
t
⩽
1
/
2, by (4), we have(32)
∫
x
b
j
k
σ
t
d
t
-
q
x
-
q
x
,
b
j
k
-
⩽
∫
x
b
j
k
σ
t
d
t
C
/
ln
⁡
∫
x
b
j
k
σ
t
d
t
⩽
C
.If ∫
x
b
j
k
σ
(
t
)
d
t
>
1
/
2, then(33)
∫
x
b
j
k
σ
t
d
t
-
q
x
-
q
x
,
b
j
k
-
⩽
2
q
x
-
q
x
,
b
j
k
-
⩽
2
q
D
+
⩽
C
.Therefore(34)
A
j
k
g
1
x
p
x
⩽
C
1
∫
x
b
j
k
σ
t
d
t
∫
x
b
j
k
g
1
t
σ
t
d
t
q
x
,
b
j
k
-
p
D
-
.Thus, by the Hölder inequality and g
1
(
t
)
⩾
1 we can get(35)
A
j
k
g
1
x
p
x
⩽
C
1
∫
x
b
j
k
σ
t
d
t
∫
x
b
j
k
g
1
t
q
x
,
b
j
k
-
σ
t
d
t
p
D
-
⩽
C
1
∫
x
b
j
k
σ
t
d
t
∫
x
b
j
k
g
1
t
q
t
σ
t
d
t
p
D
-
⩽
C
A
j
k
g
p
·
/
p
D
-
x
p
D
-
. Next, we will estimate A
j
k
g
2
(
x
). If A
j
k
g
2
(
x
)
⩽
R
(
x
), by R
(
x
)
⩽
1, we have(36)
A
j
k
g
2
x
q
x
⩽
R
x
q
x
⩽
R
x
. On the other hand, if A
j
k
g
2
(
x
)
>
R
(
x
)
>
0, noticing that A
j
k
g
2
(
x
)
⩽
1 and(37)
q
x
-
q
∞
⩾
-
q
x
-
q
∞
for
x
∈
R
,thus(38)
0
<
A
j
k
g
2
x
q
x
-
q
∞
⩽
A
j
k
g
2
x
-
q
x
-
q
∞
⩽
R
x
-
q
x
-
q
∞
.Applying (38), q
(
·
)
∈
L
H
∞
(
D
) and the Hölder inequality,(39)
A
j
k
g
2
x
q
x
⩽
A
j
k
g
2
x
-
q
x
-
q
∞
A
j
k
g
2
x
q
∞
⩽
R
x
-
q
x
-
q
∞
1
∫
x
b
j
k
σ
t
d
t
∫
x
b
j
k
g
2
t
σ
t
d
t
q
∞
⩽
C
∫
x
b
j
k
σ
t
d
t
-
1
∫
x
b
j
k
g
2
t
q
∞
σ
t
d
t
.By 0
⩽
g
2
(
t
)
⩽
1 and Lemma 17, we get(40)
A
j
k
g
2
x
q
x
⩽
C
∫
x
b
j
k
σ
t
d
t
-
1
∫
x
b
j
k
g
2
y
q
y
σ
y
d
y
+
∫
x
b
j
k
σ
y
e
+
y
m
q
x
,
b
j
k
-
d
y
⩽
C
A
j
k
g
2
q
·
x
+
A
j
k
R
x
.Hence, combining (36) and (40)(41)
A
j
k
g
2
x
p
x
⩽
C
A
j
k
g
2
q
·
x
+
A
j
k
R
x
+
R
x
p
D
-
⩽
C
A
j
k
g
p
·
/
p
D
-
x
p
D
-
+
A
j
k
R
x
p
D
-
+
R
x
p
D
-
.This completes the proof of (26) by (29), (35), and (41). Applying (26) to (24), we obtain(42)
∫
-
N
d
M
+
f
x
p
x
u
x
p
x
d
x
⩽
C
∑
k
,
j
∫
-
N
d
A
j
k
f
σ
-
1
p
·
/
p
D
-
x
p
D
-
φ
j
k
x
d
x
+
C
∑
k
,
j
∫
-
N
d
A
j
k
R
x
p
D
-
φ
j
k
x
d
x
+
C
∑
k
,
j
∫
-
N
d
R
x
p
D
-
φ
j
k
x
d
x
≕
I
1
+
I
2
+
I
3
. Let T be the following linear operator:(43)
T
h
x
=
∑
k
,
j
A
j
k
h
xand φ
(
x
)
=
∑
k
,
j
φ
j
k
(
x
). Since f
v
L
p
(
·
)
(
D
)
=
1, if we show that the operator T is bounded from L
p
D
-
(
(
-
N
,
d
)
,
σ
d
x
) to L
p
D
-
(
(
-
