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Research ArticleHardy-Littlewood-Sobolev Inequalities on ð-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; zwfu@mail.bnu.edu.cn
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 ð-adic central Morrey spaces. Furthermore, we obtain the ð-centralBMO estimates for commutators of ð-adic Riesz potential on ð-adic central Morrey spaces.
1. Introduction
Let 0 < ðŒ < ð. The Riesz potential operator ðŒðŒis defined by
setting, for all locally integrable functions ð on Rð,
ðŒðŒð (ð¥) =
1
ðŸð(ðŒ)
â«Rð
ð (ðŠ)
ð¥ â ðŠðâðŒððŠ, (1)
where ðŸð(ðŒ) = ð
ð/2
2ðŒ
Î(ðŒ/2)/Î((ð â ðŒ)/2). It is closely relatedto the Laplacian operator of fractional degree. When ð > 2and ðŒ = 2, ðŒ
ðŒð is a solution of Poisson equation âÎð¢ =
ð. 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 ð-adicfield. In the last 20 years, the field of ð-adic numbers Q
ð
has been intensively used in theoretical and mathematicalphysics (cf. [2â12]). And it has already penetrated intensivelyinto several areas of mathematics and its applications, amongwhich harmonic analysis on ð-adic field has been drawingmore and more concern (see [13â22] and references therein).
For a prime number ð, the field of ð-adic numbers Qð
is defined as the completion of the field of rational numbersQ with respect to the non-Archimedean ð-adic norm | â |
ð,
which satisfies |ð¥|ð= 0 if and only if ð¥ = 0; |ð¥ðŠ|
ð=
|ð¥|ð|ðŠ|ð; |ð¥ + ðŠ|
ð†max{|ð¥|
ð, |ðŠ|ð}. Moreover, if |ð¥|
ðÌž= |ðŠ|ð,
then |𥠱 ðŠ|ð= max{|ð¥|
ð, |ðŠ|ð}. It is well-known that Q
ð
is a typical model of non-Archimedean local fields. If anynonzero rational number ð¥ is represented as ð¥ = ððŸ(ð/ð),where ðŸ = ðŸ(ð¥) â Z and integers ð, ð are indivisible by ð,then |ð¥|
ð= ðâðŸ.
The space Qðð= Qðà Qðà â â â à Q
ðconsists of points
ð¥ = (ð¥1, ð¥2, . . . , ð¥
ð), where ð¥
ðâ Qð, ð = 1, 2, . . . , ð. The ð-
adic norm onQððis
|ð¥|ð:= max1â€ðâ€ð
ð¥ð
ð, ð¥ â Q
ð
ð. (2)
Denote by
ðµðŸ(ð) = {ð¥ â Q
ð
ð: |ð¥ â ð|
ð†ððŸ
} (3)
the ball of radius ððŸ with center at ð â Qððand by
ððŸ(ð) = ðµ
ðŸ(ð) \ ðµ
ðŸâ1(ð) = {ð¥ â Q
ð
ð: |ð¥ â ð|
ð= ððŸ
} (4)
the sphere of radius ððŸ with center at ð â Qðð, where ðŸ â Z. It
is clear that
ðµðŸ(ð) = â
ðâ€ðŸ
ðð(ð) . (5)
It is well-known that Qððis a classical kind of locally
compact Vilenkin groups. A locally compact Vilenkin groupðº is a locally compact Abelian group containing a strictlydecreasing sequence of compact open subgroups {ðº
ð}â
ð=ââ
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) âªâð=ââ
ðºð= ðº and â©â
ð=ââðºð= 0 and (2)
sup{order(ðºð/ðºð+1
: ð â Z)} < â. For several decades,parallel to the ð-adic harmonic analysis, a development wasunder way of the harmonic analysis on locally compactVilenkin groups (cf. [23â25] and references therein).
Since Qððis a locally compact commutative group under
addition, it follows from the standard analysis that there existsa Haar measure ðð¥ on Qð
ð, which is unique up to a positive
constant factor and is translation invariant.We normalize themeasure ðð¥ by the equality
â«ðµ0(0)
ðð¥ =ðµ0 (0)
ð» = 1, (6)
where |ðž|ð»denotes the Haar measure of a measurable subset
ðž ofQðð. By simple calculation, we can obtain that
ðµðŸ(ð)ð»= ððŸð
,
ððŸ(ð)ð»= ððŸð
(1 â ðâð
)
(7)
for any ð â Qðð. We should mention that the Haar measure
takes value in R; there also exist ð-adic valued measures (cf.[26, 27]). For a more complete introduction to the ð-adicfield, one can refer to [22] or [10].
