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Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013, Article ID 257537, 7 pageshttp://dx.doi.org/10.1155/2013/257537
Research ArticleHigher Order Commutators of Fractional Integral Operator onthe Homogeneous Herz Spaces with Variable Exponent
Liwei Wang,1 Meng Qu,2 and Lisheng Shu2
1 School of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, China2 School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China
Correspondence should be addressed to Liwei Wang; [email protected]
Received 28 March 2013; Accepted 20 May 2013
Academic Editor: Dachun Yang
Copyright Β© 2013 Liwei Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By decomposing functions, we establish estimates for higher order commutators generated by fractional integral with BMOfunctions or the Lipschitz functions on the homogeneous Herz spaces with variable exponent.These estimates extend some knownresults in the literatures.
1. Introduction
Let π be a locally integrable function, 0 < π½ < π, and π β N;the higher order commutators of fractional integral operatorπΌπ
π½,πare defined by
πΌπ
π½,ππ (π₯) = β«
Rπ
[π (π₯) β π (π¦)]π
π₯ β π¦πβπ½
π (π¦) ππ¦. (1)
Obviously, πΌ0π½,π
= πΌπ½and πΌ1
π½,π= [π, πΌ
π½]. The famous
Hardy-Littlewood-Sobolev theorem tells us that the frac-tional integral operator πΌ
π½is a bounded operator from the
usual Lebesgue spaces πΏπ1(Rπ) to πΏπ2(Rπ) when 0 < π1
<
π2
< β and 1/π1β 1/π
2= π½/π. Also, many generalized
results about πΌπ½and the commutator [π, πΌ
π½] on some function
spaces have been studied; see [1β3] for details.It is well known that the main motivation for studying
the spaces with variable exponent arrived in the nonlinearelasticity theory and differential equations with nonstandardgrowth. Since the fundamental paper [4] by KovaΜcΜik andRaΜkosnΔ±Μk appeared in 1991, the Lebesgue spaces with variableexponent πΏπ(β )(Rπ) have been extensively investigated. In therecent twenty years, boundedness of some important operat-ors, for example, the CalderoΜn-Zygmund operators, frac-tional integrals, and commutators, on πΏπ(β )(Rπ) has beenobtained; see [5β7]. Recently, Diening [8] extended the
(πΏπ1(Rπ), πΏπ2(Rπ)) boundedness of πΌ
π½to the Lebesgue spaces
with variable exponent. Izuki [7] first introduced the Herzspaces with variable exponent οΏ½ΜοΏ½πΌ,π
π(β )(Rπ), which is a general-
ized space of the Herz space οΏ½ΜοΏ½πΌ,ππ
(Rπ); see [9, 10], and in caseof π β BMO(Rπ), he obtained the boundedness propertiesof the commutator [π, πΌ
π½]. The paper [11] by Lu et al. indi-
cates that the commutator [π, πΌπ½] with π β BMO(Rπ) and
with π β LipπΌ(Rπ) (0 < πΌ β€ 1) has many different pro-
perties. In 2012, Zhou [12] studied the boundedness of πΌπ½
on the Herz spaces with variable exponent and proved thatthe boundedness properties of the commutator [π, πΌ
π½] also
hold in case of π β LipπΌ(Rπ) (0 < πΌ β€ 1). The higher
order commutators πΌππ½,π
are recently considered byWang et al.in the paper [13, 14]; they established the BMO and theLipschitz estimates for πΌπ
π½,πon the Lebesgue spaces with vari-
able exponent πΏπ(β )(Rπ). Motivated by [7, 12β14], in thisnote, we establish the boundedness of the higher order com-mutators πΌπ
π½,πon the Herz spaces with variable exponent.
For brevity, |πΈ| denotes the Lebesgue measure for ameasurable set πΈ β Rπ, and π
πΈdenotes the mean value of π
on πΈ (ππΈ= (1/|πΈ|) β«
πΈ
π(π₯)ππ₯). The exponent π(β )means theconjugate of π(β ), that is, 1/π(β )+1/π(β ) = 1.πΆ denotes a pos-itive constant, which may have different values even in thesame line. Let us first recall some definitions and nota-tions.