N
,
d
)
,
φ
d
x
), we obtain(44)
I
1
=
C
∫
-
N
d
T
f
σ
-
1
p
·
/
p
D
-
x
p
D
-
φ
x
d
x
⩽
C
∫
-
N
d
f
x
σ
x
-
1
p
x
σ
x
d
x
⩽
C
∫
D
f
x
p
x
v
x
p
x
d
x
⩽
C
.It is easy to see that T is bounded on L
∞
(
D
). Therefore, we only need to show that T is bounded from L
1
(
(
-
N
,
d
)
,
σ
d
x
) to L
1
,
∞
(
(
-
N
,
d
)
,
φ
d
x
) to complete our proof by the Marcinkiewicz interpolation theorem. For arbitrary λ
>
0 and each pair (
j
,
k
), define(45)
F
j
k
λ
≔
E
j
k
∩
x
∈
-
N
,
d
:
T
h
x
>
λ
.Obviously, F
j
k
(
λ
) are pairwise disjoint. Let c
j
k
(
λ
)
≔
i
n
f
F
j
k
(
λ
) and J
j
k
=
J
j
k
(
λ
)
≔
[
c
j
k
(
λ
)
,
b
j
k
). It is clear that any two intervals of J
j
k are disjoint or one is contained in the other. By the definition of J
j
k, we also have(46)
∫
J
j
k
h
t
σ
t
d
t
⩾
λ
∫
J
j
k
σ
t
d
t
.Let J
i be the maximal elements of the family J
j
k. These maximal elements exist since the intervals J
j
k have uniformly bounded length. The intervals J
i also satisfy (46). Then, by (9) and (46), we obtain (47)
∫
x
∈
-
N
,
d
:
T
h
x
>
λ
φ
x
d
x
=
∑
i
∑
k
,
j
:
J
i
⊃
J
j
k
∫
F
j
k
λ
1
b
j
k
-
x
∫
x
b
j
k
χ
J
i
t
σ
t
d
t
p
x
u
x
p
x
d
x
⩽
∑
i
∫
J
i
M
+
χ
J
i
σ
x
p
x
u
x
p
x
d
x
⩽
C
∑
i
∫
J
i
σ
x
d
x
⩽
C
∑
i
1
λ
∫
J
i
h
x
σ
x
d
x
⩽
C
λ
∫
-
N
d
h
x
σ
x
d
x
.This has proved the weak (
1,1
) inequality for T. Hence the estimate for I
1 is completed.
Since linear operatorT is bounded from L
p
D
-
(
(
-
N
,
d
)
,
σ
d
x
) to L
p
D
-
(
(
-
N
,
d
)
,
φ
d
x
), by the second inequality of (10),(48)
I
2
=
C
∫
-
N
d
T
R
x
p
D
-
φ
x
d
x
⩽
C
∫
-
N
d
R
x
p
D
-
σ
x
d
x
⩽
C
∫
D
σ
x
e
+
x
m
p
D
-
d
x
⩽
C
.Similarly, by (10), we have(49)
I
3
=
C
∑
k
,
j
∫
-
N
d
1
e
+
x
m
p
D
-
1
b
j
k
-
x
∫
x
b
j
k
σ
t
d
t
p
x
u
x
p
x
χ
E
j
k
x
d
x
⩽
C
∫
-
N
d
M
+
σ
χ
-
N
,
d
x
p
x
u
x
p
x
e
+
x
m
p
D
-
d
x
⩽
C
∫
-
N
d
R
x
p
D
-
σ
x
d
x
⩽
C
. Therefore, by (42), (44), (48), and (49), we have(50)
∫
-
N
d
M
+
f
x
p
x
u
x
p
x
d
x
⩽
C
,where C is independent of N. Let N tend to infinity and the proof of Theorem 5 is finished.Theorem6 can be proved similarly.Proof of Theorem7.