On ð-adic field, the ð-adic Riesz potential ðŒððŒ[22] is
defined by
ðŒð
ðŒð (ð¥) =
1
Îð(ðŒ)
â«Qðð
ð (ðŠ)
ð¥ â ðŠðâðŒ
ð
ððŠ, (8)
where Îð(ðŒ) = (1 â ð
ðŒâð
)/(1 â ðâðŒ
), ðŒ â C, ðŒ Ìž= 0. Whenð = 1, Haran [4, 28] obtained the explicit formula of Rieszpotentials onQ
ðand developed analytical potential theory on
Qð. Taibleson [22] gave the fundamental analytic properties
of the Riesz potentials on local fields including Qðð, as well
as the classical Hardy-Littlewood-Sobolev inequalities. Kim[18] gave a simple proof of these inequalities by using theð-adic version of the CalderoÌn-Zygmund decompositiontechnique. Volosivets [29] investigated the boundedness forRiesz potentials on generalized Morrey spaces. Like onEuclidean spaces, using the Riesz potential with ð > 2 andðŒ = 2, one can introduce the ð-adic Laplacians [13].
In this paper, we will consider the Riesz potentials andtheir commutators with ð-adic central BMO functions on ð-adic central Morrey spaces. Alvarez et al. [30] studied therelationship between central BMO spaces andMorrey spaces.Furthermore, they introduced ð-central BMO spaces andcentralMorrey spaces, respectively. In [31], we introduce theirð-adic versions.
Definition 1. Let ð â R and 1 < ð < â. The ð-adic centralMorrey space ï¿œÌï¿œð,ð(Qð
ð) is defined by
ðï¿œÌï¿œð,ð(Qð
ð):= supðŸâZ
(1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ð (ð¥)ð
ðð¥)
1/ð
< â, (9)
where ðµðŸ= ðµðŸ(0).
Remark 2. It is clear that
ð¿ð,ð
(Qð
ð) â ï¿œÌï¿œð,ð
(Qð
ð) ,
ï¿œÌï¿œð,â1/ð
(Qð
ð) = ð¿ð
(Qð
ð) .
(10)
When ð < â1/ð, the space ï¿œÌï¿œð,ð(Qðð) reduces to {0}; therefore,
we can only consider the case ð ⥠â1/ð. If 1 †ð1< ð2< â,
by HoÌlderâs inequality,
ï¿œÌï¿œð2,ð
(Qð
ð) â ï¿œÌï¿œð1,ð
(Qð
ð) (11)
for ð â R.
Definition 3. Let ð < 1/ð and 1 < ð < â. The spaceCBMOð,ð(Qð
ð) is defined by the condition
ðCBMOð,ð(Qð
ð)
:= supðŸâZ
(1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ð (ð¥) â ð
ðµðŸ
ð
ðð¥)
1/ð
< â.
(12)
Remark 4. When ð = 0, the space CBMOð,ð(Qðð) is just
CBMOð(Qðð), which is defined in [32]. If 1 †ð
1< ð2< â,
by HoÌlderâs inequality,
CBMOð2 ,ð (Qðð) â CBMOð1,ð (Qð
ð) (13)
for ð â R. By the standard proof as that inRð, we can see thatðCBMOð,ð(Qð
ð)
⌠supðŸâZ
infðâC(
1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ð (ð¥) â ðð
ðð¥)
1/ð
.
(14)
Remark 5. Formulas (9) and (12) yield that ï¿œÌï¿œð,ð(Qðð) is a
Banach space continuously included in CBMOð,ð(Qðð).
Herewe introduce theð-adicweak centralMorrey spaces.
Definition 6. Let ð â R and 1 < ð < â. The ð-adic weakcentral Morrey spaceðï¿œÌï¿œð,ð(Qð
ð) is defined by
ððï¿œÌï¿œð,ð(Qð
ð)
:= supðŸâZ
(supð¡>0ð¡ð{ð¥ â ðµ
ðŸ:ð (ð¥)
> ð¡}ð»
ðµðŸ
1+ðð
ð»
)
1/ð
< â,
(15)
where ðµðŸ= ðµðŸ(0).
