Research ArticleHardy-Littlewood-Sobolev Inequalities on π-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 Β© 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 π-adic central Morrey spaces. Furthermore, we obtain the π-centralBMO estimates for commutators of π-adic Riesz potential on π-adic central Morrey spaces.
1. Introduction
Let 0 < πΌ < π. The Riesz potential operator πΌπΌis defined by
setting, for all locally integrable functions π on Rπ,
πΌπΌπ (π₯) =
1
πΎπ(πΌ)
β«Rπ
π (π¦)
π₯ β π¦πβπΌππ¦, (1)
where πΎπ(πΌ) = π
π/2
2πΌ
Ξ(πΌ/2)/Ξ((π β πΌ)/2). It is closely relatedto the Laplacian operator of fractional degree. When π > 2and πΌ = 2, πΌ
πΌπ is a solution of Poisson equation βΞπ’ =
π. 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 π-adicfield. In the last 20 years, the field of π-adic numbers Q
π
has been intensively used in theoretical and mathematicalphysics (cf. [2β12]). And it has already penetrated intensivelyinto several areas of mathematics and its applications, amongwhich harmonic analysis on π-adic field has been drawingmore and more concern (see [13β22] and references therein).
For a prime number π, the field of π-adic numbers Qπ
is defined as the completion of the field of rational numbersQ with respect to the non-Archimedean π-adic norm | β |
π,
which satisfies |π₯|π= 0 if and only if π₯ = 0; |π₯π¦|
π=
|π₯|π|π¦|π; |π₯ + π¦|
πβ€ max{|π₯|
π, |π¦|π}. Moreover, if |π₯|
πΜΈ= |π¦|π,
then |π₯ Β± π¦|π= max{|π₯|
π, |π¦|π}. It is well-known that Q
π
is a typical model of non-Archimedean local fields. If anynonzero rational number π₯ is represented as π₯ = ππΎ(π/π),where πΎ = πΎ(π₯) β Z and integers π, π are indivisible by π,then |π₯|
π= πβπΎ.
The space Qππ= QπΓ QπΓ β β β Γ Q
πconsists of points
π₯ = (π₯1, π₯2, . . . , π₯
π), where π₯
πβ Qπ, π = 1, 2, . . . , π. The π-
adic norm onQππis
|π₯|π:= max1β€πβ€π
π₯π
π, π₯ β Q
π
π. (2)
Denote by
π΅πΎ(π) = {π₯ β Q
π
π: |π₯ β π|
πβ€ ππΎ
} (3)
the ball of radius ππΎ with center at π β Qππand by
ππΎ(π) = π΅
πΎ(π) \ π΅
πΎβ1(π) = {π₯ β Q
π
π: |π₯ β π|
π= ππΎ
} (4)
the sphere of radius ππΎ with center at π β Qππ, where πΎ β Z. It
is clear that
π΅πΎ(π) = β
πβ€πΎ
ππ(π) . (5)
It is well-known that Qππis a classical kind of locally
compact Vilenkin groups. A locally compact Vilenkin groupπΊ is a locally compact Abelian group containing a strictlydecreasing sequence of compact open subgroups {πΊ
π}β
π=ββ
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) βͺβπ=ββ
πΊπ= πΊ and β©β
π=ββπΊπ= 0 and (2)
sup{order(πΊπ/πΊπ+1
: π β Z)} < β. For several decades,parallel to the π-adic harmonic analysis, a development wasunder way of the harmonic analysis on locally compactVilenkin groups (cf. [23β25] and references therein).
Since Qππis a locally compact commutative group under
addition, it follows from the standard analysis that there existsa Haar measure ππ₯ on Qπ
π, which is unique up to a positive
constant factor and is translation invariant.We normalize themeasure ππ₯ by the equality
β«π΅0(0)
ππ₯ =π΅0 (0)
π» = 1, (6)
where |πΈ|π»denotes the Haar measure of a measurable subset
πΈ ofQππ. By simple calculation, we can obtain that
π΅πΎ(π)π»= ππΎπ
,
ππΎ(π)π»= ππΎπ
(1 β πβπ
)
(7)
for any π β Qππ. We should mention that the Haar measure
takes value in R; there also exist π-adic valued measures (cf.[26, 27]). For a more complete introduction to the π-adicfield, one can refer to [22] or [10].
