research article higher order commutators of fractional ...usual lebesgue spaces 1 (r ) to 2 (r )...
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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 ยฉ 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 ๐ be a locally integrable function, 0 < ๐ฝ < ๐, and ๐ โ N;the higher order commutators of fractional integral operator๐ผ๐
๐ฝ,๐are defined by
๐ผ๐
๐ฝ,๐๐ (๐ฅ) = โซ
R๐
[๐ (๐ฅ) โ ๐ (๐ฆ)]๐
๐ฅ โ ๐ฆ๐โ๐ฝ
๐ (๐ฆ) ๐๐ฆ. (1)
Obviously, ๐ผ0๐ฝ,๐
= ๐ผ๐ฝand ๐ผ1
๐ฝ,๐= [๐, ๐ผ
๐ฝ]. The famous
Hardy-Littlewood-Sobolev theorem tells us that the frac-tional integral operator ๐ผ
๐ฝis a bounded operator from the
usual Lebesgue spaces ๐ฟ๐1(R๐) to ๐ฟ๐2(R๐) when 0 < ๐1
<
๐2
< โ and 1/๐1โ 1/๐
2= ๐ฝ/๐. Also, many generalized
results about ๐ผ๐ฝand the commutator [๐, ๐ผ
๐ฝ] on some function
spaces have been studied; see [1โ3] 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 Kovaฬcฬik andRaฬkosnฤฑฬk appeared in 1991, the Lebesgue spaces with variableexponent ๐ฟ๐(โ )(R๐) have been extensively investigated. In therecent twenty years, boundedness of some important operat-ors, for example, the Calderoฬn-Zygmund operators, frac-tional integrals, and commutators, on ๐ฟ๐(โ )(R๐) has beenobtained; see [5โ7]. Recently, Diening [8] extended the
(๐ฟ๐1(R๐), ๐ฟ๐2(R๐)) boundedness of ๐ผ
๐ฝto the Lebesgue spaces
with variable exponent. Izuki [7] first introduced the Herzspaces with variable exponent ๏ฟฝฬ๏ฟฝ๐ผ,๐
๐(โ )(R๐), which is a general-
ized space of the Herz space ๏ฟฝฬ๏ฟฝ๐ผ,๐๐
(R๐); see [9, 10], and in caseof ๐ โ BMO(R๐), he obtained the boundedness propertiesof the commutator [๐, ๐ผ
๐ฝ]. The paper [11] by Lu et al. indi-
cates that the commutator [๐, ๐ผ๐ฝ] with ๐ โ BMO(R๐) and
with ๐ โ Lip๐ผ(R๐) (0 < ๐ผ โค 1) has many different pro-
perties. In 2012, Zhou [12] studied the boundedness of ๐ผ๐ฝ
on the Herz spaces with variable exponent and proved thatthe boundedness properties of the commutator [๐, ๐ผ
๐ฝ] also
hold in case of ๐ โ Lip๐ผ(R๐) (0 < ๐ผ โค 1). The higher
order commutators ๐ผ๐๐ฝ,๐
are recently considered byWang et al.in the paper [13, 14]; they established the BMO and theLipschitz estimates for ๐ผ๐
๐ฝ,๐on the Lebesgue spaces with vari-
able exponent ๐ฟ๐(โ )(R๐). Motivated by [7, 12โ14], in thisnote, we establish the boundedness of the higher order com-mutators ๐ผ๐
๐ฝ,๐on the Herz spaces with variable exponent.
For brevity, |๐ธ| denotes the Lebesgue measure for ameasurable set ๐ธ โ R๐, and ๐
๐ธdenotes the mean value of ๐
on ๐ธ (๐๐ธ= (1/|๐ธ|) โซ
๐ธ
๐(๐ฅ)๐๐ฅ). The exponent ๐(โ )means theconjugate of ๐(โ ), that is, 1/๐(โ )+1/๐(โ ) = 1.๐ถ 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 < ๐พ โค 1, the Lipschitz space Lip๐พ(R๐) is
the space of functions ๐ satisfying
๐Lip๐พ
= sup๐ฅ,๐ฆโR๐,๐ฅ ฬธ= ๐ฆ
๐ (๐ฅ) โ ๐ (๐ฆ)
๐ฅ โ ๐ฆ๐พ
< โ. (2)
Definition 2. For๐ โ ๐ฟ1loc(R๐
), the boundedmean oscillationspace BMO(R๐) is the space of functions ๐ satisfying
๐BMO = sup
๐ต
1
|๐ต|โซ๐ต
๐ (๐ฅ) โ ๐๐ต ๐๐ฅ < โ, (3)
where the supremum is taken over all balls ๐ต in R๐.
