# Hardy-Littlewood-Sobolev Inequalities onp-Adic Central Morrey Spaces

Authors: Qing Yan Wu; Zun Wei Fu
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

We establish the Hardy-Littlewood-Sobolev inequalities onp-adic central Morrey spaces. Furthermore, we obtain the λ-central BMO estimates for commutators of p-adic Riesz potential on p-adic central Morrey spaces.


---

## Body

## 1. Introduction

Let0
<
α
<
n. The Riesz potential operator I
α is defined by setting, for all locally integrable functions f on R
n,
(1)
I
α
f
x
=
1
γ
n
α
∫
R
n
f
y
x
-
y
n
-
α
d
y
,
where γ
n
(
α
)
=
π
n
/
2
2
α
Γ
(
α
/
2
)
/
Γ
(
(
n
-
α
)
/
2
). It is closely related to the Laplacian operator of fractional degree. When n
>
2 and α
=
2, I
α
f is a solution of Poisson equation -
Δ
u
=
f. The importance of Riesz potentials is owing to the fact that they are smooth operators and have been extensively used in various areas such as potential analysis, harmonic analysis, and partial differential equations. For more details about Riesz potentials one can refer to [1].This paper focuses on the Riesz potentials onp-adic field. In the last 20 years, the field of p-adic numbers Q
p has been intensively used in theoretical and mathematical physics (cf. [2–12]). And it has already penetrated intensively into several areas of mathematics and its applications, among which harmonic analysis on p-adic field has been drawing more and more concern (see [13–22] and references therein).For a prime numberp, the field of p-adic numbers Q
p is defined as the completion of the field of rational numbers Q with respect to the non-Archimedean p-adic norm |
·
|
p, which satisfies |
x
|
p
=
0 if and only if x
=
0; |
x
y
|
p
=
|
x
|
p
|
y
|
p; |
x
+
y
|
p
≤
max
⁡
{
|
x
|
p
,
|
y
|
p
}. Moreover, if |
x
|
p
≠
|
y
|
p, then |
x
±
y
|
p
=
max
⁡
{
|
x
|
p
,
|
y
|
p
}. It is well-known that Q
p is a typical model of non-Archimedean local fields. If any nonzero rational number x is represented as x
=
p
γ
(
m
/
n
), where γ
=
γ
(
x
)
∈
Z and integers m, n are indivisible by p, then |
x
|
p
=
p
-
γ.The spaceQ
p
n
=
Q
p
×
Q
p
×
⋯
×
Q
p consists of points x
=
(
x
1
,
x
2
,
…
,
x
n
), where x
j
∈
Q
p, j
=
1,2
,
…
,
n. The p-adic norm on Q
p
n is
(2)
x
p
≔
max
⁡
1
≤
j
≤
n
x
j
p
,
x
∈
Q
p
n
.