On π-adic field, the π-adic Riesz potential πΌππΌ[22] is
defined by
πΌπ
πΌπ (π₯) =
1
Ξπ(πΌ)
β«Qππ
π (π¦)
π₯ β π¦πβπΌ
π
ππ¦, (8)
where Ξπ(πΌ) = (1 β π
πΌβπ
)/(1 β πβπΌ
), πΌ β C, πΌ ΜΈ= 0. Whenπ = 1, Haran [4, 28] obtained the explicit formula of Rieszpotentials onQ
πand developed analytical potential theory on
Qπ. Taibleson [22] gave the fundamental analytic properties
of the Riesz potentials on local fields including Qππ, as well
as the classical Hardy-Littlewood-Sobolev inequalities. Kim[18] gave a simple proof of these inequalities by using theπ-adic version of the CalderoΜn-Zygmund decompositiontechnique. Volosivets [29] investigated the boundedness forRiesz potentials on generalized Morrey spaces. Like onEuclidean spaces, using the Riesz potential with π > 2 andπΌ = 2, one can introduce the π-adic Laplacians [13].
In this paper, we will consider the Riesz potentials andtheir commutators with π-adic central BMO functions on π-adic central Morrey spaces. Alvarez et al. [30] studied therelationship between central BMO spaces andMorrey spaces.Furthermore, they introduced π-central BMO spaces andcentralMorrey spaces, respectively. In [31], we introduce theirπ-adic versions.
Definition 1. Let π β R and 1 < π < β. The π-adic centralMorrey space οΏ½ΜοΏ½π,π(Qπ
π) is defined by
ποΏ½ΜοΏ½π,π(Qπ
π):= supπΎβZ
(1
π΅πΎ
1+ππ
π»
β«π΅πΎ
π (π₯)π
ππ₯)
1/π
< β, (9)
where π΅πΎ= π΅πΎ(0).
Remark 2. It is clear that
πΏπ,π
(Qπ
π) β οΏ½ΜοΏ½π,π
(Qπ
π) ,
οΏ½ΜοΏ½π,β1/π
(Qπ
π) = πΏπ
(Qπ
π) .
(10)
When π < β1/π, the space οΏ½ΜοΏ½π,π(Qππ) reduces to {0}; therefore,
we can only consider the case π β₯ β1/π. If 1 β€ π1< π2< β,
by HoΜlderβs inequality,
οΏ½ΜοΏ½π2,π
(Qπ
π) β οΏ½ΜοΏ½π1,π
(Qπ
π) (11)
for π β R.
Definition 3. Let π < 1/π and 1 < π < β. The spaceCBMOπ,π(Qπ
π) is defined by the condition
πCBMOπ,π(Qπ
π)
:= supπΎβZ
(1
π΅πΎ
1+ππ
π»
β«π΅πΎ
π (π₯) β π
π΅πΎ
π
ππ₯)
1/π
< β.
(12)
Remark 4. When π = 0, the space CBMOπ,π(Qππ) is just
CBMOπ(Qππ), which is defined in [32]. If 1 β€ π
1< π2< β,
by HoΜlderβs inequality,
CBMOπ2 ,π (Qππ) β CBMOπ1,π (Qπ
π) (13)
for π β R. By the standard proof as that inRπ, we can see thatπCBMOπ,π(Qπ
π)
βΌ supπΎβZ
infπβC(
1
π΅πΎ
1+ππ
π»
β«π΅πΎ
π (π₯) β ππ
ππ₯)
1/π
.
(14)
Remark 5. Formulas (9) and (12) yield that οΏ½ΜοΏ½π,π(Qππ) is a
Banach space continuously included in CBMOπ,π(Qππ).
Herewe introduce theπ-adicweak centralMorrey spaces.
Definition 6. Let π β R and 1 < π < β. The π-adic weakcentral Morrey spaceποΏ½ΜοΏ½π,π(Qπ
π) is defined by
πποΏ½ΜοΏ½π,π(Qπ
π)
:= supπΎβZ
(supπ‘>0π‘π{π₯ β π΅
πΎ:π (π₯)
> π‘}π»
π΅πΎ
1+ππ
π»
)
1/π
< β,
(15)
where π΅πΎ= π΅πΎ(0).
In Section 2, we will get the Hardy-Littlewood-Sobolevinequalities on π-adic central Morrey spaces. Namely, under
Journal of Function Spaces 3
some conditions for indexes, πΌππΌis bounded from οΏ½ΜοΏ½π,π(Qπ
π) to
οΏ½ΜοΏ½π,π
(Qππ) and is also bounded from οΏ½ΜοΏ½1,π(Qπ
π) to ποΏ½ΜοΏ½π,π(Qπ
π).
