Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 780132, 18 pages
doi:10.1155/2012/780132
Research Article
A Weighted Variant of Riemann-Liouville
Fractional Integrals on Rn
Zun Wei Fu,1 Shan Zhen Lu,2 and Wen Yuan2
1
2
Department of Mathematics, Linyi University, Shandong, Linyi 276005, China
School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematics
and Complex Systems, Ministry of Education, Beijing 100875, China
Correspondence should be addressed to Wen Yuan, wenyuan@bnu.edu.cn
Received 14 June 2012; Accepted 21 August 2012
Academic Editor: Bashir Ahmad
Copyright q 2012 Zun Wei Fu 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.
We introduce certain type of weighted variant of Riemann-Liouville fractional integral on Rn and
obtain its sharp bounds on the central Morrey and λ-central BMO spaces. Moreover, we establish a
sufficient and necessary condition of the weight functions so that commutators of weighted Hardy
operators with symbols in λ-central BMO space are bounded on the central Morrey spaces. These
results are further used to prove sharp estimates of some inequalities due to Weyl and Cesàro.
1. Introduction
Let 0 < α < 1. The well-known Riemann-Liouville fractional integral Iα is defined by
Iα fx :
1
Γα
x
0
ft
x − t1−α
dt,
x > 0,
1.1
for all locally integrable functions f on 0, ∞. The study of Riemann-Liouville fractional
integral has a very long history and number of papers involved its generalizations, variants,
and applications. For the earlier development of this kind of integrals and many important
applications in fractional calculus, we refer the interested reader to the book 1. Among
numerous material dealing with applications of fractional calculus to ordinary or partial
differential equations, we choose to refer to 2 and references therein.
2
Abstract and Applied Analysis
As the classical n-dimensional generalization of Iα , the well-known Riesz potential
the solution of Laplace equation Iα with 0 < α < n is defined by setting, for all locally
integrable functions f on Rn ,
Iα fx : Cn,α
ft
Rn
|x − t|n−α
x ∈ Rn ,
dt,
1.2
where Cn,α : π n/2 2α Γα/2/Γn − α/2. The importance of Riesz potentials lies
in the fact that they are indeed smoothing operators and have been extensively used in
many different areas such as potential analysis, harmonic analysis, and partial differential
equations. Here we refer to the paper 3, which is devoted to the sharp constant in the
Hardy-Littlewood-Sobolev inequality related to Iα .
This paper focused on another generalization, the weighted variants of RiemannLiouville fractional integrals on Rn . We investigate the boundedness of these weighted
variants on the type of central Morrey and central Campanato spaces and also give the sharp
estimates. This development begins with an equivalent definition of Iα as
x Iα fx α
1
ftx
0
1
Γα1 − t1−α
dt,
x > 0.
1.3
More generally, we use a positive function weight function ωt to replace 1/Γα1−t1−α in 1.3 and generalize the parameter x from the positive axle to the Euclidean space Rn
therein. We then derive a weighted generalization of |x|α Iα on Rn , which is called the
weighted Hardy operator originally named weighted Hardy-Littlewood avarage Hω .
More precise, let ω be a positive function on 0, 1. The weighted Hardy operator Hω is
defined by setting, for all complex-valued measurable functions f on Rn and x ∈ Rn ,
1
Hω fx :
ftxωtdt.
1.4
0
Under certain conditions on ω, Carton-Lebrun and Fosset 4 proved that Hω maps Lp Rn ,
1 < p < ∞, into itself; moreover, the operator Hω commutes with the Hilbert transform when
n 1, and with certain Calderón-Zygmund singular integrals including the Riesz transform
when n ≥ 2. Obviously, for n 1 and 0 < α < 1, if we take ωt : 1/Γα1 − t1−α , then as
mentioned above, for all x > 0,
Hω fx x−α Iα fx.
1.5
A further extension of 4 was due to Xiao 5 as follows.
Theorem A. Let 1 < p < ∞. Then, Hω is bounded on Lp Rn if and only if
A :
1
0
t−n/p ωtdt < ∞.
1.6
Abstract and Applied Analysis
3
Moreover,
Hω f p n
A.
L R → Lp Rn 1.7
Remark 1.1. Notice that the condition 1.6 implies that ω is integrable on 0, 1 since
1
1
ωtdt ≤ 0 t−n/p ωtdt. We naturally assume ω is integrable on 0, 1 throughout this paper.