Definition 3. Let ๐(โ ) : ๐ธ โ [1,โ) be a measurable func-tion.
(1) The Lebesgue space with variable exponent ๐ฟ๐(โ )(๐ธ) isdefined by
๐ฟ๐(โ )
(๐ธ) = {๐ is measurable : โซ๐ธ
(
๐ (๐ฅ)
๐)
๐(๐ฅ)
๐๐ฅ
< โ for some constant ๐ > 0} .
(4)
(2) The space with variable exponent ๐ฟ๐(โ )loc (๐ธ) is definedby
๐ฟ๐(โ )
loc (๐ธ)
= {๐ : ๐ โ ๐ฟ๐(โ )
(๐พ) for all compact subsets ๐พ โ ๐ธ } .(5)
The Lebesgue space ๐ฟ๐(โ )(๐ธ) is a Banach space with theLuxemburg norm
๐๐ฟ๐(โ )(๐ธ) = inf {๐ > 0 : โซ
๐ธ
(
๐ (๐ฅ)
๐)
๐(๐ฅ)
๐๐ฅ โค 1} . (6)
We denote
๐โ= ess inf {๐ (๐ฅ) : ๐ฅ โ ๐ธ} ,
๐+= ess sup {๐ (๐ฅ) : ๐ฅ โ ๐ธ} ,
P (๐ธ) = {๐ (โ ) : ๐โ> 1, ๐
+< โ} ,
B (๐ธ) = {๐ (โ ) : ๐ (โ ) โ P (๐ธ) ,
๐ is bounded on ๐ฟ๐(โ ) (๐ธ)} ,
(7)
where the Hardy-Littlewood maximal operator ๐ is definedby
๐๐(๐ฅ) = sup๐>0
๐โ๐
โซ๐ต(๐ฅ,๐)โฉ๐ธ
๐ (๐ฆ) ๐๐ฆ, (8)
where ๐ต(๐ฅ, ๐) = {๐ฆ โ R๐ : |๐ฅ โ ๐ฆ| < ๐}.
Proposition 4 (see [15]). If ๐(โ ) โ P(๐ธ) satisfies
๐ (๐ฅ) โ ๐ (๐ฆ) โค
โ๐ถ
log (๐ฅ โ ๐ฆ),
๐ฅ โ ๐ฆ โค
1
2,
๐ (๐ฅ) โ ๐ (๐ฆ) โค
๐ถ
log (๐ + |๐ฅ|),
๐ฆ โค |๐ฅ| ,
(9)
then one has ๐(โ ) โ B(๐ธ).
Let ๐ต๐= {๐ฅ โ R๐ : |๐ฅ| โฉฝ 2๐}, ๐
๐= ๐ต๐\๐ต๐โ1
, and ๐๐= ๐๐ ๐
be the characteristic function of the set ๐ ๐for ๐ โ Z. For
๐ โ N, we denote ๐๐
= ๐๐ ๐
if๐ โฅ 1, and ๐0= ๐๐ต0
.
Definition 5 (see [7]). For ๐ผ โ R, 0 < ๐ โค โ and ๐(โ ) โP(R๐).
