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Research ArticleHardy-Littlewood-Sobolev Inequalities on 𝑝-Adic CentralMorrey Spaces

Qing Yan Wu and Zun Wei Fu

Department of Mathematics, Linyi University, Linyi, Shandong 276005, China

Correspondence should be addressed to Zun Wei Fu; [email protected]

Received 21 October 2014; Accepted 15 December 2014

Academic Editor: Yoshihiro Sawano

Copyright © 2015 Q. Y. Wu and Z. W. Fu. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

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

1. Introduction

Let 0 < 𝛼 < 𝑛. The Riesz potential operator 𝐼𝛼is defined by

setting, for all locally integrable functions 𝑓 on R𝑛,

𝐼𝛼𝑓 (𝑥) =

1

𝛾𝑛(𝛼)

∫R𝑛

𝑓 (𝑦)

𝑥 − 𝑦𝑛−𝛼𝑑𝑦, (1)

where 𝛾𝑛(𝛼) = 𝜋

𝑛/2

2𝛼

Γ(𝛼/2)/Γ((𝑛 − 𝛼)/2). It is closely relatedto the Laplacian operator of fractional degree. When 𝑛 > 2and 𝛼 = 2, 𝐼

𝛼𝑓 is a solution of Poisson equation −Δ𝑢 =

𝑓. The importance of Riesz potentials is owing to the factthat they are smooth operators and have been extensivelyused in various areas such as potential analysis, harmonicanalysis, and partial differential equations. For more detailsabout Riesz potentials one can refer to [1].

This paper focuses on the Riesz potentials on 𝑝-adicfield. In the last 20 years, the field of 𝑝-adic numbers Q

𝑝

has been intensively used in theoretical and mathematicalphysics (cf. [2–12]). And it has already penetrated intensivelyinto several areas of mathematics and its applications, amongwhich harmonic analysis on 𝑝-adic field has been drawingmore and more concern (see [13–22] and references therein).

For a prime number 𝑝, the field of 𝑝-adic numbers Q𝑝

is defined as the completion of the field of rational numbersQ with respect to the non-Archimedean 𝑝-adic norm | ⋅ |

𝑝,

which satisfies |𝑥|𝑝= 0 if and only if 𝑥 = 0; |𝑥𝑦|

𝑝=

|𝑥|𝑝|𝑦|𝑝; |𝑥 + 𝑦|

𝑝≤ max{|𝑥|

𝑝, |𝑦|𝑝}. Moreover, if |𝑥|

𝑝̸= |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 anynonzero rational number 𝑥 is represented as 𝑥 = 𝑝𝛾(𝑚/𝑛),where 𝛾 = 𝛾(𝑥) ∈ Z and integers 𝑚, 𝑛 are indivisible by 𝑝,then |𝑥|

𝑝= 𝑝−𝛾.

The space Q𝑛𝑝= Q𝑝× Q𝑝× ⋅ ⋅ ⋅ × Q

𝑝consists of points

𝑥 = (𝑥1, 𝑥2, . . . , 𝑥

𝑛), where 𝑥

𝑗∈ Q𝑝, 𝑗 = 1, 2, . . . , 𝑛. The 𝑝-

adic norm onQ𝑛𝑝is

|𝑥|𝑝:= max1≤𝑗≤𝑛

𝑥𝑗

𝑝, 𝑥 ∈ Q

𝑛

𝑝. (2)

Denote by

𝐵𝛾(𝑎) = {𝑥 ∈ Q

𝑛

𝑝: |𝑥 − 𝑎|

𝑝≤ 𝑝𝛾

} (3)

the ball of radius 𝑝𝛾 with center at 𝑎 ∈ Q𝑛𝑝and by

𝑆𝛾(𝑎) = 𝐵

𝛾(𝑎) \ 𝐵

𝛾−1(𝑎) = {𝑥 ∈ Q

𝑛

𝑝: |𝑥 − 𝑎|

𝑝= 𝑝𝛾

} (4)

the sphere of radius 𝑝𝛾 with center at 𝑎 ∈ Q𝑛𝑝, where 𝛾 ∈ Z. It

is clear that

𝐵𝛾(𝑎) = ⋃

𝑘≤𝛾

𝑆𝑘(𝑎) . (5)

It is well-known that Q𝑛𝑝is a classical kind of locally

compact Vilenkin groups. A locally compact Vilenkin group𝐺 is a locally compact Abelian group containing a strictlydecreasing sequence of compact open subgroups {𝐺

𝑛}∞

𝑛=−∞

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015, Article ID 419532, 7 pageshttp://dx.doi.org/10.1155/2015/419532

2 Journal of Function Spaces

such that (1) ∪∞𝑛=−∞

𝐺𝑛= 𝐺 and ∩∞

𝑛=−∞𝐺𝑛= 0 and (2)

sup{order(𝐺𝑛/𝐺𝑛+1

: 𝑛 ∈ Z)} < ∞. For several decades,parallel to the 𝑝-adic harmonic analysis, a development wasunder way of the harmonic analysis on locally compactVilenkin groups (cf. [23–25] and references therein).