In Section 2, we will get the Hardy-Littlewood-Sobolevinequalities on ð-adic central Morrey spaces. Namely, under
Journal of Function Spaces 3
some conditions for indexes, ðŒððŒis bounded from ï¿œÌï¿œð,ð(Qð
ð) to
ï¿œÌï¿œð,ð
(Qðð) and is also bounded from ï¿œÌï¿œ1,ð(Qð
ð) to ðï¿œÌï¿œð,ð(Qð
ð).
In Section 3, we establish the boundedness for commutatorsgenerated by ðŒð
ðŒand ð-central BMO functions on ð-adic
central Morrey spaces.Throughout this paper the letter ð¶ 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 ð-adic central Morrey spaces.
Theorem7. LetðŒ be a complex numberwith 0 < ReðŒ < ð andlet 1 †ð < ð/ReðŒ, 0 < 1/ð = 1/ð â ReðŒ/ð, ð < âReðŒ/ð,and ð = ð + ReðŒ/ð.
(i) If ð > 1, then ðŒððŒis bounded from ï¿œÌï¿œð,ð(Qð
ð) to ï¿œÌï¿œð,ð(Qð
ð).
(ii) If ð = 1, then ðŒððŒ
is bounded from ï¿œÌï¿œ1,ð(Qðð) to
ðï¿œÌï¿œð,ð
(Qðð).
In order to give the proof of this theorem, we need thefollowing result.
Lemma 8 (see [22]). Let ðŒ be a complex number with 0 <ReðŒ < ð and let 1 †ð < ð < â satisfy 1/ð = 1/ð â ReðŒ/ð.
(i) If ð â ð¿ð(Qðð), ð > 1, then
ðŒð
ðŒðð¿ð(Qð
ð)†ðŽðð
ðð¿ð(Qð
ð), (16)
where ðŽððis independent of ð.
(ii) If ð â ð¿1(Qðð), ð > 0, then
{ð¥ â Q
ð
ð:ðŒð
ðŒð (ð¥)
> ð }ð»â€ (ðŽ
ð
ðð¿1(Qð
ð)
ð )
ð
, (17)
where ðŽð> 0 is independent of ð.
Proof ofTheorem 7. Let ð be a function in ï¿œÌï¿œð,ð(Qðð). For fixed
ðŸ â Z, denote ðµðŸ(0) by ðµ
ðŸ.
(i) If ð > 1, write
(1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ðŒð
ðŒð (ð¥)
ð
ðð¥)
1/ð
†(1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ðŒð
ðŒ(ðððµðŸ
) (ð¥)
ð
ðð¥)
1/ð
+ (1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ðŒð
ðŒ(ðððµð
ðŸ
) (ð¥)
ð
ðð¥)
1/ð
:= ðŒ + ðŒðŒ.
(18)
For ðŒ, since 1/ð = 1/ð â ReðŒ/ð and ð = ð + ReðŒ/ð, byLemma 8,
ðŒ = (1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ðŒð
ðŒ(ðððµðŸ
) (ð¥)
ð
ðð¥)
1/ð
â€ðµðŸ
â1/ðâð
ð»
(â«ðµðŸ
ðððµðŸ
(ð¥)
ð
ðð¥)
1/ð
â€ðï¿œÌï¿œð,ð(Qð
ð).
(19)
For ðŒðŒ, we firstly give the following estimate. For ð¥ â ðµðŸ,
by HoÌlderâs inequality, we have
ðŒð
ðŒ(ðððµð
ðŸ
) (ð¥)
=
1
Îð(ðŒ)
â«ðµð
ðŸ
ð (ðŠ)
ð¥ â ðŠðâðŒ
ð
ððŠ
â€1
Îð(ðŒ)
â«ðµð
ðŸ
ð (ðŠ)
ð¥ â ðŠðâReðŒð
ððŠ
=1
Îð(ðŒ)
â
â
ð=ðŸ+1
â«ðð
ð (ðŠ)
ð¥ â ðŠðâReðŒð
ððŠ
=1
Îð(ðŒ)
â
â
ð=ðŸ+1
â«ðð
ðâð(ðâReðŒ) ð (ðŠ)
ððŠ
â€1
Îð(ðŒ)
â
â
ð=ðŸ+1
ðâð(ðâReðŒ)
(â«ðµð
ð (ðŠ)ð
ððŠ)
1/ð
ðµð1â1/ð
ð»
â€1
Îð(ðŒ)
ðï¿œÌï¿œð,ð(Qð
ð)
â
â
ð=ðŸ+1
ðâð(ðâReðŒ) ðµð
1+ð
ð»
†ð¶ðµðŸ
ð
ð»
ðï¿œÌï¿œð,ð(Qð
ð).