(1) The homogeneous Herz spaces ๏ฟฝฬ๏ฟฝ๐ผ,๐๐(โ )
(R๐) are definedby
๏ฟฝฬ๏ฟฝ๐ผ,๐
๐(โ )(R๐
) = {๐ โ ๐ฟ๐(โ )
loc (R๐
\ {0}) :๐
๏ฟฝฬ๏ฟฝ๐ผ,๐
๐(โ )(R๐)
< โ} , (10)
where๐
๏ฟฝฬ๏ฟฝ๐ผ,๐
๐(โ )(R๐)
={2๐ผ๐๐๐๐
๐ฟ๐(โ )(R๐)}โ
๐=โโ
โ๐(Z). (11)
(2) The nonhomogeneous Herz spaces ๐พ๐ผ,๐๐(โ )
(R๐) are de-fined by
๐พ๐ผ,๐
๐(โ )(R๐
) = {๐ โ ๐ฟ๐(โ )
loc (R๐
) :๐
๐พ๐ผ,๐
๐(โ )(R๐)
< โ} , (12)
where๐
๐พ๐ผ,๐
๐(โ )(R๐)
={2๐ผ๐๐๐๐
๐ฟ๐(โ )(R๐)}โ
๐=0
โ๐(N). (13)
In this note, we obtain the following results.
Theorem 6. Suppose that ๐ โ Lip๐ฝ1
(R๐) (0 < ๐ฝ1
< 1),๐2(โ ) โ P(R๐) satisfies conditions (9) in Proposition 4. If
0 < ๐ < min {1/(๐1)+, 1/(๐
2)+}, 0 < ๐ฝ + ๐๐ฝ
1< ๐๐, 0 <
๐ผ < ๐๐ โ ๐ฝ โ ๐๐ฝ1, 0 < ๐
1โค ๐2
< โ, and 1/๐1(๐ฅ) โ
1/๐2(๐ฅ) = (๐ฝ + ๐๐ฝ
1)/๐, then the higher order commutators
๐ผ๐
๐ฝ,๐are bounded from ๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐) to ๏ฟฝฬ๏ฟฝ๐ผ,๐2
๐2(โ )(R๐).
Theorem 7. Suppose that ๐ โ BMO(Rn), ๐2(โ ) โ P(R๐)
satisfies conditions (9) in Proposition 4. If 0 < ๐ <min {1/(๐
1)+, 1/(๐
2)+}, 0 < ๐ฝ < ๐๐, 0 < ๐ผ < ๐๐ โ ๐ฝ,
0 < ๐1
โค ๐2
< โ, and 1/๐1(๐ฅ) โ 1/๐
2(๐ฅ) = ๐ฝ/๐, then the
higher order commutators ๐ผ๐๐ฝ,๐
are bounded from ๏ฟฝฬ๏ฟฝ๐ผ,๐1๐1(โ )(R๐) to
๏ฟฝฬ๏ฟฝ๐ผ,๐2
๐2(โ )(R๐).
Remark A. The previous main results generalize the(๐ฟ๐(โ )
(R๐), ๐ฟ๐(โ )(R๐)) boundedness of the higher ordercommutators ๐ผ๐
๐ฝ,๐in [13] to the case of the Herz spaces with
variable exponent. If ๐ = 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 ๐(โ ) โ P(R๐); if ๐ โ ๐ฟ๐(โ )(R๐) and๐ โ ๐ฟ๐
(โ )
(R๐), then
โซR๐
๐ (๐ฅ) ๐ (๐ฅ) ๐๐ฅ โค ๐๐
๐๐ฟ๐(โ )(R๐)
๐๐ฟ๐(โ )(R๐)
, (14)
where ๐๐= 1 + 1/๐
โโ 1/๐
+.
Lemma 9 (see [7]). Let ๐(โ ) โ B(R๐); then for all balls ๐ต inR๐,
1
|๐ต|
๐๐ต๐ฟ๐(โ )(R๐)
๐๐ต๐ฟ๐(โ )(R๐)
โค ๐ถ. (15)
Lemma 10 (see [7]). Let ๐2(โ ) โ B(R๐); then for all balls ๐ต in
R๐ and all measurable subsets ๐ โ ๐ต, one can take a constant0 < ๐ < 1/(๐
2)+, so that
๐๐๐ฟ๐
2(โ )
(R๐)๐๐ต
๐ฟ๐
2(โ )
(R๐)
โค ๐ถ(|๐|
|๐ต|)
๐
. (16)
Lemma 11 (see [8]). Suppose that ๐1(โ ) โ P(R๐) satisfies
conditions (9) in Proposition 4, 0 < ๐ฝ < ๐/(๐1)+and 1/๐
1(๐ฅ)โ
1/๐2(๐ฅ) = ๐ฝ/๐; then
๐ผ๐ฝ(๐)
๐ฟ๐2(โ )(R๐)โค ๐ถ
๐๐ฟ๐1(โ )(R๐). (17)
Lemma 12 (see [13]). Suppose that ๐1(โ ), ๐2(โ ) โ P(R๐).