Since Q𝑛𝑝is a locally compact commutative group under

addition, it follows from the standard analysis that there existsa Haar measure 𝑑𝑥 on Q𝑛

𝑝, which is unique up to a positive

constant factor and is translation invariant.We normalize themeasure 𝑑𝑥 by the equality

∫𝐵0(0)

𝑑𝑥 =𝐵0 (0)

𝐻 = 1, (6)

where |𝐸|𝐻denotes the Haar measure of a measurable subset

𝐸 ofQ𝑛𝑝. By simple calculation, we can obtain that

𝐵𝛾(𝑎)𝐻= 𝑝𝛾𝑛

,

𝑆𝛾(𝑎)𝐻= 𝑝𝛾𝑛

(1 − 𝑝−𝑛

)

(7)

for any 𝑎 ∈ Q𝑛𝑝. 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-adicfield, one can refer to [22] or [10].

On 𝑝-adic field, the 𝑝-adic Riesz potential 𝐼𝑝𝛼[22] is

defined by

𝐼𝑝

𝛼𝑓 (𝑥) =

1

Γ𝑛(𝛼)

∫Q𝑛𝑝

𝑓 (𝑦)

𝑥 − 𝑦𝑛−𝛼

𝑝

𝑑𝑦, (8)

where Γ𝑛(𝛼) = (1 − 𝑝

𝛼−𝑛

)/(1 − 𝑝−𝛼

), 𝛼 ∈ C, 𝛼 ̸= 0. When𝑛 = 1, Haran [4, 28] obtained the explicit formula of Rieszpotentials onQ

𝑝and developed analytical potential theory on

Q𝑝. Taibleson [22] gave the fundamental analytic properties

of the Riesz potentials on local fields including Q𝑛𝑝, as well

as the classical Hardy-Littlewood-Sobolev inequalities. Kim[18] gave a simple proof of these inequalities by using the𝑝-adic version of the Calderón-Zygmund decompositiontechnique. Volosivets [29] investigated the boundedness forRiesz potentials on generalized Morrey spaces. Like onEuclidean spaces, using the Riesz potential with 𝑛 > 2 and𝛼 = 2, one can introduce the 𝑝-adic Laplacians [13].

In this paper, we will consider the Riesz potentials andtheir commutators with 𝑝-adic central BMO functions on 𝑝-adic central Morrey spaces. Alvarez et al. [30] studied therelationship between central BMO spaces andMorrey spaces.Furthermore, they introduced 𝜆-central BMO spaces andcentralMorrey spaces, respectively. In [31], we introduce their𝑝-adic versions.

Definition 1. Let 𝜆 ∈ R and 1 < 𝑞 < ∞. The 𝑝-adic centralMorrey space �̇�𝑞,𝜆(Q𝑛

𝑝) is defined by

𝑓�̇�𝑞,𝜆(Q𝑛

𝑝):= sup𝛾∈Z

(1

𝐵𝛾

1+𝜆𝑞

𝐻

∫𝐵𝛾

𝑓 (𝑥)𝑞

𝑑𝑥)

1/𝑞

< ∞, (9)

where 𝐵𝛾= 𝐵𝛾(0).

Remark 2. It is clear that

𝐿𝑞,𝜆

(Q𝑛

𝑝) ⊂ �̇�𝑞,𝜆

(Q𝑛

𝑝) ,

�̇�𝑞,−1/𝑞

(Q𝑛

𝑝) = 𝐿𝑞

(Q𝑛

𝑝) .

(10)

When 𝜆 < −1/𝑞, the space �̇�𝑞,𝜆(Q𝑛𝑝) reduces to {0}; therefore,

we can only consider the case 𝜆 ≥ −1/𝑞. If 1 ≤ 𝑞1< 𝑞2< ∞,

by Hölder’s inequality,

�̇�𝑞2,𝜆

(Q𝑛

𝑝) ⊂ �̇�𝑞1,𝜆

(Q𝑛

𝑝) (11)

for 𝜆 ∈ R.