Denote by
(3)
B
γ
a
=
x
∈
Q
p
n
:
x
-
a
p
≤
p
γ
the ball of radius p
γ with center at a
∈
Q
p
n and by
(4)
S
γ
a
=
B
γ
a
∖
B
γ
-
1
a
=
x
∈
Q
p
n
:
x
-
a
p
=
p
γ
the sphere of radius p
γ with center at a
∈
Q
p
n, where γ
∈
Z. It is clear that
(5)
B
γ
a
=
⋃
k
≤
γ
S
k
a
.It is well-known thatQ
p
n is a classical kind of locally compact Vilenkin groups. A locally compact Vilenkin group G is a locally compact Abelian group containing a strictly decreasing sequence of compact open subgroups {
G
n
}
n
=
-
∞
∞ such that (1) ∪
n
=
-
∞
∞
G
n
=
G and ∩
n
=
-
∞
∞
G
n
=
0 and (2) sup
⁡
{
order
(
G
n
/
G
n
+
1
:
n
∈
Z
)
}
<
∞. For several decades, parallel to the p-adic harmonic analysis, a development was under way of the harmonic analysis on locally compact Vilenkin groups (cf. [23–25] and references therein).SinceQ
p
n is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure d
x on Q
p
n, which is unique up to a positive constant factor and is translation invariant. We normalize the measure d
x by the equality
(6)
∫
B
0
0
d
x
=
B
0
0
H
=
1
,
where |
E
|
H denotes the Haar measure of a measurable subset E of Q
p
n. By simple calculation, we can obtain that
(7)
B
γ
a
H
=
p
γ
n
,
S
γ
a
H
=
p
γ
n
1
-
p
-
n
for any a
∈
Q
p
n. We should mention that the Haar measure takes value in R; there also exist p-adic valued measures (cf. [26, 27]). For a more complete introduction to the p-adic field, one can refer to [22] or [10].Onp-adic field, the p-adic Riesz potential I
α
p [22] is defined by
(8)
I
α
p
f
x
=
1
Γ
n
α
∫
Q
p
n
f
y
x
-
y
p
n
-
α
d
y
,
where Γ
n
(
α
)
=
(
1
-
p
α
-
n
)
/
(
1
-
p
-
α
), α
∈
C, α
≠
0. When n
=
1, Haran [4, 28] obtained the explicit formula of Riesz potentials on Q
p and developed analytical potential theory on Q
p. Taibleson [22] gave the fundamental analytic properties of the Riesz potentials on local fields including Q
p
n, as well as the classical Hardy-Littlewood-Sobolev inequalities. Kim [18] gave a simple proof of these inequalities by using the p-adic version of the Calderón-Zygmund decomposition technique. Volosivets [29] investigated the boundedness for Riesz potentials on generalized Morrey spaces. Like on Euclidean spaces, using the Riesz potential with n
>
2 and α
=
2, one can introduce the p-adic Laplacians [13].In this paper, we will consider the Riesz potentials and their commutators withp-adic central BMO functions on p-adic central Morrey spaces. Alvarez et al. [30] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced λ-central BMO spaces and central Morrey spaces, respectively. In [31], we introduce their p-adic versions.Definition 1.
Letλ
∈
R and 1
<
q
<
∞. The p-adic central Morrey space B
˙
q
,
λ
(
Q
p
n
) is defined by
(9)
f
B
˙
q
,
λ
Q
p
n
≔
sup
⁡
γ
∈
Z
1
B
γ
H
1
+
λ
q
∫
B
γ
f
x
q
d
x
1
/
q
<
∞
,
where B
γ
=
B
γ
(
0
).Remark 2.
It is clear that(10)
L
q
,
λ
Q
p
n
⊂
B
˙
q
,
λ
Q
p
n
,
B
˙
q
,
-
1
/
q
Q
p
n
=
L
q
Q
p
n
.
When λ
<
-
1
/
q, the space B
˙
q
,
λ
(
Q
p
n
) reduces to {
0
}; therefore, we can only consider the case λ
≥
-
1
/
q. If 1
≤
q
1
<
q
2
<
∞, by Hölder’s inequality,
(11)
B
˙
q
2
,
λ
Q
p
n
⊂
B
˙
q
1
,
λ
Q
p
n
for λ
∈
R.Definition 3.
Letλ
<
1
/
n and 1
<
q
<
∞. The space CBM
O
q
,
λ
(
Q
p
n
) is defined by the condition
(12)
f
CBM
O
q
,
λ
Q
p
n
≔
sup
⁡
γ
∈
Z
1
B
γ
H
1
+
λ
q
∫
B
γ
f
x
-
f
B
γ
q
d
x
1
/
q
<
∞
.Remark 4.
Whenλ
=
0, the space CBM
O
q
,
λ
(
Q
p
n
) is just CBM
O
q
(
Q
p
n
), which is defined in [32]. If 1
≤
q
1
<
q
2
<
∞, by Hölder’s inequality,
(13)
CBM
O
q
2
,
λ
Q
p
n
⊂
CBM
O
q
1
,
λ
Q
p
n
for λ
∈
R. By the standard proof as that in R
n, we can see that
(14)
f
CBM
O
q
,
λ
Q
p
n
~
sup
⁡
γ
∈
Z
inf
⁡
c
∈
C
1
B
γ
H
1
+
λ
q
∫
B
γ
f
x
-
c
q
d
x
1
/
q
.Remark 5.
