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607

Documenta Math.

Some Sharp Weighted Estimates
for Multilinear Operators1
Liu Lanzhe

Received: August 8, 2004
Revised: November 25, 2004
Communicated by Heinz Siedentop

Abstract. In this paper, we establish a sharp inequality for
some multilinear operators related to certain integral operators.
The operators include Calder´on-Zygmund singular integral operator, Littlewood-Paley operator, Marcinkiewicz operator and BochnerRiesz operator. As application, we obtain the weighted norm inequalities and L log L type estimate for the multilinear operators.
2000 Mathematics Subject Classification: 42B20, 42B25.
Keywords and Phrases: Multilinear Operator; Calder´on-Zygmund operator; Littlewood-Paley operator; Marcinkiewicz operator; BochnerRiesz operator; Sharp estimate; BMO.
1. Introduction
Let T be a singular integral operator. In[1][2][3], Cohen and Gosselin studied
the Lp (p > 1) boundedness of the multilinear singular integral operator T A
defined by

Z
Rm+1 (A; x, y)
A
K(x, y)f (y)dy.
T (f )(x) =
|x − y|m
n
R
In[6], Hu and Yang obtain a variant sharp estimate for the multilinear singular
integral operators. The main purpose of this paper is to prove a sharp inequality
for some multilinear operators related to certain non-convolution type integral
operators. In fact, we shall establish the sharp inequality for the multilinear
operators only under certain conditions on the size of the integral operators.
The integral operators include Calder´on-Zygmund singular integral operator,
1 Supported

by the NNSF (Grant: 10271071)
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Liu Lanzhe

Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator. As applications, we obtain weighted norm inequalities and L log L type
estimates for these multilinear operators.
2. Notations and Results
First, let us introduce some notations(see[6][12-14]). Throughout this paper,
Q will denote a cube of Rn with side parallel to the axes. For any locally
integrable function f , the sharp function of f is defined by
Z
1
#
f (x) = sup
|f (y) − fQ |dy,
x∈Q |Q| Q
where, and in what follows, fQ = |Q|−1

R

Q

1
f (x) = sup inf
c∈C
|Q|
x∈Q
#

f (x)dx. It is well-known that(see[6])
Z

Q

|f (y) − c|dy.

We say that f belongs to BM O(Rn ) if f # belongs to L∞ (Rn ) and ||f ||BM O =
||f # ||L∞ . For 0 < r < ∞, we denote fr# by
fr# (x) = [(|f |r )# (x)]1/r .
Let M be the
R Hardy-Littlewood maximal operator defined by M (f )(x) =

supx∈Q |Q|−1 Q |f (y)|dy, we write Mp (f ) = (M (f p ))1/p for 0 < p < ∞; For
k ∈ N , we denote by M k the operator M iterated k times, i.e., M 1 (f )(x) =
M (f )(x) and M k (f )(x) = M (M k−1 (f ))(x) for k ≥ 2. Let B be a Young
˜ be the complementary associated to B, we denote that, for a
function and B
function f
¶
¾
½
µ
Z
1
|f (y)|
dy ≤ 1
||f ||B, Q = inf λ > 0 :
B
|Q| Q
λ
and the maximal function by
MB (f )(x) = sup ||f ||B,

x∈Q

Q;

The main Young function to be using in this paper is B(t) = t(1 + log + t) and
˜ = expt, the corresponding maximal denoted by MLlogL
its complementary B(t)
and MexpL . We have the generalized H¨older’s inequality(see[12])
Z
1
|f (y)g(y)|dy ≤ ||f ||B, Q ||g||B, Q
|Q| Q
and the following inequality (in fact they are equivalent), for any x ∈ Rn ,
MLlogL (f )(x) ≤ CM 2 (f )(x)
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Weighted Estimates for Multilinear Operators

609

and the following inequalities, for all cubes Q any b ∈ BM O(Rn ),
||b − bQ ||exp L,

Q

≤ C||b||BM O , |b2k+1 Q − b2Q | ≤ 2k||b||BM O .

We denote the Muckenhoupt weights by Ap for 1 ≤ p < ∞(see[6]).
We are going to consider some integral operators as following.
Let m be a positive integer and A be a function on Rn . We denote that
Rm+1 (A; x, y) = A(x) −

X 1
Dα A(y)(x − y)α .
α!

|α|≤m

Definition 1. Let S and S ′ be Schwartz space and its dual and T : S → S ′
be a linear operator. Suppose there exists a locally integrable function K(x, y)

on Rn × Rn such that
Z
T (f )(x) =
K(x, y)f (y)dy
Rn

for every bounded and compactly supported function f . The multilinear operator related to the integral operator T is defined by
Z
Rm+1 (A; x, y)
K(x, y)f (y)dy.
T A (f )(x) =
|x − y|m
Rn
Definition 2.