Definition 3. Let 𝜆 < 1/𝑛 and 1 < 𝑞 < ∞. The spaceCBMO𝑞,𝜆(Q𝑛

𝑝) is defined by the condition

𝑓CBMO𝑞,𝜆(Q𝑛

𝑝)

:= sup𝛾∈Z

(1

𝐵𝛾

1+𝜆𝑞

𝐻

∫𝐵𝛾

𝑓 (𝑥) − 𝑓

𝐵𝛾

𝑞

𝑑𝑥)

1/𝑞

< ∞.

(12)

Remark 4. When 𝜆 = 0, the space CBMO𝑞,𝜆(Q𝑛𝑝) is just

CBMO𝑞(Q𝑛𝑝), which is defined in [32]. If 1 ≤ 𝑞

1< 𝑞2< ∞,

by Hölder’s inequality,

CBMO𝑞2 ,𝜆 (Q𝑛𝑝) ⊂ CBMO𝑞1,𝜆 (Q𝑛

𝑝) (13)

for 𝜆 ∈ R. By the standard proof as that inR𝑛, we can see that𝑓CBMO𝑞,𝜆(Q𝑛

𝑝)

∼ sup𝛾∈Z

inf𝑐∈C(

1

𝐵𝛾

1+𝜆𝑞

𝐻

∫𝐵𝛾

𝑓 (𝑥) − 𝑐𝑞

𝑑𝑥)

1/𝑞

.

(14)

Remark 5. Formulas (9) and (12) yield that �̇�𝑞,𝜆(Q𝑛𝑝) is a

Banach space continuously included in CBMO𝑞,𝜆(Q𝑛𝑝).

Herewe introduce the𝑝-adicweak centralMorrey spaces.

Definition 6. Let 𝜆 ∈ R and 1 < 𝑞 < ∞. The 𝑝-adic weakcentral Morrey space𝑊�̇�𝑞,𝜆(Q𝑛

𝑝) is defined by

𝑓𝑊�̇�𝑞,𝜆(Q𝑛

𝑝)

:= sup𝛾∈Z

(sup𝑡>0𝑡𝑞{𝑥 ∈ 𝐵

𝛾:𝑓 (𝑥)

> 𝑡}𝐻

𝐵𝛾

1+𝜆𝑞

𝐻

)

1/𝑞

< ∞,

(15)

where 𝐵𝛾= 𝐵𝛾(0).

In Section 2, we will get the Hardy-Littlewood-Sobolevinequalities on 𝑝-adic central Morrey spaces. Namely, under

Journal of Function Spaces 3

some conditions for indexes, 𝐼𝑝𝛼is bounded from �̇�𝑞,𝜆(Q𝑛

𝑝) to

�̇�𝑟,𝜇

(Q𝑛𝑝) and is also bounded from �̇�1,𝜆(Q𝑛

𝑝) to 𝑊�̇�𝑟,𝜇(Q𝑛

𝑝).

In Section 3, we establish the boundedness for commutatorsgenerated by 𝐼𝑝

𝛼and 𝜆-central BMO functions on 𝑝-adic

central Morrey spaces.Throughout this paper the letter 𝐶 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 inequalitieson 𝑝-adic central Morrey spaces.

Theorem7. Let𝛼 be a complex numberwith 0 < Re𝛼 < 𝑛 andlet 1 ≤ 𝑞 < 𝑛/Re𝛼, 0 < 1/𝑟 = 1/𝑞 − Re𝛼/𝑛, 𝜆 < −Re𝛼/𝑛,and 𝜇 = 𝜆 + Re𝛼/𝑛.

(i) If 𝑞 > 1, then 𝐼𝑝𝛼is bounded from �̇�𝑞,𝜆(Q𝑛

𝑝) to �̇�𝑟,𝜇(Q𝑛

𝑝).

(ii) If 𝑞 = 1, then 𝐼𝑝𝛼

is bounded from �̇�1,𝜆(Q𝑛𝑝) to

𝑊�̇�𝑟,𝜇

(Q𝑛𝑝).

In order to give the proof of this theorem, we need thefollowing result.

Lemma 8 (see [22]). Let 𝛼 be a complex number with 0 <Re𝛼 < 𝑛 and let 1 ≤ 𝑞 < 𝑟 < ∞ satisfy 1/𝑟 = 1/𝑞 − Re𝛼/𝑛.

(i) If 𝑓 ∈ 𝐿𝑞(Q𝑛𝑝), 𝑞 > 1, then

𝐼𝑝

𝛼𝑓𝐿𝑟(Q𝑛

𝑝)≤ 𝐴𝑞𝑟

𝑓𝐿𝑞(Q𝑛

𝑝), (16)

where 𝐴𝑞𝑟is independent of 𝑓.

