Volume 15, 2011
35
λ-CENTRAL BMO ESTIMATES FOR MULTILINEAR COMMUTATORS
OF LITTLEWOOD-PALEY OPERATOR ON CENTRAL MORREY
SPACES
Ying Shen, Liu Lanzhe
College of Mathematics
Changsha University of Science and Technology
Changsha, 410077
P.R.of China
E-mail: lanzheliu@163.com
Abstract: In this paper, we establish λ-central BMO estimates for the multilinear commutator related to the Littlewood-Paley operator in central Morrey spaces, and so does the multilinear commutator of the fractional maximal
operator.
Key words: λ-central space; Multilinear commutator; Littlewood-Paley operator; Central Morrey spaces.
MSC 2010: 42B20, 42B35.
1. Introduction
Let b ∈ BM O(Rn ) and T be the Calderón-Zygmund operator, the commutator [b, T ] generated by b and T is defined by
[b, T ](f ) = bT (f ) − T (bf ).
A classical result of Coifman, Rochberb and Weiss (see [2]) proved that the
commutator [b, T ] is bounded on Lp (Rn ), (1 < p < ∞). Since BM O ⊂
∩
q
q
q>1 CBM O (see [3]), if we only assume b ∈ CBM O , or more generally
b ∈ CBM Oq,λ with q > 1, then [b, T ] may not be a bounded operator on
Lp (Rn ). However, it has some boundedness properties on other spaces. As
a matter of fact, Grafakos, Li and Yang (see [4]) considered the commutator with b ∈ CBM Oq on Herz spaces for the first time. Alvarez, GuzmánPartida and Lakey (see [1]) and Komori (see [6]) have obtained the λ-central
BMO estimates for the commutators of a class of singular integral operators
on central Morrey spaces. Motivated by these results, in this paper, we will
establish λ-central BMO estimates for the multilinear commutator related to
the Littlewood-Paley operator in central Morrey spaces. And the multilinear
commutator of the fractional maximal operator is also discussed.
2. Preliminaries and Theorem
First, let us introduce some notations. Let Mδ be the fractional maximal
operator, which is
∫
δ/n−1
Mδ (f )(x) = sup |B|
|f (y)|dy, 0 < δ/n < 1,
(1)
B∋x
B
36
Bulletin of TICMI
⃗
and Mδb be the multilinear commutator of the fractional maximal operator
which is defined as follows:
∫ ∏
m
⃗b
δ/n−1
Mδ (f )(x) = sup |B|
| (bj (x) − bj (y))f (y)|dy.
(2)
x∈B
B
j=1
For bj ∈ CBM Opj+1 ,λj+1 (Rn )(j = 1, · · · , m), set
||⃗b||CBM Op⃗,⃗λ =
m
∏
||bj ||CBM Opj+1 ,λj+1 .
j=1
Given a positive integer m and 1 ≤ j ≤ m, we denote by Cjm the family of
all finite subsets σ = {σ(1), · · ·, σ(j)} of {1, · · ·, m} of j different elements.
For σ ∈ Cjm , set σ c = {1, · · ·, m} \ σ. For ⃗b = (b1 , · · ·, bm ) and σ = {σ(1), · ·
·, σ(j)} ∈ Cjm , set ⃗bσ = (bσ(1) , · · ·, bσ(j) ), bσ = bσ(1) · · · bσ(j) and ||⃗bσ ||CBM Op⃗,⃗λ =
||bσ(1) ||CBM Op2 ,λ2 · · · ||bσ(j) ||CBM Opj+1 ,λj+1 .
Definition 1. Let 0 < λ < δ/n, 0 < δ < n and 1 < q < ∞. A function
f ∈ Lqloc (Rn ) is said to belong to the λ-central bounded mean oscillation space
CBM Oq,λ (Rn ) if
(
||f ||CBM Oq,λ = sup
r>0
1
|B(0, r)|1+λq
)1/q
∫
|f (x) − fB(0,r) | dx
< ∞.
q
(3)
B(0,r)
where, B = B(0, r) = {x ∈ Rn : |x| < r} and fB(0,r) is the mean value of f on
B(0, r).