In Section 3, we establish the boundedness for commutatorsgenerated by πΌπ
πΌand π-central BMO functions on π-adic
central Morrey spaces.Throughout this paper the letter πΆ 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 π-adic central Morrey spaces.
Theorem7. LetπΌ be a complex numberwith 0 < ReπΌ < π andlet 1 β€ π < π/ReπΌ, 0 < 1/π = 1/π β ReπΌ/π, π < βReπΌ/π,and π = π + ReπΌ/π.
(i) If π > 1, then πΌππΌis bounded from οΏ½ΜοΏ½π,π(Qπ
π) to οΏ½ΜοΏ½π,π(Qπ
π).
(ii) If π = 1, then πΌππΌ
is bounded from οΏ½ΜοΏ½1,π(Qππ) to
ποΏ½ΜοΏ½π,π
(Qππ).
In order to give the proof of this theorem, we need thefollowing result.
Lemma 8 (see [22]). Let πΌ be a complex number with 0 <ReπΌ < π and let 1 β€ π < π < β satisfy 1/π = 1/π β ReπΌ/π.
(i) If π β πΏπ(Qππ), π > 1, then
πΌπ
πΌππΏπ(Qπ
π)β€ π΄ππ
ππΏπ(Qπ
π), (16)
where π΄ππis independent of π.
(ii) If π β πΏ1(Qππ), π > 0, then
{π₯ β Q
π
π:πΌπ
πΌπ (π₯)
> π }π»β€ (π΄
π
ππΏ1(Qπ
π)
π )
π
, (17)
where π΄π> 0 is independent of π.
Proof ofTheorem 7. Let π be a function in οΏ½ΜοΏ½π,π(Qππ). For fixed
πΎ β Z, denote π΅πΎ(0) by π΅
πΎ.
(i) If π > 1, write
(1
π΅πΎ
1+ππ
π»
β«π΅πΎ
πΌπ
πΌπ (π₯)
π
ππ₯)
1/π
β€ (1
π΅πΎ
1+ππ
π»
β«π΅πΎ
πΌπ
πΌ(πππ΅πΎ
) (π₯)
π
ππ₯)
1/π
+ (1
π΅πΎ
1+ππ
π»
β«π΅πΎ
πΌπ
πΌ(πππ΅π
πΎ
) (π₯)
π
ππ₯)
1/π
:= πΌ + πΌπΌ.
(18)
For πΌ, since 1/π = 1/π β ReπΌ/π and π = π + ReπΌ/π, byLemma 8,
πΌ = (1
π΅πΎ
1+ππ
π»
β«π΅πΎ
πΌπ
πΌ(πππ΅πΎ
) (π₯)
π
ππ₯)
1/π
β€π΅πΎ
β1/πβπ
π»
(β«π΅πΎ
πππ΅πΎ
(π₯)
π
ππ₯)
1/π
β€ποΏ½ΜοΏ½π,π(Qπ
π).
(19)
For πΌπΌ, we firstly give the following estimate. For π₯ β π΅πΎ,
by HoΜlderβs inequality, we have
πΌπ
πΌ(πππ΅π
πΎ
) (π₯)
=
1
Ξπ(πΌ)
β«π΅π
πΎ
π (π¦)
π₯ β π¦πβπΌ
π
ππ¦
β€1
Ξπ(πΌ)
β«π΅π
πΎ
π (π¦)
π₯ β π¦πβReπΌπ
ππ¦
=1
Ξπ(πΌ)
β
β
π=πΎ+1
β«ππ
π (π¦)
π₯ β π¦πβReπΌπ
ππ¦
=1
Ξπ(πΌ)
β
β
π=πΎ+1
β«ππ
πβπ(πβReπΌ) π (π¦)
ππ¦
β€1
Ξπ(πΌ)
β
β
π=πΎ+1
πβπ(πβReπΌ)
(β«π΅π
π (π¦)π
ππ¦)
1/π
π΅π1β1/π
π»
β€1
Ξπ(πΌ)
ποΏ½ΜοΏ½π,π(Qπ
π)
β
β
π=πΎ+1
πβπ(πβReπΌ) π΅π
1+π
π»
β€ πΆπ΅πΎ
π
π»
ποΏ½ΜοΏ½π,π(Qπ
π).
