Paper-12-Acta24.2010. 131KB Jun 04 2011 12:03:10 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 24/2010
pp. 119-129

SHARP WEIGHTED INEQUALITIES FOR VECTOR-VALUED
MULTILINEAR COMMUTATORS OF MARCINKIEWICZ
OPERATOR

FENG Qiufen
Abstract. In this paper, we prove the sharp inequality for the vector-valued
multilinear commutators related to the Marcinkiewicz operator. By using the sharp
inequality, we obtain the weighted Lp -norm inequality for the vector-valued multilinear commutators.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Introduction
Let T be the Calder´on-Zygmund singular integral operator, we know that the
commutator [b, T ](f ) = T (bf )−bT (f ) (where b ∈ BM O(Rn )) is bounded on Lp (Rn )
for 1 < p < ∞ (see [3]). In [9], the sharp estimates for some multilinear commutators
of the Calder´on-Zygmund singular integral operators are obtained. The main purpose of this paper is to prove a sharp inequality for some vector-valued multilinear
commutators related to the Marcinkiewicz operator. By using the sharp inequality, we obtain the weighted Lp -norm inequality for the vector-valued multilinear

commutators.
2. Notations and Results
First let us introduce some notations(see [4][9][10]). In this paper, Q will
R denote
n
−1
a cube of R with sides parallel to the axes, and for a cue Q let fQ = |Q|
Q f (z)dz
and the sharp function of f is defined by
Z
1
f # (x) = sup
|f (y) − fQ |dy.
Q∋x |Q| Q
It is well-known that(see [4])

119

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...

1
f (x) = sup inf
c∈C
|Q|
Q∋x
#

Z

|f (y) − c|dy.

Q

We say that b belongs to BM O(Rn ) if b# belongs to L∞ (Rn ) and define ||b||BM O =
−
→
||b# ||L∞ . If b = (b1 , · · · , bm ), bj ∈ BM O for (j = 1, · · · , m), then
||~b||BM O =

m

Y

||bj ||BM O .

j=1

Let M be the Hardy-Littlewood maximal operator, that is that
Z
−1
|f (y)|dy;
M (f )(x) = sup |Q|
x∈Q

Q

we write that Mp (f ) = (M (|f |p ))1/p for 0 < p < ∞.
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 ||BM O = ||bσ(1) ||BM O · · · ||bσ(j) ||BM O .

We denote the Muckenhoupt weights by Ap , let Ω ∈ Ap and 1 ≤ p < ∞, ω satisfy
the inverse H¨
older inequality, there exists a constant C and 1 < q < ∞, for any cube
Q, we get(see [10])


1
|Q|

1/q
Z
C
ω(x) dx
ω(x)dx.
≤
|Q| Q
Q

Z

q

In this paper, we will study some vector-valued multilinear commutators as following.
Definition.
Let 0 < γ ≤ 1 and Ω be homogeneous of degree zero on Rn such
R
that S n−1 Ω(x′ )dσ(x′ ) = 0. Assume that Ω ∈ Lipγ (S n−1 ), that is there exists a
constant M > 0 such that for any x, y ∈ S n−1 , |Ω(x) − Ω(y)| ≤ M |x − y|γ . Set
bj (j = 1, · · · , m) as a fixed locally integrable function of Rn , then when 1 < r < ∞,
The vector-valued Marcinkiewicz multilinear commutators is defined by
∞
X
~
~
|µbΩ (f )(x)| = ( (µbΩ (fi )(x))r )1/r ,
i=1

where
~
µbΩ (f )(x)

=

Z

0

∞

dt
~
|Ftb (f )(x)|2 3
t

120

1/2

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...

and
~

Ftb (f )(x) =
Set

Z

|x−y|≤t



Ω(x − y) 
|x − y|n−1

Ft (f )(x) =

m
Y

j=1

Z

|x−y|≤t



(bj (x) − bj (y)) f (y)dy.

Ω(x − y)
f (y)dy,
|x − y|n−1

we also define that
µΩ (f )(x) =

Z

∞

0

dt
|Ft (f )(x)|2 3
t

1/2

,

which is the Marcinkiewicz operator(see
[11]).
n
o

1/2
R∞
Let H be the space H = h : ||h|| = 0 |h(t)|2 dt/t3
< ∞ . Then, it is clear
that

˜
˜
µΩ (f )(x) = ||Ft (f )(x)|| and µbΩ (f )(x) = ||Ftb (f )(x)||.
~

Note that when b1 = · · · = bm , |µbΩ (f )(x)| is just the m order vector-valued
Marcinkiewicz operator multilinear commutators. It is well known that commutators
are of great interest in harmonic analysis and have been widely studied by many
authors (see [1][4-8][10]). Our main purpose is to establish the sharp inequality for
the vector-valued Marcinkiewicz operator multilinear commutators.
Now we state our main results as following.
3. Main Theorem and proof
First, we will establish the following theorem.
Theorem 1. Let 1 < r < ∞, bj ∈ BM O(Rn ) for j = 1, · · · , m. Then for any
1 < s < ∞, there exists a constant C > 0 such that for any f ∈ C0∞ (Rn ) and any
x
e ∈ Rn ,


m X

X
~ c
~
x) +
||~bσ ||BM O Ms (|µbΩσ (f )|r )(e
x) .
(|µbΩ (f )|r )# (e
x) ≤ C ||~b||BM O Ms (|f |r )(e
j=1 σ∈Cjm