0
Obviously, Theorem A implies the celebrated result of Hardy et al. 6, Theorem 329,
namely, for all 0 < α < 1 and 1 < p < ∞,
Iα Lp dx → Lp x−pα dx
Γ 1 − 1/p
.
Γ 1 α − 1/p
1.8
The constant A in 1.6 also seems to be of interest as it equals to p/p − 1 if ω ≡ 1 and n 1.
In this case, Hω is precisely reduced to the classical Hardy operator H defined by
1
x
Hfx x
ftdt,
x > 0,
1.9
0
which is the most fundamental integral averaging operator in analysis. Also, a celebrated
operator norm estimate due to Hardy et al. 6, that is,
HLp R
→ Lp R
p
p−1
1.10
with 1 < p < ∞, can be deduced from Theorem A immediately.
Recall that BMORn is defined to be the space of all b ∈ Lloc Rn such that
1
|B|
n
B⊂R
bBMO : sup
|bx − bB | dx < ∞,
B
1.11
where bB 1/|B| B b and the supremum is taken over all balls B in Rn with sides parallel
to the axes. It is well known that L∞ Rn BMORn , since BMORn contains unbounded
functions such as log |x|. Another interesting result of Xiao in 5 is that the weighted Hardy
operator Hω is bounded on BMORn , if and only if
1
1.12
ωtdt < ∞.
0
Moreover,
Hω BMORn → BMORn 1
ϕtdt.
1.13
0
In recent years, several authors have extended and considered the action of weighted Hardy
operators on various spaces. We mention here, the work of Rim and Lee 7, Kuang 8, Krulić
et al. 9, Tang and Zhai 10, Tang and Zhou 11.
4
Abstract and Applied Analysis
The main purpose of this paper is to make precise the mapping properties of weighted
Hardy operators on the central Morrey and λ-central BMO spaces. The study of the central
Morrey and λ-central BMO spaces are traced to the work of Wiener 12, 13 on describing the
behavior of a function at the infinity. The conditions he considered are related to appropriate
weighted Lq 1 < q < ∞ spaces. Beurling 14 extended this idea and defined a pair of dual
Banach spaces Aq and Bq , where 1/q 1/q
1. To be precise, Aq is a Banach algebra with
respect to the convolution, expressed as a union of certain weighted Lq spaces. The space Bq is
expressed as the intersection of the corresponding weighted Lq spaces. Later, Feichtinger 15
observed that the space Bq can be equivalently described by the set of all locally q
-integrable
functions f satisfying that
f q
sup 2−kn/q
fχk < ∞,
B
q
1.14
k≥0
where χ0 is the characteristic function of the unit ball {x ∈ Rn : |x| ≤ 1}, χk is the characteristic
function of the annulus {x ∈ Rn : 2k−1 < |x| ≤ 2k }, k 1, 2, 3, . . ., and · q
is the norm in Lq .
By duality, the space Aq , called Beurling algebra now, can be equivalently described by the
set of all locally q-integrable functions f satisfying that
∞
f q 2kn/q fχk q < ∞.
A
1.15
k0
Based on these, Chen and Lau 16 and Garcı́a-Cuerva 17 introduced an atomic space HAq
associated with the Beurling algebra Aq and identified its dual as the space CMOq , which is
defined to be the space of all locally q-integrable functions f satisfying that
sup
R≥1
1
|B0, R|
fx − fB0,R q dx
1/q
< ∞.
1.16
B0,R
By replacing k ∈ N ∪ {0} with k ∈ Z in 1.3 and 1.6, we obtain the spaces Ȧq and
Ḃ , which are the homogeneous version of the spaces Aq and Bq , and the dual space of Ȧq
is just Ḃ q . Related to these homogeneous spaces, in 18, 19, Lu and Yang introduced the
˙ q and CṀOq , respectively.
homogeneous counterparts of HAq and CMOq , denoted by HA
q
These spaces were originally denoted by HK and CBMOq in 18, 19. Recall that the space
q
CṀO is defined to be the space of all locally q-integrable functions f satisfying that
q
sup
R>0
1
|B0, R|
fx − fB0,R q dx
1/q
< ∞.
1.17
B0,R
˙ q is just CṀOq .
It was also proved by Lu and Yang that the dual space of HA
In 2000, Alvarez et al. 20 introduced the following λ-central bounded mean
oscillation spaces and the central Morrey spaces, respectively.