(1) Let 0 < ๐ฝ < ๐/(๐1)+, ๐ โ BMO(Rn). If ๐
2(โ ) satisfies
conditions (9) in Proposition 4 and 1/๐1(๐ฅ)โ1/๐
2(๐ฅ) =
๐ฝ/๐, then๐ผ๐
๐ฝ,๐(๐)
๐ฟ๐2(โ )(R๐)โค ๐ถโ๐โ
๐
BMO๐
๐ฟ๐1(โ )(R๐). (18)
(2) Let 0 < ๐ฝ + ๐๐ฝ1< ๐/(๐
1)+, ๐ โ Lip
๐ฝ1
(R๐) (0 < ๐ฝ1<
1). If ๐2(โ ) satisfies conditions (9) in Proposition 4 and
1/๐1(๐ฅ) โ 1/๐
2(๐ฅ) = (๐ฝ + ๐๐ฝ
1)/๐, then
๐ผ๐
๐ฝ,๐(๐)
๐ฟ๐2(โ )(R๐)โค ๐ถโ๐โ
๐
Lip๐ฝ1
๐๐ฟ๐1(โ )(R๐). (19)
Lemma 13 (see [16]). Let ๐ โ BMO(Rn), ๐ > ๐ (๐, ๐ โ N);one has
(1) ๐ถโ1||๐||๐BMO โค sup๐ตโR๐(1/||๐๐ต||๐ฟ๐(โ )(R๐))||(๐ โ๐๐ต)๐
๐๐ต||๐ฟ๐(โ )(R๐) โค ๐ถ||๐||
๐
BMO;(2) ||(๐ โ ๐
๐ต๐
)๐
๐๐ต๐
||๐ฟ๐(โ )(R๐) โค ๐ถ(๐ โ ๐)
๐
||๐||๐
BMOร||๐๐ต๐
||๐ฟ๐(โ )(R๐).
Proof of Theorem 6. Let ๐ โ ๏ฟฝฬ๏ฟฝ๐ผ,๐1๐1(โ )(R๐); we can write
๐ (๐ฅ) =
โ
โ
๐=โโ
๐ (๐ฅ) ๐๐(๐ฅ) =
โ
โ
๐=โโ
๐๐(๐ฅ) . (20)
For 0 < ๐1/๐2โค 1, applying the inequality
(
โ
โ
๐=1
๐๐)
๐1/๐2
โค
โ
โ
๐=1
๐๐1/๐2
๐(๐๐> 0, ๐ = 1, 2 . . .) , (21)
we obtain๐ผ๐
๐ฝ,๐(๐)
๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐2
๐2(โ )(R๐)
= ๐ถ(
โ
โ
๐=โโ
2๐ผ๐2๐๐ผ๐
๐ฝ,๐(๐) ๐๐
๐2
๐ฟ๐2(โ )(R๐)
)
๐1/๐2
โค ๐ถ
โ
โ
๐=โโ
2๐ผ๐1๐
(
๐โ2
โ
๐=โโ
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐))
๐1
+ ๐ถ
โ
โ
๐=โโ
2๐ผ๐1๐
(
โ
โ
๐=๐โ1
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐))
๐1
= ๐1+ ๐2.