~

Theorem 2. Let 1 < r < ∞, bj ∈ BM O(Rn ) for j = 1, · · · , m. Then |µbΩ |r is
bounded on Lp (Rn ) for 1 < p < ∞.
To proof the theorem,we need the following lemmas.
121

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...

Lemma 1. (see [11]) Let w ∈ Ap and 1 < r < ∞, 1 < p < ∞. When
Ω ∈ Lipr (S n−1 ) for 0 < γ ≤ 1, then |µΩ |r is bounded on Lp (w).
Lemma 2. Let 1 < r < ∞, bj ∈ BM O for j = 1, · · · , k and k ∈ N . Then, we
have
Z Y
k
k
Y
1
|bj (y) − (bj )Q |dy ≤ C
||bj ||BM O
|Q| Q
j=1

and



 1
|Q|

Z Y
k

Q j=1

j=1

1/r

|bj (y) − (bj )Q |r dy 

≤C

k
Y

||bj ||BM O .

j=1

Proof. For σ ∈ Ckm , where k ≤ m and m ∈ N, we have
Z
1
|(b(y) − (bj )Q )σ |dy ≤ C||bσ ||BM O
|Q| Q
and



1
|Q|

Z

r

|(b(y) − (bj )Q )σ | dy

Q

1/r

≤ C||bσ ||BM O .

We just need to choose pj > 1 and qj > 1 , where 1 ≤ j ≤ k, such that 1/p1 + · · · +
1/pk = 1 and 1/q1 + · · · + 1/qk = 1/r. After that,using the H¨
older’s inequality with
exponent 1/p1 + · · · + 1/pk = 1 and 1/q1 + · · · + 1/qk = 1/r respectively, we may
get the conclusions.
Lemma
0 < γ ≤ 1 and Ω be homogeneous of degree zero on Rn
R 3.(see ′[11]) Let
such that S n−1 Ω(x )dσ(x′ ) = 0. Assume that Ω ∈ Lipγ (S n−1 ), if Q = Q(x0 , d), y ∈
(2Q)c , then




 Ω(x − y)
Ω(x0 − y) 
|x − x0 |γ
|x − x0 |

 |x − y|n−1 − |x0 − y|n−1  ≤ C |x0 − y|n + |x0 − y|n−1+γ .
Proof of Theorem 1. It suffices to prove for f ∈ C0∞ (Rn ) and some constant C0 ,
the following inequality holds:




Z
m X
X
1
~
~
x) +
Ms (|µbΩ (f )|r )(e
x) .
||µb (f )(x)|r − C0 |dx ≤ C Ms (|f |r )(e
|Q| Q Ω
m
j=1 σ∈Cj

122

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...

Fix a cube Q = Q(x0 , d) and x
˜ ∈ Q. We write, f = g + h = gi +R hi , where
gi = fi χ2Q and hi = fi χ(2Q)c . If let ˜b = (b1 , ..., bm ), where (bj )Q = |Q|−1 Q bj (y)dy,
for 1 ≤ j ≤ m. We have


Z
m
Y
˜
 (bj (x) − bj (y)) fi (y) Ω(x − y) dy
Ftb (fi )(x) =
|x − y|n−1
|x−y|≤t j=1


Z
m
Y
 ((bj (x) − (bj )2Q ) − (bj (y) − (bj )2Q )) fi (y) Ω(x − y) dy
=
|x − y|n−1
|x−y|≤t
j=1

=

m X
X

m−j

(−1)

j=0 σ∈Cjm

(b(x) − (b)2Q )σ

Z

(b(y) − (b)2Q )σc fi (y)

|x−y|≤t

= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (fi )(x)
+(−1)m Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )fi )(x)
+

m−1
X

X

˜

(−1)m−j (b(x) − (b)2Q )σ Ftbσ (fi )(x),

j=1 σ∈Cjm

123

c

Ω(x − y)
dy
|x − y|n−1

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...