Remark 1. If two functions which differ by a constant are regarded
as a function in the space CBM Oq,λ becomes a Banach space. The space
CBM Oq,λ (Rn ) when λ = 0 is just the space CBM O(Rn ) defined as follows:
(
||f ||CBM Oq = sup
r>0
1
|B(0, r)|
)1/q
|f (x) − fB(0,r) | dx
< ∞.
∫
q
B(0,r)
Apparently, (3) is equivalent to the following condition (see [6]):
(
||f ||CBM Oq,λ = sup inf
r>0 c∈C
)1/q
∫
1
|B(0, r)|1+λq
|f (x) − c| dx
q
< ∞.
B(0,r)
Definition 2. Let λ ∈ R and 1 < q < ∞. The central Morrey space
Ḃ (Rn ) is defined by
q,λ
(
||f ||Ḃ q,λ = sup
r>0
1
|B(0, r)|1+λq
∫
)1/q
< ∞.
|f (x)| dx
q
(4)
B(0,r)
Remark 2. It follows from (3) and (4) that Ḃ q,λ (Rn ) is a Banach space
continuously included in CBM Oq,λ (Rn ). We denote by CM Oq,λ (Rn ) and
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37
B q,λ (Rn ) the inhomogeneous versions of the λ-central bounded mean oscillation
space and the central Morrey space by taking the supremum over r ≥ 1 in Definition 1 and Definition 2 instead of r > 0 there. Obviously, CBM Oq,λ (Rn ) ⊂
CM Oq,λ (Rn ) for λ < δ/n and 1 < q < ∞, and Ḃ q,λ (Rn ) ⊂ B q,λ (Rn ) for λ ∈ R
and 1 < q < ∞.
Remark 3. When λ1 < λ2 , it follows from the property of monotone
functions that B q,λ1 (Rn ) ⊂ B q,λ2 (Rn ) and CM Oq,λ1 (Rn ) ⊂ CM Oq,λ2 (Rn )
for 1 < q < ∞. If 1 < q1 < q2 < ∞, then by Hölder’s inequality, we
know that Ḃ q2 ,λ (Rn ) ⊂ Ḃ q1 ,λ (Rn ) for λ ∈ R and CBM Oq2 ,λ ⊂ CBM Oq1 ,λ ,
CM Oq2 ,λ (Rn ) ⊂ CM Oq1 ,λ (Rn ) for 0 < λ < δ/n.
Definition 3. Let 1 ≤ q < ∞, α ∈ R. The central Campanato space is
defined by(see [17])
CLα, q (Rn ) = {f ∈ Lqloc (Rn ) : ||f ||CLα, q < ∞},
where
(
−α
||f ||CLα, q = sup |B(0, r)|
r>0
1
|B(0, r)|
)1/q
|f (x) − fB(0,r) | dx
.
∫
q
B(0,r)
Definition 4. Fix δ > 0. Let ψ be a fixed function which satisfies the
following
∫ properties:
(1) Rn ψ(x)dx = 0;
(2)|ψ(x)| ≤ C(1 + |x|)−(n+1−δ) ;
(3)|ψ(x + y) − ψ(x)| ≤ C|y|ϵ (1 + |x|)−(n+ϵ−δ) when 2|y| < |x|.
n+1
We denote that Γ(x) = {(y, t) ∈ R+
: |x − y| < t} and the characteristic
function of Γ(x) by χΓ(x) . The Littlewood-Paley multilinear commutator is
defined by
[∫ ∫
]1/2
⃗b
⃗b
2 dydt
|Ft (f )(x, y)| n+1
Sδ (f )(x) =
,
(5)
t
Γ(x)
where
⃗
Ftb (f )(x, y)
∫
=
m
∏
(bj (x) − bj (z))ψt (y − z)f (z)dz,
(6)
Rn j=1
and ψt (x) = t−n+δ ψ(x/t) for t > 0. We also define that
(∫ ∫
|f ∗
Sδ (f )(x) =
Γ(x)
dydt
ψt (y)|2 n+1
t
)1/2
,
(7)
which is the Littlewood-Paley operator(see
∫ ∫[16]).