(22)
We first estimate ๐1. Noting that if ๐ฅ โ ๐
๐, ๐ฆ โ ๐
๐, and
๐ โค ๐ โ 2, then |๐ฅ โ ๐ฆ| โผ |๐ฅ| โผ 2๐, we get
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
โค โซ๐ ๐
๐ (๐ฅ) โ ๐ (๐ฆ)๐
๐ฅ โ ๐ฆ๐โ๐ฝ
๐๐(๐ฆ)
๐๐ฆ โ ๐
๐(๐ฅ)
โค ๐ถ2๐(๐ฝโ๐)
โซ๐ ๐
๐ (๐ฅ)โ ๐ (๐ฆ)๐
๐๐(๐ฆ)
๐๐ฆ โ ๐
๐(๐ฅ)
โค ๐ถ2๐(๐ฝ+๐๐ฝ
1โ๐)
โ๐โ๐
Lip๐ฝ1
โซ๐ ๐
๐๐(๐ฆ)
๐๐ฆ โ ๐
๐(๐ฅ) .
(23)
By Hoฬlderโs inequality, Lemmas 9 and 10, we have๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐)
โค ๐ถ2๐(๐ฝ+๐๐ฝ
1โ๐)
โ๐โ๐
Lip๐ฝ1
๐๐
๐ฟ๐1(โ )(R๐)
ร๐๐ต๐
๐ฟ๐2(โ )(R๐)
๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
โค ๐ถ2๐(๐ฝ+๐๐ฝ
1)
โ๐โ๐
Lip๐ฝ1
๐๐
๐ฟ๐1(โ )(R๐)
ร๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐)
โค ๐ถ2๐(๐ฝ+๐๐ฝ
1)
โ๐โ๐
Lip๐ฝ1
๐๐
๐ฟ๐1(โ )(R๐)
ร๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐)
๐๐ต๐
๐ฟ๐
2(โ )
(R๐)๐๐ต๐
๐ฟ๐
2(โ )
(R๐)
โค ๐ถ2๐(๐ฝ+๐๐ฝ
1)
2๐๐(๐โ๐)
โ๐โ๐
Lip๐ฝ1
๐๐
๐ฟ๐1(โ )(R๐)
ร๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐).
(24)
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4 Journal of Function Spaces and Applications
Note that
๐ผ๐ฝ+๐๐ฝ
1
(๐๐ต๐
) (๐ฅ) โฅ ๐ผ๐ฝ+๐๐ฝ
1
(๐๐ต๐
) (๐ฅ) โ ๐๐ต๐
(๐ฅ)
= โซ๐ต๐
๐๐ฆ
๐ฅ โ ๐ฆ๐โ๐ฝโ๐๐ฝ
1
โ ๐๐ต๐
(๐ฅ)
โฅ ๐ถ2๐(๐ฝ+๐๐ฝ
1)
โ ๐๐ต๐
(๐ฅ) .
(25)
By Lemmas 8 and 11, we obtain
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐)โค ๐ถ2โ๐๐
๐๐ต๐
๐ฟ๐2(โ )(R๐)
โค ๐ถ2โ๐๐
2โ๐(๐ฝ+๐๐ฝ
1)๐ผ๐ฝ+๐๐ฝ
1
(๐๐ต๐
)๐ฟ๐2(โ )(R๐)
โค ๐ถ2โ๐(๐ฝ+๐๐ฝ
1)
2โ๐๐
๐๐ต๐
๐ฟ๐1(โ )(R๐)
โค ๐ถ2โ๐(๐ฝ+๐๐ฝ
1)๐๐ต๐
โ1
๐ฟ๐
1(โ )
(R๐).
(26)
Combining (24) and (26), we have the estimate
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐)โค ๐ถ2(๐โ๐)(๐ฝ+๐๐ฝ
1โ๐๐)
ร โ๐โ๐
Lip๐ฝ1
๐๐
๐ฟ๐1(โ )(R๐).
(27)
Thus,
๐1โค ๐ถโ๐โ
๐๐1
Lip๐ฝ1
ร
โ
โ
๐=โโ
(
๐โ2
โ
๐=โโ
2๐ผ๐๐๐
๐ฟ๐1(โ )(R๐)2(๐โ๐)(๐ฝ+๐๐ฝ
1โ๐๐+๐ผ)
)
๐1
.