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2 Journal of Function Spaces and Applications
Definition 1. For 0 < πΎ β€ 1, the Lipschitz space LipπΎ(Rπ) is
the space of functions π satisfying
πLipπΎ
= supπ₯,π¦βRπ,π₯ ΜΈ= π¦
π (π₯) β π (π¦)
π₯ β π¦πΎ
< β. (2)
Definition 2. Forπ β πΏ1loc(Rπ
), the boundedmean oscillationspace BMO(Rπ) is the space of functions π satisfying
πBMO = sup
π΅
1
|π΅|β«π΅
π (π₯) β ππ΅ ππ₯ < β, (3)
where the supremum is taken over all balls π΅ in Rπ.
Definition 3. Let π(β ) : πΈ β [1,β) be a measurable func-tion.
(1) The Lebesgue space with variable exponent πΏπ(β )(πΈ) isdefined by
πΏπ(β )
(πΈ) = {π is measurable : β«πΈ
(
π (π₯)
π)
π(π₯)
ππ₯
< β for some constant π > 0} .
(4)
(2) The space with variable exponent πΏπ(β )loc (πΈ) is definedby
πΏπ(β )
loc (πΈ)
= {π : π β πΏπ(β )
(πΎ) for all compact subsets πΎ β πΈ } .(5)
The Lebesgue space πΏπ(β )(πΈ) is a Banach space with theLuxemburg norm
ππΏπ(β )(πΈ) = inf {π > 0 : β«
πΈ
(
π (π₯)
π)
π(π₯)
ππ₯ β€ 1} . (6)
We denote
πβ= ess inf {π (π₯) : π₯ β πΈ} ,
π+= ess sup {π (π₯) : π₯ β πΈ} ,
P (πΈ) = {π (β ) : πβ> 1, π
+< β} ,
B (πΈ) = {π (β ) : π (β ) β P (πΈ) ,
π is bounded on πΏπ(β ) (πΈ)} ,
(7)
where the Hardy-Littlewood maximal operator π is definedby
ππ(π₯) = supπ>0
πβπ
β«π΅(π₯,π)β©πΈ
π (π¦) ππ¦, (8)
where π΅(π₯, π) = {π¦ β Rπ : |π₯ β π¦| < π}.
Proposition 4 (see [15]). If π(β ) β P(πΈ) satisfies
π (π₯) β π (π¦) β€
βπΆ
log (π₯ β π¦),
π₯ β π¦ β€
1
2,
π (π₯) β π (π¦) β€
πΆ
log (π + |π₯|),
π¦ β€ |π₯| ,
(9)
then one has π(β ) β B(πΈ).
Let π΅π= {π₯ β Rπ : |π₯| β©½ 2π}, π
π= π΅π\π΅πβ1
, and ππ= ππ π
be the characteristic function of the set π πfor π β Z. For
π β N, we denote ππ
= ππ π
ifπ β₯ 1, and π0= ππ΅0
.
Definition 5 (see [7]). For πΌ β R, 0 < π β€ β and π(β ) βP(Rπ).
(1) The homogeneous Herz spaces οΏ½ΜοΏ½πΌ,ππ(β )
(Rπ) are definedby
οΏ½ΜοΏ½πΌ,π
π(β )(Rπ
) = {π β πΏπ(β )
loc (Rπ
\ {0}) :π
οΏ½ΜοΏ½πΌ,π
π(β )(Rπ)
< β} , (10)
whereπ
οΏ½ΜοΏ½πΌ,π
π(β )(Rπ)
={2πΌππππ
πΏπ(β )(Rπ)}β
π=ββ
βπ(Z). (11)
(2) The nonhomogeneous Herz spaces πΎπΌ,ππ(β )
(Rπ) are de-fined by
πΎπΌ,π
π(β )(Rπ
) = {π β πΏπ(β )
loc (Rπ
) :π
πΎπΌ,π
π(β )(Rπ)
< β} , (12)
whereπ
πΎπΌ,π
π(β )(Rπ)
={2πΌππππ
πΏπ(β )(Rπ)}β
π=0
βπ(N). (13)
In this note, we obtain the following results.