Abstract and Applied Analysis
5
Definition 1.2. Let λ ∈ R and 1 < q < ∞. The central Morrey space Ḃ q,λ Rn is defined to be the
space of all locally q-integrable functions f satisfying that
f Ḃq,λ
sup
R>0
1
|B0, R|1
λq
fxq dx
1/q
1.18
< ∞.
B0,R
q
Definition 1.3. Let λ < 1/n and 1 < q < ∞. A function f ∈ Lloc Rn is said to belong to the
q,λ
λ-central bounded mean oscillation space CṀO Rn if
f q,λ
CṀO
sup
1
|B0, R|1
λq
R>0
fx − fB0,R q dx
1/q
< ∞.
1.19
B0,R
We remark that if two functions which differ by a constant are regarded as a function
q,λ
q,λ
in the space CṀO , then CṀO becomes a Banach space. Apparently, 1.19 is equivalent
to the following condition:
sup inf
R>0 c∈C
1
|B0, R|1
λq
fx − cq dx
1/q
1.20
< ∞.
B0,R
q,λ
Remark 1.4. Ḃq,λ is a Banach space which is continuously included in CṀO . One can easily
check Ḃq,λ Rn {0} if λ < −1/q, Ḃq,0 Rn Ḃq Rn , Ḃq,−1/q Rn Lq Rn , and Ḃq,λ Rn Lq Rn if λ > −1/q. Similar to the classical Morrey space, we only consider the case −1/q <
λ ≤ 0 in this paper.
q,λ
q
Remark 1.5. The space CṀO when λ 0 is just the space CṀO . It is easy to see that
q
q,λ
BMO ⊂ CṀO for all 1 < q < ∞. When λ ∈ 0, 1/n, then the space CṀO is just the central
version of the Lipschitz space Lipλ Rn .
Remark 1.6. If 1 < q1 < q2 < ∞, then by Hölder’s inequality, we know that Ḃ q2 ,λ ⊂ Ḃq1 ,λ for
q ,λ
q ,λ
λ ∈ R, and CṀO 2 ⊂ CṀO 1 for λ < 1/n.
For more recent generalization about central Morrey and Campanato space, we refer
to 21. We also remark that in recent years, there exists an increasing interest in the study of
Morrey-type spaces and the related theory of operators; see, for example, 22.
In this paper, we give sufficient and necessary conditions on the weight ω which
ensure that the corresponding weighted Hardy operator Hω is bounded on Ḃ q,λ Rn and
q,λ
CṀO Rn . Meanwhile, we can work out the corresponding operator norms. Moreover, we
establish a sufficient and necessary condition of the weight functions so that commutators
of weighted Hardy operators with symbols in central Campanato-type space are bounded
on the central Morrey-type spaces. These results are further used to prove sharp estimates of
some inequalities due to Weyl and Cesàro.
6
Abstract and Applied Analysis
2. Sharp Estimates of Hω
Let us state our main results.
Theorem 2.1. Let 1 < q < ∞ and −1/q < λ ≤ 0. Then Hω is a bounded operator on Ḃ q,λ Rn if and
only if
B :
1
2.1
tnλ ωtdt < ∞.
0
Moreover, when 2.1 holds, the operator norm of Hω on Ḃq,λ Rn is given by
Hω Ḃq,λ Rn → Ḃq,λ Rn B.
2.2
Proof. Suppose 2.1 holds. For any R > 0, using Minkowski’s inequality, we have
1
|B0, R|1
λq
1 ≤
1 1/q
B0,R
1
|B0, R|1
λq
0
Hω f xq dx
ftxq dx
1
λq
|B0, tR|
B0,tR
1
≤ f Ḃq,λ Rn tnλ ωtdt.
0
ωtdt
B0,R
1
1/q
fxq dx
2.3
1/q
tnλ ωtdt
0
It implies that
Hω Ḃq,λ Rn → Ḃq,λ Rn ≤
1
tnλ ωtdt.
2.4
0
Thus Hω maps Ḃq,λ Rn into itself.
The proof of the converse comes from a standard calculation. If Hω is a bounded
operator on Ḃq,λ Rn , take
f0 x |x|nλ ,
x ∈ Rn .
2.5
Then
f0 q,λ n Ω−λ
n Ḃ R 1
1/q ,
nqλ n
where Ωn π n/2 /Γ1 n/2 is the volume of the unit ball in Rn .
2.6
Abstract and Applied Analysis
7
We have
Hω f0 f0
1
2.7
tnλ ωtdt,
0
Hω Ḃq,λ Rn → Ḃq,λ Rn ≥
1
2.8
tnλ ωtdt.