(28)
If 1 < ๐1
< โ, noting that ๐ฝ + ๐๐ฝ1โ ๐๐ + ๐ผ < 0, by
Hoฬlderโs inequality, we have
๐1โค ๐ถโ๐โ
๐๐1
Lip๐ฝ1
ร
โ
โ
๐=โโ
(
๐โ2
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
2(๐โ๐)(๐ฝ+๐๐ฝ
1โ๐๐+๐ผ)๐
1/2
)
ร (
๐โ2
โ
๐=โโ
2(๐โ๐)(๐ฝ+๐๐ฝ
1โ๐๐+๐ผ)๐
1/2
)
๐1/๐
1
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
โ
โ
๐=๐+2
2(๐โ๐)(๐ฝ+๐๐ฝ
1โ๐๐+๐ผ)๐
1/2
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
๐๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(29)
If 0 < ๐1โค 1, by inequality (21), we have
๐1โค ๐ถโ๐โ
๐๐1
Lip๐ฝ1
ร
โ
โ
๐=โโ
๐โ2
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
2(๐โ๐)(๐ฝ+๐๐ฝ
1โ๐๐+๐ผ)๐
1
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
โ
โ
๐=๐+2
2(๐โ๐)(๐ฝ+๐๐ฝ
1โ๐๐+๐ผ)๐
1
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
๐๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(30)
Next, we estimate ๐2. By Lemma 12(2), we obtain
๐2โค ๐ถโ๐โ
๐๐1
Lip๐ฝ1
โ
โ
๐=โโ
(
โ
โ
๐=๐โ1
2๐ผ๐๐๐
๐ฟ๐1(โ )(R๐)2๐ผ(๐โ๐)
)
๐1
. (31)
If 1 < ๐1< โ, by Hoฬlderโs inequality, we have
๐2โค ๐ถโ๐โ
๐๐1
Lip๐ฝ1
โ
โ
๐=โโ
(
โ
โ
๐=๐โ1
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
2๐ผ(๐โ๐)๐
1/2
)
ร (
โ
โ
๐=๐โ1
2๐ผ(๐โ๐)๐
1/2
)
๐1/๐
1
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
๐+1
โ
๐=โโ
2๐ผ(๐โ๐)๐
1/2
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
๐๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(32)
If 0 < ๐1โค 1, by inequality (21), we have
๐2โค ๐ถโ๐โ
๐๐1
Lip๐ฝ1
โ
โ
๐=โโ
โ
โ
๐=๐โ1
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
2๐ผ(๐โ๐)๐
1
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
๐+1
โ
๐=โโ
2๐ผ(๐โ๐)๐
1
โค ๐ถโ๐โ๐๐1
Lip๐ฝ1
๐๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(33)
Combining the estimates for ๐1and ๐
2, the proof of
Theorem 6 is completed.
Proof of Theorem 7. Let ๐ โ ๏ฟฝฬ๏ฟฝ๐ผ,๐1๐1(โ )(R๐); we can write
๐ (๐ฅ) =
โ
โ
๐=โโ
๐ (๐ฅ) ๐๐(๐ฅ) =
โ
โ
๐=โโ
๐๐(๐ฅ) . (34)
-
Journal of Function Spaces and Applications 5
By inequality (21), we obtain
๐ผ๐
๐ฝ,๐(๐)
๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐2
๐2(โ )(R๐)
= ๐ถ(
โ
โ
๐=โโ
2๐ผ๐2๐๐ผ๐
๐ฝ,๐(๐) ๐๐
๐2
๐ฟ๐2(โ )(R๐)
)
๐1/๐2
โค ๐ถ
โ
โ
๐=โโ
2๐ผ๐1๐
(
๐โ2
โ
๐=โโ
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐))
๐1
+ ๐ถ
โ
โ
๐=โโ
2๐ผ๐1๐
(
โ
โ
๐=๐โ1
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐))
๐1
= ๐1+ ๐2.