(20)
The last inequality is due to the fact that π < βReπΌ/π.Consequently,
πΌπΌ = (1
π΅πΎ
1+ππ
π»
β«π΅πΎ
πΌπ
πΌ(πππ΅π
πΎ
) (π₯)
π
ππ₯)
1/π
β€ πΆποΏ½ΜοΏ½π,π(Qπ
π).
(21)
The above estimates imply that
πΌπ
πΌποΏ½ΜοΏ½π,π(Qπ
π)β€ πΆ
ποΏ½ΜοΏ½π,π(Qπ
π). (22)
4 Journal of Function Spaces
(ii) If π = 1, set π1= πππ΅πΎ
and π2= π β π
1; by Lemma 8,
we have{π₯ β π΅
πΎ:πΌπ
πΌπ1(π₯) > π‘}
π»
β€ πΆ(
π1πΏ1(Qπ
π)
π‘)
π
= πΆπ‘βπ
(β«π΅πΎ
π (π₯) ππ₯)
π
β€ πΆπ‘βππ΅πΎ
(1+π)π
π»
ππ
οΏ½ΜοΏ½1,π(Qππ)
= πΆπ‘βππ΅πΎ
1+ππ
π»
ππ
οΏ½ΜοΏ½1,π(Qππ).
(23)
On the other hand, by the same estimate as (30), we have
πΌπ
πΌπ2(π₯) β€ πΆ
π΅πΎ
π
π»
π2οΏ½ΜοΏ½1,π(Qπ
π). (24)
Then using Chebyshevβs inequality, we obtain
{π₯ β π΅
πΎ:πΌπ
πΌπ2(π₯) > π‘}
π»β€ π‘βπ
β«π΅πΎ
πΌπ
πΌπ2(π₯)π
ππ₯
β€ πΆπ‘βππ΅πΎ
1+ππ
π»
π2π
οΏ½ΜοΏ½1,π(Qππ)
β€ πΆπ‘βππ΅πΎ
1+ππ
π»
ππ
οΏ½ΜοΏ½1,π(Qππ).
(25)
SinceπΌπ
πΌπ (π₯)
β€πΌπ
πΌπ1(π₯) +πΌπ
πΌπ2(π₯) , (26)
we get
{π₯ β π΅
πΎ:πΌπ
πΌπ (π₯)
> π‘}π»β€{π₯ β π΅
πΎ:πΌπ
πΌπ1(π₯) >
π‘
2}π»
+{π₯ β π΅
πΎ:πΌπ
πΌπ2(π₯) >
π‘
2}π»
β€ πΆπ‘βππ΅πΎ
1+ππ
π»
ππ
οΏ½ΜοΏ½1,π(Qππ).
(27)
Therefore,
(π‘π{π₯ β π΅
πΎ:πΌπ
πΌπ (π₯)
> π‘}π»
π΅πΎ
1+ππ
π»
)
1/π
β€ πΆποΏ½ΜοΏ½1,π(Qπ
π), (28)
for any π‘ > 0 and πΎ β Z. This completes the proof.
For application, we now introduce a pseudo-differentialoperatorπ·πΌ defined by Vladimirov in [33].
The operator π·πΌ : π β π·πΌπ is defined as convolutionof generalized functions π
βπΌand π:
π·πΌ
π = πβπΌβ π, πΌ ΜΈ= β1, (29)
where ππΌ= |π₯|πΌβ1
π/Ξ(πΌ) and Ξ(πΌ) = (1 β ππΌβ1)/(1 β πβπΌ).
Let us consider the equation
π·πΌ
π = π, π β E
, (30)
where E is the space of linear continuous functionals on Eand here E denotes the set of locally constant functions onQπ. A complex-valued function π(π₯) defined onQ
πis called
locally constant if for any point π₯ β Qπthere exists an integer
π(π₯) β Z such that
π (π₯ + π₯
) = π (π₯) ,
π₯πβ€ ππ(π₯)
.
(31)
The following lemma (page 154 in [10]) gives solutions of(30).
Lemma 9. For πΌ > 0 any solution of (30) is expressed by theformula
π = π·βπΌ
π + πΆ, (32)
where πΆ is an arbitrary constant; for πΌ < 0 a solution of (30) isunique and it is expressed by formula (32) for πΆ = 0.