Formulas9 and 12 yield that B
˙
q
,
λ
(
Q
p
n
) is a Banach space continuously included in CBM
O
q
,
λ
(
Q
p
n
).Here we introduce thep-adic weak central Morrey spaces.Definition 6.
Letλ
∈
R and 1
<
q
<
∞. The p-adic weak central Morrey space W
B
˙
q
,
λ
(
Q
p
n
) is defined by
(15)
f
W
B
˙
q
,
λ
Q
p
n
≔
sup
⁡
γ
∈
Z
sup
⁡
t
>
0
t
q
x
∈
B
γ
:
f
x
>
t
H
B
γ
H
1
+
λ
q
1
/
q
<
∞
,
where B
γ
=
B
γ
(
0
).In Section2, we will get the Hardy-Littlewood-Sobolev inequalities on p-adic central Morrey spaces. Namely, under some conditions for indexes, I
α
p is bounded from B
˙
q
,
λ
(
Q
p
n
) to B
˙
r
,
μ
(
Q
p
n
) and is also bounded from B
˙
1
,
λ
(
Q
p
n
) to W
B
˙
r
,
μ
(
Q
p
n
). In Section 3, we establish the boundedness for commutators generated by I
α
p and λ-central BMO functions on p-adic central Morrey spaces.Throughout this paper the letterC will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.


## 2. Hardy-Littlewood-Sobolev Inequalities

We get the following Hardy-Littlewood-Sobolev inequalities onp-adic central Morrey spaces.Theorem 7.
Letα be a complex number with 0
<
Re
α
<
n and let 1
≤
q
<
n
/
Re
α, 0
<
1
/
r
=
1
/
q
-
Re
α
/
n, λ
<
-
Re
α
/
n, and μ
=
λ
+
Re
α
/
n.(i)
Ifq
>
1, then I
α
p is bounded from B
˙
q
,
λ
(
Q
p
n
) to B
˙
r
,
μ
(
Q
p
n
).
(ii)
Ifq
=
1, then I
α
p is bounded from B
˙
1
,
λ
(
Q
p
n
) to W
B
˙
r
,
μ
(
Q
p
n
).In order to give the proof of this theorem, we need the following result.Lemma 8 (see [22]).
Letα be a complex number with 0
<
Re
α
<
n and let 1
≤
q
<
r
<
∞ satisfy 1
/
r
=
1
/
q
-
Re
α
/
n.(i)
Iff
∈
L
q
(
Q
p
n
), q
>
1, then
(16)
I
α
p
f
L
r
Q
p
n
≤
A
q
r
f
L
q
Q
p
n
,
where A
q
r is independent of f.
(ii)
Iff
∈
L
1
(
Q
p
n
), s
>
0, then
(17)
x
∈
Q
p
n
:
I
α
p
f
x
>
s
H
≤
A
r
f
L
1
Q
p
n
s
r
,
where A
r
>
0 is independent of f.Proof of Theorem7.
Letf be a function in B
˙
q
,
λ
(
Q
p
n
). For fixed γ
∈
Z, denote B
γ
(
0
) by B
γ.(i)
Ifq
>
1, write
(18)
1
B
γ
H
1
+
μ
r
∫
B
γ
I
α
p
f
x
r
d
x
1
/
r
≤
1
B
γ
H
1
+
μ
r
∫
B
γ
I
α
p
f
χ
B
γ
x
r
d
x
1
/
r
+
1
B
γ
H
1
+
μ
r
∫
B
γ
I
α
p
f
χ
B
γ
c
x
r
d
x
1
/
r
≔
I
+
I
I
.