(35)
For ๐1, using Hoฬlderโs inequality and Lemma 8, we have
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
โค ๐ถ2๐(๐ฝโ๐)
โซ๐ ๐
๐ (๐ฅ) โ ๐ (๐ฆ)๐
๐๐(๐ฆ)
๐๐ฆ โ ๐
๐(๐ฅ)
โค ๐ถ2๐(๐ฝโ๐)
๐
โ
๐=0
๐ถ๐
๐
๐ (๐ฅ) โ ๐
๐ต๐
๐โ๐
ร โซ๐ ๐
๐๐ต๐
โ ๐ (๐ฆ)
๐ ๐๐(๐ฆ)
๐๐ฆ
โค ๐ถ2๐(๐ฝโ๐)
๐๐
๐ฟ๐1(โ )(R๐)
ร
๐
โ
๐=0
๐ถ๐
๐
๐ (๐ฅ) โ ๐
๐ต๐
๐โ๐(๐๐ต๐
โ ๐)๐
๐๐
๐ฟ๐
1(โ )
(R๐).
(36)
By Lemmas 9, 10, and 13, we have
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐)
โค ๐ถ2๐(๐ฝโ๐)
๐๐
๐ฟ๐1(โ )(R๐)
ร
๐
โ
๐=0
๐ถ๐
๐
(๐ (๐ฅ) โ ๐
๐ต๐
)๐โ๐
๐๐
๐ฟ๐2(โ )(R๐)
ร
(๐๐ต๐
โ ๐)๐
๐๐
๐ฟ๐
1(โ )
(R๐)
โค ๐ถ2๐(๐ฝโ๐)
๐๐
๐ฟ๐1(โ )(R๐)
ร
๐
โ
๐=0
๐ถ๐
๐(๐ โ ๐)
๐โ๐
โ๐โ๐โ๐
BMO๐๐ต๐
๐ฟ๐2(โ )(R๐)
ร โ๐โ๐
BMO๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
= ๐ถ(๐ โ ๐ + 1)๐
โ๐โ๐
BMO๐๐
๐ฟ๐1(โ )(R๐)
ร 2๐(๐ฝโ๐)
๐๐ต๐
๐ฟ๐2(โ )(R๐)
๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
โค ๐ถ(๐ โ ๐ + 1)๐
โ๐โ๐
BMO๐๐
๐ฟ๐1(โ )(R๐)
ร 2๐๐ฝ๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐)
โค ๐ถ(๐ โ ๐ + 1)๐
โ๐โ๐
BMO๐๐
๐ฟ๐1(โ )(R๐)
ร 2๐๐ฝ๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐)
๐๐ต๐
๐ฟ๐
2(โ )
(R๐)๐๐ต๐
๐ฟ๐
2(โ )
(R๐)
โค ๐ถ(๐ โ ๐ + 1)๐
โ๐โ๐
BMO๐๐
๐ฟ๐1(โ )(R๐)
ร 2๐๐ฝ
2๐๐(๐โ๐)
๐๐ต๐
๐ฟ๐
1(โ )
(R๐)
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐).
(37)
Note that
๐ผ๐ฝ(๐๐ต๐
) (๐ฅ) โฅ โซ๐ต๐
๐๐ฆ
๐ฅ โ ๐ฆ๐โ๐ฝ
โ ๐๐ต๐
(๐ฅ) โฅ ๐ถ2๐๐ฝ
โ ๐๐ต๐
(๐ฅ) .
(38)
By Lemmas 8 and 11, we obtain
๐๐ต๐
โ1
๐ฟ๐
2(โ )
(R๐)โค ๐ถ2โ๐๐
๐๐ต๐
๐ฟ๐2(โ )(R๐)
โค ๐ถ2โ๐๐
2โ๐๐ฝ
๐ผ๐ฝ(๐๐ต๐
)๐ฟ๐2(โ )(R๐)
โค ๐ถ2โ๐๐ฝ
2โ๐๐
๐๐ต๐
๐ฟ๐1(โ )(R๐)
โค ๐ถ2โ๐๐ฝ
๐๐ต๐
โ1
๐ฟ๐
1(โ )
(R๐).