Combining with Theorem 7, we obtain the followingregular property of the solution.
Corollary 10. Let 0 < πΌ < 1 and let 1 β€ π < 1/πΌ, 0 < 1/π =1/π β πΌ, π < βπΌ, and π = π + πΌ. If π β E β© οΏ½ΜοΏ½π,π(Qπ
π), then
(i) when π > 1, (30) has a solution in οΏ½ΜοΏ½π,π(Qππ),
(ii) when π = 1, (30) has a solution inποΏ½ΜοΏ½π,π(Qππ).
3. Commutators of π-Adic Riesz Potential
In this section, we will establish the π-central BMO estimatesfor commutators πΌπ,π
πΌof π-adic Riesz potential which is
defined by
πΌπ,π
πΌπ = ππΌ
π
πΌπ β πΌπ
πΌ(ππ) , (33)
for some suitable functions π.
Theorem 11. Suppose 0 < ReπΌ < π, 1 < π1< π/ReπΌ, π
1<
π2< β, and 1/π = 1/π
1+ 1/π2β ReπΌ/π. Let 0 β€ π
2< 1/π,
π1satisfies π
1< βπ2β ReπΌ/π, and π = π
1+ π2+ ReπΌ/π. If
π β πΆπ΅πππ2,π2(Qππ), then πΌπ,π
πΌis bounded from οΏ½ΜοΏ½π1 ,π1(Qπ
π) to
οΏ½ΜοΏ½π,π
(Qππ), and the following inequality holds:
πΌπ,π
πΌποΏ½ΜοΏ½π,π(Qπ
π)
β€ πΆ βπβπΆπ΅ππ
π2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π). (34)
Before proving this theorem,we need the following result.
Lemma 12 (see [31]). Suppose that π β πΆπ΅πππ,π(Qππ) and
π, π β Z, π β₯ 0. Thenππ΅π
β ππ΅π
β€ ππ π β π
βπβπΆπ΅πππ,π(Qππ)max {π΅π
π
π»
,π΅ππ
π»} .
(35)
Journal of Function Spaces 5
Proof of Theorem 11. Suppose that π is a function inοΏ½ΜοΏ½π1,π1(Qππ). For fixed πΎ β Z, denote π΅
πΎ(0) by π΅
πΎ. We write
(1
π΅πΎ
π»
β«π΅πΎ
πΌπ,π
πΌπ (π₯)
π
ππ₯)
1/π
β€ (1
π΅πΎ
π»
β«π΅πΎ
(π (π₯) β π
π΅πΎ
) (πΌπ
πΌπππ΅πΎ
) (π₯)
π
ππ₯)
1/π
+ (1
π΅πΎ
π»
β«π΅πΎ
(π (π₯) β π
π΅πΎ
) (πΌπ
πΌπππ΅π
πΎ
) (π₯)
π
ππ₯)
1/π
+ (1
π΅πΎ
π»
β«π΅πΎ
πΌπ
πΌ((π β π
π΅πΎ
)πππ΅πΎ
) (π₯)
π
ππ₯)
1/π
+ (1
π΅πΎ
π»
β«π΅πΎ
πΌπ
πΌ((π β π
π΅πΎ
)πππ΅π
πΎ
) (π₯)
π
ππ₯)
1/π
:= π½1+ π½2+ π½3+ π½4.
(36)
Set 1/π = 1/π1β ReπΌ/π; then 1/π = 1/π
2+ 1/π; by
Lemma 8 and HoΜlderβs inequality, we have
π½1= (
1π΅πΎ
π»
β«π΅πΎ
(π (π₯) β π
π΅πΎ
) (πΌπ
πΌπππ΅πΎ
) (π₯)
π
ππ₯)
1/π
β€π΅πΎ
β1/π
π»
(β«π΅πΎ
π (π₯) β π
π΅πΎ
π2
ππ₯)
1/π2
β (β«π΅πΎ
πΌπ
πΌ(πππ΅πΎ
) (π₯)
π
ππ₯)
1/π
β€ πΆπ΅πΎ
β1/π+π2
π»
βπβCBMOπ2,π2 (Qππ)
β (β«π΅πΎ
πππ΅πΎ
(π₯)
π1
ππ₯)
1/π1
β€ πΆπ΅πΎ
π
π»
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π).