(ii) If 𝑓 ∈ 𝐿1(Q𝑛𝑝), 𝑠 > 0, then

{𝑥 ∈ Q

𝑛

𝑝:𝐼𝑝

𝛼𝑓 (𝑥)

> 𝑠}𝐻≤ (𝐴

𝑟

𝑓𝐿1(Q𝑛

𝑝)

𝑠)

𝑟

, (17)

where 𝐴𝑟> 0 is independent of 𝑓.

Proof ofTheorem 7. Let 𝑓 be a function in �̇�𝑞,𝜆(Q𝑛𝑝). For fixed

𝛾 ∈ Z, denote 𝐵𝛾(0) by 𝐵

𝛾.

(i) If 𝑞 > 1, write

(1

𝐵𝛾

1+𝜇𝑟

𝐻

∫𝐵𝛾

𝐼𝑝

𝛼𝑓 (𝑥)

𝑟

𝑑𝑥)

1/𝑟

≤ (1

𝐵𝛾

1+𝜇𝑟

𝐻

∫𝐵𝛾

𝐼𝑝

𝛼(𝑓𝜒𝐵𝛾

) (𝑥)

𝑟

𝑑𝑥)

1/𝑟

+ (1

𝐵𝛾

1+𝜇𝑟

𝐻

∫𝐵𝛾

𝐼𝑝

𝛼(𝑓𝜒𝐵𝑐

𝛾

) (𝑥)

𝑟

𝑑𝑥)

1/𝑟

:= 𝐼 + 𝐼𝐼.

(18)

For 𝐼, since 1/𝑟 = 1/𝑞 − Re𝛼/𝑛 and 𝜇 = 𝜆 + Re𝛼/𝑛, byLemma 8,

𝐼 = (1

𝐵𝛾

1+𝜇𝑟

𝐻

∫𝐵𝛾

𝐼𝑝


) (𝑥)

𝑟

𝑑𝑥)

1/𝑟

≤𝐵𝛾

−1/𝑟−𝜇

𝐻

(∫𝐵𝛾

𝑓𝜒𝐵𝛾

(𝑥)

𝑞

𝑑𝑥)

1/𝑞

≤𝑓�̇�𝑞,𝜆(Q𝑛

𝑝).

(19)

For 𝐼𝐼, we firstly give the following estimate. For 𝑥 ∈ 𝐵𝛾,

by Hölder’s inequality, we have

𝐼𝑝


𝛾

) (𝑥)

=

1

Γ𝑛(𝛼)

∫𝐵𝑐

𝛾

𝑓 (𝑦)


𝑝

𝑑𝑦

≤1

Γ𝑛(𝛼)

∫𝐵𝑐

𝛾

𝑓 (𝑦)

𝑥 − 𝑦𝑛−Re𝛼𝑝

𝑑𝑦

=1

Γ𝑛(𝛼)

∞

∑

𝑘=𝛾+1

∫𝑆𝑘

𝑓 (𝑦)


𝑑𝑦

=1

Γ𝑛(𝛼)

∞

∑

𝑘=𝛾+1

∫𝑆𝑘

𝑝−𝑘(𝑛−Re𝛼) 𝑓 (𝑦)

𝑑𝑦

≤1

Γ𝑛(𝛼)

∞

∑

𝑘=𝛾+1

𝑝−𝑘(𝑛−Re𝛼)

(∫𝐵𝑘

𝑓 (𝑦)𝑞

𝑑𝑦)

1/𝑞

𝐵𝑘1−1/𝑞

𝐻

≤1

Γ𝑛(𝛼)


𝑝)

∞

∑

𝑘=𝛾+1

𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘

1+𝜆

𝐻

≤ 𝐶𝐵𝛾

𝜇

𝐻


𝑝).

(20)

The last inequality is due to the fact that 𝜆 < −Re𝛼/𝑛.Consequently,

𝐼𝐼 = (1

𝐵𝛾

1+𝜇𝑟

𝐻

∫𝐵𝛾

𝐼𝑝


𝛾

) (𝑥)

𝑟

𝑑𝑥)

1/𝑟

≤ 𝐶𝑓�̇�𝑞,𝜆(Q𝑛

𝑝).

