Volume 10, 2006
1
MAIN ARTICLES
SHARP FUNCTION INEQUALITY FOR THE MULTILINEAR
COMMUTATOR OF THE LITTLEWOOD-PALEY OPERATOR
Changhong Wu, Lanzhe Liu
Department of Mathematics
Changsha University of Science and Technology
Changsha, 410076
P.R.of China
E-mail: lanzheliu@163.com
Abstract: In this paper, we obtain the sharp inequality for the multilinear
commutator related to the Littlewood-Paley operator. Using the sharp inequality, the weighted Lp -norm inequality for the multilinear commutator for
1 < p < ∞ is proved.
Key words: Multilinear commutator; Littlewood-Paley operator; Sharp inequality. MSC 1991: 42B20, 42B25.
1. Introduction
The commutators of singular integral operators have been well studied (see
[1-4]). Let T be the Calderón-Zygmund singular integral operator. A classical result of Coifman, Rocherberg and Weiss (see [3]) states that commutator
[b, T ](f ) = T (bf ) − bT (f ) (where b ∈ BM O(Rn ) (see below)) is bounded
on Lp (Rn )) for 1 < p < ∞. In [8-10], the sharp estimates for some multilinear commutators of the Calderón-Zygmund singular integral operators are
obtained. The Littlewood-Paley operator is an important operator in the harmonic analysis (see [12]). The main purpose of this paper is to prove the sharp
inequality for the multilinear commutator related to the Littlewood-Paley operator. By using the sharp inequality, we obtain the weighted Lp -norm inequality for the multilinear commutator for 1 < p < ∞.
2. Preliminaries and Theorems
First let us introduce some notations (see [4], [9], [11]). In this paper, Q
will denote a cube of Rn with sides parallel Rto the axes. For a cube Q and a
locally integrable function b, let bQ = |Q|−1 b(x)dx, the sharp function of b
Q
is defined by
1
b (x) = sup
Q(x∈Q) |Q|
Z
#
|b(y) − bQ |dy.
Q
It is well-known that (see [4])
1
b (x) = sup inf
Q(x∈Q) c∈C |Q|
Z
#
|b(y) − c|dy.
Q
2
Bulletin of TICMI
We say that b belongs to BM O(Rn ) if b# belongs to L∞ (Rn ) and define
kbkBM O := kb# kL∞ . It is known as well (see [11]) that
kb − b2k Q kBM O ≤ CkkbkBM O .
For bj ∈ BM O(Rn ) (j = 1, · · · , m), set b̃ = (b1 , · · · , bm ) and
kb̃kBM O :=
m
Y
kbj kBM O .
j=1
For a 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 b̃ = (b1 , · · · , bm ) and
σ := {σ(1), · · · , σ(j)} ∈ Cjm , set b̃σ = (bσ(1) , · · · , bσ(j) ), bσ := bσ(1) · · · bσ(j) and
kb̃σ kBM O := kbσ(1) kBM O · · · kbσ(j) kBM O .
Let M be the Hardy-Littlewood maximal operator, i.e.,
Z
1
|f (y)|dy;
M (f )(x) := sup
Q(x∈Q) |Q|
Q
1
further Mp (f ) := (M (|f |p )) p for 1 < p < ∞.
We denote the Muckenhoupt weights by A1 (see [4]), i.e.,
A1 := {w : M (w)(x) ≤ Cw(x), a.e.}.
Throughout this paper, we will study some multilinear commutators.
Definition. Let bj (j = 1, · · · , m) be the fixed locally integrable functions on Rn . Let further ε > 0 and ψ be a fixed function which satisfies the
following Rproperties:
(1)
ψ(x)dx = 0,
Rn
(2) |ψ(x)| ≤ C(1 + |x|)−(n+1) ,
(3) |ψ(x + y) − ψ(x)| ≤ C|y|ε (1 + |x|)−(n+1+ε) when 2|y| < |x|;
n+1
: |x − y| ≤ t} and the characteristic function
Denote Γ(x) := {(y, t) ∈ R+
of Γ(x) by χΓ(x) . The Littlewood-Paley multilinear commutator is defined by