Theorem 6. Suppose that π β Lipπ½1
(Rπ) (0 < π½1
< 1),π2(β ) β P(Rπ) satisfies conditions (9) in Proposition 4. If
0 < π < min {1/(π1)+, 1/(π
2)+}, 0 < π½ + ππ½
1< ππ, 0 <
πΌ < ππ β π½ β ππ½1, 0 < π
1β€ π2
< β, and 1/π1(π₯) β
1/π2(π₯) = (π½ + ππ½
1)/π, then the higher order commutators
πΌπ
π½,πare bounded from οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ) to οΏ½ΜοΏ½πΌ,π2
π2(β )(Rπ).
Theorem 7. Suppose that π β BMO(Rn), π2(β ) β P(Rπ)
satisfies conditions (9) in Proposition 4. If 0 < π <min {1/(π
1)+, 1/(π
2)+}, 0 < π½ < ππ, 0 < πΌ < ππ β π½,
0 < π1
β€ π2
< β, and 1/π1(π₯) β 1/π
2(π₯) = π½/π, then the
higher order commutators πΌππ½,π
are bounded from οΏ½ΜοΏ½πΌ,π1π1(β )(Rπ) to
οΏ½ΜοΏ½πΌ,π2
π2(β )(Rπ).
Remark A. The previous main results generalize the(πΏπ(β )
(Rπ), πΏπ(β )(Rπ)) boundedness of the higher ordercommutators πΌπ
π½,πin [13] to the case of the Herz spaces with
variable exponent. If π = 1, our conclusions coincide withthe corresponding results in [7, 12]. Moreover, the sameboundedness also holds for the nonhomogeneous case.
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Journal of Function Spaces and Applications 3
2. Proof of Theorems 6 and 7
To prove our main results, we need the following lemmas.
Lemma 8 (see [4]). Let π(β ) β P(Rπ); if π β πΏπ(β )(Rπ) andπ β πΏπ
(β )
(Rπ), then
β«Rπ
π (π₯) π (π₯) ππ₯ β€ ππ
ππΏπ(β )(Rπ)
ππΏπ(β )(Rπ)
, (14)
where ππ= 1 + 1/π
ββ 1/π
+.
Lemma 9 (see [7]). Let π(β ) β B(Rπ); then for all balls π΅ inRπ,
1
|π΅|
ππ΅πΏπ(β )(Rπ)
ππ΅πΏπ(β )(Rπ)
β€ πΆ. (15)
Lemma 10 (see [7]). Let π2(β ) β B(Rπ); then for all balls π΅ in
Rπ and all measurable subsets π β π΅, one can take a constant0 < π < 1/(π
2)+, so that
πππΏπ
2(β )
(Rπ)ππ΅
πΏπ
2(β )
(Rπ)
β€ πΆ(|π|
|π΅|)
π
. (16)
Lemma 11 (see [8]). Suppose that π1(β ) β P(Rπ) satisfies
conditions (9) in Proposition 4, 0 < π½ < π/(π1)+and 1/π
1(π₯)β
1/π2(π₯) = π½/π; then
πΌπ½(π)
πΏπ2(β )(Rπ)β€ πΆ
ππΏπ1(β )(Rπ). (17)
Lemma 12 (see [13]). Suppose that π1(β ), π2(β ) β P(Rπ).