(21)

The above estimates imply that

𝐼𝑝

𝛼𝑓�̇�𝑟,𝜇(Q𝑛

𝑝)≤ 𝐶


𝑝). (22)


(ii) If 𝑞 = 1, set 𝑓1= 𝑓𝜒𝐵𝛾

and 𝑓2= 𝑓 − 𝑓

1; by Lemma 8,

we have{𝑥 ∈ 𝐵

𝛾:𝐼𝑝

𝛼𝑓1(𝑥) > 𝑡}

𝐻

≤ 𝐶(

𝑓1𝐿1(Q𝑛

𝑝)

𝑡)

𝑟

= 𝐶𝑡−𝑟

(∫𝐵𝛾

𝑓 (𝑥) 𝑑𝑥)

𝑟

≤ 𝐶𝑡−𝑟𝐵𝛾

(1+𝜆)𝑟

𝐻

𝑓𝑟

�̇�1,𝜆(Q𝑛𝑝)

= 𝐶𝑡−𝑟𝐵𝛾

1+𝜇𝑟

𝐻

𝑓𝑟

�̇�1,𝜆(Q𝑛𝑝).

(23)

On the other hand, by the same estimate as (30), we have

𝐼𝑝

𝛼𝑓2(𝑥) ≤ 𝐶

𝐵𝛾

𝜇

𝐻

𝑓2�̇�1,𝜆(Q𝑛

𝑝). (24)

Then using Chebyshev’s inequality, we obtain

{𝑥 ∈ 𝐵

𝛾:𝐼𝑝

𝛼𝑓2(𝑥) > 𝑡}

𝐻≤ 𝑡−𝑟

∫𝐵𝛾

𝐼𝑝

𝛼𝑓2(𝑥)𝑟

𝑑𝑥


1+𝜇𝑟

𝐻

𝑓2𝑟

�̇�1,𝜆(Q𝑛𝑝)


1+𝜇𝑟

𝐻

𝑓𝑟

�̇�1,𝜆(Q𝑛𝑝).

(25)

Since𝐼𝑝

𝛼𝑓 (𝑥)

≤𝐼𝑝

𝛼𝑓1(𝑥) +𝐼𝑝

𝛼𝑓2(𝑥) , (26)

we get

{𝑥 ∈ 𝐵

𝛾:𝐼𝑝

𝛼𝑓 (𝑥)

> 𝑡}𝐻≤{𝑥 ∈ 𝐵

𝛾:𝐼𝑝

𝛼𝑓1(𝑥) >

𝑡

2}𝐻

+{𝑥 ∈ 𝐵

𝛾:𝐼𝑝

𝛼𝑓2(𝑥) >

𝑡

2}𝐻


1+𝜇𝑟

𝐻

𝑓𝑟

�̇�1,𝜆(Q𝑛𝑝).

(27)

Therefore,

(𝑡𝑟{𝑥 ∈ 𝐵

𝛾:𝐼𝑝

𝛼𝑓 (𝑥)

> 𝑡}𝐻

𝐵𝛾

1+𝜇𝑟

𝐻

)

1/𝑟

≤ 𝐶𝑓�̇�1,𝜆(Q𝑛

𝑝), (28)

for any 𝑡 > 0 and 𝛾 ∈ Z. This completes the proof.

For application, we now introduce a pseudo-differentialoperator𝐷𝛼 defined by Vladimirov in [33].

The operator 𝐷𝛼 : 𝜓 → 𝐷𝛼𝜓 is defined as convolutionof generalized functions 𝑓

−𝛼and 𝜓:

𝐷𝛼

𝜓 = 𝑓−𝛼∗ 𝜓, 𝛼 ̸= −1, (29)

where 𝑓𝛼= |𝑥|𝛼−1

𝑝/Γ(𝛼) and Γ(𝛼) = (1 − 𝑝𝛼−1)/(1 − 𝑝−𝛼).

Let us consider the equation

𝐷𝛼

𝜓 = 𝑔, 𝑔 ∈ E

, (30)

where E is the space of linear continuous functionals on Eand here E denotes the set of locally constant functions onQ𝑝. A complex-valued function 𝑓(𝑥) defined onQ

𝑝is called

locally constant if for any point 𝑥 ∈ Q𝑝there exists an integer

𝑙(𝑥) ∈ Z such that

𝑓 (𝑥 + 𝑥

) = 𝑓 (𝑥) ,

𝑥𝑝≤ 𝑝𝑙(𝑥)

.

(31)

The following lemma (page 154 in [10]) gives solutions of(30).

Lemma 9. For 𝛼 > 0 any solution of (30) is expressed by theformula

𝜓 = 𝐷−𝛼

𝑔 + 𝐶, (32)

where 𝐶 is an arbitrary constant; for 𝛼 < 0 a solution of (30) isunique and it is expressed by formula (32) for 𝐶 = 0.

Combining with Theorem 7, we obtain the followingregular property of the solution.