Let F (x, y, t) defined on Rn × Rn × [0, +∞). Set
Z
Ft (f )(x) =
F (x, y, t)f (y)dy
Rn

for every bounded and compactly supported function f and
Z
Rm+1 (A; x, y)
F (x, y, t)f (y)dy.
FtA (f )(x) =
|x − y|m
n
R
Let H be a Banach space of functions h : [0, +∞) → R. For each fixed x ∈ Rn ,
we view Ft (f )(x) and FtA (f )(x) as a mapping from [0, +∞) to H. Then, the
multilinear operators related to Ft is defined by
S A (f )(x) = ||FtA (f )(x)||;
We also define that S(f )(x) = ||Ft (f )(x)||.
Note that when m = 0, T A and S A are just the commutators of T , S and A.
While when m > 0, it is non-trivial generalizations of the commutators. It is
well known that multilinear operators are of great interest in harmonic analysis
and have been widely studied by many authors (see [1-5][7]). The main purpose
of this paper is to prove a sharp inequality for the multilinear operators T A
and S A . We shall prove the following theorems in Section 3.

Theorem 1. Let Dα A ∈ BM O(Rn ) for all α with |α| = m. Suppose that T
is the same as in Definition 1 such that T is bounded on Lp (w) for all w ∈ Ap
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Liu Lanzhe

with 1 < p < ∞ and weak bounded of (L1 (w), L1 (w)) for all w ∈ A1 . If T A
satisfies the following size condition:
|T A (f )(x) − T A (f )(x0 )| ≤ C

X

|α|=m

||Dα A||BM O M 2 (f )(x)

for any cube Q = Q(x0 , d) with suppf ⊂ (2Q)c , x ∈ Q = Q(x0 , d). Then for
any 0 < r < 1, there exists a constant C > 0 such that for any f ∈ C0∞ (Rn )

and any x ∈ Rn ,
(T A (f ))#
r (x) ≤ C

X

|α|=m

||Dα A||BM O M 2 (f )(x).

Theorem 2. Let Dα A ∈ BM O(Rn ) for all α with |α| = m. Suppose that S
is the same as in Definition 2 such that S is bounded on Lp (w) for all w ∈ Ap ,
1 < p < ∞ and weak bounded of (L1 (w), L1 (w)) for all w ∈ A1 . If S A satisfies
the following size condition:
||FtA (f )(x) − FtA (f )(x0 )|| ≤ C

X

|α|=m

||Dα A||BM O M 2 (f )(x)

for any cube Q = Q(x0 , d) with suppf ⊂ (2Q)c , x ∈ Q = Q(x0 , d). Then for
any 0 < r < 1, there exists a constant C > 0 such that for any f ∈ C0∞ (Rn )
and any x ∈ Rn ,
(S A (f ))#
r (x) ≤ C

X

|α|=m

||Dα A||BM O M 2 (f )(x).

From the theorems, we get the following
Corollary. Let Dα A ∈ BM O(Rn ) for all α with |α| = m. Suppose that
T A , T and S A , S satisfy the conditions of Theorem 1 and Theorem 2.
(a). If w ∈ Ap for 1 < p < ∞. Then T A and S A are all bounded on Lp (w),
that is
X
||T A (f )||Lp (w) ≤ C
||Dα A||BM O ||f ||Lp (w)
|α|=m

and
||S A (f )||Lp (w) ≤ C
(b).

X

|α|=m

||Dα A||BM O ||f ||Lp (w) .

If w ∈ A1 . Then there exists a constant C > 0 such that for each λ > 0,

w({x ∈ Rn : |T A (f )(x)| > λ})
Z
X
α
≤C
||D A||BM O
|α|=m

Rn

|f (x)|
λ

µ

1 + log

+

µ

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|f (x)|
λ

¶¶

w(x)dx

Weighted Estimates for Multilinear Operators

611

and
w({x ∈ Rn : |S A (f )(x)| > λ})
Z
X
α
≤C
||D A||BM O

Rn

|α|=m

|f (x)|
λ

µ

1 + log

+

µ

|f (x)|
λ

¶¶

w(x)dx.