ForI, since 1
/
r
=
1
/
q
-
Re
α
/
n and μ
=
λ
+
Re
α
/
n, by Lemma 8,
(19)
I
=
1
B
γ
H
1
+
μ
r
∫
B
γ
I
α
p
f
χ
B
γ
x
r
d
x
1
/
r
≤
B
γ
H
-
1
/
r
-
μ
∫
B
γ
f
χ
B
γ
x
q
d
x
1
/
q
≤
f
B
˙
q
,
λ
Q
p
n
.
ForI
I, we firstly give the following estimate. For x
∈
B
γ, by Hölder’s inequality, we have
(20)
I
α
p
f
χ
B
γ
c
x
=
1
Γ
n
α
∫
B
γ
c
f
y
x
-
y
p
n
-
α
d
y
≤
1
Γ
n
α
∫
B
γ
c
f
y
x
-
y
p
n
-
Re
α
d
y
=
1
Γ
n
α
∑
k
=
γ
+
1
∞
∫
S
k
f
y
x
-
y
p
n
-
Re
α
d
y
=
1
Γ
n
α
∑
k
=
γ
+
1
∞
∫
S
k
p
-
k
n
-
Re
α
f
y
d
y
≤
1
Γ
n
α
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
∫
B
k
f
y
q
d
y
1
/
q
B
k
H
1
-
1
/
q
≤
1
Γ
n
α
f
B
˙
q
,
λ
Q
p
n
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
B
k
H
1
+
λ
≤
C
B
γ
H
μ
f
B
˙
q
,
λ
Q
p
n
.
The last inequality is due to the fact that λ
<
-
Re
α
/
n. Consequently,
(21)
I
I
=
1
B
γ
H
1
+
μ
r
∫
B
γ
I
α
p
f
χ
B
γ
c
x
r
d
x
1
/
r
≤
C
f
B
˙
q
,
λ
Q
p
n
.
The above estimates imply that(22)
I
α
p
f
B
˙
r
,
μ
Q
p
n
≤
C
f
B
˙
q
,
λ
Q
p
n
.
(ii)
Ifq
=
1, set f
1
=
f
χ
B
γ and f
2
=
f
-
f
1; by Lemma 8, we have
(23)
x
∈
B
γ
:
I
α
p
f
1
x
>
t
H
≤
C
f
1
L
1
Q
p
n
t
r
=
C
t
-
r
∫
B
γ
f
x
d
x
r
≤
C
t
-
r
B
γ
H
1
+
λ
r
f
B
˙
1
,
λ
Q
p
n
r
=
C
t
-
r
B
γ
H
1
+
μ
r
f
B
˙
1
,
λ
Q
p
n
r
.
On the other hand, by the same estimate as30, we have
(24)
I
α
p
f
2
x
≤
C
B
γ
H
μ
f
2
B
˙
1
,
λ
Q
p
n
.
Then using Chebyshev’s inequality, we obtain
(25)
x
∈
B
γ
:
I
α
p
f
2
x
>
t
H
≤
t
-
r
∫
B
γ
I
α
p
f
2
x
r
d
x
≤
C
t
-
r
B
γ
H
1
+
μ
r
f
2
B
˙
1
,
λ
Q
p
n
r
≤
C
t
-
r
B
γ
H
1
+
μ
r
f
B
˙
1
,
λ
Q
p
n
r
.
Since(26)
I
α
p
f
x
≤
I
α
p
f
1
x
+
I
α
p
f
2
x
,
we get
(27)
x
∈
B
γ
:
I
α
p
f
x
>
t
H
≤
x
∈
B
γ
:
I
α
p
f
1
x
>
t
2
H
+
x
∈
B
γ
:
I
α
p
f
2
x
>
t
2
H
≤
C
t
-
r
B
γ
H
1
+
μ
r
f
B
˙
1
,
λ
Q
p
n
r
.