Corollary 10. Let 0 < 𝛼 < 1 and let 1 ≤ 𝑞 < 1/𝛼, 0 < 1/𝑟 =1/𝑞 − 𝛼, 𝜆 < −𝛼, and 𝜇 = 𝜆 + 𝛼. If 𝑔 ∈ E ∩ �̇�𝑞,𝜆(Q𝑛

𝑝), then

(i) when 𝑞 > 1, (30) has a solution in �̇�𝑟,𝜇(Q𝑛𝑝),

(ii) when 𝑞 = 1, (30) has a solution in𝑊�̇�𝑟,𝜇(Q𝑛𝑝).

3. Commutators of 𝑝-Adic Riesz Potential

In this section, we will establish the 𝜆-central BMO estimatesfor commutators 𝐼𝑝,𝑏

𝛼of 𝑝-adic Riesz potential which is

defined by

𝐼𝑝,𝑏

𝛼𝑓 = 𝑏𝐼

𝑝

𝛼𝑓 − 𝐼𝑝

𝛼(𝑏𝑓) , (33)

for some suitable functions 𝑓.

Theorem 11. Suppose 0 < Re𝛼 < 𝑛, 1 < 𝑞1< 𝑛/Re𝛼, 𝑞

1<

𝑞2< ∞, and 1/𝑞 = 1/𝑞

1+ 1/𝑞2− Re𝛼/𝑛. Let 0 ≤ 𝜆

2< 1/𝑛,

𝜆1satisfies 𝜆

1< −𝜆2− Re𝛼/𝑛, and 𝜆 = 𝜆

1+ 𝜆2+ Re𝛼/𝑛. If

𝑏 ∈ 𝐶𝐵𝑀𝑂𝑞2,𝜆2(Q𝑛𝑝), then 𝐼𝑝,𝑏

𝛼is bounded from �̇�𝑞1 ,𝜆1(Q𝑛

𝑝) to

�̇�𝑞,𝜆

(Q𝑛𝑝), and the following inequality holds:

𝐼𝑝,𝑏

𝛼𝑓�̇�𝑞,𝜆(Q𝑛

𝑝)

≤ 𝐶 ‖𝑏‖𝐶𝐵𝑀𝑂

𝑞2,𝜆2 (Q𝑛𝑝)

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝). (34)

Before proving this theorem,we need the following result.

Lemma 12 (see [31]). Suppose that 𝑏 ∈ 𝐶𝐵𝑀𝑂𝑞,𝜆(Q𝑛𝑝) and

𝑗, 𝑘 ∈ Z, 𝜆 ≥ 0. Then𝑏𝐵𝑗

− 𝑏𝐵𝑘

≤ 𝑝𝑛 𝑗 − 𝑘

‖𝑏‖𝐶𝐵𝑀𝑂𝑞,𝜆(Q𝑛𝑝)max {𝐵𝑗

𝜆

𝐻

,𝐵𝑘𝜆

𝐻} .

(35)


Proof of Theorem 11. Suppose that 𝑓 is a function in�̇�𝑞1,𝜆1(Q𝑛𝑝). For fixed 𝛾 ∈ Z, denote 𝐵

𝛾(0) by 𝐵

𝛾. We write

(1

𝐵𝛾

𝐻

∫𝐵𝛾

𝐼𝑝,𝑏

𝛼𝑓 (𝑥)

𝑞

𝑑𝑥)

1/𝑞

≤ (1

𝐵𝛾

𝐻

∫𝐵𝛾

(𝑏 (𝑥) − 𝑏

𝐵𝛾

) (𝐼𝑝

𝛼𝑓𝜒𝐵𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

+ (1

𝐵𝛾

𝐻

∫𝐵𝛾

(𝑏 (𝑥) − 𝑏

𝐵𝛾

) (𝐼𝑝

𝛼𝑓𝜒𝐵𝑐

𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

+ (1

𝐵𝛾

𝐻

∫𝐵𝛾

𝐼𝑝

𝛼((𝑏 − 𝑏

𝐵𝛾

)𝑓𝜒𝐵𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

+ (1

𝐵𝛾

𝐻

∫𝐵𝛾

𝐼𝑝

𝛼((𝑏 − 𝑏

𝐵𝛾

)𝑓𝜒𝐵𝑐

𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

:= 𝐽1+ 𝐽2+ 𝐽3+ 𝐽4.