 21
Z Z
dydt 

|Ftb̃ (f )(x, y)|2 n+1  ,
Sψb̃ (f )(x) := 
t
Γ(x)
where
Ftb̃ (f )(x) =
Z "Y
m
Rn
j=1
#
(bj (x) − bj (z)) ψt (y − z)f (z)dz
Volume 10, 2006
and ψt (x) = t−n ψ
³x´
t
for t > 0. Set Ft (f )(x) :=
3
R
ψt (x − y)f (y)dy, we also
Rn
introduce


Sψ (f )(x) := 
 12
Z Z
|Ft (f )(x)|2
dydt 
 ,
tn+1
Γ(x)
which is the Littlewood-Paley
S operator(see [12]).

 12




Z Z


dydt


2
|h(y, t)| n+1  < ∞ , then
Let H be the space H = h : khk = 


t




Rn+1
+
n
Ftb̃ (f )(x, y)
for each fixed x ∈ R ,
to H, and it is clear that
may be viewed as a mapping from [0, +∞)
°
°
Sψ (f )(x) = °χΓ(x) Ft (f )(y)°
and
°
°
°
°
Sψb̃ (f )(x) = °χΓ(x) Ftb̃ (f )(x, y)° .
Note that when b1 = · · · = bm , Sψb̃ is just the m order commutator(see[1],
[6], [7]). It is well known that commutators are of great interest in harmonic
analysis and have been widely studied by many authors (see [1-3], [5-10]). Our
main purpose is to establish the sharp inequality for the multilinear commutator.
Now we state our theorems as follows.
Theorem 1. Let bj ∈ BM O(Rn ) for j = 1, · · · , m. Then for any 1 <
r < ∞, there exists a constant C > 0 such that for any f ∈ C0∞ (Rn ) and any
x ∈ Rn ,


m
³
´
X X
(Sψb̃ (f ))# (x) ≤ Ckb̃kBM O Mr (f )(x) +
Mr Sψb̃σc (f ) (x) .
j=1 σ∈Cjm
Theorem 2. Let bj ∈ BM O(Rn ) for j = 1, · · · , m. Then Sψb̃ is bounded
on Lp (w) for w ∈ A1 and 1 < p < ∞.
3. Proof of Theorems
To prove the theorems, we need the following lemmas.
Lemma 1. (see [12]) Let w ∈ Ap and 1 < p < ∞. Then Sψ is bounded on
Lp (w).
Lemma 2. Let 1 < r < ∞, bj ∈ BM O(Rn ) for j = 1, · · · , k and k ∈ N .
Then, we have
Z Y
k
k
Y
1
|bj (y) − (bj )Q |dy ≤ C
kbj kBM O
|Q| j=1
j=1
Q
4
Bulletin of TICMI
and

 r1
Z Y
k
k
Y
r
 1

|bj (y) − (bj )Q | dy
≤C
kbj kBM O .
|Q| j=1
j=1
Q
Proof. Choose 1 < pj < ∞, j = 1, · · · , m, such that
1
1
+ ··· +
= 1,
p1
pm
by virtue of Hölder’s inequality, we obtain

 p1
j
Z Y
Z
k
k
Y
1
1
pj


|bj (y) − (bj )Q |dy ≤
|bj (y) − (bj )Q | dy
|Q| j=1
|Q|
j=1
Q
Q
≤ C
k
Y
kbj kBM O
j=1
and


 r1
 p1 r
j
Z Y
Z
k
k
Y
1
1
r
p
r
j


|bj (y) − (bj )Q | dy  ≤
|bj (y) − (bj )Q | dy 
|Q| j=1
|Q|
j=1
Q
Q
≤ C
k
Y
||bj ||BM O .
j=1
Proof of Theorem 1. It suffices to prove that for f ∈ C0∞ (Rn ) and some
constant C0 , the following inequality:


Z
m X
³
´
X
1
|Sψb̃ (f )(x)−C0 |dx ≤ C||b||BM O Mr (f )(x) +
Mr Sψb̃σc (f )(x)  .
|Q|
j=1 σ∈C m
Q
j
Fix a cube Q = Q(x0 , d) and x̃ ∈ Q. Set f1 = f χ2Q and f2 = f χ2Qc .
We first consider the case m = 1. Evidently,
Ftb1 (f )(x, y) = (b1 (x) − (b1 )2Q )Ft (f )(y) − Ft ((b1 − (b1 )2Q )f1 )(y)
−Ft ((b1 − (b1 )2Q )f2 )(y).
Then,
¯ b
¯
¯S 1 (f )(x) − Sψ (((b1 )2Q − b1 ) f2 ) (x0 )¯
ψ
¯°
° °
°¯
= ¯°χΓ(x) Ftb1 (f )(x, y)° − °χΓ(x0 ) Ft (((b1 )2Q − b1 ) f2 ) (y)°¯
°
°
≤ °χΓ(x) Ftb1 (f )(x, y) − χΓ(x0 ) Ft (((b1 )2Q − b1 ) f2 ) (y)°
°
° °
°
≤ °χΓ(x) (b1 (x) − (b1 )2Q )Ft (f )(y)° + °χΓ(x) Ft ((b1 − (b1 )2Q )f1 )(y)°
°
°
+ °χΓ(x) Ft ((b1 − (b1 )2Q )f2 ) (y) − χΓ(x0 ) Ft ((b1 − (b1 )2Q )f2 )(y)°
= A(x) + B(x) + C(x).
Volume 10, 2006
5
For A(x), in view of Hölder’s inequality with the exponent 1/r + 1/r0 = 1, we
get
Z
1
A(x)dx
|Q|
Q
Z
1
=
|b1 (x) − (b1 )2Q ||Sψ (f )(x)|dx
|Q|
Q

≤ C
Z
1
|2Q|
 10 
r
1
0
|b1 (x) − (b1 )2Q |r dx 
|Q|
2Q
 1r
Z
|Sψ (f )(x)|r dx
Q
≤ Ckb1 kBM O Mr (Sψ (f ))(x̃).
For B(x), choose 1 < p < r, by the boundedness of Sψ on Lp (Rn ) and because
of Hölder’s inequality, we obtain
Z
1
B(x)dx
|Q|
Q

1
≤ 
|Q|

≤ C

≤ C
 p1
Z
|Sψ ((b1 − (b1 )2Q )f1 )(x)|p dx
Rn
1
|Q|
 p1
Z
|(b1 (x) − (b1 )2Q )f (x)|p dx
2Q
1
|2Q|
Z
 r1 
1
|f (x)|r dx 
|2Q|
2Q
 r−p
rp
Z
|b1 (x) − (b1 )2Q |rp/(r−p) dx
2Q
≤ Ckb1 kBM O Mr (f )(x̃).
For C(x), by virtue of Minkowski’s inequality, we obtain
C(x) ≤


Z
Z Z




n+1
R+
Z
≤C
(2Q)c
(2Q)c
2
 21
 dydt 
|χΓ(x) − χΓ(x0 ) ||b1 (z) − (b1 )2Q ||ψt (y − z)||f (z)| n+1 
t
¯
¯Z Z
¯
t1−n dydt
¯
|b1 (z) − (b1 )2Q ||f (z)| ¯
¯
(t + |y − z|)2n+2
¯|x−y|≤t
6
Bulletin of TICMI
¯1
¯2
¯
t1−n dydt
¯
−
¯ dz
2n+2
¯
(t + |y − z|)
¯
|x0 −y|≤t