3. Proof of Theorem
To prove the theorems, we need the following lemmas.
Lemma 1 (Kolmogorov, [6, p.485]). Let 0 < p < q < ∞ and for any function
f ≥ 0. We define that, for 1/r = 1/p − 1/q
||f ||W Lq = sup λ|{x ∈ Rn : f (x) > λ}|1/q , Np,q (f ) = sup ||f χE ||Lp /||χE ||Lr ,
E

λ>0

where the sup is taken for all measurable sets E with 0 < |E| < ∞. Then
||f ||W Lq ≤ Np,q (f ) ≤ (q/(q − p))1/p ||f ||W Lq .
Lemma 2([12, p.165]) Let w ∈ A1 . Then there exists a constant C > 0 such
that for any function f and for all λ > 0,
Z
w({y ∈ Rn : M 2 f (y) > λ}) ≤ Cλ−1
|f (y)|(1 + log+ (λ−1 |f (y)|))w(y)dy.
Rn

Lemma 3.([3, p.448]) Let A be a function on Rn and Dα A ∈ Lq (Rn ) for all
α with |α| = m and some q > n. Then
m

|Rm (A; x, y)| ≤ C|x − y|

X

|α|=m

Ã

1
˜
|Q(x, y)|

Z

˜
Q(x,y)

α

q

|D A(z)| dz

!1/q

,

˜ is the cube centered at x and having side length 5√n|x − y|.
where Q
Proof of Theorem 1.
It suffices to prove for f ∈ C0∞ (Rn ) and some
constant C0 , the following inequality holds:
µ
¶1/r
Z
1
A
r
|T (f )(x) − C0 | dx
≤ CM 2 (f ).
|Q| Q

˜
˜ = 5√nQ and A(x)
=
Fix a cube
˜ ∈ Q. Let Q
P Q1 = αQ(x0 , αd) and x
α ˜
˜
A(x) −
(D
A)
x
,
then
R
(A;
x,
y)
=
R
(
A;
x,
y)
and
D
A
=
˜
m
m
Q
α!
|α|=m

Dα A − (Dα A)Q˜ for |α| = m. We write, for f1 = f χQ˜ and f2 = f χRn \Q˜ ,
T A (f )(x)

=

Z

Rn

=

Z

Rn

Rm+1 (A; x, y)
K(x, y)f (y)dy
|x − y|m
Rm+1 (A; x, y)
K(x, y)f2 (y)dy
|x − y|m

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Liu Lanzhe
˜ x, y)
Rm (A;
K(x, y)f1 (y)dy
|x − y|m
Rn
Z
X 1
K(x, y)(x − y)α α ˜
−
D A(y)f1 (y)dy
α! Rn
|x − y|m
+

Z

|α|=m

then
¯ A
¯
¯T (f )(x) − T A (f2 )(x0 )¯
¯
¯ Ã
!
¯
¯
¶
¯
¯
X 1 ¯ µ (x − ·)α
˜ x, ·)
¯
Rm (A;
¯
¯
α ˜
¯
¯
+
(x)
(x)
f
D
Af
T
¯
¯T
1
1
¯
¯
m
m
¯
¯
|x − ·|
α!
|x − ·|
|α|=m
¯
¯
+ ¯T A (f2 )(x) − T A (f2 )(x0 )¯

≤

:= I(x) + II(x) + III(x),
thus,
µ
≤
:=

1
|Q|

µ

Z

Q

¯ A
¯
¯T (f )(x) − T A (f2 )(x0 )¯r dx

C
I(x)r dx
|Q| Q
I + II + III.
Z

¶1/r

¶1/r µ
¶1/r µ
¶1/r
Z
Z
C
C
II(x)r dx +
III(x)r dx
+
|Q| Q
|Q| Q

˜
Now, let us estimate I, II and III, respectively. First, for x ∈ Q and y ∈ Q,
using Lemma 3, we get
X
˜ x, y) ≤ C|x − y|m
Rm (A;
||Dα A||BM O ,
|α|=m

thus, by Lemma 1 and the weak type (1,1) of T , we get
X
||T (f1 )χQ ||Lr
I ≤ C
||Dα A||BM O |Q|−1
||χQ ||Lr/(1−r)
|α|=m
X
≤ C
||Dα A||BM O |Q|−1 ||T (f1 )||W L1
|α|=m

≤ C

X

|α|=m

˜ −1
||Dα A||BM O |Q|

Z

˜
Q

|f (y)|dy ≤ C

X

|α|=m

||Dα A||BM O M (f )(˜
x);

For II, similar to the proof of I, we get
X
X
˜ 1 )χQ ||Lr
||T (Dα Af
˜ 1 )||W L1
II ≤ C
≤C
|Q|−1
|Q|−1 ||T (Dα Af
||χQ ||Lr/(1−r)
|α|=m
|α|=m
Z
X
X
˜ −1
˜
≤ C
|Q|
|Dα A(y)||f
(y)|dy ≤ C
||Dα A||exp L,Q˜ ||f ||LlogL,Q˜
|α|=m

