λ-CENTRAL BMO ESTIMATES FOR
COMMUTATORS OF N -DIMENSIONAL HARDY
OPERATORS
λ-Central BMO Estimates
ZUN-WEI FU
Department of Mathematics
Linyi Normal University
Linyi Shandong, 276005, P.R. of China
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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EMail: lyfzw@tom.com
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Received:
12 April, 2008
Accepted:
13 October, 2008
Communicated by:
R.N. Mohapatra
2000 AMS Sub. Class.:
26D15, 42B25, 42B99.
Key words:
Commutator, N -dimensional Hardy operator, λ-central BMO space, Central
Morrey space.
Abstract:
This paper gives the λ-central BMO estimates for commutators of n-dimensional
Hardy operators on central Morrey spaces.
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Acknowledgements:
The research is supported by the NNSF (Grant No. 10571014; 10871024) of
People’s Republic of China.
The author would like to express his thanks to Prof. Shanzhen Lu for his constant
encourage. This paper is dedicated to him for his 70th birthday. The author also
would like to express his gratitude to the referee for his valuable comments.
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Contents
1
Introduction and Main Results
3
2
Proofs of Theorems
7
λ-Central BMO Estimates
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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1.
Introduction and Main Results
Let f be a locally integrable function on Rn . The n-dimensional Hardy operators
are defined by
Z
Z
1
f (t)
∗
Hf (x) := n
f (t)dt, H f (x) :=
dt,
x∈Rn \ {0}.
n
|x| |t|≤|x|
|t|
|t|>|x|
In [4], Christ and Grafakos obtained results for the boundedness of H on Lp (Rn )
spaces. They also found the exact operator norms of H on Lp (Rn ) spaces, where
1 < p < ∞.
It is easy to see that H and H∗ satisfy
Z
Z
g(x)Hf (x) dx =
f (x)H∗ g(x) dx.
(1.1)
Rn
λ-Central BMO Estimates
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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Rn
We have
|Hf (x)| ≤ Cn M f (x),
where M is the Hardy-Littlewood maximal operator which is defined by
Z
1
|f (t)|dt,
(1.2)
M f (x) = sup
Q3x |Q| Q
where the supremum is taken over all balls containing x.
Recently, Fu et al. [2] gave the definition of commutators of n-dimensional Hardy
operators.
Definition 1.1. Let b be a locally integrable function on Rn . We define the commutators of n-dimensional Hardy operators as follows:
Hb f := bHf − H(f b),
Hb∗ f := bH∗ f − H∗ (f b).
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In [2], Fu et al. gave the central BMO estimates for commutators of n-dimensional
Hardy operators. In 2000, Alvarez, Guzmán-Partida and Lakey [1] studied the relationship between central BMO spaces and Morrey spaces. Furthermore, they introduced λ-central bounded mean oscillation spaces and central Morrey spaces, respectively.
Definition 1.2 (λ-central BMO space). Let 1 < q < ∞ and − 1q < λ < n1 . A
function f ∈ Lqloc (Rn ) is said to belong to the λ-central bounded mean oscillation
space C Ṁ Oq, λ (Rn ) if
λ-Central BMO Estimates
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
(1.3) kf kC Ṁ Oq, λ (Rn ) = sup
R>0
1
|B(0, R)|1+λq
Z
q
1q
|f (x) − fB(0, R) | dx
< ∞.
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B(0,R)
Remark 1. If two functions which differ by a constant are regarded as a function in
the space C Ṁ Oq, λ (Rn ), then C Ṁ Oq, λ (Rn ) becomes a Banach space. Apparently,
(1.3) is equivalent to the following condition (see [1]):
1q
Z
1
q
sup inf
|f (x) − c| dx
< ∞.
|B(0, R)|1+λq B(0,R)
R>0 c∈C
Definition 1.3 (Central Morrey spaces, see [1]). Let 1 < q < ∞ and − 1q < λ < 0.
The central Morrey space Ḃ q, λ (Rn ) is defined by
(1.4)
kf kḂ q, λ (Rn ) = sup
R>0
1
|B(0, R)|1+λq
Z
|f (x)|q dx
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1q
< ∞.
B(0,R)
Remark 2. It follows from (1.3) and (1.4) that Ḃ q, λ (Rn ) is a Banach space continuously included in C Ṁ Oq, λ (Rn ).