Let H be the space H = h : ||h|| = ( Rn+1 |h(y, t)|2 t(dydt
)1/2 < ∞, then,
n+1)
+
⃗
Rn , Ftb (f )(x, y)
for each fixed x ∈
to H, and it is clear that
may be viewed as a mapping from [0, +∞)
Sδ (f )(x) = ||χΓ(x) Ft (f )(x)||
38
Bulletin of TICMI
and
⃗
⃗
Sδb (f )(x) = ||χΓ(x) Ftb (f )(x, y)||.
⃗
Note that when b1 = · · · = bm , Sδb is just the commutator of order m.
It is well known that commutators are of great interest in harmonic analysis
and have been widely studied by many authors(see [5][7-10][12-15]). Our main
purpose is to study the boundedness properties for the multilinear commutator
on central Morrey spaces.
Now we state our theorems as following.
Theorem 1. Let 0 < δ < n, 1 < p < n/δ, 1/q = 1/p−δ/n. If λ1 < −δ/n
and λ2 = λ1 + δ/n, then Sδ is bounded from Ḃ p,λ1 (Rn ) to Ḃ q,λ2 (Rn ).
Theorem 2. Let 0 < δ < n, 1 < p < n/δ, 1/q = 1/p−δ/n. If λ1 < −δ/n
and λ2 = λ1 + δ/n, then Mδ is bounded from Ḃ p,λ1 (Rn ) to Ḃ q,λ2 (Rn ).
Theorem 3. Let 0 < δ < n, 1 < pu < n/δ(1 ≤ u ≤ m + 1), 1/p1 +
1/p2 + · · · 1/pm+1 < 1 and 1/q = 1/p1 + 1/p2 + · · · + 1/pm+1 − δ/n. Suppose
0 < λi < δ/n(i = 2, 3, · · · , m + 1), λ1 < −λ2 − λ3 − · · · − λm+1 − δ/n and
λ = λ1 + λ2 + · · · λm+1 + δ/n. If bj ∈ CBM Opj+1 ,λj+1 (Rn ), for j = 1, · · · , m,
⃗
then Sδb is bounded from Ḃ p1 ,λ1 (Rn ) to Ḃ q,λ (Rn ), and the following inequality
holds:
⃗
||Sδb (f )||Ḃ q,λ ≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 .
Theorem 4. Let 0 < δ < n, 1 < pu < n/δ(1 ≤ u ≤ m + 1), 1/p1 +
1/p2 + · · · 1/pm+1 < 1 and 1/q = 1/p1 + 1/p2 + · · · + 1/pm+1 − δ/n. Suppose
0 < λi < δ/n(i = 2, 3, · · · , m + 1), λ1 < −λ2 − λ3 − · · · − λm+1 − δ/n and
λ = λ1 + λ2 + · · · λm+1 + δ/n. If bj ∈ CBM Opj+1 ,λj+1 (Rn ), for j = 1, · · · , m,
⃗
then Mδb is bounded from Ḃ p1 ,λ1 (Rn ) to Ḃ q,λ (Rn ), and the following inequality
holds:
⃗
||Mδb (f )||Ḃ q,λ ≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 .
3. Proof of Theorems.
To prove the theorems, we need the following lemmas.
Lemma 1.(see [16]) Let 0 < δ < n, 1 < p < n/δ and 1/q = 1/p − δ/n.
Then Sδ is bounded from Lp (Rn ) to Lq (Rn ).
Lemma 2.(see [11]) Let 0 < δ < n, 1 < p < n/δ and 1/q = 1/p − δ/n.