0
2.8 together with 2.4 yields the desired result.
Corollary 2.2. i For 0 < α < 1, 1 < q < ∞, and −1/q < λ ≤ 0,
Iα Ḃq,λ dx → Ḃq,λ x−qα dx Γ1 λ
.
Γ1 α λ
2.9
ii For 1 < q < ∞ and −1/q < λ ≤ 0,
HḂq,λ → Ḃq,λ 1
.
1
λ
2.10
q,λ
Next, we state the corresponding conclusion for the space CṀO Rn .
q,λ
Theorem 2.3. Let 1 < q < ∞ and 0 ≤ λ < 1/n. Then Hω is a bounded operator on CṀO Rn q,λ
if and only if 2.1 holds. Moreover, when 2.1 holds, the operator norm of Hω on CṀO Rn is
given by
Hω CṀOq,λ Rn → CṀOq,λ Rn B.
2.11
q,λ
Proof. Suppose 2.1 holds. If f ∈ CṀO Rn , then for any R > 0 and ball B0, R, using
Fubini’s theorem, we see that
Hω f
B0,R
1
0
1
|B0, R|
ftxdx ωtdt B0,R
1
fB0,tR ωtdt.
0
2.12
8
Abstract and Applied Analysis
Using Minkowski’s inequality, we have
1
|B0, R|1
λq
≤
B0,R
q 1/q
1
ftx − fB0,tR dt dx
B0,R 0
|B0, R|1
λq
1 1/q
1
|B0, R|1
λq
1 1
0
q
Hω f x − Hω f B0,R dx
ftx − fB0,tR q dx
1/q
ωtdt
2.13
B0,R
1
fx − fB0,tR q dx
1/q
|B0, tR|1
λq B0,tR
1
≤ f CṀOq,λ Rn tnλ ωtdt,
tnλ ωtdt
0
0
q,λ
which implies Hω is bounded on CṀO Rn and
Hω CṀOq,λ Rn → CṀOq,λ Rn ≤ B.
2.14
q,λ
Conversely, if Hω is a bounded operator on CṀO Rn , take
f0 x |x|nλ ,
−|x|nλ ,
x ∈ Rnr ,
x ∈ Rnl ,
2.15
where Rnr and Rnl denote the right and the left halves of Rn , separated by the hyperplane
x1 0, and x1 is the first coordinate of x ∈ Rn .
Thus, by a standard calculation, we see that f0 B0,R 0 and
f0 q,λ
CṀO
Rn Ω−λ
n Hω f0 f0
1
1/q ,
nqλ n
1
2.16
nλ
t ωtdt.
0
From this formula we have
Hω CṀOq,λ Rn → CṀOq,λ Rn ≥ B.
The proof is complete.
2.17
Abstract and Applied Analysis
9
Corollary 2.4. i For 1 < q < ∞ and 0 ≤ λ < 1, we have
HCṀOq,λ → CṀOq,λ 1
.
1
λ
2.18
ii For 1 < q < ∞, we have HCṀOq → CṀOq 1.
3. A Characterization of Weight Functions via Commutators
A well-known result of Coifman et al. 23 states that the commutator generated by CalderónZygmund singular integrals and BMO functions is bounded on Lp Rn , 1 < p < ∞. Recently,
we introduced the commutators of weighted Hardy operators and BMO functions introduced
in 24. For any locally integrable function b on Rn and integrable function ω : 0, 1 →
0, ∞, the commutator of the weighted Hardy operator Hωb is defined by
Hωb f : bHω f − Hω bf .
3.1
It is easy to see that when b ∈ L∞ Rn and ω satisfies the condition 1.6, then the
commutator Hωb is bounded on Lp Rn , 1 < p < ∞. An interesting choice of b is that
it belongs to the class of BMORn . When symbols b ∈ BMORn , the condition 1.6 on
weight functions ω can not ensure the boundedness of Hωb on Lp Rn . Via controlling Hωb
by the Hardy-Littlewood maximal operators instead of sharp maximal functions, we 24
established a sufficient and necessary more stronger condition on weight functions ω which
ensures that Hωb is bounded on Lp Rn , where 1 < p < ∞. More recently, Fu and Lu 25
studied the boundedness of Hωb on the classical Morrey spaces. Tang et al. 26 and Tang and
Zhou 11 obtained the corresponding result on some Herz-type and Triebel-Lizorkin-type
spaces. We also refer to the work 27 for more general m-linear Hardy operators.
