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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
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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
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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
𝛼(𝑓𝜒𝐵𝛾
) (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
𝛼(𝑓𝜒𝐵𝑐
𝛾
) (𝑥)
=
1
Γ𝑛(𝛼)
∫𝐵𝑐
𝛾
𝑓 (𝑦)
𝑥 − 𝑦𝑛−𝛼
𝑝
𝑑𝑦
≤1
Γ𝑛(𝛼)
∫𝐵𝑐
𝛾
𝑓 (𝑦)
𝑥 − 𝑦𝑛−Re𝛼𝑝
𝑑𝑦
=1
Γ𝑛(𝛼)
∞
∑
𝑘=𝛾+1
∫𝑆𝑘
𝑓 (𝑦)
𝑥 − 𝑦𝑛−Re𝛼𝑝
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)
𝑓�̇�𝑞,𝜆(Q𝑛
𝑝)
∞
∑
𝑘=𝛾+1
𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
1+𝜆
𝐻
≤ 𝐶𝐵𝛾
𝜇
𝐻
𝑓�̇�𝑞,𝜆(Q𝑛
𝑝).
(20)
The last inequality is due to the fact that 𝜆 <
−Re𝛼/𝑛.Consequently,
𝐼𝐼 = (1
𝐵𝛾
1+𝜇𝑟
𝐻
∫𝐵𝛾
𝐼𝑝
𝛼(𝑓𝜒𝐵𝑐
𝛾
) (𝑥)
𝑟
𝑑𝑥)
1/𝑟
≤ 𝐶𝑓�̇�𝑞,𝜆(Q𝑛
𝑝).
(21)
The above estimates imply that
𝐼𝑝
𝛼𝑓�̇�𝑟,𝜇(Q𝑛
𝑝)≤ 𝐶
𝑓�̇�𝑞,𝜆(Q𝑛
𝑝). (22)
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4 Journal of Function Spaces
(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
≤ 𝐶𝑡−𝑟𝐵𝛾
1+𝜇𝑟
𝐻
𝑓2𝑟
�̇�1,𝜆(Q𝑛𝑝)
≤ 𝐶𝑡−𝑟𝐵𝛾
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
≤ 𝐶𝑡−𝑟𝐵𝛾
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)
-
Journal of Function Spaces 5
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
𝛼(𝑓𝜒𝐵𝛾
) (𝑥)
𝑟
𝑑𝑥)
1/𝑟
≤ 𝐶𝐵𝛾
−1/𝑟+𝜆2
𝐻
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
⋅ (∫𝐵𝛾
𝑓𝜒𝐵𝛾
(𝑥)
𝑞1
𝑑𝑥)
1/𝑞1
≤ 𝐶𝐵𝛾
𝜆
𝐻
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
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
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
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
𝛼(𝑓𝜒𝐵𝑐
𝛾
) (𝑥)
=
1
Γ𝑛(𝛼)
∫𝐵𝑐
𝛾
𝑓 (𝑦)
𝑥 − 𝑦𝑛−𝛼
𝑝
𝑑𝑦
≤1
Γ𝑛(𝛼)
∫𝐵𝑐
𝛾
𝑓 (𝑦)
𝑥 − 𝑦𝑛−Re𝛼𝑝
𝑑𝑦
=1
Γ𝑛(𝛼)
∞
∑
𝑘=𝛾+1
∫𝑆𝑘
𝑓 (𝑦) 𝑝−𝑘(𝑛−Re𝛼)
𝑑𝑦
≤1
Γ𝑛(𝛼)
∞
∑
𝑘=𝛾+1
𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
1−1/𝑞
1
𝐻(∫𝑆𝑘
𝑓 (𝑦)𝑞1
𝑑𝑦)
1/𝑞1
≤1
Γ𝑛(𝛼)
𝑓�̇�𝑞1,𝜆1 (Q𝑛
𝑝)
∞
∑
𝑘=𝛾+1
𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
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)
𝑥 − 𝑦𝑛−𝛼
𝑝
𝑑𝑦
≤1
Γ𝑛(𝛼)
∫𝐵𝑐
𝛾
𝑏 (𝑦) − 𝑏
𝐵𝛾
𝑓 (𝑦)
𝑥 − 𝑦𝑛−Re𝛼𝑝
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
𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
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
𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
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
𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
1−1/𝑞
2+𝜆1
𝐻
× [(∫𝐵𝑘
𝑏 (𝑦) − 𝑏
𝐵𝑘
𝑞2
𝑑𝑦)
1/𝑞2
+𝑏𝐵𝑘
− 𝑏𝐵𝛾
𝐵𝑘1/𝑞2
𝐻] .
(40)
-
6 Journal of Function Spaces
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)
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
𝑓�̇�𝑞1,𝜆1 (Q𝑛
𝑝)
×
∞
∑
𝑘=𝛾+1
𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
1−1/𝑞
2+𝜆1
𝐻
⋅ [𝐵𝑘1/𝑞2+𝜆2
𝐻+ 𝑝𝑛
(𝑘 − 𝛾)𝐵𝑘1/𝑞2+𝜆2
𝐻]
≤𝐶
Γ𝑛(𝛼)
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
𝑓�̇�𝑞1,𝜆1 (Q𝑛
𝑝)
⋅
∞
∑
𝑘=𝛾+1
(𝑘 − 𝛾) 𝑝−𝑘(𝑛−Re𝛼) 𝐵𝑘
1+𝜆1+𝜆2
𝐻
≤𝐶
Γ𝑛(𝛼)
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
𝑓�̇�𝑞1,𝜆1 (Q𝑛
𝑝)
∞
∑
𝑘=𝛾+1
(𝑘 − 𝛾) 𝑝𝑘𝑛𝜆
= 𝐶𝐵𝛾
𝜆
𝐻
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
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
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
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
‖𝑏‖CBMO𝑞2,𝜆2 (Q𝑛𝑝)
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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