Then Mδ is bounded from Lp (Rn ) to Lq (Rn ).
Lemma 3. Let 0 < δ < n, 1 < p < n/δ, λ > 0. Suppose b ∈
CBM Op,λ (Rn ), then
|b2k+1 B − bB | ≤ C||b||CBM Op,λ k|2k+1 B|λ for k ≥ 1 .
Proof.
|b2k+1 B − bB | ≤
k
∑
j=0
|b2j+1 B − b2j B | ≤
k
∑
j=0
1
j
|2 B|
∫
|b(y) − b2j+1 B |dy
2j B
Volume 15, 2011
≤C
k (
∑
j=0
)1/p
∫
1
|b(y) − b2j+1 B | dy
p
|2j+1 B|
39
≤ C||b||CBM Op,λ
2j+1 B
≤ C||b||CBM Op,λ (k + 1)|2
k
∑
|2j+1 B|λ
j=0
k+1
B| ≤ C||b||CBM Op,λ k|2
λ
k+1
λ
B| .
Proof of Theorem 1. Let f be a function in Ḃ p,λ1 (Rn ). For fixed r > 0,
set B = B(0, r). We consider
(
)1/q
∫
1
q
|Sδ (f )(x)| dx
|B|1+λ2 q B
(
)1/q (
)1/q
∫
∫
1
1
q
q
≤
|Sδ (f χB )(x)| dx
+
|Sδ (f χB c )(x)| dx
|B|1+λ2 q B
|B|1+λ2 q B
= I + II.
For I, by the boundedness of Sδ from Lp (Rn ) to Lq (Rn ), we have
)1/p
(∫
−1/q−λ2
p
I ≤ C|B|
|f (x)| dx
−1/q−λ2
B
1/p+λ1
≤ C|B|
|B|
≤ C||f ||Ḃ p,λ1 .
||f ||Ḃ p,λ1
For II, using Minkowski’s inequality, we have
)1/2
(∫ ∫
∫
t1−n
Sδ (f χB c )(x) ≤
dydt
dz
f (z)
2(n+1−δ)
Bc
Γ(x) (t + |x − z|)
)1/2
(∫ ∞
∫
tdt
dz
≤
f (z)
(t + |x − z|)2(n+1−δ)
c
0
∫B
≤
f (z)|x − z|−n+δ dz,
Bc
thus, using Hölder’s inequality and note that x ∈ B, we get
∞ ∫
∑
|Sδ (f χB c )(x)| ≤
|x − z|−n+δ |f (z)|dz
k+1 B\2k B
k=1 2
∞
∑
≤ C
k=0
≤ C
∞
∑
(∫
|2 B|
k
)1/p
|f (z)| dz
δ/n−1
p
|2k+1 B|1−1/p
2k+1 B
|2k B|δ/n−1 |2k+1 B|1/p+λ1 ||f ||Ḃ p,λ1 |2k+1 B|1−1/p
k=0
≤ C||f ||Ḃ p,λ1 |B|δ/n+λ1 ,
therefore, we deduce
II ≤ C||f ||Ḃ p,λ1 |B|δ/n+λ1 |B|−1/q−λ2 |B|1/q ≤ C||f ||Ḃ p,λ1 .
40
Bulletin of TICMI
This completes the proof of Theorem 1.
Proof of Theorem 2. Set
∫
Seδ (f )(x) =
|x − z|−n+δ |f (z)|dz,
Rn
it is easy to know the Theorem 1 is also true for the operator Seδ (f )(x). Since
Mδ (f )(x) ≤ Seδ (f )(x),
thus, Theorem 2 can be easily deduced. We omit the details here.
Proof of Theorem
3. Let f be a function in Ḃ p1 ,λ1 (Rn ). When m = 1,
∫
set (b1 )B = |B|−1 B b1 (x)dx and note that
Sδb1 (f )(x) = (b1 (x) − (b1 )B )Sδ (f )(x) − Sδ ((b1 − (b1 )B )f )(x).