Similar to 24, we are devoted to the construction of a sufficient and necessary
condition which is stronger than B ∞ in Theorem 2.1 on the weight functions so that
commutators of weighted Hardy operators with symbols in λ-central BMO space are
bounded on the central Morrey spaces. For the boundedness of commutators with symbols
in central BMO spaces, we refer the interested reader to 28, 29 and Mo 30.
Theorem 3.1. Let 1 < q1 < q < ∞, 1/q1 1/q 1/q2 , −1/q < λ < 0. Assume further that ω
is a positive integrable function on 0, 1. Then, the commutator Hωb is bounded from Ḃq,λ Rn to
q
Ḃq1 ,λ Rn , for any b ∈ CṀO 2 Rn , if and only if
C :
1
0
2
tnλ ωt log dt < ∞.
t
3.2
Remark 3.2. The condition 2.1, that is, B < ∞, is weaker than C < ∞. In fact, let
D :
1
0
1
tnλ ωt log dt < ∞.
t
3.3
10
Abstract and Applied Analysis
By C B log 2 D, we know that C < ∞ implies B < ∞. But the following example shows
that B < ∞ does not imply C < ∞. For 0 < β < 1, if we take
⎧
−1
β
⎪
, 0 < s ≤ 1,
⎪
⎨s
s−nλ−1
ωs
s−1−β , 1 < s < ∞,
e
⎪
⎪
⎩0,
s 0, ∞,
3.4
and ωt ωlog1/t,
where 0 ≤ t ≤ 1, then B < ∞ and C ∞.
Proof. i Let R ∈ 0, ∞. Denote B0, R by B and B0, tR by tB. Assume C < ∞. We get
1
|B|
q1 1/q1
1
q1 1/q1
1
b
≤
dx
bx − btxftx ωtdt
Hω fx dx
|B| B
0
B
≤
1
|B|
1
1
|B|
1/q1
dx
0
B
1
|B|
q1
bx − bB ftxωtdt
1
bB − btB ftxωtdt
q1
1/q1
3.5
dx
0
B
1
btx − btB ftxωtdt
q1
1/q1
dx
0
B
: I1 I2 I3 .
By the Minkowski inequality and the Hölder inequality with 1/q1 1/q 1/q2 , we
have
I1 ≤
1 0
≤
1 0
λ
1
|B|
1
|B|
bx − bB ftxq1 dx
1/q1
ωtdt
B
|bx − bB |q2 dx
1/q2 B
≤ |B| bCṀOq2
1 1
0
ftxq dx
1/q
B
fxq dx
|tB|1
qλ tB
1
λ
q
tnλ ωtdt.
≤ |B| bCṀO 2 f Ḃq,λ
0
1
|B|
ωtdt
3.6
1/q
nλ
t ωtdt
Abstract and Applied Analysis
11
Similarly, we have
I3 ≤
1 0
≤
1 0
1
|B|
1
|tB|
btx − btB ftxq1 dx
1/q1
ωtdt
B
q2
|bx − btB | dx
1/q2 tB
λ
≤ |B| bCṀOq2
1 1
1
|tB|
fxq dx
1/q
ωtdt
tB
fxq dx
3.7
1/q
nλ
t ωtdt
|tB|1
qλ tB
1
tnλ ωtdt.
≤ C|B|λ bCṀOq2 f Ḃq,λ
0
0
Now we estimate I2 ,
1 1/q1
1
ftxq1 dx
I2 ≤
|bB − btB |ωtdt
|B| B
0
1
≤ f Ḃq,λ
|tB|λ |bB − btB |ωtdt
0
∞
f Ḃq,λ
k0
∞
≤ f Ḃq,λ
k0
2−k
2−k−1
2−k
2−k−1
3.8
λ
|tB| |bB − btB |ωtdt
|tB|
λ
k
|b2−i B − b2−i−1 B |
|b2−k−1 B − btB | ωtdt.
i0
We see that
k
i0
|b2−i B − b2−i−1 B | ≤ C
k i0
1
|2−i B|
2−i B
b y − b2−i B q2 dy
1/q2
3.9
≤ CbCṀOq2 k 1.