Then by Minkowski’s inequality, we get
Z
1
~
||µbΩ (f )(x)|r − |µΩ ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q ))h)(x0 )|r |dx
|Q| Q
Z
1
~
|||Ftb (f )(x)||r − ||Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q ))h)(x0 )||r |dx
≤
|Q| Q
1/r
Z X
∞
1
~b
r
≤
||Ft (fi )(x) − Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )hi )(x0 )||
dx
|Q| Q
i=1
1/r
Z X
∞
1
r
≤
||(b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )Ft (fi )(x)||
dx
|Q| Q
i=1
1/r
Z X
∞ m−1
X X
˜bσc
1
r
dx
||(b(x) − (b)2Q )σ Ft (fi )(x)||
+
|Q| Q
m
i=1 j=1 σ∈Cj

1/r
Z X
∞
1
r
||Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )gi )(x)||
+
|Q| Q
i=1
Z X
∞
1
||Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )hi )(x)
+
|Q| Q
i=1
1/r
r
dx
−Ft ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )hi )(x0 )||

= I1 + I2 + I3 + I4 .

For I1 , by H¨
older’s inequality with exponent 1/s′ + 1/s = 1 and lemma 2, we get
I1

1
≤ C
|Q|


Z

|

m
Y

(bj (x) − (bj )2Q )||µΩ (f )(x)|r dx

Q j=1

1
≤ C
|2Q|

Z

|

m
Y

2Q j=1

s′

1/s′

(bj (x) − (bj )2Q )| dx

≤ C||~b||BM O Ms (|µΩ (f )|r )(˜
x).

124



1
|Q|

Z

Q

|µΩ (f )(x)|sr dx

1/s

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...
For I2 , by H¨
older’s inequality with exponent 1/s′ + 1/s = 1, we have
I2 =

≤

1
|Q|

Z m−1
X X

m−1
X

Q j=1 σ∈C m
j

X

j=1 σ∈Cjm

≤ C

Z

1
|Q|

Q

~

c

~

c

||(b(x) − (b)2Q )σ Ftbσ (f )(x)||r dx
|(b(x) − (b)2Q )σ ||µbΩσ (f )(x)|r dx

m−1
X

1/s′ 
1/s
Z
X  1 Z
~bσc
1
s
s′
|(b(x) − (b)2Q )σ | dx
|µ (f )(x)|r dx
|2Q| 2Q
|Q| Q Ω
m

m−1
X

X

j=1 σ∈Cj

≤ C

~ c
x).
||~bσ ||BM O Ms (|µbΩσ (f )|r )(˜

j=1 σ∈Cjm

For I3 , we choose
p
L (Rn )(see lemma 1)
I3

some p, such that 1 < p < s, by the boundness of |µΩ |r on
and H¨
older’s inequality, we gain

Z
m
Y
1
=
||Ft ( (bj (y) − (bj )2Q )g)(x)||r dx
|Q| Q
j=1

1/p
Z
m
Y
1
≤ 
|µΩ ( (bj (y) − (bj )2Q )f χ2Q )(x)|pr dx
|Q| Rn
j=1



≤ C


1
|Q|

Z

1
≤ C
|2Q|

|

m
Y

Rn j=1

Z

|

1/p

(bj (y) − (bj )2Q )|p |f χ2Q |pr dx

m
Y

sp/(s−p)

(bj (y) − (bj )2Q )|

2Q j=1

≤ C||~b||BM O Ms (|f |r )(˜
x).

125

(s−p)/sp

dx



1
|2Q|

Z

2Q

1/s

|f (x)|sr dx

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...

For I4 , we have
||Ft (

=



−

m
Y

(bj − (bj )2Q )h)(x) − Ft (

j=1

Z

∞

0

|

m
Y

(bj − (bj )2Q )h)(x0 )||r

j=1

Z

Z

|x−y|≤t

|x0 −y|≤t



m
Ω(x − y)|h(y)|r  Y
(bj (y) − (bj )2Q ) dy
|x − y|n−1
j=1



Ω(x0 − y)|h(y)|r 
|x0 − y|n−1

m
Y

j=1



1/2

dt
(bj (y) − (bj )2Q ) dy|2 3 
t

 2 1/2


Y
m


|Ω(x − y)||h(y)|r 
 dy  dt 
(b
(y)
−
(b
)
)
j
j
2Q


n−1
|x − y|
t3
|x0 −y|≤t,|x0 −y|>t
0

j=1
 2 1/2




Y
Z ∞ Z
m


|Ω(x0 − y)||h(y)|r 
 dy  dt 


+
(b
(y)
−
(b
)
)
j
j
2Q


n−1
|x0 − y|
t3
0
|x−y|>t,|x0 −y|≤t

j=1


Z ∞ "Z
 |Ω(x − y)| |Ω(x0 − y)| 


−
+

n−1
|x0 − y|n−1 
|x−y|≤t,|x0 −y|≤t |x − y|
0


2 1/2

Y
m


dt
×  (bj (y) − (bj )2Q ) |h(y)|r dy  3 
t




Z ∞ Z

≤ 


j=1

≡ J1 + J2 + J3 .

126

F. Qiufen - Sharp weighted inequalities for vector-valued multilinear comutators ...

For J1 , we get
J1 ≤

≤

≤

≤

≤

≤



!1/2

m
Z
 |f (y)|r
Y
dt
 (bj (y) − (bj )2Q )
C
dy
 |x − y|n−1

3
|x−y|≤t