Inspired by [2], [3] and [5], we will establish the λ-central BMO estimates for
commutators of n-dimensional Hardy operators on central Morrey spaces.
Theorem 1.4. Let Hb be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞,
1
= p11 + p12 , − 1q < λ < 0, 0 ≤ λ2 < n1 and λ = λ1 + λ2 . If b ∈ C Ṁ Op2 , λ2 (Rn ),
q
then the commutator Hb is bounded from Ḃ p1 , λ (Rn ) to Ḃ q, λ (Rn ) and satisfies the
following inequality:
λ-Central BMO Estimates
kHb f kḂ q, λ (Rn ) ≤ CkbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn ) .
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
Let λ2 = 0 in Theorem 1.4. We can obtain the central BMO estimates for commutators of n-dimensional Hardy operators, Hb , on central Morrey spaces.
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Corollary 1.5. Let Hb be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞,
1
= p11 + p12 and − 1q < λ < 0. If b ∈ C Ṁ Op2 (Rn ), then the commutator Hb is
q
bounded from Ḃ p1 , λ (Rn ) to Ḃ q, λ (Rn ) and satisfies the following inequality:
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II
kHb f kḂ q, λ (Rn ) ≤ CkbkC Ṁ Op2 (Rn ) kf kḂ p1 , λ (Rn ) .
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Similar to Theorem 1.4, we have:
Theorem 1.6. Let Hb∗ be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞,
1
= p11 + p12 , − 1q < λ < 0, 0 ≤ λ2 < n1 and λ = λ1 + λ2 . If b ∈ C Ṁ Op2 , λ2 (Rn ),
q
then the commutator Hb∗ is bounded from Ḃ p1 , λ1 (Rn ) to Ḃ q, λ (Rn ) and satisfies the
following inequality:
kHb∗ f kḂ q, λ (Rn ) ≤ CkbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn ) .
Let λ2 = 0 in Theorem 1.6. We can get the central BMO estimates for commutators of n-dimensional Hardy operators, Hb∗ , on central Morrey spaces.
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Corollary 1.7. Let Hb∗ be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞,
1
= p11 + p12 and − 1q < λ < 0. If b ∈ C Ṁ Op2 (Rn ), then the commutator Hb∗ is
q
bounded from Ḃ p1 , λ (Rn ) to Ḃ q, λ (Rn ) and satisfies the following inequality:
kHb∗ f kḂ q, λ (Rn ) ≤ CkbkC Ṁ Op2 (Rn ) kf kḂ p1 , λ (Rn ) .
λ-Central BMO Estimates
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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2.
Proofs of Theorems
Proof of Theorem 1.4. Let f be a function in Ḃ p1 , λ1 (Rn ). For fixed R > 0, denote
B(0, R) by B. Write
1q
Z
1
q
|Hb f (x)| dx
|B| B
q 1
Z Z
q
1
1
dx
f
(y)(b(x)
−
b(y))
dy
=
|B| B |x|n B(0, |x|)
q 1
Z Z
q
1
1
dx
≤
f
(y)(b(x)
−
b
)
dy
B
|B| B |x|n B(0, |x|)
q 1
Z Z
q
1
1
dx
+
f
(y)(b(y)
−
b
)
dy
B
|B| B |x|n B(0, |x|)
:= I + J.
1
q
For =
we have
1
p1
+
1
,
p2
p1
p1
by Hölder’s inequality and the boundedness of H from L to L ,
− 1q
Z
I ≤ |B|
p2
p1 Z
2
|b(x) − bB | dx
B
p1
p1
|H(f χB )(x)| dx
B
1
1
≤ C|B|− q kbkC Ṁ Op2 , λ2 (Rn ) |B| p2
+λ2
Z
|f (x)|p1 dx
p1
1
B
λ
= C|B| kbkC Ṁ Op2 , λ2 (Rn )
1
Z
|B|1+p1 λ1 B
≤ C|B|λ kbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn ) .