(36)

Set 1/𝑟 = 1/𝑞1− Re𝛼/𝑛; then 1/𝑞 = 1/𝑞

2+ 1/𝑟; by

Lemma 8 and Hölder’s inequality, we have

𝐽1= (

1𝐵𝛾

𝐻

∫𝐵𝛾

(𝑏 (𝑥) − 𝑏

𝐵𝛾

) (𝐼𝑝

𝛼𝑓𝜒𝐵𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

≤𝐵𝛾

−1/𝑞

𝐻

(∫𝐵𝛾

𝑏 (𝑥) − 𝑏

𝐵𝛾

𝑞2

𝑑𝑥)

1/𝑞2

⋅ (∫𝐵𝛾

𝐼𝑝


) (𝑥)

𝑟

𝑑𝑥)

1/𝑟

≤ 𝐶𝐵𝛾

−1/𝑟+𝜆2

𝐻

‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)

⋅ (∫𝐵𝛾

𝑓𝜒𝐵𝛾

(𝑥)

𝑞1

𝑑𝑥)

1/𝑞1

≤ 𝐶𝐵𝛾

𝜆

𝐻


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝).

(37)

Similarly, denote 1/𝑙 = 1/𝑞1+ 1/𝑞

2; then 1/𝑞 = 1/𝑙 −

Re𝛼/𝑛, and by Hölder’s inequality and Lemma 8, we get

𝐽3= (

1𝐵𝛾

𝐻

∫𝐵𝛾

𝐼𝑝

𝛼((𝑏 − 𝑏

𝐵𝛾

)𝑓𝜒𝐵𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

≤ 𝐶𝐵𝛾

−1/𝑞

𝐻

(∫𝐵𝛾

(𝑏 (𝑥) − 𝑏

𝐵𝛾

)𝑓 (𝑥)

𝑙

𝑑𝑥)

1/𝑙

≤ 𝐶𝐵𝛾

−1/𝑞

𝐻

(∫𝐵𝛾

𝑏 (𝑥) − 𝑏

𝐵𝛾

𝑞2

𝑑𝑥)

1/𝑞2

⋅ (∫𝐵𝛾

𝑓 (𝑥)𝑞1

𝑑𝑥)

1/𝑞1

≤ 𝐶𝐵𝛾

𝜆

𝐻


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝).

(38)

To estimate 𝐽2and 𝐽

4, we firstly give the following

estimates. For 𝑥 ∈ 𝐵𝛾, by Hölder’s inequality, we obtain

𝐼𝑝


𝛾

) (𝑥)

=

1

Γ𝑛(𝛼)

∫𝐵𝑐

𝛾

𝑓 (𝑦)


𝑝

𝑑𝑦

≤1

Γ𝑛(𝛼)

∫𝐵𝑐

𝛾

𝑓 (𝑦)


𝑑𝑦

=1

Γ𝑛(𝛼)

∞

∑

𝑘=𝛾+1

∫𝑆𝑘

𝑓 (𝑦) 𝑝−𝑘(𝑛−Re𝛼)

𝑑𝑦

≤1

Γ𝑛(𝛼)

∞

∑

𝑘=𝛾+1


1−1/𝑞

1

𝐻(∫𝑆𝑘

𝑓 (𝑦)𝑞1

𝑑𝑦)

1/𝑞1

≤1

Γ𝑛(𝛼)

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

∞

∑

𝑘=𝛾+1


1+𝜆1

𝐻

=1

Γ𝑛(𝛼)

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

𝑝(𝛾+1)(𝑛𝜆

1+Re𝛼)

1 − 𝑝𝑛𝜆1+Re𝛼

= 𝐶𝐵𝛾

𝜆1+Re𝛼/𝑛𝐻

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝),

(39)

where the penultimate “=” is due to the fact that 𝜆1+Re𝛼/𝑛 <

−𝜆2≤ 0. Similarly,

𝐼𝑝

𝛼((𝑏 − 𝑏

𝐵𝛾

)𝑓𝜒𝐵𝑐

𝛾

) (𝑥)

=

1

Γ𝑛(𝛼)

∫𝐵𝑐

𝛾

(𝑏 (𝑦) − 𝑏𝐵𝛾

)𝑓 (𝑦)


𝑝

𝑑𝑦

≤1

Γ𝑛(𝛼)

∫𝐵𝑐

𝛾

𝑏 (𝑦) − 𝑏

𝐵𝛾

𝑓 (𝑦)


𝑑𝑦

=1

Γ𝑛(𝛼)

∞

∑

𝑘=𝛾+1

∫𝑆𝑘

𝑏 (𝑦) − 𝑏

𝐵𝛾

𝑓 (𝑦) 𝑝−𝑘(𝑛−Re𝛼)

𝑑𝑦

=1

Γ𝑛(𝛼)