We have
(
(
1
|B|1+λq
)1/q
∫
|Sδb1 (f )(x)|q dx
B
)1/q
1
q
≤
|(b1 (x) − (b1 )B )(Sδ (f χB ))(x)| dx
|B|1+λq B
(
)1/q
∫
1
q
+
|(b1 (x) − (b1 )B )(Sδ (f χ(B)c ))(x)| dx
|B|1+λq B
)1/q
(
∫
1
q
|Sδ ((b1 − (b1 )B )f χB )(x)| dx
+
|B|1+λq B
(
)1/q
∫
1
q
+
|Sδ ((b1 − (b1 )B )f χ(B)c )(x)| dx
|B|1+λq B
= J1 + J2 + J3 + J4 .
∫
For J1 , taking 1 < p1 < n/δ and t such that 1/t = 1/p1 − δ/n, choosing
1/q = 1/p2 + 1/t, by Hölder’s inequality and the boundedness of Sδ from
Lp1 (Rn ) to Lt (Rn ), we know
(∫
)1/p2 (∫
)1/t
−1/q−λ
p2
t
J1 ≤ |B|
|b1 (x) − (b1 )B | dx
|Sδ (f χB )(x)|
B
−1/q−λ
≤ C|B|
−1/q−λ
(∫
|B|
1/p2 +λ2
||b1 ||CBM Op2 ,λ2
≤ C|B|
|B|
||b1 ||CBM Op2 ,λ2 |B|
≤ C||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 .
1/p2 +λ2
B
)1/p1
|f (x)| dx
p1
B
1/p1 +λ1
||f ||Ḃ p1 ,λ1
For J2 , using the fact |Sδ (f χB c )(x)| ≤ C||f ||Ḃ p,λ1 |B|δ+λ1 from the proof of
Theorem 1 and by Hölder’ inequality, we get
(∫
)1/p2
−1/q−λ
δ/n+λ1
p2
J2 ≤ C|B|
|B|
||f ||Ḃ p,λ1
|b1 (x) − (b1 )B | dx
|B|1/q−1/p2
B
Volume 15, 2011
41
≤ C|B|−1/q−λ |B|δ/n+λ1 ||f ||Ḃ p,λ1 |B|1/p2 +λ2 ||b1 ||CBM Op2 ,λ2 |B|1/q−1/p2
≤ C||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 .
For J3 , taking 1 < l < n/δ and q such that 1/q = 1/l − δ/n, choosing 1/l =
1/p1 + 1/p2 , by the boundedness of Sδ from Ll to Lq and Hölder’s inequality,
we have
(∫
)1/l
−1/q−λ
l
J3 ≤ C|B|
|(b1 (x) − (b1 )B f (x)| dx
B
−1/q−λ
(∫
≤ C|B|
)1/p2 (∫
)1/p1
p1
|b1 (x) − (b1 )B | dx
|f (x)| dx
p2
B
1/p2 +λ2
−1/q−λ
B
≤ C|B|
|B|
||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 |B|1/p1 +λ1
≤ C||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 .
For J4 , note that x ∈ B, using Hölder’s inequality, Lemma 3, noticing that
λ2 > 0 and λ1 < −λ2 − δ/n, we have
|Sδ ((b1 − (b1 )B )f χ(B)c )(x)|
)1/p2
(∫
∞
∑
p2
k
δ/n−1
|b1 (z) − (b1 )B | dz
≤ C
|2 B|
2k+1 B
k=0
)1/p1
(∫
|f (z)|p1 dz
×
2k+1 B
≤ C||f ||Ḃ p1 ,λ1
[( ∫
×
∞
∑
|2k+1 B|1−1/p1 −1/p2
|2k B|δ/n−1 |2k+1 B|1/p1 +λ1 |2k+1 B|1−1/p1 −1/p2
k=0
|b1 (z) − (b1 )2k+1 B |p2 dz
2k+1 B
≤ C||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1
∞
∑
)1/p2
]
k2kn(λ1 +λ2 +δ/n) |B|λ1 +λ2 +δ/n
k=0
λ1 +λ2 +δ/n
≤ C||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 |B|
+ |(b1 )2k+1 B − (b1 )B ||2k+1 B|1/p2
,
thus, we get
J4 ≤ C||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 |B|λ1 +λ2 +δ/n |B|−1/q−λ |B|1/q
≤ C||b1 ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 .