p1
|f (x)| dx
p1
1
1
λ-Central BMO Estimates
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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For J, we have
q
Z Z
1
1
q
dx
f
(y)(b(y)
−
b
)
dy
J =
B
|B| B |x|n B(0, |x|)
q
Z
Z
0
1
1 X
dx
=
f
(y)(b(y)
−
b
)
dy
B
|B| k=−∞ 2k B\2k−1 B |x|n B(0, |x|)
k Z
q
Z
0
X
1
C X
f
(y)(b(y)
−
b
)
dy
≤
dx
B
k
q
|B| k=−∞ |2 B| 2k B\2k−1 B i=−∞ 2i B\2i−1 B
q
Z
Z
0
k
X
X
1
C
≤
f (y)(b(y) − b2i B ) dy dx
k
q
|B| k=−∞ |2 B| 2k B\2k−1 B i=−∞ 2i B\2i−1 B
q
Z
Z
0
k
X
1
C X
i
+
f
(y)(b
−
b
)
dy
dx
B
2B
k
q
|B|
|2 B| 2k B\2k−1 B 2i B\2i−1 B
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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JJ
II
J
I
i=−∞
k=−∞
:= J1 + J2
By Hölder’s inequality ( p11 +
λ-Central BMO Estimates
1
p2
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= 1q ), we have
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( k
Z
p1
0
X
1
1
C X |2k B|
p1
i
0
q
J1 ≤
|2 B|
|f (y)| dy
k
q
|B| k=−∞ |2 B| i=−∞
2i B
)
Z
p1 q
2
p2
×
|b(y) − b2i B | dy
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2i B
0
X
C
|2k B|
q
q
≤
kbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn )
|B|
|2k B|q
k=−∞
(
k
X
i=−∞
)q
|2i B|λ+1
≤ C|B|qλ kbkqC Ṁ Op2 , λ2 (Rn ) kf kqḂ p1 , λ1 (Rn ) .
To estimate J2 , the following fact is applied.
For λ2 ≥ 0,
|b2i B − bB | ≤
−1
X
|b2j+1 B − b2j B |
j=i
λ-Central BMO Estimates
−1
X
Z
1
|b(y) − b2j+1 B |dy
≤
j B|
|2
jB
2
j=i
p1
Z
−1 X
2
1
p2
j+1 B | dy
≤C
|b(y)
−
b
2
|2j+1 B| 2j+1 B
j=i
λ2
≤ CkbkC Ṁ Op2 , λ2 (Rn ) |B|
−1
X
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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(j+1)nλ2
2
JJ
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J
I
j=i
≤ CkbkC Ṁ Op2 , λ2 (Rn ) |i||B|λ2 .
By Hölder’s inequality ( p11 +
1
p01
= 1), we have
q
Z
Z
0
k
X
X
C
1
J2 =
f (y)(b2i B − bB ) dy dx
k
q
|B| k=−∞ |2 B| 2k B\2k−1 B i=−∞ 2i B\2i−1 B
(
)q
0
k
k
qλ2
X
X
C
|2
B||B|
kbkqC Ṁ Op2 , λ2 (Rn ) kf kqḂ p1 , λ1 (Rn )
|i||2i B|λ1 +1
≤
k B|q
|B|
|2
i=−∞
k=−∞
0
X
C
|2k B||B|qλ2 |k|q |2k B|(λ1 +1)q
q
q
≤
kbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn )
|B|
|2k B|q
k=−∞
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≤ C|B|qλ kbkqC Ṁ Op2 , λ2 (Rn ) kf kqḂ p1 , λ1 (Rn ) .
Combining the estimates of I, J1 and J2 , we get the required estimate for Theorem
1.4.
Proof of Theorem 1.6. We omit the details here.
λ-Central BMO Estimates
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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References
[1] J. ALVARAREZ, M. GUZMAN-PARTIDA AND J. LAKEY, Spaces of bounded
λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures,
Collect. Math., 51 (2000), 1–47.
[2] Z.W. FU, Z.G. LIU, S.Z. LU AND H.B. WANG, Characterization for commutators of N-dimensional fractional Hardy operators. Science in China (Ser. A), 10
(2007), 1418–1426.
[3] Z.W. FU, Y. LIN AND S.Z. LU, λ-Central BMO estimates for commutators of
singular integral operators with rough kernels. Acta Math. Sinica (English Ser.),
3 (2008), 373–386.
[4] M. CHRIST AND L. GRAFAKOS, Best constants for two non-convolution inequalities, Proc. Amer. Math. Soc., 123 (1995), 1687–1693.
[5] S.C. LONG AND J. WANG, Commutators of Hardy operators, J. Math. Anal.
Appl., 274 (2002), 626–644.
λ-Central BMO Estimates
Zun-Wei Fu
vol. 9, iss. 4, art. 111, 2008
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