(37)
Similarly, denote 1/π = 1/π1+ 1/π
2; then 1/π = 1/π β
ReπΌ/π, and by HoΜlderβs inequality and Lemma 8, we get
π½3= (
1π΅πΎ
π»
β«π΅πΎ
πΌπ
πΌ((π β π
π΅πΎ
)πππ΅πΎ
) (π₯)
π
ππ₯)
1/π
β€ πΆπ΅πΎ
β1/π
π»
(β«π΅πΎ
(π (π₯) β π
π΅πΎ
)π (π₯)
π
ππ₯)
1/π
β€ πΆπ΅πΎ
β1/π
π»
(β«π΅πΎ
π (π₯) β π
π΅πΎ
π2
ππ₯)
1/π2
β (β«π΅πΎ
π (π₯)π1
ππ₯)
1/π1
β€ πΆπ΅πΎ
π
π»
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π).
(38)
To estimate π½2and π½
4, we firstly give the following
estimates. For π₯ β π΅πΎ, by HoΜlderβs inequality, we obtain
πΌπ
πΌ(πππ΅π
πΎ
) (π₯)
=
1
Ξπ(πΌ)
β«π΅π
πΎ
π (π¦)
π₯ β π¦πβπΌ
π
ππ¦
β€1
Ξπ(πΌ)
β«π΅π
πΎ
π (π¦)
π₯ β π¦πβReπΌπ
ππ¦
=1
Ξπ(πΌ)
β
β
π=πΎ+1
β«ππ
π (π¦) πβπ(πβReπΌ)
ππ¦
β€1
Ξπ(πΌ)
β
β
π=πΎ+1
πβπ(πβReπΌ) π΅π
1β1/π
1
π»(β«ππ
π (π¦)π1
ππ¦)
1/π1
β€1
Ξπ(πΌ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
β
β
π=πΎ+1
πβπ(πβReπΌ) π΅π
1+π1
π»
=1
Ξπ(πΌ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
π(πΎ+1)(ππ
1+ReπΌ)
1 β πππ1+ReπΌ
= πΆπ΅πΎ
π1+ReπΌ/ππ»
ποΏ½ΜοΏ½π1,π1 (Qπ
π),
(39)
where the penultimate β=β is due to the fact that π1+ReπΌ/π <
βπ2β€ 0. Similarly,
πΌπ
πΌ((π β π
π΅πΎ
)πππ΅π
πΎ
) (π₯)
=
1
Ξπ(πΌ)
β«π΅π
πΎ
(π (π¦) β ππ΅πΎ
)π (π¦)
π₯ β π¦πβπΌ
π
ππ¦
β€1
Ξπ(πΌ)
β«π΅π
πΎ
π (π¦) β π
π΅πΎ
π (π¦)
π₯ β π¦πβReπΌπ
ππ¦
=1
Ξπ(πΌ)
β
β
π=πΎ+1
β«ππ
π (π¦) β π
π΅πΎ
π (π¦) πβπ(πβReπΌ)
ππ¦
=1
Ξπ(πΌ)
β
β
π=πΎ+1
πβπ(πβReπΌ) π΅π
1β1/π
1β1/π2
π»
β (β«ππ
π (π¦)π1
ππ¦)
1/π1
(β«ππ
π (π¦) β π
π΅πΎ
π2
ππ¦)
1/π2
β€1
Ξπ(πΌ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
β
β
π=πΎ+1
πβπ(πβReπΌ) π΅π
1β1/π
2+π1
π»
β (β«π΅π
π (π¦) β π
π΅πΎ
π2
ππ¦)
1/π2
β€1
Ξπ(πΌ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
β
β
π=πΎ+1
πβπ(πβReπΌ) π΅π
1β1/π
2+π1
π»
Γ [(β«π΅π
π (π¦) β π
π΅π
π2
ππ¦)
1/π2
+ππ΅π
β ππ΅πΎ
π΅π1/π2
π»] .
(40)
6 Journal of Function Spaces
Since π β₯ πΎ + 1, by Lemma 12, we haveππ΅π
β ππ΅πΎ
β€ ππ
(π β πΎ) βπβCBMOπ2,π2 (Qππ)
π΅ππ2
π». (41)
ThusπΌπ
πΌ((π β π
π΅πΎ
)πππ΅π
πΎ
) (π₯)
β€1
Ξπ(πΌ)
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
Γ
β
β
π=πΎ+1
πβπ(πβReπΌ) π΅π
1β1/π
2+π1
π»
β [π΅π1/π2+π2
π»+ ππ
(π β πΎ)π΅π1/π2+π2
π»]
β€πΆ
Ξπ(πΌ)
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
β
β
β
π=πΎ+1
(π β πΎ) πβπ(πβReπΌ) π΅π
1+π1+π2
π»
β€πΆ
Ξπ(πΌ)
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
β
β
π=πΎ+1
(π β πΎ) ππππ
= πΆπ΅πΎ
π
π»
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π).
