Volume 9 (2008), Issue 4, Article 111, 5 pp.
λ-CENTRAL BMO ESTIMATES FOR COMMUTATORS OF N -DIMENSIONAL
HARDY OPERATORS
ZUN-WEI FU
D EPARTMENT OF M ATHEMATICS
L INYI N ORMAL U NIVERSITY
L INYI S HANDONG , 276005
P EOPLE ’ S R EPUBLIC OF C HINA
lyfzw@tom.com
Received 12 April, 2008; accepted 13 October, 2008
Communicated by R.N. Mohapatra
A BSTRACT. This paper gives the λ-central BMO estimates for commutators of n-dimensional
Hardy operators on central Morrey spaces.
Key words and phrases: Commutator, N -dimensional Hardy operator, λ-central BMO space, Central Morrey space.
2000 Mathematics Subject Classification. 26D15, 42B25, 42B99.
1. I NTRODUCTION AND M AIN R ESULTS
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
(1.1)
g(x)Hf (x) dx =
f (x)H∗ g(x) dx.
Rn
Rn
We have
|Hf (x)| ≤ Cn M f (x),
where M is the Hardy-Littlewood maximal operator which is defined by
Z
1
(1.2)
M f (x) = sup
|f (t)|dt,
Q3x |Q| Q
where the supremum is taken over all balls containing x.
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.
118-08
2
Z UN -W EI F U
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).
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
1q
Z
1
q
(1.3)
kf kC Ṁ Oq, λ (Rn ) = sup
|f (x) − fB(0, R) | dx
< ∞.
|B(0, R)|1+λq B(0,R)
R>0
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
1q
Z
1
q
(1.4)
kf kḂ q, λ (Rn ) = sup
|f (x)| dx
< ∞.
|B(0, R)|1+λq B(0,R)
R>0
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 1.1. Let Hb be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞, 1q = p11 + p12 ,
− 1q < λ < 0, 0 ≤ λ2 < n1 and λ = λ1 + λ2 . If b ∈ C Ṁ Op2 , λ2 (Rn ), then the commutator Hb is
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 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn ) .
Let λ2 = 0 in Theorem 1.1. We can obtain the central BMO estimates for commutators of
n-dimensional Hardy operators, Hb , on central Morrey spaces.
Corollary 1.2. Let Hb be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞, 1q = p11 + p12
and − 1q < λ < 0. If b ∈ C Ṁ Op2 (Rn ), then the commutator Hb is 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 ) .
Similar to Theorem 1.1, we have:
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 111, 5 pp.
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λ-C ENTRAL BMO E STIMATES
3
Theorem 1.3. Let Hb∗ be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞, 1q = p11 + p12 ,
− 1q < λ < 0, 0 ≤ λ2 < n1 and λ = λ1 + λ2 . If b ∈ C Ṁ Op2 , λ2 (Rn ), 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 1.3. We can get the central BMO estimates for commutators of
n-dimensional Hardy operators, Hb∗ , on central Morrey spaces.
Corollary 1.4. Let Hb∗ be defined as above. Suppose 1 < p1 < ∞, p01 < p2 < ∞, 1q = p11 + p12
and − 1q < λ < 0. If b ∈ C Ṁ Op2 (Rn ), then the commutator Hb∗ is 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 ) .
2. P ROOFS OF T HEOREMS
Proof of Theorem 1.1. 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.
For
1
q
=
1
p1
+
1
,
p2
by Hölder’s inequality and the boundedness of H from Lp1 to Lp1 , we have
− 1q
Z
I ≤ |B|
p2
p1 Z
|b(x) − bB | dx
2
|H(f χB )(x)| dx
B
− 1q
≤ C|B|
p1
p1
1
B
kbkC Ṁ Op2 , λ2 (Rn ) |B|
1
+λ2
p2
Z
p1
p1
|f (x)| dx
1
B
λ
= C|B| kbkC Ṁ Op2 , λ2 (Rn )
1
|B|1+p1 λ1
Z
p1
p1
|f (x)| dx
1
B
λ
≤ C|B| kbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn ) .
For J, we have
q
Z Z
1
1
q
dx
J =
f
(y)(b(y)
−
b
)
dy
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|)
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 111, 5 pp.
http://jipam.vu.edu.au/
4
Z UN -W EI F U
q
k Z
Z
0
X
C X
1
≤
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
1
C X
≤
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
|B|
|2k B|q 2k B\2k−1 B 2i B\2i−1 B
i=−∞
k=−∞
:= J1 + J2
By Hölder’s inequality ( p11 +
1
p2
= 1q ), we have
( k
Z
p1
0
X
1
1
C X |2k B|
p
i
1
q0
J1 ≤
|2
B|
|f
(y)|
dy
|B| k=−∞ |2k B|q i=−∞
2i B
)
p1 q
Z
2
p2
×
|b(y) − b2i B | dy
2i B
0
X
C
|2k B|
q
q
≤
kbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn )
|B|
|2k B|q
k=−∞
qλ
≤ C|B|
(
k
X
)q
|2i B|λ+1
i=−∞
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
−1
X
Z
1
≤
|b(y) − b2j+1 B |dy
j B|
|2
jB
2
j=i
p1
Z
−1 X
2
1
p2
≤C
|b(y) − b2j+1 B | dy
j+1
|2 B| 2j+1 B
j=i
λ2
≤ CkbkC Ṁ Op2 , λ2 (Rn ) |B|
−1
X
2(j+1)nλ2
j=i
λ2
≤ CkbkC Ṁ Op2 , λ2 (Rn ) |i||B| .
By Hölder’s inequality ( p11 +
1
p01
= 1), we have
q
Z
Z
0
k
X
C X
1
i
J2 =
f
(y)(b
−
b
)
dy
dx
B
2B
k
q
|B| k=−∞ |2 B| 2k B\2k−1 B i=−∞ 2i B\2i−1 B
( k
)q
0
X
C
|2k B||B|qλ2 X
q
q
i
λ1 +1
≤
kbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn )
|i||2 B|
|B|
|2k B|q
i=−∞
k=−∞
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 111, 5 pp.
http://jipam.vu.edu.au/
λ-C ENTRAL BMO E STIMATES
5
0
X
|2k B||B|qλ2 |k|q |2k B|(λ1 +1)q
C
q
q
≤
kbkC Ṁ Op2 , λ2 (Rn ) kf kḂ p1 , λ1 (Rn )
|B|
|2k B|q
k=−∞
≤ 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 1.1.
Proof of Theorem 1.3. We omit the details here.
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[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.
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http://jipam.vu.edu.au/