Z

≤
|b1 (z) − (b1 )2Q ||f (z)| 
Z Z
(2Q)c
¯
¯
1
¯
¯ (t + |x + y − z|)2n+2
Z Z
|y|≤t,|x+y−z|≤t
¯
¶1
¯ dydt 2
1
¯
−
dz
(t + |x0 + y − z|)2n+2 ¯ tn−1
Z
≤
|b1 (z) − (b1 )2Q ||f (z)|
(2Q)c


×
 12
Z Z
|x − x0 |t1−n

dydt dz.
2n+3
(t + |x + y − z|)
|y|≤t,|x+y−z|≤t
Note that 2t + |x + y − z| ≥ 2t + |x − z| − |y| ≥ t + |x − z| when |y| ≤ t and
Z∞
tdt
= C|x − z|−2n−1 .
(t + |x − z|)2n+3
0
Then, for x ∈ Q,
Z
C(x) ≤ C
|b1 (z) − (b1 )2Q ||f (z)|
(2Q)c


×
Z Z
Z
 12
22n+3 |x0 − x|t1−n dydt 
 dz
(2t + 2|x + y − z|)2n+3
|y|≤t
1
≤ C
|b1 (z) − (b1 )2Q ||f (z)||x − x0 | 2
(2Q)c


×
Z Z
Z
 12
t1−n dydt

 dz
(2t + |x + y − z|)2n+3
|y|≤t
1
≤ C
|b1 (z) − (b1 )2Q ||f (z)||x − x0 | 2
(2Q)c


×
Z Z
|y|≤t
 12
t1−n dydt

 dz
(t + |x − z|)2n+3
Volume 10, 2006
7
Z
1
≤ C
|b1 (z) − (b1 )2Q ||f (z)||x − x0 | 2
(2Q)c

Z∞
×
Z
tdt
 dz
(t + |x − z|)2n+3
0
1
|b1 (z) − (b1 )2Q ||f (z)|
≤ C
≤ C
 12
(2Q)c
∞
X
Z
|x0 − x| 2
dz
1
|x0 − z|n+ 2
|x0 − x| 2 |x0 − z|−(n+ 2 ) |b1 (z) − (b1 )2Q ||f (z)|dz
1
1
k=1
≤ C
2k+1 Q\2k Q
∞
X
−k/2 k+1
2
|2
Z
Q|
−1
k=1
2k+1 Q