We can assumeD to be a bounded open interval and f to be nonnegative with f
v
L
p
(
·
)
(
D
)
=
1. It is sufficient to prove that(51)
∫
D
M
+
f
x
p
x
u
x
p
x
d
x
⩽
C
.For k
∈
Z, define(52)
Ω
k
=
x
∈
D
:
M
+
f
x
>
2
k
.Let I
j
k
=
(
a
j
k
,
b
j
k
) be the component intervals of Ω
k and E
j
k
=
I
j
k
∖
Ω
k
+
1, where j is an integer. Using the same procedure as (24), we obtain(53)
∫
D
M
+
f
x
p
x
u
x
p
x
d
x
⩽
2
p
+
∑
k
,
j
∫
D
A
j
k
f
σ
-
1
x
p
x
φ
j
k
x
d
x
,where(54)
A
j
k
g
x
=
1
∫
x
b
j
k
σ
t
d
t
∫
x
b
j
k
g
t
σ
t
d
t
χ
E
j
k
x
,
φ
j
k
x
=
1
b
j
k
-
x
∫
x
b
j
k
σ
t
d
t
p
x
u
x
p
x
χ
E
j
k
x
.Let g
(
x
)
=
f
(
x
)
σ
-
1
(
x
)
,
g
1
(
x
)
=
g
(
x
)
χ
{
x
∈
D
:
g
(
x
)
>
1
}
(
x
) and g
2
(
x
)
=
g
(
x
)
χ
{
x
∈
D
:
g
(
x
)
⩽
1
}
(
x
). The estimate for A
j
k
g
1
(
x
) is the same as (35). Since A
j
k
g
2
(
x
)
⩽
1, we have(55)
A
j
k
g
x
p
x
⩽
C
A
j
k
g
1
x
p
x
+
A
j
k
g
2
x
p
x
⩽
C
A
j
k
g
p
·
/
p
D
-
x
p
D
-
+
1
.Combining (53) and (55), we obtain(56)
∫
D
M
+
f
x
p
x
u
x
p
x
d
x
⩽
C
∑
k
,
j
∫
D
A
j
k
f
σ
-
1
p
·
/
p
D
-
x
p
D
-
φ
j
k
x
d
x
+
C
∑
k
,
j
∫
D
φ
j
k
x
d
x
≕
I
4
+
I
5
.Similarly as I
1 we can estimate I
4 to get I
4
⩽
C. Applying (9) to I
5, we have(57)
I
5
=
C
∑
k
,
j
∫
D
1
b
j
k
-
x
∫
x
b
j
k
σ
t
d
t
p
x
u
x
p
x
χ
E
j
k
x
d
x
⩽
C
∫
D
M
+
σ
χ
D
x
p
x
u
x
p
x
d
x
⩽
C
∫
D
σ
x
d
x
⩽
C
.The Corollary8 can be obtained by the results of Theorems 5, 6, and 7 directly.Proof of Corollary9.
Without loss of generality, we can assume thatf is nonnegative and bounded with a compact support and f
v
p
(
·
)
=
1. Let H
=
R
∖
[
-
c
,
c
]. Due to Theorem 7 we only need to prove(58)
∫
H
M
+
f
x
p
x
u
x
p
x
d
x
⩽
C
.Let N
>
c be a positive integer and D
=
(
-
N
,
c
)
∪
(
c
,
N
). For k
∈
Z, define(59)
Ω
k
=
x
∈
H
:
M
+
f
x
>
2
k
∩
D
.Let I
j
k
=
(
a
j
k
,
b
j
k
) be the component intervals of Ω
k and E
j
k
=
I
j
k
∖
Ω
k
+
1, where j is an integer. Using the same procedure as (24), we obtain(60)
∫
D
M
+
f
x
p
x
u
x
p
x
d
x
⩽
2
p
+
∑
k
,
j
∫
D
A
j
k
f
σ
-
1
x
p
x
φ
j
k
x
d
x
, where A
j
k and φ
j
k have the similar definitions as those in the proof of Theorem 5. Let g
(
x
)
=
f
(
x
)
σ
-
1
(
x
)
,
g
1
(
x
)
=
g
(
x
)
χ
{
x
∈
D
:
g
(
x
)
>
1
}
(
x
) and g
2
(
x
)
=
g
(
x
)
χ
{
x
∈
D
:
g
(
x
)
⩽
1
}
(
x
). The estimate for A
j
k
g
1
(
x