(1) Let 0 < π½ < π/(π1)+, π β BMO(Rn). If π
2(β ) satisfies
conditions (9) in Proposition 4 and 1/π1(π₯)β1/π
2(π₯) =
π½/π, thenπΌπ
π½,π(π)
πΏπ2(β )(Rπ)β€ πΆβπβ
π
BMOπ
πΏπ1(β )(Rπ). (18)
(2) Let 0 < π½ + ππ½1< π/(π
1)+, π β Lip
π½1
(Rπ) (0 < π½1<
1). If π2(β ) satisfies conditions (9) in Proposition 4 and
1/π1(π₯) β 1/π
2(π₯) = (π½ + ππ½
1)/π, then
πΌπ
π½,π(π)
πΏπ2(β )(Rπ)β€ πΆβπβ
π
Lipπ½1
ππΏπ1(β )(Rπ). (19)
Lemma 13 (see [16]). Let π β BMO(Rn), π > π (π, π β N);one has
(1) πΆβ1||π||πBMO β€ supπ΅βRπ(1/||ππ΅||πΏπ(β )(Rπ))||(π βππ΅)π
ππ΅||πΏπ(β )(Rπ) β€ πΆ||π||
π
BMO;(2) ||(π β π
π΅π
)π
ππ΅π
||πΏπ(β )(Rπ) β€ πΆ(π β π)
π
||π||π
BMOΓ||ππ΅π
||πΏπ(β )(Rπ).
Proof of Theorem 6. Let π β οΏ½ΜοΏ½πΌ,π1π1(β )(Rπ); we can write
π (π₯) =
β
β
π=ββ
π (π₯) ππ(π₯) =
β
β
π=ββ
ππ(π₯) . (20)
For 0 < π1/π2β€ 1, applying the inequality
(
β
β
π=1
ππ)
π1/π2
β€
β
β
π=1
ππ1/π2
π(ππ> 0, π = 1, 2 . . .) , (21)
we obtainπΌπ
π½,π(π)
π1
οΏ½ΜοΏ½πΌ,π2
π2(β )(Rπ)
= πΆ(
β
β
π=ββ
2πΌπ2ππΌπ
π½,π(π) ππ
π2
πΏπ2(β )(Rπ)
)
π1/π2
β€ πΆ
β
β
π=ββ
2πΌπ1π
(
πβ2
β
π=ββ
πΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ))
π1
+ πΆ
β
β
π=ββ
2πΌπ1π
(
β
β
π=πβ1
πΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ))
π1
= π1+ π2.
(22)
We first estimate π1. Noting that if π₯ β π
π, π¦ β π
π, and
π β€ π β 2, then |π₯ β π¦| βΌ |π₯| βΌ 2π, we get
πΌπ
π½,π(ππ) ππ
β€ β«π π
π (π₯) β π (π¦)π
π₯ β π¦πβπ½
ππ(π¦)
ππ¦ β π
π(π₯)
β€ πΆ2π(π½βπ)
β«π π
π (π₯)β π (π¦)π
ππ(π¦)
ππ¦ β π
π(π₯)
β€ πΆ2π(π½+ππ½
1βπ)
βπβπ
Lipπ½1
β«π π
ππ(π¦)
ππ¦ β π
π(π₯) .
(23)
By HoΜlderβs inequality, Lemmas 9 and 10, we haveπΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ)
β€ πΆ2π(π½+ππ½
1βπ)
βπβπ
Lipπ½1
ππ
πΏπ1(β )(Rπ)
Γππ΅π
πΏπ2(β )(Rπ)
ππ΅π
πΏπ
1(β )
(Rπ)
β€ πΆ2π(π½+ππ½
1)
βπβπ
Lipπ½1
ππ
πΏπ1(β )(Rπ)
Γππ΅π
πΏπ
1(β )
(Rπ)
ππ΅π
β1
πΏπ
2(β )
(Rπ)
β€ πΆ2π(π½+ππ½
1)
βπβπ
Lipπ½1
ππ
πΏπ1(β )(Rπ)
Γππ΅π
πΏπ
1(β )
(Rπ)
ππ΅π
β1
πΏπ
2(β )
(Rπ)
ππ΅π
πΏπ
2(β )
(Rπ)ππ΅π
πΏπ
2(β )
(Rπ)
β€ πΆ2π(π½+ππ½
1)
2ππ(πβπ)
βπβπ
Lipπ½1
ππ
πΏπ1(β )(Rπ)
Γππ΅π
πΏπ
1(β )
(Rπ)
ππ΅π
β1
πΏπ
2(β )
(Rπ).