≤ C
∞
X

2−k/2 
k=1
1
Z
|2k+1 Q|
 r1

|f (z)|r dz 
2k+1 Q


×
|b1 (z) − (b1 )2Q ||f (z)|dz
Z
1
 10
r

|b1 (z) − (b1 )2Q | dz 
r0
|2k+1 Q|
2k+1 Q
≤ C
∞
X
2−k/2 kkb1 kBM O Mr (f )(x̃)
k=1
≤ Ckb1 kBM O Mr (f )(x̃).
Thus,
1
|Q|
Z
C(x)dx ≤ C||b1 ||BM O Mr (f )(x̃).
Q
Now, we consider the case m ≥ 2. For b = (b1 , · · · , bm ) we have
#
Z "Y
m
Ftb̃ (f )(x, y) =
(bj (x) − bj (z)) ψt (y − z)f (z)dz
=
Z Y
m
Rn
j=1
[(bj (x) − (bj )2Q ) − (bj (z) − (bj )2Q )] ψt (y − z)f (z)dz
j=1
=
Rn
m
X
X
j=0
σ∈Cjm
Z
m−j
(−1)
(b(x) − (b)2Q )σ
(b(z) − (b)2Q )σc ψt (y − z)f (z)dz
Rn
= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (f )(y)
+(−1)m Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(y)
8
Bulletin of TICMI
+
m−1
X
Z
X
m−j
(−1)
(b(x) − (b)2Q )σ
j=1 σ∈Cjm
(b(z) − b(x))σc ψt (y − z)f (z)dz
Rn
= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (f )(y)
+(−1)m Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(y)
m−1
X X
+
cm,j (b(x) − (b)2Q )σ Ftb̃σc (f )(x, y).
j=1 σ∈Cjm
Thus,
¯
¯
¯
¯ b̃
S
(f
)(x)
−
S
(((b
)
−
b
)
·
·
·
((b
)
−
b
))
f
)
(x
)
¯ ψ
ψ
1 2Q
1
m 2Q
m
2
0 ¯
°
°
°
°
b̃
≤ °χΓ(x) Ft (f )(x, y) − χΓ(x0 ) Ft (((b1 )2Q − b1 ) · · · ((bm )2Q − bm ) f2 ) (y)°
°
°
≤ °χΓ(x) (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (f )(y)°
m−1
°
X X °
°
°
b̃σc
+
°χΓ(x) (b̃(x) − (b)2Q )σ Ft (f )(x, y)°
j=1 σ∈Cjm
°
°
+ °χΓ(x) Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f1 )(y)°
°
Ãm
!
Ãm
! °
°
°
Y
Y
°
°
+ °χΓ(x) Ft
(bj − (bj )2Q )f2 (y) − χΓ(x0 ) Ft
(bj − (bj )2Q )f2 (y)°
°
°
j=1
j=1
= I1 (x) + I2 (x) + I3 (x) + I4 (x).
For I1 (x), according to Hölder’s inequality with the exponent 1/p1 + · · · +
1/pm + 1/r = 1, where 1 < pj < ∞, j = 1, · · · , m, we get
Z
1
I1 (x)dx
|Q|
Q
Z
1
=
|b1 (x) − (b1 )2Q | · · · |bm (x) − (bm )2Q ||Sψ (f )(x)|dx
|Q|
Q

1
≤ 
|Q|
µ
×
 p1
Z
1
|b1 (x) − (b1 )2Q |
Q
1
|Q|
p1 
Z
|Sψ (f )(x)|r dx
¶ 1r

1
···
|Q|
 p1
Z
m
|bm (x) − (bm )2Q | dx
pm
Q
Q
≤ Ckb̃kBM O Mr (Sψ (f ))(x̃).
For I2 (x), in view of Minkowski’s inequality and Lemma 2, we get
Z
1
I2 (x)dx
|Q|
Q
Volume 10, 2006
≤
m−1
X
j=1
≤ C
9
X 1 Z
|(b(x) − (b)2Q )σ ||Sψb̃σc (f )(x)|dx
|Q|
σ∈C m
Q
j
m−1
X
j=1

X
 1
|2Q|
σ∈C m
j

×
1
|Q|
 10
Z
r
r0
|(b(x) − (b)2Q )σ | dx
2Q
 1r
Z
|Sψb̃σc (f )(x)|r dx
Q
≤ C
m−1
X
X
||b̃σ ||BM O Mr (Sψb̃σc (f ))(x̃).
j=1 σ∈Cjm
For I3 (x), choose 1 < p < r, 1 < pj < ∞, j = 1, · · · , m such that
1
+ ··· +
p1
1
p
+ = 1, by the boundedness of Sψ on Lp (Rn ) and Hölder’s inequality, we
pm r
get
Z
1
I3 (x)dx
|Q|

Q
1
≤ 
|Q|

≤ C
|Sψ ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f1 )(x)|p dx
Rn
1
|Q|

≤ C
 p1
Z
 p1
Z
|(b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )|p |f1 (x)|p dx
Rn
1
|2Q|
 1r
Z
|f (x)|r dx
2Q