) is also the same as (35). Let q
(
x
)
=
p
(
x
)
/
p
-, then q
(
x
)
∈
L
H
∞
(
H
) and q
(
x
)
⩾
q
∞
=
p
∞
/
p
- for x
∈
H. Since A
j
k
g
2
(
x
)
⩽
1 for every j and k, by the Hölder inequality and Lemma 17,(61)
A
j
k
g
2
x
q
x
⩽
A
j
k
g
2
x
q
∞
⩽
C
∫
x
b
j
k
σ
t
d
t
-
1
∫
x
b
j
k
g
2
t
q
∞
σ
t
d
t
⩽
C
∫
x
b
j
k
σ
t
d
t
-
1
∫
x
b
j
k
g
2
t
q
t
σ
t
d
t
+
∫
x
b
j
k
σ
t
e
+
t
m
q
x
,
b
j
k
-
d
t
⩽
C
A
j
k
g
2
q
·
x
+
A
j
k
e
+
x
-
m
x
.By (35) and (61), we get(62)
A
j
k
g
x
p
x
⩽
C
A
j
k
g
1
x
q
x
+
A
j
k
g
2
x
q
x
p
-
⩽
C
A
j
k
g
p
·
/
p
-
x
p
-
+
A
j
k
e
+
x
-
m
x
p
-
. Therefore we have(63)
∫
D
M
+
f
x
p
x
u
x
p
x
d
x
⩽
C
∑
k
,
j
∫
D
A
j
k
f
σ
-
1
p
·
/
p
-
x
p
-
φ
j
k
x
d
x
+
C
∑
k
,
j
∫
D
A
j
k
e
+
x
-
m
x
p
-
φ
j
k
x
d
x
≕
I
6
+
I
7
. By the similar estimates of I
1 and I
2, we get I
6
⩽
C and(64)
I
7
⩽
C
∫
D
σ
x
e
+
x
m
p
-
d
x
⩽
C
∫
-
N
-
c
σ
x
e
+
x
m
p
-
d
x
+
∫
c
N
σ
x
e
+
x
m
p
-
d
x
⩽
C
.Let N tend to infinity and the proof is complete.


## 3. Applications

In this section, we assume thatI
=
(
0
,
b
) with 0
<
b
<
+
∞
. Let(65)
R
α
·
f
x
=
∫
0
x
f
t
x
-
t
α
x
-
1
d
t
,
x
∈
0
,
b
,
W
α
·
f
x
=
∫
x
b
f
t
x
-
t
α
x
-
1
d
t
,
x
∈
0
,
b
,where 0
<
α
(
x
)
<
1. If α
(
x
) is a constant function, R
α is the classical Riemann-Liouville operator and W
α is the classical Weyl operator. In [17], Andersen and Sawyer obtained the two weighted norm inequalities of R
α and W
α from L
p
(
R
,
u
d
x
) to L
q
(
R
,
v
d
x
). Other results about R
α and W
α on L
p can be seen in [18–21] and so forth.Edmunds et al. [9] studied the boundedness of R
α, W
α, R
α
(
·
), and W
α
(
·
) in variable Lebesgue spaces. Kokilashvili et al. [11] discussed the two weighted norm inequalities of R
α and W
α from L
w
p
(
·
)
(
R
+
) to L
v
q
(
·
)
(
R
+
). In this section, we will discuss the two-weight inequalities of R
α
(
·
) and W
α
(
·
) in L
p
(
·
)
(
I
).Definition 18 (see [9]).
Givenp
(
·
)
:
E
→
[
1
,
+
∞
), we say that p
(
·
)
∈
P
-
(
E
) if there exists a constant C
>
0 such that for all x
,
y
∈
E, with 0
<
x
-
y
⩽
1
/
2, (66)
p
x
-
p
y
⩽
C
-
ln
⁡
x
-
y
.We also say that p
(
·
)
∈
P
+
(
E
) if there exists a constant C
>
0 such that for all x
,
y
∈
E with 0
<
y
-
x
⩽
1
/
2, (67)
p
x
-
p
y
⩽
C
-
ln
⁡
y
-
x
.Our results in this section are the following theorems:Theorem 19.