(24)
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4 Journal of Function Spaces and Applications
Note that
πΌπ½+ππ½
1
(ππ΅π
) (π₯) β₯ πΌπ½+ππ½
1
(ππ΅π
) (π₯) β ππ΅π
(π₯)
= β«π΅π
ππ¦
π₯ β π¦πβπ½βππ½
1
β ππ΅π
(π₯)
β₯ πΆ2π(π½+ππ½
1)
β ππ΅π
(π₯) .
(25)
By Lemmas 8 and 11, we obtain
ππ΅π
β1
πΏπ
2(β )
(Rπ)β€ πΆ2βππ
ππ΅π
πΏπ2(β )(Rπ)
β€ πΆ2βππ
2βπ(π½+ππ½
1)πΌπ½+ππ½
1
(ππ΅π
)πΏπ2(β )(Rπ)
β€ πΆ2βπ(π½+ππ½
1)
2βππ
ππ΅π
πΏπ1(β )(Rπ)
β€ πΆ2βπ(π½+ππ½
1)ππ΅π
β1
πΏπ
1(β )
(Rπ).
(26)
Combining (24) and (26), we have the estimate
πΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ)β€ πΆ2(πβπ)(π½+ππ½
1βππ)
Γ βπβπ
Lipπ½1
ππ
πΏπ1(β )(Rπ).
(27)
Thus,
π1β€ πΆβπβ
ππ1
Lipπ½1
Γ
β
β
π=ββ
(
πβ2
β
π=ββ
2πΌπππ
πΏπ1(β )(Rπ)2(πβπ)(π½+ππ½
1βππ+πΌ)
)
π1
.
(28)
If 1 < π1
< β, noting that π½ + ππ½1β ππ + πΌ < 0, by
HoΜlderβs inequality, we have
π1β€ πΆβπβ
ππ1
Lipπ½1
Γ
β
β
π=ββ
(
πβ2
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
2(πβπ)(π½+ππ½
1βππ+πΌ)π
1/2
)
Γ (
πβ2
β
π=ββ
2(πβπ)(π½+ππ½
1βππ+πΌ)π
1/2
)
π1/π
1
β€ πΆβπβππ1
Lipπ½1
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
β
β
π=π+2
2(πβπ)(π½+ππ½
1βππ+πΌ)π
1/2
β€ πΆβπβππ1
Lipπ½1
ππ1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(29)
If 0 < π1β€ 1, by inequality (21), we have
π1β€ πΆβπβ
ππ1
Lipπ½1
Γ
β
β
π=ββ
πβ2
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
2(πβπ)(π½+ππ½
1βππ+πΌ)π
1
β€ πΆβπβππ1
Lipπ½1
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
β
β
π=π+2
2(πβπ)(π½+ππ½
1βππ+πΌ)π
1
β€ πΆβπβππ1
Lipπ½1
ππ1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(30)
Next, we estimate π2. By Lemma 12(2), we obtain
π2β€ πΆβπβ
ππ1
Lipπ½1
β
β
π=ββ
(
β
β
π=πβ1
2πΌπππ
πΏπ1(β )(Rπ)2πΌ(πβπ)
)
π1
. (31)
If 1 < π1< β, by HoΜlderβs inequality, we have
π2β€ πΆβπβ
ππ1
Lipπ½1
β
β
π=ββ
(
β
β
π=πβ1
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
2πΌ(πβπ)π
1/2
)
Γ (
β
β
π=πβ1
2πΌ(πβπ)π
1/2
)
π1/π
1
β€ πΆβπβππ1
Lipπ½1
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
π+1
β
π=ββ
2πΌ(πβπ)π
1/2
β€ πΆβπβππ1
Lipπ½1
ππ1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(32)
If 0 < π1β€ 1, by inequality (21), we have
π2β€ πΆβπβ
ππ1
Lipπ½1
β
β
π=ββ
β
β
π=πβ1
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
2πΌ(πβπ)π
1
β€ πΆβπβππ1
Lipπ½1
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
π+1
β
π=ββ
2πΌ(πβπ)π
1
β€ πΆβπβππ1
Lipπ½1
ππ1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(33)
Combining the estimates for π1and π
2, the proof of
Theorem 6 is completed.