(42)
Now by (39) and HoΜlderβs inequality, we obtain
π½2= (
1π΅πΎ
π»
β«π΅πΎ
(π (π₯) β π
π΅πΎ
) (πΌπ
πΌπππ΅π
πΎ
) (π₯)
π
ππ₯)
1/π
β€ πΆπ΅πΎ
π1+ReπΌ/πβ1/ππ»
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
β (β«π΅πΎ
π (π₯) β π
π΅πΎ
π
ππ₯)
1/π
β€ πΆπ΅πΎ
π1+ReπΌ/πβ1/π
2
π»
ποΏ½ΜοΏ½π1,π1 (Qπ
π)
β (β«π΅πΎ
π (π₯) β π
π΅πΎ
π2
ππ₯)
1/π2
β€ πΆπ΅πΎ
π
π»
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π).
(43)
It follows from (42) that
π½4= (
1π΅πΎ
π»
β«π΅πΎ
πΌπ
πΌ((π β π
π΅πΎ
)πππ΅π
πΎ
) (π₯)
π
ππ₯)
1/π
β€ πΆπ΅πΎ
π
π»
βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π).
(44)
The above estimates imply thatπΌπ,π
πΌποΏ½ΜοΏ½π,π(Qπ
π)
β€ πΆ βπβCBMOπ2,π2 (Qππ)
ποΏ½ΜοΏ½π1,π1 (Qπ
π). (45)
This completes the proof of the theorem.
Remark 13. Since π-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).
References
[1] L. Grafakos,Modern Fourier Analysis, vol. 250 ofGraduate Textsin Mathematics, Springer, New York, NY, USA, 2nd edition,2008.
[2] S. Albeverio and W. Karwowski, βA random walk on p-adicsβthe generator and its spectrum,β Stochastic Processes and theirApplications, vol. 53, no. 1, pp. 1β22, 1994.
[3] V.A.Avetisov, A.H. Bikulov, S. V. Kozyrev, andV.A.Osipov, βπ-adicmodels of ultrametric diffusion constrained by hierarchicalenergy landscapes,β Journal of Physics. A. Mathematical andGeneral, vol. 35, no. 2, pp. 177β189, 2002.
[4] S. Haran, βRiesz potentials and explicit sums in arithmetic,βInventiones Mathematicae, vol. 101, no. 3, pp. 697β703, 1990.
[5] A. Khrennikov, p-Adic Valued Distributions in MathematicalPhysics, Kluwer Academic Publishers, Dordrecht, The Nether-lands, 1994.
[6] A. Khrennikov, Non-Archimedean Analysis: Quantum Para-doxes, Dynamical Systems and Biological Models, Kluwer Aca-demic Publishers, Dordrecht, The Netherlands, 1997.
[7] A. N. Kochubei, βA non-Archimedean wave equation,β PacificJournal of Mathematics, vol. 235, no. 2, pp. 245β261, 2008.
[8] V. S. Varadarajan, βPath integrals for a class of π-adicSchroΜdinger equations,β Letters inMathematical Physics, vol. 39,no. 2, pp. 97β106, 1997.
[9] V. S. Vladimirov and I. V. Volovich, βπ-adic quantum mechan-ics,β Communications in Mathematical Physics, vol. 123, no. 4,pp. 659β676, 1989.
[10] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-AdicAnalysis and Mathematical Physics. Volume I, Series on Sovietand East European Mathematics, World Scientific, Singapore,1992.
[11] I. V. Volovich, βπ-adic space-time and string theory,βAkademiya Nauk SSSR: Teoreticheskaya i MatematicheskayaFizika, vol. 71, no. 3, pp. 337β340, 1987.
[12] I. V. Volovich, βπ-adic string,β Classical and Quantum Gravity,vol. 4, no. 4, pp. L83βL87, 1987.
[13] S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, βHar-monic analysis in the p-adicLizorkinspaces: fractional opera-tors, pseudo-differential equations, p-adic wavelets, Tauberiantheorems,β Journal of Fourier Analysis and Applications, vol. 12,pp. 393β425, 2006.