(20)
The last inequality is due to the fact that ð < âReðŒ/ð.Consequently,
ðŒðŒ = (1
ðµðŸ
1+ðð
ð»
â«ðµðŸ
ðŒð
ðŒ(ðððµð
ðŸ
) (ð¥)
ð
ðð¥)
1/ð
†ð¶ðï¿œÌï¿œð,ð(Qð
ð).
(21)
The above estimates imply that
ðŒð
ðŒðï¿œÌï¿œð,ð(Qð
ð)†ð¶
ðï¿œÌï¿œð,ð(Qð
ð). (22)
4 Journal of Function Spaces
(ii) If ð = 1, set ð1= ðððµðŸ
and ð2= ð â ð
1; by Lemma 8,
we have{ð¥ â ðµ
ðŸ:ðŒð
ðŒð1(ð¥) > ð¡}
ð»
†ð¶(
ð1ð¿1(Qð
ð)
ð¡)
ð
= ð¶ð¡âð
(â«ðµðŸ
ð (ð¥) ðð¥)
ð
†ð¶ð¡âððµðŸ
(1+ð)ð
ð»
ðð
ï¿œÌï¿œ1,ð(Qðð)
= ð¶ð¡âððµðŸ
1+ðð
ð»
ðð
ï¿œÌï¿œ1,ð(Qðð).
(23)
On the other hand, by the same estimate as (30), we have
ðŒð
ðŒð2(ð¥) †ð¶
ðµðŸ
ð
ð»
ð2ï¿œÌï¿œ1,ð(Qð
ð). (24)
Then using Chebyshevâs inequality, we obtain
{ð¥ â ðµ
ðŸ:ðŒð
ðŒð2(ð¥) > ð¡}
ð»â€ ð¡âð
â«ðµðŸ
ðŒð
ðŒð2(ð¥)ð
ðð¥
†ð¶ð¡âððµðŸ
1+ðð
ð»
ð2ð
ï¿œÌï¿œ1,ð(Qðð)
†ð¶ð¡âððµðŸ
1+ðð
ð»
ðð
ï¿œÌï¿œ1,ð(Qðð).
(25)
SinceðŒð
ðŒð (ð¥)
â€ðŒð
ðŒð1(ð¥) +ðŒð
ðŒð2(ð¥) , (26)
we get
{ð¥ â ðµ
ðŸ:ðŒð
ðŒð (ð¥)
> ð¡}ð»â€{ð¥ â ðµ
ðŸ:ðŒð
ðŒð1(ð¥) >
ð¡
2}ð»
+{ð¥ â ðµ
ðŸ:ðŒð
ðŒð2(ð¥) >
ð¡
2}ð»
†ð¶ð¡âððµðŸ
1+ðð
ð»
ðð
ï¿œÌï¿œ1,ð(Qðð).
(27)
Therefore,
(ð¡ð{ð¥ â ðµ
ðŸ:ðŒð
ðŒð (ð¥)
> ð¡}ð»
ðµðŸ
1+ðð
ð»
)
1/ð
†ð¶ðï¿œÌï¿œ1,ð(Qð
ð), (28)
for any ð¡ > 0 and ðŸ â Z. This completes the proof.
For application, we now introduce a pseudo-differentialoperatorð·ðŒ defined by Vladimirov in [33].
The operator ð·ðŒ : ð â ð·ðŒð is defined as convolutionof generalized functions ð
âðŒand ð:
ð·ðŒ
ð = ðâðŒâ ð, ðŒ Ìž= â1, (29)
where ððŒ= |ð¥|ðŒâ1
ð/Î(ðŒ) and Î(ðŒ) = (1 â ððŒâ1)/(1 â ðâðŒ).