Proof of Theorem 7. Let π β οΏ½ΜοΏ½πΌ,π1π1(β )(Rπ); we can write
π (π₯) =
β
β
π=ββ
π (π₯) ππ(π₯) =
β
β
π=ββ
ππ(π₯) . (34)
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Journal of Function Spaces and Applications 5
By inequality (21), we obtain
πΌπ
π½,π(π)
π1
οΏ½ΜοΏ½πΌ,π2
π2(β )(Rπ)
= πΆ(
β
β
π=ββ
2πΌπ2ππΌπ
π½,π(π) ππ
π2
πΏπ2(β )(Rπ)
)
π1/π2
β€ πΆ
β
β
π=ββ
2πΌπ1π
(
πβ2
β
π=ββ
πΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ))
π1
+ πΆ
β
β
π=ββ
2πΌπ1π
(
β
β
π=πβ1
πΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ))
π1
= π1+ π2.
(35)
For π1, using HoΜlderβs inequality and Lemma 8, we have
πΌπ
π½,π(ππ) ππ
β€ πΆ2π(π½βπ)
β«π π
π (π₯) β π (π¦)π
ππ(π¦)
ππ¦ β π
π(π₯)
β€ πΆ2π(π½βπ)
π
β
π=0
πΆπ
π
π (π₯) β π
π΅π
πβπ
Γ β«π π
ππ΅π
β π (π¦)
π ππ(π¦)
ππ¦
β€ πΆ2π(π½βπ)
ππ
πΏπ1(β )(Rπ)
Γ
π
β
π=0
πΆπ
π
π (π₯) β π
π΅π
πβπ(ππ΅π
β π)π
ππ
πΏπ
1(β )
(Rπ).
(36)
By Lemmas 9, 10, and 13, we have
πΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ)
β€ πΆ2π(π½βπ)
ππ
πΏπ1(β )(Rπ)
Γ
π
β
π=0
πΆπ
π
(π (π₯) β π
π΅π
)πβπ
ππ
πΏπ2(β )(Rπ)
Γ
(ππ΅π
β π)π
ππ
πΏπ
1(β )
(Rπ)
β€ πΆ2π(π½βπ)
ππ
πΏπ1(β )(Rπ)
Γ
π
β
π=0
πΆπ
π(π β π)
πβπ
βπβπβπ
BMOππ΅π
πΏπ2(β )(Rπ)
Γ βπβπ
BMOππ΅π
πΏπ
1(β )
(Rπ)
= πΆ(π β π + 1)π
βπβπ
BMOππ
πΏπ1(β )(Rπ)
Γ 2π(π½βπ)
ππ΅π
πΏπ2(β )(Rπ)
ππ΅π
πΏπ
1(β )
(Rπ)
β€ πΆ(π β π + 1)π
βπβπ
BMOππ
πΏπ1(β )(Rπ)
Γ 2ππ½ππ΅π
πΏπ
1(β )
(Rπ)
ππ΅π
β1
πΏπ
2(β )
(Rπ)
β€ πΆ(π β π + 1)π
βπβπ
BMOππ
πΏπ1(β )(Rπ)
Γ 2ππ½ππ΅π
πΏπ
1(β )
(Rπ)
ππ΅π
β1
πΏπ
2(β )
(Rπ)
ππ΅π
πΏπ
2(β )
(Rπ)ππ΅π
πΏπ
2(β )
(Rπ)
β€ πΆ(π β π + 1)π
βπβπ
BMOππ
πΏπ1(β )(Rπ)
Γ 2ππ½
2ππ(πβπ)
ππ΅π
πΏπ
1(β )
(Rπ)
ππ΅π
β1
πΏπ
2(β )
(Rπ).
(37)
Note that
πΌπ½(ππ΅π
) (π₯) β₯ β«π΅π
ππ¦
π₯ β π¦πβπ½
β ππ΅π
(π₯) β₯ πΆ2ππ½
β ππ΅π
(π₯) .