≤ C

X

|α|=m

˜
Q

||Dα A||BM O ML log L (f )(˜
x) ≤ C

|α|=m

X

|α|=m

||Dα A||BM O M 2 (f )(˜
x);

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Weighted Estimates for Multilinear Operators

613

For III, using H¨
older’ inequality and the size condition of T , we have
X
III ≤ C
||Dα A||BM O M 2 (f )(˜
x).
|α|=m

This completes the proof of Theorem 1.
Proof of Theorem 2.
It is only to prove for f ∈ C0∞ (Rn ) and some
constant C0 , the following inequality holds:
¶1/r
µ
Z
1
A
r
|S (f )(x) − C0 | dx
≤ CM 2 (f ).
|Q| Q
˜ and A(x)
˜
Fix a cube Q = Q(x0 , d) and x
˜ ∈ Q. Let Q
be the same as the proof
of Theorem 1. We write, for f1 = f χQ˜ and f2 = f χRn \Q˜ ,
FtA (f )(x)

=

˜ x, y)
Rm (A;
F (x, y, t)f1 (y)dy
|x − y|m
Rn
Z
X 1
F (x, y, t)(x − y)α α ˜
−
D A(y)f1 (y)dy
α! Rn
|x − y|m
|α|=m
Z
Rm+1 (A; x, y)
F (x, y, t)f2 (y)dy,
+
|x − y|m
Rn

Z

then
¯
¯
|S A (f )(x) − S A (f2 )(x0 )| = ¯||FtA (f )(x)|| − ||FtA (f2 )(x0 )||¯

||F A (f )(x) − FtA (f2 )(x0 )||
¯¯ t Ã
¯¯
!
¯¯
¯¯
¶
¯¯
¯¯
X 1 ¯¯ µ (x − ·)α
˜ x, ·)
¯¯
Rm (A;
¯¯
¯¯
α ˜
¯
¯
Ft
f1 (x)¯¯ +
D Af1 (x)¯¯¯¯
¯ ¯ Ft
¯
¯
m
m
¯
¯¯
¯
|x − ·|
α!
|x − ·|

≤
≤

|α|=m

+||FtA (f2 )(x)

FtA (f2 )(x0 )||

−
:= J(x) + JJ(x) + JJJ(x),
thus,
µ

1
|Q|

Z

Q

A

A

r

|S (f )(x) − S (f2 )(x0 )| dx

¶1/r

¶1/r µ
¶1/r µ
¶1/r
Z
Z
Z
C
C
C
r
r
r
J(x) dx +
JJ(x) dx +
JJJ(x) dx
≤
|Q| Q
|Q| Q
|Q| Q
:= J + JJ + JJJ.
µ

Now, similar to the proof of Theorem 1, we have
Z
X
X
1
α
|f (x)|dx ≤ C
||Dα A||BM O M (f )(˜
x)
J ≤C
||D A||BM O
˜ Q˜
|
Q|
|α|=m
|α|=m
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Liu Lanzhe

and
JJ

X
˜ 1 )χQ ||Lr
||S(Dα Af
˜ 1 )||W L1
≤C
|Q|−1 ||S(Dα Af
||χQ ||Lr/(1−r)
|α|=m
|α|=m
Z
X
X
˜
˜ −1
≤ C
|Dα A(y)||f
(y)|dy ≤ C
|Q|
||Dα A||BM O M 2 (f )(˜
x);

≤ C

X

|Q|−1

|α|=m

˜
Q

|α|=m

For JJJ, using the size condition of S, we have
X
JJJ ≤ C
||Dα A||BM O M 2 (f )(˜
x).
|α|=m

This completes the proof of Theorem 2.
From Theorem 1, 2 and the weighted boundedness of T and S, we may obtain
the conclusion of Corollary(a).
From Theorem 1, 2 and Lemma 2, we may obtain the conclusion of Corollary(b).
4. Applications
In this section we shall apply the Theorem 1, 2 and Corollary of the paper to
some particular operators such as the Calder´on-Zygmund singular integral operator, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz
operator.
Application 1. Calder´
on-Zygmund singular integral operator.
Let T be the Calder´
on-Zygmund operator(see[6][14][15]), the multilinear operator related to T is defined by
Z
Rm+1 (A; x, y)
K(x, y)f (y)dy.
T A (f )(x) =
|x − y|m
Then it is easily to see that T satisfies the conditions in Theorem 1 and Corollary. In fact, it is only to verify that T A satisfies the size condition in Theorem
1, which has done in [6](see also [12][13]). Thus the conclusions of Theorem 1
and Corollary hold for T A .
Application 2. Littlewood-Paley operator.
Let εR> 0 and ψ be a fixed function which satisfies the following properties:
(1) Rn ψ(x)dx = 0,
(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|;
The multilinear Littlewood-Paley operator is defined by(see[8])
gψA (f )(x)
where
FtA (f )(x) =