Let us consider the equation
ð·ðŒ
ð = ð, ð â E
, (30)
where E is the space of linear continuous functionals on Eand here E denotes the set of locally constant functions onQð. A complex-valued function ð(ð¥) defined onQ
ðis called
locally constant if for any point ð¥ â Qðthere exists an integer
ð(ð¥) â Z such that
ð (ð¥ + ð¥
) = ð (ð¥) ,
ð¥ð†ðð(ð¥)
.
(31)
The following lemma (page 154 in [10]) gives solutions of(30).
Lemma 9. For ðŒ > 0 any solution of (30) is expressed by theformula
ð = ð·âðŒ
ð + ð¶, (32)
where ð¶ is an arbitrary constant; for ðŒ < 0 a solution of (30) isunique and it is expressed by formula (32) for ð¶ = 0.
Combining with Theorem 7, we obtain the followingregular property of the solution.
Corollary 10. Let 0 < ðŒ < 1 and let 1 †ð < 1/ðŒ, 0 < 1/ð =1/ð â ðŒ, ð < âðŒ, and ð = ð + ðŒ. If ð â E â© ï¿œÌï¿œð,ð(Qð
ð), then
(i) when ð > 1, (30) has a solution in ï¿œÌï¿œð,ð(Qðð),
(ii) when ð = 1, (30) has a solution inðï¿œÌï¿œð,ð(Qðð).
3. Commutators of ð-Adic Riesz Potential
In this section, we will establish the ð-central BMO estimatesfor commutators ðŒð,ð
ðŒof ð-adic Riesz potential which is
defined by
ðŒð,ð
ðŒð = ððŒ
ð
ðŒð â ðŒð
ðŒ(ðð) , (33)
for some suitable functions ð.
Theorem 11. Suppose 0 < ReðŒ < ð, 1 < ð1< ð/ReðŒ, ð
1<
ð2< â, and 1/ð = 1/ð
1+ 1/ð2â ReðŒ/ð. Let 0 †ð
2< 1/ð,
ð1satisfies ð
1< âð2â ReðŒ/ð, and ð = ð
1+ ð2+ ReðŒ/ð. If
ð â ð¶ðµððð2,ð2(Qðð), then ðŒð,ð
ðŒis bounded from ï¿œÌï¿œð1 ,ð1(Qð
ð) to
ï¿œÌï¿œð,ð
(Qðð), and the following inequality holds:
ðŒð,ð
ðŒðï¿œÌï¿œð,ð(Qð
ð)
†ð¶ âðâð¶ðµðð
ð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð). (34)
Before proving this theorem,we need the following result.
Lemma 12 (see [31]). Suppose that ð â ð¶ðµððð,ð(Qðð) and
ð, ð â Z, ð ⥠0. Thenððµð
â ððµð
†ðð ð â ð
âðâð¶ðµððð,ð(Qðð)max {ðµð
ð
ð»
,ðµðð
ð»} .
(35)
Journal of Function Spaces 5
Proof of Theorem 11. Suppose that ð is a function inï¿œÌï¿œð1,ð1(Qðð). For fixed ðŸ â Z, denote ðµ
ðŸ(0) by ðµ
ðŸ. We write
(1
ðµðŸ
ð»
â«ðµðŸ
ðŒð,ð
ðŒð (ð¥)
ð
ðð¥)
1/ð
†(1
ðµðŸ
ð»
â«ðµðŸ
(ð (ð¥) â ð
ðµðŸ
) (ðŒð
ðŒðððµðŸ
) (ð¥)
ð
ðð¥)
1/ð
+ (1
ðµðŸ
ð»
â«ðµðŸ
(ð (ð¥) â ð
ðµðŸ
) (ðŒð
ðŒðððµð
ðŸ
) (ð¥)
ð
ðð¥)
1/ð
+ (1
ðµðŸ
ð»
â«ðµðŸ
ðŒð
ðŒ((ð â ð
ðµðŸ
)ðððµðŸ
) (ð¥)
ð
ðð¥)
1/ð
+ (1
ðµðŸ
ð»
â«ðµðŸ
ðŒð
ðŒ((ð â ð
ðµðŸ
)ðððµð
ðŸ
) (ð¥)
ð
ðð¥)
1/ð
:= ðœ1+ ðœ2+ ðœ3+ ðœ4.