(38)
By Lemmas 8 and 11, we obtain
ππ΅π
β1
πΏπ
2(β )
(Rπ)β€ πΆ2βππ
ππ΅π
πΏπ2(β )(Rπ)
β€ πΆ2βππ
2βππ½
πΌπ½(ππ΅π
)πΏπ2(β )(Rπ)
β€ πΆ2βππ½
2βππ
ππ΅π
πΏπ1(β )(Rπ)
β€ πΆ2βππ½
ππ΅π
β1
πΏπ
1(β )
(Rπ).
(39)
Combining (37) and (39), we have the estimate
πΌπ
π½,π(ππ) ππ
πΏπ2(β )(Rπ)β€ πΆ(π β π + 1)
π
Γ βπβπ
BMO2(πβπ)(π½βππ)
ππ
πΏπ1(β )(Rπ).
(40)
Thus,
π1β€ πΆβπβ
ππ1
BMO
β
β
π=ββ
(
πβ2
β
π=ββ
2πΌπππ
πΏπ1(β )(Rπ)
Γ(π β π + 1)π
2(πβπ)(π½βππ+πΌ)
)
π1
.
(41)
-
6 Journal of Function Spaces and Applications
In case of 1 < π1
< β, noting that π½ β ππ + πΌ < 0, byHoΜlderβs inequality, we have
π1β€ πΆβπβ
ππ1
BMO
Γ
β
β
π=ββ
(
πβ2
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
2(πβπ)(π½βππ+πΌ)π
1/2
)
Γ (
πβ2
β
π=ββ
(π β π + 1)ππ
2(πβπ)(π½βππ+πΌ)π
1/2
)
π1/π
1
β€ πΆβπβππ1
BMO
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
β
β
π=π+2
2(πβπ)(π½βππ+πΌ)π
1/2
β€ πΆβπβππ1
BMOπ
π1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(42)
In case of 0 < π1β€ 1, by inequality (21), we have
π1β€ πΆβπβ
ππ1
BMO
Γ
β
β
π=ββ
πβ2
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
Γ (π β π + 1)ππ1
2(πβπ)(π½βππ+πΌ)π
1
β€ πΆβπβππ1
BMO
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
Γ
β
β
π=π+2
(π β π + 1)ππ1
2(πβπ)(π½βππ+πΌ)π
1
β€ πΆβπβππ1
BMOπ
π1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(43)
For π2, by Lemma 12(1), we obtain
π2β€ πΆβπβ
ππ1
BMO
β
β
π=ββ
(
β
β
π=πβ1
2πΌπππ
πΏπ1(β )(Rπ)2πΌ(πβπ)
)
π1
. (44)
If 1 < π1< β, by HoΜlderβs inequality, we have
π2β€ πΆβπβ
ππ1
BMO
β
β
π=ββ
(
β
β
π=πβ1
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
2πΌ(πβπ)π
1/2
)
Γ (
β
β
π=πβ1
2πΌ(πβπ)π
1/2
)
π1/π
1
β€ πΆβπβππ1
BMO
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
π+1
β
π=ββ
2πΌ(πβπ)π
1/2
β€ πΆβπβππ1
BMOπ
π1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(45)
If 0 < π1β€ 1, by inequality (21), we have
π2β€ πΆβπβ
ππ1
BMO
β
β
π=ββ
β
β
π=πβ1
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
2πΌ(πβπ)π
1
β€ πΆβπβππ1
BMO
β
β
π=ββ
2πΌππ1
ππ
π1
πΏπ1(β )(Rπ)
π+1
β
π=ββ
2πΌ(πβπ)π
1
β€ πΆβπβππ1
BMOπ
π1
οΏ½ΜοΏ½πΌ,π1
π1(β )(Rπ)
.
(46)
Combining the estimates for π1and π
2, consequently, we
have provedTheorem 7.
Acknowledgments
The authors thank the referees for their valuable commentsto the original version of this note. This paper is supportedby the NSF of China (no. 11201003); the Natural ScienceFoundation of Anhui Higher Education Institutions of China(no. KJ2011A138; no. KJ2013B034).
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