1
×
|2Q|

 pp1
Z
1
|b1 (x) − (b1 )2Q | dx
p1
2Q
1
×···
|2Q|
 pp1
Z
m
|bm (x) − (bm )2Q |
2Q
≤ Ckb̃kBM O Mr (f1 )(x̃).
ppm
dx
10
Bulletin of TICMI
For I4 (x), similar to the proof in the case m = 1, we obtain
¯m
¯
Z
¯Y
¯
1
1 ¯
¯
−(n+ 2 )
2
I4 (x) ≤ C
|x0 − x| |x0 − z|
¯ (bj (z) − (bj )2Q )¯ |f (z)|dz,
¯
¯
j=1
(2Q)c
taking 1 < pj < ∞ j = 1, · · · , m such that 1/p1 + · · · + 1/pm + 1/r = 1, then,
for x ∈ Q
Z
∞
X
1
1
I4 (x) ≤ C
|x0 − x| 2 |x0 − z|−(n+ 2 )
k=1
2k+1 Q\2k Q
¯
¯
m
¯
¯Y
¯
¯
× ¯ (bj (z) − (bj )2Q )¯ |f (z)|dz
¯
¯
j=1
¯
Z ¯¯Y
m
∞
¯
X
¯
¯
≤ C
2−k/2 |2k+1 Q|−1
¯ (bj (z) − (bj )2Q )¯ |f (z)|dz
¯
¯
k=1
2k+1 Q

≤ C
∞
X

2−k/2 
k=1
1
|2k+1 Q|
×


j=1
≤ C
∞
X
 r1

|f (z)|r dz 
2k+1 Q

m
Y
Z
j=1
Z
1
|2k+1 Q|
 p1
j

|bj (z) − (bj )2Q |pj dz 
2k+1 Q
m −km
k 2
m
Y
kbj kBM O Mr (f )(x̃)
j=1
k=1
≤ Ckb̃kBM O Mr (f )(x̃).
Thus
1
|Q|
Z
I4 (x)dx ≤ Ckb̃kBM O Mr (f )(x̃).
Q
This completes the proof of the theorem.
Proof of Theorem 2. Choose 1 < r < p in Theorem 1 and using Lemma
1, we may get the conclusion of Theorem 2. This finishes the proof.
Acknowledgement. The authors would like to express their gratitude to
the referee for his comments and suggestions.
References
[1] Alvarez, J., Babgy, R. J., Kurtz, D. S., and Pérez, C., Weighted estimates for commutators of linear operators. Studia Math., 104 (1993), 195-209.
Volume 10, 2006
11
[2] Coifman, R., and Meyer, Y., Wavelets, Calderón-Zygmund and Multilinear operators.
Cambridge Studies in Advanced Math., Cambridge University Press, Cambridge, 48 (1997).
[3] Coifman, R., Rochberg, R., and Weiss, G., Factorization theorems for Hardy spaces
in several variables. Ann. of Math., 103 (1976), 611-635.
[4] Garcia-Cuerva, J. and Rubio de Francia, J. L., Weighted Norm Inequalities and
Related Topics. North-Holland Math., 16, Amsterdam, 1985.
[5] Hu, G. and Yang, D. C., A variant sharp estimate for multilinear singular integral
operators, Studia Math., 141 (2000), 25-42.
[6] Liu, L. Z., Weighted weak type estimates for commutators of Littlewood-Paley operator. Japanese J. of Math., 29 , 1(2003), 1-13.
[7] Liu, L. Z., The continuity of commutators on Triebel-Lizorkin spaces. Integral Equations and Operator Theory, 49, 1 (2004), 65-76.
[8] Pérez, C., Endpoint estimate for commutators of singular integral operators. J. Func.
Anal., 128 (1995), 163-185.
[9] Pérez, C., and Pradolini, G., Sharp weighted endpoint estimates for commutators
of singular integral operators. Michigan Math. J., 49 (2001), 23-37. [10] Pérez, C. and
Trujillo-Gonzalez, R., Sharp weighted estimates for multilinear commutators. J. London
Math. Soc., 65 (2002), 672-692.
[11] Stein, E. M., Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press, Princeton NJ, 1993.
[12] Torchinsky, A., The Real Variable Methods in Harmonic Analysis. Pure and Applied
Math., 123, Academic Press, New York, 1986.
Received March, 14, 2006; revised June, 21, 2006; accepted July, 26, 2006.