(36)
Set 1/ð = 1/ð1â ReðŒ/ð; then 1/ð = 1/ð
2+ 1/ð; by
Lemma 8 and HoÌlderâs inequality, we have
ðœ1= (
1ðµðŸ
ð»
â«ðµðŸ
(ð (ð¥) â ð
ðµðŸ
) (ðŒð
ðŒðððµðŸ
) (ð¥)
ð
ðð¥)
1/ð
â€ðµðŸ
â1/ð
ð»
(â«ðµðŸ
ð (ð¥) â ð
ðµðŸ
ð2
ðð¥)
1/ð2
â (â«ðµðŸ
ðŒð
ðŒ(ðððµðŸ
) (ð¥)
ð
ðð¥)
1/ð
†ð¶ðµðŸ
â1/ð+ð2
ð»
âðâCBMOð2,ð2 (Qðð)
â (â«ðµðŸ
ðððµðŸ
(ð¥)
ð1
ðð¥)
1/ð1
†ð¶ðµðŸ
ð
ð»
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð).
(37)
Similarly, denote 1/ð = 1/ð1+ 1/ð
2; then 1/ð = 1/ð â
ReðŒ/ð, and by HoÌlderâs inequality and Lemma 8, we get
ðœ3= (
1ðµðŸ
ð»
â«ðµðŸ
ðŒð
ðŒ((ð â ð
ðµðŸ
)ðððµðŸ
) (ð¥)
ð
ðð¥)
1/ð
†ð¶ðµðŸ
â1/ð
ð»
(â«ðµðŸ
(ð (ð¥) â ð
ðµðŸ
)ð (ð¥)
ð
ðð¥)
1/ð
†ð¶ðµðŸ
â1/ð
ð»
(â«ðµðŸ
ð (ð¥) â ð
ðµðŸ
ð2
ðð¥)
1/ð2
â (â«ðµðŸ
ð (ð¥)ð1
ðð¥)
1/ð1
†ð¶ðµðŸ
ð
ð»
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð).
(38)
To estimate ðœ2and ðœ
4, we firstly give the following
estimates. For ð¥ â ðµðŸ, by HoÌlderâs inequality, we obtain
ðŒð
ðŒ(ðððµð
ðŸ
) (ð¥)
=
1
Îð(ðŒ)
â«ðµð
ðŸ
ð (ðŠ)
ð¥ â ðŠðâðŒ
ð
ððŠ
â€1
Îð(ðŒ)
â«ðµð
ðŸ
ð (ðŠ)
ð¥ â ðŠðâReðŒð
ððŠ
=1
Îð(ðŒ)
â
â
ð=ðŸ+1
â«ðð
ð (ðŠ) ðâð(ðâReðŒ)
ððŠ
â€1
Îð(ðŒ)
â
â
ð=ðŸ+1
ðâð(ðâReðŒ) ðµð
1â1/ð
1
ð»(â«ðð
ð (ðŠ)ð1
ððŠ)
1/ð1
â€1
Îð(ðŒ)
ðï¿œÌï¿œð1,ð1 (Qð
ð)
â
â
ð=ðŸ+1
ðâð(ðâReðŒ) ðµð
1+ð1
ð»
=1
Îð(ðŒ)
ðï¿œÌï¿œð1,ð1 (Qð
ð)
ð(ðŸ+1)(ðð
1+ReðŒ)
1 â ððð1+ReðŒ
= ð¶ðµðŸ
ð1+ReðŒ/ðð»
ðï¿œÌï¿œð1,ð1 (Qð
ð),
(39)
where the penultimate â=â is due to the fact that ð1+ReðŒ/ð <
âð2†0. Similarly,
ðŒð
ðŒ((ð â ð
ðµðŸ
)ðððµð
ðŸ
) (ð¥)
=
1
Îð(ðŒ)
â«ðµð
ðŸ
(ð (ðŠ) â ððµðŸ
)ð (ðŠ)
ð¥ â ðŠðâðŒ
ð
ððŠ
â€1
Îð(ðŒ)
â«ðµð
ðŸ
ð (ðŠ) â ð
ðµðŸ
ð (ðŠ)
ð¥ â ðŠðâReðŒð
ððŠ
=1
Îð(ðŒ)
â
â
ð=ðŸ+1
â«ðð
ð (ðŠ) â ð
ðµðŸ
ð (ðŠ) ðâð(ðâReðŒ)
ððŠ
=1
Îð(ðŒ)
â
â
ð=ðŸ+1
ðâð(ðâReðŒ) ðµð
1â1/ð
1â1/ð2
ð»
â (â«ðð
ð (ðŠ)ð1