This completes the proof of the case m = 1.
∫
When m > 1, set ⃗bB = ((b1 )B , · · · , (bm )B ), where (bj )B = |B|−1 B |bj (x)|dx,
1 ≤ j ≤ m, we have
⃗
Ftb (f )(x, y)
=
m ∑
∑
j=0 σ∈Cjm
(−1)m−j (b(x) − (b)B )σ
42
Bulletin of TICMI
∫
×
(b(z) − (b)B )σc ψt (y − z)f (z)dz
Rn
m
∏
m
∏
(bj (x) − (bj )B )Ft (f )(y) + (−1) Ft ( (bj − (bj )B ))f )(y)
=
m
j=1
+
j=1
∫
m−1
∑
∑
j=1
σ∈Cjm
(−1)m−j (b(x) − (b)B )σ
(b(z) − b(x))σc ψt (y − z)f (z)dz
Rn
m
m
∏
∏
m
=
(bj (x) − (bj )B )Ft (f )(y) + (−1) Ft ( (bj − (bj )B ))f )(y)
j=1
j=1
m−1
∑
+
∑
⃗
(b(x) − (b)B )σ Ftbσc (f )(x, y),
j=1 σ∈Cjm
thus,
⃗
⃗
Sδb (f )(x) = ||χΓ(x) Ftb (f )(x)||
m
∏
≤ ||χΓ(x) (bj (x) − (bj )B )Ft (f )(x)||
j=1
+
m−1
∑
∑
⃗
||χΓ(x) (b(x) − (b)B )σ Ftbσc (f )(x)||
j=1 σ∈Cjm
m
∏
+||χΓ(x) Ft ( (bj − (bj )B )f )(x)||
j=1
≤
m
∏
m
∏
(bj (x) − (bj )B )Sδ (f )(x) + (−1) Sδ ( (bj − (bj )B ))B )f )(x)
m
j=1
+
m−1
∑
j=1
∑
(−1)m−j (b(x) − bB )σ Sδ ((b − bB )σc f )(x).
j=1 σ∈Cjm
We consider
(
(
≤
1
|B|1+λq
)1/q
∫
⃗
|Sδb (f )(x)|q dx
B
∫
m
∏
)1/q
1
| (bj (x) − (bj )B )(Sδ (f χB ))(x)|q dx
1+λq
|B|
B j=1
(
)1/q
∫ ∏
m
1
q
+
| (bj (x) − (bj )B )(Sδ (f χB c ))(x)| dx
|B|1+λq B j=1
(
)1/q
∫
m
∏
1
+
|Sδ ( (bj − (bj )B )f χB )(x)|q dx
|B|1+λq B
j=1
Volume 15, 2011
(
+
1
|B|1+λq

+
1
|B|1+λq

+
1
|B|1+λq
∫
43
m
∏
|Sδ ( (bj − (bj )B )f χB c )(x)|q dx
B
)1/q
j=1
1/q
∫ m−1
∑ ∑
q
(b(x) − bB )σ Sδ ((b − bB )σc f χB )(x) dx
B
j=1 σ∈Cjm
1/q
∫ m−1
q
∑
∑
(b(x) − bB )σ Sδ ((b − bB )σc f χB c )(x) dx
B
j=1 σ∈Cjm
= ν1 + ν2 + ν3 + ν4 + ν5 + ν6 .