(39)
Combining (37) and (39), we have the estimate
๐ผ๐
๐ฝ,๐(๐๐) ๐๐
๐ฟ๐2(โ )(R๐)โค ๐ถ(๐ โ ๐ + 1)
๐
ร โ๐โ๐
BMO2(๐โ๐)(๐ฝโ๐๐)
๐๐
๐ฟ๐1(โ )(R๐).
(40)
Thus,
๐1โค ๐ถโ๐โ
๐๐1
BMO
โ
โ
๐=โโ
(
๐โ2
โ
๐=โโ
2๐ผ๐๐๐
๐ฟ๐1(โ )(R๐)
ร(๐ โ ๐ + 1)๐
2(๐โ๐)(๐ฝโ๐๐+๐ผ)
)
๐1
.
(41)
-
6 Journal of Function Spaces and Applications
In case of 1 < ๐1
< โ, noting that ๐ฝ โ ๐๐ + ๐ผ < 0, byHoฬlderโs inequality, we have
๐1โค ๐ถโ๐โ
๐๐1
BMO
ร
โ
โ
๐=โโ
(
๐โ2
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
2(๐โ๐)(๐ฝโ๐๐+๐ผ)๐
1/2
)
ร (
๐โ2
โ
๐=โโ
(๐ โ ๐ + 1)๐๐
2(๐โ๐)(๐ฝโ๐๐+๐ผ)๐
1/2
)
๐1/๐
1
โค ๐ถโ๐โ๐๐1
BMO
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
โ
โ
๐=๐+2
2(๐โ๐)(๐ฝโ๐๐+๐ผ)๐
1/2
โค ๐ถโ๐โ๐๐1
BMO๐
๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(42)
In case of 0 < ๐1โค 1, by inequality (21), we have
๐1โค ๐ถโ๐โ
๐๐1
BMO
ร
โ
โ
๐=โโ
๐โ2
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
ร (๐ โ ๐ + 1)๐๐1
2(๐โ๐)(๐ฝโ๐๐+๐ผ)๐
1
โค ๐ถโ๐โ๐๐1
BMO
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
ร
โ
โ
๐=๐+2
(๐ โ ๐ + 1)๐๐1
2(๐โ๐)(๐ฝโ๐๐+๐ผ)๐
1
โค ๐ถโ๐โ๐๐1
BMO๐
๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(43)
For ๐2, by Lemma 12(1), we obtain
๐2โค ๐ถโ๐โ
๐๐1
BMO
โ
โ
๐=โโ
(
โ
โ
๐=๐โ1
2๐ผ๐๐๐
๐ฟ๐1(โ )(R๐)2๐ผ(๐โ๐)
)
๐1
. (44)
If 1 < ๐1< โ, by Hoฬlderโs inequality, we have
๐2โค ๐ถโ๐โ
๐๐1
BMO
โ
โ
๐=โโ
(
โ
โ
๐=๐โ1
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
2๐ผ(๐โ๐)๐
1/2
)
ร (
โ
โ
๐=๐โ1
2๐ผ(๐โ๐)๐
1/2
)
๐1/๐
1
โค ๐ถโ๐โ๐๐1
BMO
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
๐+1
โ
๐=โโ
2๐ผ(๐โ๐)๐
1/2
โค ๐ถโ๐โ๐๐1
BMO๐
๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(45)
If 0 < ๐1โค 1, by inequality (21), we have
๐2โค ๐ถโ๐โ
๐๐1
BMO
โ
โ
๐=โโ
โ
โ
๐=๐โ1
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
2๐ผ(๐โ๐)๐
1
โค ๐ถโ๐โ๐๐1
BMO
โ
โ
๐=โโ
2๐ผ๐๐1
๐๐
๐1
๐ฟ๐1(โ )(R๐)
๐+1
โ
๐=โโ
2๐ผ(๐โ๐)๐
1
โค ๐ถโ๐โ๐๐1
BMO๐
๐1
๏ฟฝฬ๏ฟฝ๐ผ,๐1
๐1(โ )(R๐)
.
(46)
Combining the estimates for ๐1and ๐
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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Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
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Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
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