Therefore,
I2 ≤ C|B|λ bCṀOq2 f Ḃq,λ
1
0
1
tnλ ωt log dt.
t
3.10
Combining the estimates of I1 , I2 , and I3 , we conclude that Hωb is bounded from
Ḃ R to Ḃq1 Rn .
q
Conversely, assume that for any b ∈ CṀO 2 , Hωb is bounded from Ḃ q,λ Rn to Ḃq2 ,λ Rn .
We need to show that C < ∞. Since C B log 2 D, we will prove that B < ∞ and D < ∞,
respectively. To this end, let
q,λ
n
b0 x log|x|
3.11
12
Abstract and Applied Analysis
q
for all x ∈ Rn . Then it follows from Remark 1.5 that b0 ∈ BMO ⊂ CṀO 2 , and
b0 Hω Ḃq,λ → Ḃq1 ,λ
< ∞.
3.12
Let f0 x |x|nλ , x ∈ Rn . Then
f0 q,λ Ω−λ
n Ḃ
1
1/q ,
nqλ n
Hωb0 f0 x
|x|nλ
1
0
3.13
1
tnλ ωt log dt.
t
For λ > −1/q > −1/q1 , we obtain
b0 Hω f0 Ḃq1 ,λ
1
1
Ω−λ
n 1/q1
nq1 λ n
0
1
tnλ ωt log dt.
t
3.14
So,
b0 Hω Ḃq1 ,λ
→ Ḃq,λ
≥ Cn,λ,q,q1
1
0
1
tnλ ωt log dt.
t
3.15
Therefore, we have
D < ∞.
3.16
On the other hand,
1/2
0
tnλ ωtdt ≤ C
1
1/2
0
1
tnλ ωt log dt < ∞,
t
3.17
tnλ ωtdt < ∞,
1/2
since tnλ and ωt are integrable functions on 1/2, 1. Combining the above estimates, we get
B < ∞.
3.18
Combining 3.18 and 3.16, we then obtain the desired result.
Notice that comparing with Theorems 2.1 and 2.3, we need a priori assumption
in Theorem 3.1 that ω is integrable on 0, 1. However, by Remark 1.1, this assumption is
reasonable in some sense.
Abstract and Applied Analysis
13
q ,λ
When b ∈ CṀO 2 2 Rn with λ2 > 0, namely, b is a central λ-Lipschitz function, we
have the following conclusion. The proof is similar to that of Theorem 3.1. We give some
details here.
Theorem 3.3. Let 1 < q1 < q < ∞, 1/q1 1/q 1/q2 , −1/q < λ < 0, −1/q1 < λ1 < 0,
q ,λ
0 < λ2 < 1/n, and λ1 λ λ2 . If 2.1 holds true, then for all b ∈ CṀO 2 2 Rn , the corresponding
commutator Hωb is bounded from Ḃq,λ Rn to Ḃq1 ,λ1 Rn .
Proof. Let I1 , I2 , and I3 be as in the proof of Theorem 3.1. Then, following the estimates of I1
and I3 in the proof of Theorem 3.1, we see that
1
I1 ≤ |B|λ1 bCṀOq2 ,λ2 f Ḃq,λ
tnλ ωtdt,
0
1
I3 ≤ |B| bCṀOq2 ,λ2 f Ḃq,λ
λ1
3.19
tnλ1 ωtdt
0
1
≤ |B|λ1 bCṀOq2 ,λ2 f Ḃq,λ
tnλ ωtdt.
0
For I2 , we also have
∞
I2 ≤ f Ḃq,λ
k0
2−k
2−k−1
|tB|
λ
k
|b2−k−1 B − btB | ωtdt.
|b2−i B − b2−i−1 B |
3.20
i0
Since now 0 < λ2 < 1/n, we see that
k
i0
|b2−i B − b2−i−1 B | ≤ C
k i0
1
|2−i B|
2−i B
≤ CbCṀOq2 ,λ2 |B|λ2
b y − b2−i B q2 dy
k
2−inλ2
1/q2
3.21
i0
≤ CbCṀOq2 ,λ2 |B|λ2 .
Therefore,
I2 ≤ C|B|λ1 bCṀOq2 .λ2 f Ḃq,λ
Ḃ
q,λ
1
tnλ ωtdt.
3.22
0
Combining the estimates of I1 , I2 , and I3 , we conclude that Hωb is bounded from
R to Ḃq1 ,λ1 Rn .
n
Different from Theorem 3.1, it is still unknown whether the condition 2.1 in
Theorem 3.3 is sharp. That is, whether the fact that Hωb is bounded from Ḃq,λ Rn to Ḃq1 ,λ1 Rn q ,λ
for all b ∈ CṀO 2 2 Rn induces 2.1?