For ν1 , taking 1 < p1 < n/δ and t such that 1/t = 1/p1 − δ/n, choosing
1/q = 1/p2 + · · · + 1/pm+1 + 1/t, by Hölder’s inequality and the boundedness
of Sδ from Lp1 (Rn ) to Lt (Rn ), we have
−1/q−λ
ν1 ≤ |B|
m (∫
∏
j=1
m
∏
−1/q−λ
≤ C|B|
≤ C|B|−1/q−λ
j=1
m
∏
|bj (x) − (bj )B |
pj+1
)1/t
)1/pj+1 (∫
t
|(Sδ (f χB ))(x)| dx
dx
B
B
(∫
|B|
1/pj+1 +λj+1
||bj ||CBM Opj+1 ,λj+1
)1/p1
|f (x)| dx
p1
B
|B|1/pj+1 +λj+1 ||bj ||CBM Opj+1 ,λj+1 |B|1/p1 +λ1 ||f ||Ḃ p1 ,λ1
j=1
≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 .
For ν2 , using the fact |Sδ (f χ(B)c )(x)| ≤ C||f ||Ḃ p,λ1 |B|δ/n+λ1 from the proof of
Theorem 1 and the Hölder’s inequality, we get
)1/pi+1
m (∫
∏
−1/q−λ
pi+1
δ/n+λ1
ν2 ≤ C|B|
||f ||Ḃ p1 ,λ1 |B|
|bi (x) − (bi )B | dx
i=1
× |B|1/q−1/p2 −···−1/pm+1
−1/q−λ
≤ C|B|
||f ||Ḃ p1 ,λ1 |B|
δ/n+λ1
m
∏
B
|B|1/pi+1 +λi+1 ||bi ||CBM Opi+1 ,λi+1
i=1
1/q−1/p2 −···−1/pm+1
× |B|
≤ C||⃗b||
CBM O p⃗,⃗λ
||f ||Ḃ p1 ,λ1 .
For ν3 , taking 1 < p1 < n/δ and q such that 1/q = 1/l − δ/n, choosing
1/l = 1/p1 + · · · + 1/pm+1 , by the boundedness of Sδ from Ll to Lq and
Hölder’s inequality, we have
(∫
)1/l
m
∏
ν3 ≤ C|B|−1/q−λ
| (bj (x) − (bj )B )f (x)|l dx
B
j=1
44
Bulletin of TICMI
−1/q−λ
≤ C|B|
≤ C|B|−1/q−λ
m (∫
∏
j=1
m
∏
|bj (x) − (bj )B |
pj+1
)1/pj+1 (∫
)1/p1
p1
dx
|f (x)| dx
B
B
|B|1/pj+1 +λj+1 ||bi ||CBM Opj+1 ,λj+1 |B|1/p1 +λ1 ||f ||Ḃ p1 ,λ1
j=1
≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 .
For ν4 , note that x ∈ B, by Hölder’s inequality, Lemma 3 and noticing that
λj > 0(2 ≤ j ≤ m + 1), λ1 < −λ2 − · · · − λm+1 < δ/n, we have
m
∏
|Sδ ( (bj − (bj )B )f χ(B)c )(x)|
≤
j=1
∫
∞
∑
|
2k+1 B\2k B
≤
k=0
∞
∑
C
m
∏
j=1
|b1 (z) − (b1 )B | dz
|2 B|
p2
δ/n−1
2k+1 B
(∫
|bm (z) − (bm )B |
···
pm+1
(∫
2k+1 B
)1/p1
|f (z)| dz
×
p1
2k+1 B
×
≤
)1/p2
(∫
k
k=0
≤
(bj (z) − (bj )B )||x − z|−n+δ |f (z)|dz
C||f ||Ḃ p1 ,λ1
k+1
∞
∑
k=0
1/p2 +λ2
k|2 B|
C||⃗b||
CBM O p⃗,⃗λ
)1/pm+1
dz
|2k+1 B|1−1/p1 −1/p2 ···−1/pm+1
|2k B|δ/n−1 |2k+1 B|1/p1 +λ1 |2k+1 B|1−1/p1 −1/p2 −···1/pm+1
||b1 ||CBM Op2 ,λ2 · · · k|2k+1 B|1/pm+1 +λm+1 ||bm ||CBM Opm+1 ,λm+1
||f ||Ḃ p1 ,λ1 |B|λ1 +λ2 +···+λm+1 +δ/n ,
thus, we obtain
ν4 ≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 |B|λ1 +λ2 +···+λm+1 +δ/n |B|−1/q−λ |B|1/q
≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 .