Therefore,
(28)
t
r
x
∈
B
γ
:
I
α
p
f
x
>
t
H
B
γ
H
1
+
μ
r
1
/
r
≤
C
f
B
˙
1
,
λ
Q
p
n
,
for any t
>
0 and γ
∈
Z. This completes the proof.For application, we now introduce a pseudo-differential operatorD
α defined by Vladimirov in [33].The operatorD
α
:
ψ
→
D
α
ψ is defined as convolution of generalized functions f
-
α and ψ:
(29)
D
α
ψ
=
f
-
α
*
ψ
,
α
≠
-
1
,
where f
α
=
|
x
|
p
α
-
1
/
Γ
(
α
) and Γ
(
α
)
=
(
1
-
p
α
-
1
)
/
(
1
-
p
-
α
).Let us consider the equation(30)
D
α
ψ
=
g
,
g
∈
E
′
,
where E
′ is the space of linear continuous functionals on E and here E denotes the set of locally constant functions on Q
p. A complex-valued function f
(
x
) defined on Q
p is called locally constant if for any point x
∈
Q
p there exists an integer l
(
x
)
∈
Z such that
(31)
f
x
+
x
′
=
f
x
,
x
′
p
≤
p
l
x
.The following lemma (page 154 in [10]) gives solutions of 30.Lemma 9.
Forα
>
0 any solution of 30 is expressed by the formula
(32)
ψ
=
D
-
α
g
+
C
,
where C is an arbitrary constant; for α
<
0 a solution of 30 is unique and it is expressed by formula 32 for C
=
0.Combining with Theorem7, we obtain the following regular property of the solution.Corollary 10.
Let0
<
α
<
1 and let 1
≤
q
<
1
/
α, 0
<
1
/
r
=
1
/
q
-
α, λ
<
-
α, and μ
=
λ
+
α. If g
∈
E
′
∩
B
˙
q
,
λ
(
Q
p
n
), then(i)
whenq
>
1, 30 has a solution in B
˙
r
,
μ
(
Q
p
n
),
(ii)
whenq
=
1, 30 has a solution in W
B
˙
r
,
μ
(
Q
p
n
).


## 3. Commutators ofp-Adic Riesz Potential

In this section, we will establish theλ-central BMO estimates for commutators I
α
p
,
b of p-adic Riesz potential which is defined by
(33)
I
α
p
,
b
f
=
b
I
α
p
f
-
I
α
p
b
f
,
for some suitable functions f.Theorem 11.
Suppose0
<
Re
α
<
n, 1
<
q
1
<
n
/
Re
α, q
1
′
<
q
2
<
∞, and 1
/
q
=
1
/
q
1
+
1
/
q
2
-
Re
α
/
n. Let 0
≤
λ
2
<
1
/
n, λ
1 satisfies λ
1
<
-
λ
2
-
Re
α
/
n, and λ
=
λ
1
+
λ
2
+
Re
α
/
n. If b
∈
C
B
M
O
q
2
,
λ
2
(
Q
p
n
), then I
α
p
,
b is bounded from B
˙
q
1
,
λ
1
(
Q
p
n
) to B
˙
q
,
λ
(
Q
p
n
), and the following inequality holds:
(34)
I
α
p
,
b
f
B
˙
q
,
λ
Q
p
n
≤
C
b
C
B
M
O
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
.Before proving this theorem, we need the following result.Lemma 12 (see [31]).
Suppose thatb
∈
C
B
M
O
q
,
λ
(
Q
p
n
) and j
,
k
∈
Z, λ
≥
0. Then
(35)
b
B
j
-
b
B
k
≤
p
n
j
-
k
b
C
B
M
O
q
,
λ
Q
p
n
max
⁡
B
j
H
λ
,
B
k
H
λ
.Proof of Theorem11.