14
Abstract and Applied Analysis
More general, we may extend the previous results to the kth order commutator of the
weighted Hardy operator. Given k ≥ 1 and a vector b b1 , . . . , bk , we define the higher
order commutator of the weighted Hardy operator as
Hωb fx 1
0
⎞
⎛
k
⎝
bj x − bj tx ⎠ftxωtdt,
x ∈ Rn .
3.23
j1
When k 0, we understand that Hωb Hω . Notice that if k 1, then Hωb Hωb .
Using the method in the proof of Theorems 3.1 and 3.3, we can also get the following
Theorem 3.4. For the sake of convenience, we give the sketch of the proof of Theorem 3.4i
here.
Theorem 3.4. Let k ≥ 2, 1 < q1 < q, q2 , . . . , qk < ∞, 1/q1 1/q ki2 1/qi , −1/q < λ < 0,
k
−1/q1 < λ1 < 0, 0 ≤ λ2 , . . . , λk < 1/n, and λ1 λ i2 λi .
i Assume further that ω is a positive integrable function on 0, 1. The commutator Hωb is
q
q
bounded from Ḃq,λ Rn to Ḃ q1 ,λ Rn , for any b b2 , . . . , bk ∈ CṀO 2 Rn × · · · × CṀO k Rn , if
and only if
1
0
2 k−1
tnλ ωt log
dt < ∞.
t
3.24
q ,λ
ii Let λ2 , . . . , λk > 0 and b b2 , . . . , bk ∈ CṀO 2 2 Rn × · · · × CṀOqk ,λk Rn . If 2.1 holds
b
true, then the corresponding commutator Hω is bounded from Ḃq,λ Rn to Ḃq1 ,λ1 Rn .
Proof. Let R ∈ 0, ∞. Denote B0, R by B and B0, tR by tB. Assume C < ∞. We get
1
|B|
q1 1/q1
b
Hω fx dx
B
⎧
⎞
⎤q1 ⎫1/q1
⎡ ⎛
k
⎨ 1 1 ⎬
⎣ ⎝
≤
bj x − bj tx ⎠ftxωtdt⎦ dx
⎩ |B| B
⎭
0
j2
⎧
⎡ ⎛
⎨ 1 1 ⎣ ⎝
≤C
b x − bi tx
⎩ |B| B 0 i∈I j∈J m∈{2,...,k}\I∪J i
I⊂{2,...,k}J⊂{2,...,k},J∩I∅
⎤q1 ⎫1/q1
⎞
⎬
× bj x − bj B bm tx − bm tB ⎠ftxωtdt⎦ dx
.
⎭
3.25
Abstract and Applied Analysis
15
Then, applying the Minkowski inequality and the Hölder inequality with 1/q1 1/q k
b
i2 1/qi , and repeating the arguments in the proof of Theorem 3.1, Hω is bounded from
q
q
Ḃq,λ Rn to Ḃq1 Rn for any b b2 , . . . , bk ∈ CṀO 2 Rn × · · · × CṀO k Rn , provided
1
0
2 k−1
t ωt log
dt < ∞.
t
nλ
3.26
Conversely, assume that Hωb is bounded from Ḃq,λ Rn to Ḃq1 Rn for any b q2
q
b2 , . . . , bk ∈ CṀO Rn × · · · × CṀO k Rn . We choose b b2 , . . . , bk with bj x log |x| for
q
q
all x ∈ Rn and j ∈ {2, . . . , k}. Then b ∈ CṀO 2 Rn × · · · × CṀO k Rn . Repeating the argument
in the proof of Theorem 3.1 then yields the desired conclusion.
We point out that, it is still unknown whether the condition 2.1 in Theorem 3.4ii is
sharp.
4. Adjoint Operators and Related Results
In this section, we focus on the corresponding results for the adjoint operators of weighted
Hardy operators.
Recall that the weighted Cesàro operator Gω is defined by
Gω fx 1−α
·
1 x −n
t ωtdt,
f
t
0
x ∈ Rn .
4.1
If 0 < α < 1, n 1, and ωt 1/Γα1/t − 11−α , then Gω f· is reduced to
Jα f·, where Jα is a variant of Weyl integral operator and defined by
1
Jα fx Γα
∞
x
ft
1−α
t − x
dt
t
4.2
for all x ∈ 0, ∞. When ω ≡ 1 and n 1, Gω is the classical Cesàro operator:
⎧ ∞ f y
⎪
⎪
⎪
dy,
⎨
y x
Gfx x
f y
⎪
⎪
⎪−
dy,
⎩
y
−∞
x > 0,
4.3
x < 0.