∞

∑

𝑘=𝛾+1


1−1/𝑞

1−1/𝑞2

𝐻

⋅ (∫𝑆𝑘

𝑓 (𝑦)𝑞1

𝑑𝑦)

1/𝑞1

(∫𝑆𝑘

𝑏 (𝑦) − 𝑏

𝐵𝛾

𝑞2

𝑑𝑦)

1/𝑞2

≤1

Γ𝑛(𝛼)

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

∞

∑

𝑘=𝛾+1


1−1/𝑞

2+𝜆1

𝐻

⋅ (∫𝐵𝑘

𝑏 (𝑦) − 𝑏

𝐵𝛾

𝑞2

𝑑𝑦)

1/𝑞2

≤1

Γ𝑛(𝛼)

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

∞

∑

𝑘=𝛾+1


1−1/𝑞

2+𝜆1

𝐻

× [(∫𝐵𝑘

𝑏 (𝑦) − 𝑏

𝐵𝑘

𝑞2

𝑑𝑦)

1/𝑞2

+𝑏𝐵𝑘

− 𝑏𝐵𝛾

𝐵𝑘1/𝑞2

𝐻] .

(40)


Since 𝑘 ≥ 𝛾 + 1, by Lemma 12, we have𝑏𝐵𝑘

− 𝑏𝐵𝛾

≤ 𝑝𝑛

(𝑘 − 𝛾) ‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)

𝐵𝑘𝜆2

𝐻. (41)

Thus𝐼𝑝

𝛼((𝑏 − 𝑏

𝐵𝛾

)𝑓𝜒𝐵𝑐

𝛾

) (𝑥)

≤1

Γ𝑛(𝛼)


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

×

∞

∑

𝑘=𝛾+1


1−1/𝑞

2+𝜆1

𝐻

⋅ [𝐵𝑘1/𝑞2+𝜆2

𝐻+ 𝑝𝑛

(𝑘 − 𝛾)𝐵𝑘1/𝑞2+𝜆2

𝐻]

≤𝐶

Γ𝑛(𝛼)


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

⋅

∞

∑

𝑘=𝛾+1

(𝑘 − 𝛾) 𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘

1+𝜆1+𝜆2

𝐻

≤𝐶

Γ𝑛(𝛼)


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

∞

∑

𝑘=𝛾+1

(𝑘 − 𝛾) 𝑝𝑘𝑛𝜆

= 𝐶𝐵𝛾

𝜆

𝐻


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝).

(42)

Now by (39) and Hölder’s inequality, we obtain

𝐽2= (

1𝐵𝛾

𝐻

∫𝐵𝛾

(𝑏 (𝑥) − 𝑏

𝐵𝛾

) (𝐼𝑝

𝛼𝑓𝜒𝐵𝑐

𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

≤ 𝐶𝐵𝛾

𝜆1+Re𝛼/𝑛−1/𝑞𝐻

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

⋅ (∫𝐵𝛾

𝑏 (𝑥) − 𝑏

𝐵𝛾

𝑞

𝑑𝑥)

1/𝑞

≤ 𝐶𝐵𝛾

𝜆1+Re𝛼/𝑛−1/𝑞

2

𝐻

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝)

⋅ (∫𝐵𝛾

𝑏 (𝑥) − 𝑏

𝐵𝛾

𝑞2

𝑑𝑥)

1/𝑞2

≤ 𝐶𝐵𝛾

𝜆

𝐻


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝).

(43)

It follows from (42) that

𝐽4= (

1𝐵𝛾

𝐻

∫𝐵𝛾

𝐼𝑝

𝛼((𝑏 − 𝑏

𝐵𝛾

)𝑓𝜒𝐵𝑐

𝛾

) (𝑥)

𝑞

𝑑𝑥)

1/𝑞

≤ 𝐶𝐵𝛾

𝜆

𝐻


𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝).

(44)

The above estimates imply that𝐼𝑝,𝑏

𝛼𝑓�̇�𝑞,𝜆(Q𝑛

𝑝)

≤ 𝐶 ‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)

𝑓�̇�𝑞1,𝜆1 (Q𝑛

𝑝). (45)

This completes the proof of the theorem.

Remark 13. Since 𝑝-adic field is a kind of locally com-pact Vilenkin groups, we can further consider the Hardy-Littlewood-Sobolev inequalities on such groups, which ismore complicated and will appear elsewhere.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was partially supported by NSF of China (Grantnos. 11271175, 11171345, and 11301248) and AMEP (DYSP)of Linyi University and Macao Science and TechnologyDevelopment Fund, MSAR (Ref. 018/2014/A1).

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