ððŠ)
1/ð1
(â«ðð
ð (ðŠ) â ð
ðµðŸ
ð2
ððŠ)
1/ð2
â€1
Îð(ðŒ)
ðï¿œÌï¿œð1,ð1 (Qð
ð)
â
â
ð=ðŸ+1
ðâð(ðâReðŒ) ðµð
1â1/ð
2+ð1
ð»
â (â«ðµð
ð (ðŠ) â ð
ðµðŸ
ð2
ððŠ)
1/ð2
â€1
Îð(ðŒ)
ðï¿œÌï¿œð1,ð1 (Qð
ð)
â
â
ð=ðŸ+1
ðâð(ðâReðŒ) ðµð
1â1/ð
2+ð1
ð»
à [(â«ðµð
ð (ðŠ) â ð
ðµð
ð2
ððŠ)
1/ð2
+ððµð
â ððµðŸ
ðµð1/ð2
ð»] .
(40)
6 Journal of Function Spaces
Since ð ⥠ðŸ + 1, by Lemma 12, we haveððµð
â ððµðŸ
†ðð
(ð â ðŸ) âðâCBMOð2,ð2 (Qðð)
ðµðð2
ð». (41)
ThusðŒð
ðŒ((ð â ð
ðµðŸ
)ðððµð
ðŸ
) (ð¥)
â€1
Îð(ðŒ)
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð)
Ã
â
â
ð=ðŸ+1
ðâð(ðâReðŒ) ðµð
1â1/ð
2+ð1
ð»
â [ðµð1/ð2+ð2
ð»+ ðð
(ð â ðŸ)ðµð1/ð2+ð2
ð»]
â€ð¶
Îð(ðŒ)
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð)
â
â
â
ð=ðŸ+1
(ð â ðŸ) ðâð(ðâReðŒ) ðµð
1+ð1+ð2
ð»
â€ð¶
Îð(ðŒ)
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð)
â
â
ð=ðŸ+1
(ð â ðŸ) ðððð
= ð¶ðµðŸ
ð
ð»
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð).
(42)
Now by (39) and HoÌlderâs inequality, we obtain
ðœ2= (
1ðµðŸ
ð»
â«ðµðŸ
(ð (ð¥) â ð
ðµðŸ
) (ðŒð
ðŒðððµð
ðŸ
) (ð¥)
ð
ðð¥)
1/ð
†ð¶ðµðŸ
ð1+ReðŒ/ðâ1/ðð»
ðï¿œÌï¿œð1,ð1 (Qð
ð)
â (â«ðµðŸ
ð (ð¥) â ð
ðµðŸ
ð
ðð¥)
1/ð
†ð¶ðµðŸ
ð1+ReðŒ/ðâ1/ð
2
ð»
ðï¿œÌï¿œð1,ð1 (Qð
ð)
â (â«ðµðŸ
ð (ð¥) â ð
ðµðŸ
ð2
ðð¥)
1/ð2
†ð¶ðµðŸ
ð
ð»
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð).
(43)
It follows from (42) that
ðœ4= (
1ðµðŸ
ð»
â«ðµðŸ
ðŒð
ðŒ((ð â ð
ðµðŸ
)ðððµð
ðŸ
) (ð¥)
ð
ðð¥)
1/ð
†ð¶ðµðŸ
ð
ð»
âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð).
(44)
The above estimates imply thatðŒð,ð
ðŒðï¿œÌï¿œð,ð(Qð
ð)
†ð¶ âðâCBMOð2,ð2 (Qðð)
ðï¿œÌï¿œð1,ð1 (Qð
ð). (45)
This completes the proof of the theorem.
Remark 13. Since ð-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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