Journal of Function Spaces 7
[14] N. M. Chuong and H. D. Hung, βMaximal functions andweighted norm inequalities on local fields,β Applied and Com-putational Harmonic Analysis, vol. 29, no. 3, pp. 272β286, 2010.
[15] N. M. Chuong, Y. V. Egorov, A. Khrennikov, Y. Meyer, andD. Mumford, Harmonic, Waveletand p-Adic Analysis, WorldScientific Publishers, Singapore, 2007.
[16] Y.-C. Kim, βCarleson measures and the BMO space on the π-adic vector space,β Mathematische Nachrichten, vol. 282, no. 9,pp. 1278β1304, 2009.
[17] Y.-C. Kim, βWeak type estimates of square functions associ-ated with quasiradial Bochner-Riesz means on certain Hardyspaces,β Journal of Mathematical Analysis and Applications, vol.339, no. 1, pp. 266β280, 2008.
[18] Y.-C. Kim, βA simple proof of the π-adic version of theSobolev embedding theorem,β Communications of the KoreanMathematical Society, vol. 25, no. 1, pp. 27β36, 2010.
[19] S. Z. Lu and D. C. Yang, βThe decomposition of Herz spaceson local fields and its applications,β Journal of MathematicalAnalysis and Applications, vol. 196, no. 1, pp. 296β313, 1995.
[20] K. M. Rogers, βA van der Corput lemma for the π-adicnumbers,β Proceedings of the American Mathematical Society,vol. 133, no. 12, pp. 3525β3534, 2005.
[21] K. M. Rogers, βMaximal averages along curves over the p-adicnumbers,β Bulletin of the Australian Mathematical Society, vol.70, no. 3, pp. 357β375, 2004.
[22] M. H. Taibleson, Fourier Analysis on Local Fields, PrincetonUniversity Press, Princeton,NJ, USA,University of Tokyo Press,Tokyo, Japan, 1975.
[23] S.-h. Lan, βThe commutators on Herz spaces over locallycompact Vilenkin groups,βAdvances inMathematics, vol. 35, no.5, pp. 539β550, 2006.
[24] C. Tang, βThe boundedness of multilinear commutators onlocally compact Vilenkin groups,β Journal of Function Spacesand Applications, vol. 4, no. 3, pp. 261β273, 2006.
[25] J. Wu, βBoundedness of commutators on homogeneousMorrey-Herz spaces over locally compact Vilenkin groups,βAnalysis in Theory and Applications, vol. 25, no. 3, pp. 283β296,2009.
[26] A. Khrennikov, βπ-adic valued probability measures,β Indaga-tiones Mathematicae, vol. 7, no. 3, pp. 311β330, 1996.
[27] A. Khrennikov and M. Nilsson, p-Adic Deterministic and Ran-dom Dynamical Systems, Kluwer, Dordreht, The Netherlands,2004.
[28] S. Haran, βAnalytic potential theory over the π-adics,β Annalesde lβinstitut Fourier, vol. 43, no. 4, pp. 905β944, 1993.
[29] S. S. Volosivets, βMaximal function and Riesz potential on π-adic linear spaces,β p-Adic Numbers, Ultrametric Analysis, andApplications, vol. 5, no. 3, pp. 226β234, 2013.
[30] J. Alvarez, J. Lakey, and M. GuzmaΜn-Partida, βSpaces ofbounded π-central mean oscillation, Morrey spaces, and π-central Carleson measures,β Universitat de Barcelona. Col-lectanea Mathematica, vol. 51, no. 1, pp. 1β47, 2000.
[31] Q. Y. Wu, L. Mi, and Z. W. Fu, βBoundedness of p-adic Hardyoperators and their commutatorson p-adic central Morrey andBMO spaces,β Journal of Function Spaces and Applications, vol.2013, Article ID 359193, 10 pages, 2013.
[32] Z. W. Fu, Q. Y. Wu, and S. Z. Lu, βSharp estimates of p-adichardy andHardy-Littlewood-PoΜlya operators,βActaMathemat-ica Sinica, vol. 29, no. 1, pp. 137β150, 2013.
[33] V. S. Vladimirov, βGeneralized functions over p-adic numberfield,β Uspekhi Matematicheskikh Nauk, vol. 43, pp. 17β53, 1988.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of