Suppose thatf is a function in B
˙
q
1
,
λ
1
(
Q
p
n
). For fixed γ
∈
Z, denote B
γ
(
0
) by B
γ. We write
(36)
1
B
γ
H
∫
B
γ
I
α
p
,
b
f
x
q
d
x
1
/
q
≤
1
B
γ
H
∫
B
γ
b
x
-
b
B
γ
I
α
p
f
χ
B
γ
x
q
d
x
1
/
q
+
1
B
γ
H
∫
B
γ
b
x
-
b
B
γ
I
α
p
f
χ
B
γ
c
x
q
d
x
1
/
q
+
1
B
γ
H
∫
B
γ
I
α
p
b
-
b
B
γ
f
χ
B
γ
x
q
d
x
1
/
q
+
1
B
γ
H
∫
B
γ
I
α
p
b
-
b
B
γ
f
χ
B
γ
c
x
q
d
x
1
/
q
≔
J
1
+
J
2
+
J
3
+
J
4
.
Set1
/
r
=
1
/
q
1
-
Re
α
/
n; then 1
/
q
=
1
/
q
2
+
1
/
r; by Lemma 8 and Hölder’s inequality, we have
(37)
J
1
=
1
B
γ
H
∫
B
γ
b
x
-
b
B
γ
I
α
p
f
χ
B
γ
x
q
d
x
1
/
q
≤
B
γ
H
-
1
/
q
∫
B
γ
b
x
-
b
B
γ
q
2
d
x
1
/
q
2
·
∫
B
γ
I
α
p
f
χ
B
γ
x
r
d
x
1
/
r
≤
C
B
γ
H
-
1
/
r
+
λ
2
b
CBM
O
q
2
,
λ
2
Q
p
n
·
∫
B
γ
f
χ
B
γ
x
q
1
d
x
1
/
q
1
≤
C
B
γ
H
λ
b
CBM
O
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
.
Similarly, denote1
/
l
=
1
/
q
1
+
1
/
q
2; then 1
/
q
=
1
/
l
-
Re
α
/
n, and by Hölder’s inequality and Lemma 8, we get
(38)
J
3
=
1
B
γ
H
∫
B
γ
I
α
p
b
-
b
B
γ
f
χ
B
γ
x
q
d
x
1
/
q
≤
C
B
γ
H
-
1
/
q
∫
B
γ
b
x
-
b
B
γ
f
x
l
d
x
1
/
l
≤
C
B
γ
H
-
1
/
q
∫
B
γ
b
x
-
b
B
γ
q
2
d
x
1
/
q
2
·
∫
B
γ
f
x
q
1
d
x
1
/
q
1
≤
C
B
γ
H
λ
b
CBM
O
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
.
To estimateJ
2 and J
4, we firstly give the following estimates. For x
∈
B
γ, by Hölder’s inequality, we obtain
(39)
I
α
p
f
χ
B
γ
c
x
=
1
Γ
n
α
∫
B
γ
c
f
y
x
-
y
p
n
-
α
d
y
≤
1
Γ
n
α
∫
B
γ
c
f
y
x
-
y
p
n
-
Re
α
d
y
=
1
Γ
n
α
∑
k
=
γ
+
1
∞
∫
S
k
f
y
p
-
k
n
-
Re
α
d
y
≤
1
Γ
n
α
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
B
k
H
1
-
1
/
q
1
∫
S
k
f
y
q
1
d
y
1
/
q
1
≤
1
Γ
n
α
f
B
˙
q
1
,
λ
1
Q
p
n
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
B
k
H
1
+
λ
1
=
1
Γ
n
α
f
B
˙
q
1
,
λ
1
Q
p
n
p
γ
+
1
n
λ
1
+
Re
α
1
-
p
n
λ
1
+
Re
α
=
C
B
γ
H
λ
1
+
Re
α
/
n
f
B
˙
q
1
,
λ
1
Q
p
n
,
where the penultimate “=” is due to the fact that λ
1
+
Re
α
/
n
<
-
λ
2
≤
0. Similarly,
(40)
I
α
p
b
-
b
B
γ
f
χ
B
γ
c
x
=
1
Γ
n
α
∫
B
γ
c
b
y
-
b
B
γ
f
y
x
-
y
p
n
-
α
d
y
≤
1
Γ
n
α
∫
B
γ
c
b
y
-
b
B
γ
f
y
x
-
y
p
n
-
Re
α
d
y
=
1
Γ
n
α
∑
k
=
γ
+
1
∞
∫
S
k
b