=

µZ

0

Z

Rn

∞

dt
|FtA (f )(x)|2
t

¶1/2

,

Rm+1 (A; x, y)
ψt (x − y)f (y)dy
|x − y|m

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Weighted Estimates for Multilinear Operators

615

and ψt (x) = t−n ψ(x/t) for t > 0. We write Ft (f ) = ψt ∗ f . We also define that
gψ (f )(x) =

µZ

∞

0

2 dt

|Ft (f )(x)|

t

¶1/2

,

which is the Littlewood-Paley operator(see [15]);
Let H be a space of functions h : [0, +∞) → R, normed by ||h|| =
¡R ∞
¢1/2
|h(t)|2 dt/t
< ∞. Then, for each fixed x ∈ Rn , FtA (f )(x) may be
0
viewed as a mapping from [0, +∞) to H, and it is clear that
gψ (f )(x) = ||Ft (f )(x)|| and gψA (f )(x) = ||FtA (f )(x)||.
It is known that gψ is bounded on Lp (w) for all w ∈ Ap , 1 < p < ∞ and
weak (L1 (w), L1 (w)) bounded for all w ∈ A1 . Thus it is only to verify that
gψA satisfies the size condition in Theorem 2. In fact, we write, for a cube
˜ c , x ∈ Q = Q(x0 , d),
Q = Q(x0 , d) with suppf ⊂ (Q)
FtA (f )(x) − FtA (f )(x0 )
=

Z

Rn

+

Z

µ

Rn

−

ψt (x0 − y)
ψt (x − y)
−
|x − y|m
|x0 − y|m

˜ x, y)f (y)dy
Rm (A;

ψt (x0 − y)
˜ x, y) − Rm (A;
˜ x0 , y))f (y)dy
(Rm (A;
|x0 − y|m

X 1 Z

|α|=m

:=

¶

α!

Rn

µ

(x0 − y)α ψt (x0 − y)
(x − y)α ψt (x − y)
−
|x − y|m
|x0 − y|m

¶

˜
Dα A(y)f
(y)dy

I1 + I2 + I3 .

By Lemma 3 and the following inequality(see[14])
|bQ1 − bQ2 | ≤ C log(|Q2 |/|Q1 |)||b||BM O , f orQ1 ⊂ Q2 ,
we know that, for x ∈ Q and y ∈ 2k+1 Q \ 2k Q with k ≥ 1,
X
˜ x, y)| ≤ C|x − y|m
|Rm (A;
(||Dα A||BM O + |(Dα A)Q(x,y)
− (Dα A)Q˜ |)
˜
|α|=m

m

≤ Ck|x − y|

X

|α|=m

||Dα A||BM O .

Note that |x − y| ∼ |x0 − y| for x ∈ Q and y ∈ Rn \ Q. By the condition on ψ
and Minkowski’ inequality , we obtain
Z
||I1 ||

≤

C

Rn

˜ x, y)||f (y)|
|Rm (A;
|x0 − y|m

"Z

0

∞

µ

t|x − x0 |ε
t|x − x0 |
+
n+1
|x0 − y|(t + |x0 − y|)
(t + |x0 − y|)n+1+ε

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¶2

dt
t

#1/2

dy

616

Liu Lanzhe
≤

C

Z

(2Q)c

≤

X

C

µ

|x − x0 |ε
|x − x0 |
+
m+n+1
|x0 − y|
|x0 − y|m+n+ε

¶

˜ x, y)||f (y)|dy
|Rm (A;

||Dα A||BM O

|α|=m
∞ Z
X

2k+1 Q\2k Q

k=1

≤

X

C

k

||Dα A||BM O

C

X

|x − x0 |ε
|x − x0 |
+
n+1
|x0 − y|
|x0 − y|n+ε

∞
X

k(2−k + 2−εk )|2k+1 Q|−1

¶

|f (y)|dy

Z

|f (y)|dy

2k+1 Q

k=1

|α|=m

≤

µ

||Dα A||BM O M (f )(x);

|α|=m

For I2 , by the formula (see [3]):
˜ x, y) − Rm (A;
˜ x0 , y) =
Rm (A;

X 1
˜ x, x0 )(x − y)β
Rm−|β| (Dβ A;
β!

|β|