For ν5 , taking 1 < s < n/δ and t such that 1/t = 1/s − δ/n, choosing 1/q =
1/p3 + 1/t, 1/s = 1/p1 + 1/p2 , by the boundedness of Sδ from Lt to Ls and
Hölder’s inequality, we have
−1/q−λ
ν5 ≤ C|B|
(∫
×
m−1
∑
∑ (∫
j=1 σ∈Cjm
)1/p3
|(b(x) − bB )σ | dx
p3
B
)1/t
|Sδ ((b − bB )σc f χB )(x)| dx
t
B
Volume 15, 2011
−1/q−λ
≤ C|B|
(∫
×
≤ C
m−1
∑
∑ (∫
j=1 σ∈Cjm
45
)1/p3
|(b(x) − bB )σ | dx
p3
B
)1/p2 (∫
)1/p1
p1
|(b − bB )σc | dx
|f (x)| dx
p2
B
m−1
∑ ∑
B
|B|−1/q−λ |B|1/p3 +λ3 ||⃗bσ ||CBM Op3 ,λ3 |B|1/p2 +λ2
j=1 σ∈Cjm
×||⃗bσc ||CBM Op2 ,λ2 |B|1/p1 +λ1 ||f ||Ḃ p1 ,λ1 ≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 .
For ν6 , note that x ∈ B, by Hölder’s inequality , Lemma 3 and noticing that
λj > 0(2 ≤ j ≤ m + 1), λ1 < −λ2 − δ/n, we have
|Sδ ((b − bB )σc f χB c )(x)|
(∫
)1/p2
∞
∑
k
δ/n−1
p2
≤ C
|2 B|
|(b(z) − bB )σc | dz
2k+1 B
k=0
)1/p1
(∫
|f (z)| dz
×
p1
2k+1 B
≤ C||⃗bσc ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1
|2k+1 B|1−1/p1 −1/p2
∞
∑
|2k B|δ/n−1 |2k+1 B|1/p1 +λ1 k m |2k+1 B|1/p2 +λ2
k=1
×|2
k+1
B|
1−1/p1 −1/p2
≤ C||⃗bσc ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 |B|λ1 +λ2 +δ/n ,
thus, we get
ν6 ≤ C
m−1
∑
∑
|B|−1/q−λ ||⃗bσc ||CBM Op2 ,λ2 ||f ||Ḃ p1 ,λ1 |B|λ1 +λ2 +δ/n
j=1 σ∈Cjm
(∫
×
)1/p3
|(b(x) − bB )σ | dx
|B|1/q−1/p3
p3
B
≤ C||⃗b||CBM Op⃗,⃗λ ||f ||Ḃ p1 ,λ1 .
This completes the total proof of the Theorem 3.
Proof of Theorem 4. Set
∫
m
∏
f⃗b
−n+δ
Sδ (f )(x) =
|x − z|
| (bj (x) − bj (z))f (z)|dz,
Rn
j=1
f⃗
it is easy to know the Theorem 3 is also true for the commutator Sδb (f )(x).
Since
⃗
f⃗
Mδb (f )(x) ≤ Sδb (f )(x),
46
Bulletin of TICMI
thus, Theorem 4 can be easily deduced. We omit the details here.
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Received February, 2, 2011; Accepted December, 23, 2011.