y
-
b
B
γ
f
y
p
-
k
n
-
Re
α
d
y
=
1
Γ
n
α
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
B
k
H
1
-
1
/
q
1
-
1
/
q
2
·
∫
S
k
f
y
q
1
d
y
1
/
q
1
∫
S
k
b
y
-
b
B
γ
q
2
d
y
1
/
q
2
≤
1
Γ
n
α
f
B
˙
q
1
,
λ
1
Q
p
n
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
B
k
H
1
-
1
/
q
2
+
λ
1
·
∫
B
k
b
y
-
b
B
γ
q
2
d
y
1
/
q
2
≤
1
Γ
n
α
f
B
˙
q
1
,
λ
1
Q
p
n
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
B
k
H
1
-
1
/
q
2
+
λ
1
×
∫
B
k
b
y
-
b
B
k
q
2
d
y
1
/
q
2
+
b
B
k
-
b
B
γ
B
k
H
1
/
q
2
.
Sincek
≥
γ
+
1, by Lemma 12, we have
(41)
b
B
k
-
b
B
γ
≤
p
n
k
-
γ
b
CBM
O
q
2
,
λ
2
Q
p
n
B
k
H
λ
2
.
Thus
(42)
I
α
p
b
-
b
B
γ
f
χ
B
γ
c
x
≤
1
Γ
n
α
b
CBMO
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
×
∑
k
=
γ
+
1
∞
p
-
k
n
-
Re
α
B
k
H
1
-
1
/
q
2
+
λ
1
·
B
k
H
1
/
q
2
+
λ
2
+
p
n
k
-
γ
B
k
H
1
/
q
2
+
λ
2
≤
C
Γ
n
α
b
CBMO
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
·
∑
k
=
γ
+
1
∞
k
-
γ
p
-
k
n
-
Re
α
B
k
H
1
+
λ
1
+
λ
2
≤
C
Γ
n
α
b
CBMO
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
∑
k
=
γ
+
1
∞
k
-
γ
p
k
n
λ
=
C
B
γ
H
λ
b
CBMO
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
.
Now by39 and Hölder’s inequality, we obtain
(43)
J
2
=
1
B
γ
H
∫
B
γ
b
x
-
b
B
γ
I
α
p
f
χ
B
γ
c
x
q
d
x
1
/
q
≤
C
B
γ
H
λ
1
+
Re
α
/
n
-
1
/
q
f
B
˙
q
1
,
λ
1
Q
p
n
·
∫
B
γ
b
x
-
b
B
γ
q
d
x
1
/
q
≤
C
B
γ
H
λ
1
+
Re
α
/
n
-
1
/
q
2
f
B
˙
q
1
,
λ
1
Q
p
n
·
∫
B
γ
b
x
-
b
B
γ
q
2
d
x
1
/
q
2
≤
C
B
γ
H
λ
b
CBM
O
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
.
It follows from42 that
(44)
J
4
=
1
B
γ
H
∫
B
γ
I
α
p
b
-
b
B
γ
f
χ
B
γ
c
x
q
d
x
1
/
q
≤
C
B
γ
H
λ
b
CBMO
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
.
The above estimates imply that(45)
I
α
p
,
b
f
B
˙
q
,
λ
Q
p
n
≤
C
b
CBM
O
q
2
,
λ
2
Q
p
n
f
B
˙
q
1
,
λ
1
Q
p
n
.
This completes the proof of the theorem.Remark 13.
Sincep-adic field is a kind of locally compact Vilenkin groups, we can further consider the Hardy-Littlewood-Sobolev inequalities on such groups, which is more complicated and will appear elsewhere.



---

*Source: 419532-2015-04-22.xml*