It is given thatp
(
·
)
:
I
→
[
1
,
+
∞
) such that 1
<
p
I
-
⩽
p
I
+
<
+
∞ and 0
<
α
I
-
⩽
α
(
x
)
<
1
/
p
I
+. If (
u
q
(
·
)
/
p
(
·
)
,
v
)
∈
S
p
(
·
)
+
(
I
) and p
(
·
)
∈
P
σ
+
(
I
)
∩
P
+
(
I
), where σ
(
x
)
=
v
(
x
)
-
p
′
(
x
) and q
(
x
)
=
p
(
x
)
/
1
-
α
(
x
)
p
(
x
), then(68)
W
α
·
f
u
L
q
·
I
⩽
C
f
v
L
p
·
I
.Theorem 20.
Givenp
(
·
)
:
I
→
[
1
,
+
∞
) such that 1
<
p
I
-
⩽
p
I
+
<
+
∞ and 0
<
α
I
-
⩽
α
(
x
)
<
1
/
p
I
+. If (
u
q
(
·
)
/
p
(
·
)
,
v
)
∈
S
p
(
·
)
-
(
I
) and p
(
·
)
∈
P
σ
-
(
I
)
∩
P
-
(
I
), where σ
(
x
)
=
v
(
x
)
-
p
′
(
x
) and q
(
x
)
=
p
(
x
)
/
1
-
α
(
x
)
p
(
x
), then (69)
R
α
·
f
u
L
q
·
I
⩽
C
f
v
L
p
·
I
.Remark 21.
The setsP
σ
+
(
I
)
∩
P
+
(
I
) and P
σ
-
(
I
)
∩
P
-
(
I
) are not empty. Let p
(
·
)
∈
P
+
(
I
) (resp., p
(
·
)
∈
P
-
(
I
)). If σ is lower Ahlfors Q-regular (Q
>
0), which means there exists a constant C
=
C
(
Q
)
>
0 such that ∫
a
b
σ
(
t
)
d
t
⩾
C
(
b
-
a
)
Q for every bounded interval (
a
,
b
)
⊂
I, then p
(
·
)
∈
P
σ
+
(
I
) (resp., p
(
·
)
∈
P
σ
-
(
I
)).In order to prove Theorem19, we need the following lemma.Lemma 22 (see [9]).
It is given thatp
(
·
)
∈
P
+
(
I
), 0
<
α
I
-
⩽
α
(
x
)
<
1
/
p
I
+, and q
(
x
)
=
p
(
x
)
/
1
-
α
(
x
)
p
(
x
). If f
L
p
(
·
)
(
I
)
⩽
1, then there exists a positive constant C depending only on p
(
·
) and α
(
·
) such that(70)
W
α
·
f
x
⩽
C
M
+
f
x
p
x
/
q
x
,
x
∈
I
.Proof of Theorem19.
By the homogeneity of norm, we can assume thatf
v
L
p
(
·
)
(
I
)
⩽
1 and f
L
p
(
·
)
(
I
)
⩽
1. It is sufficient to prove(71)
∫
I
W
α
x
f
x
q
x
u
x
q
x
d
x
⩽
C
.Applying (70) and Theorem 7 we have(72)
∫
I
W
α
x
f
x
q
x
u
x
q
x
d
x
⩽
C
∫
I
M
+
f
x
p
x
u
x
q
x
/
p
x
p
x
d
x
⩽
C
.Using the similar proving method as that of Lemma3.1 in [9], we can prove the following lemma.Lemma 23.
It is given thatp
(
·
)
∈
P
-
(
I
), 0
<
α
I
-
⩽
α
(
x
)
<
1
/
p
I
+, and q
(
x
)
=
p
(
x
)
/
1
-
α
(
x
)
p
(
x
). If f
L
p
(
·
)
(
I
)
⩽
1, then there exists a positive constant C depending only on p
(
·
) and α
(
·
) such that(73)
R
α
·
f
x
⩽
C
M
-
f
x
p
x
/
q
x
,
x
∈
I
.By this lemma and the two weighted results ofM
-, we can get the result of Theorem 20 directly.



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*Source: 1648281-2016-12-05.xml*