It was pointed out in 5 that the weighted Hardy operator Hω and the weighted
Cesàro operator Gω are adjoint mutually, namely,
Rn
gxHω fxdx for all admissible pairs f and g.
Rn
fxGω gxdx
4.4
16
Abstract and Applied Analysis
Since Ȧq and Ḃq are a pair of dual Banach spaces, it follows from Theorem 2.1 the
following.
Theorem 4.1. Let 1 < q < ∞. Then Gω is bounded on Ȧq Rn if and only if
E :
1
4.5
ωtdt < ∞.
0
Moreover, when 4.5 holds, the operator norm of Gω on Ȧq Rn is given by
Gω Ȧq Rn → Ȧq Rn E.
4.6
Corollary 4.2. i For 0 < α < 1 and 1 < q < ∞,
Jα Ȧq dx → Ȧq xq1−α dx Γ1
.
Γ1 α
4.7
ii For 1 < q < ∞, we have
GȦq Rn → Ȧq Rn 1.
4.8
Since the dual space of H Ȧq 1 < q < ∞ is isomorphic to CṀO
Theorem 2.3 implies the following result.
q
see 18, 19,
Theorem 4.3. Let 1 < q < ∞. Then Gω is a bounded operator on H Ȧq Rn if and only if 4.5 holds.
Moreover, when 4.5 holds, the operator norm of Gω on H Ȧq Rn is given by
Gω H Ȧq Rn → H Ȧq Rn E.
4.9
Corollary 4.4. For 1 < q < ∞, we have
GH Ȧq → H Ȧq 1.
4.10
Following the idea in Section 3, we define the higher order commutator of the
weighted Cesàro operator as
⎞
k x
x
bj
− bj x ⎠f
t−n ωtdt,
Gbω fx ⎝
t
t
0
j1
1
⎛
x ∈ Rn .
4.11
When k 0, Gbω is understood as Gω . Notice that if k 1, then Gbω Gbω . Similar to the proofs
of Theorems 3.1 and 3.3, we have the following result.
Theorem 4.5. Let k ≥ 2, 1 < q1 < q, q2 , . . . , qk < ∞, 1/q1 1/q −1/q1 < λ1 < 0, 0 ≤ λ2 , . . . , λk < 1/n, and λ1 λ ki2 λi .
k
i2
1/qi , −1/q < λ < 0,
Abstract and Applied Analysis
17
i Assume further that ω is a positive integrable function on 0, 1. The commutator Gbω is
q
q
bounded from Ḃq,λ Rn to Ḃ q1 ,λ Rn , for any b b2 , . . . , bk ∈ CṀO 2 Rn × · · · × CṀO k Rn , if
and only if
1
0
2 k−1
t−nλ
1 ωt log
dt < ∞.
t
4.12
q ,λ
ii Let λ2 , . . . , λk > 0 and b b2 , . . . , bk ∈ CṀOq2 ,λ2 Rn × · · · × CṀO k k Rn . Then the
corresponding commutator Gbω is bounded from Ḃq,λ Rn to Ḃq1 ,λ1 Rn , provided that
1
t−nλ
1 ωtdt < ∞.
4.13
0
We conclude this paper with some comments on the discrete version of the weighted
Hardy and Cesàro operators.
Let N0 be the set of all nonnegative integers and 2−N0 denote the set {2−j : j ∈ N0 }.
Let now ϕ be a nonnegative function defined on 2−N0 and f be a complex-valued measurable
#ω is defined by
function on Rn . The discrete weighted Hardy operator H
∞
#ω f x H
2−k f 2−k x ω 2−k ,
x ∈ Rn ,
4.14
k0
and the corresponding discrete weighted Cesàro operator is defined by setting, for all x ∈ Rn ,
∞ ω f x f 2k x 2kn−1 ω 2−k .
G
4.15
k0
We remark that, by the same argument as above with slight modifications, all the results
#ω
related to the operators Hω and Gω in Sections 1–4 are also true for their discrete versions H
ω.
and G
Acknowledgments
This work is partially supported by the Laboratory of Mathematics and Complex Systems,
Ministry of Education of China and the National Natural Science Foundation Grant nos.
10901076, 11101038, 11171345, and 10931001.
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