Open Access

Commutators for multilinear singular integrals on weighted Morrey spaces

Journal of Inequalities and Applications20142014:109

https://doi.org/10.1186/1029-242X-2014-109

Received: 6 September 2013

Accepted: 18 February 2014

Published: 4 March 2014

Abstract

In this paper we study the iterated commutators for multilinear singular integrals on weighted Morrey spaces. A strong type estimate and a weak endpoint estimate for the commutators are obtained. In the last section we present a problem for the multilinear Fourier multiplier with limited smooth condition.

MSC: 42B20, 42B25.

Keywords

multilinear singular integralsmultiple weightscommutators

1 Introduction

As an important direction of harmonic analysis, the theory of multilinear Calderón-Zygmund singular integral operators has attracted more and more attention, which originated from the work of Coifman and Meyer [1], and it systematically was studied by Grafakos and Torres [2, 3]. The literature of the standard theory of multilinear Calderón-Zygmund singular integrals is by now quite vast, for example see [2, 46]. In 2009, the authors [7] introduced the new multiple weights and new maximal functions and obtained some weighted estimates for multilinear Calderón-Zygmund singular integrals. They also resolved some problems opened up in [8] and [9].

Let S ( R n ) and S ( R n ) be the Schwartz spaces of all rapidly decreasing functions and tempered distributions, respectively. Having fixed m N , let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions,
T : S ( R n ) × × S ( R n ) S ( R n ) .
Following [2], the m-multilinear Calderón-Zygmund operator T satisfies the following conditions:
  1. (S1)

    there exist q i < ( i = 1 , , m ), it extends to a bounded multilinear operator from L q 1 × × L q m to L q , where 1 q = 1 q 1 + + 1 q m ;

     
  2. (S2)
    there exists a function K, defined off the diagonal x = y 1 = = y m in ( R n ) m + 1 , satisfying
    T ( f ) ( x ) = T ( f 1 , , f m ) ( x ) = ( R m ) n K ( x , y 1 , , y m ) f 1 ( y 1 ) f m ( y m ) d y 1 d y m
    (1)
     
for all x j = 1 m supp f j and f 1 , , f m S ( R n ) , where
| K ( y 0 , y 1 , , y m ) | A ( l , k = 0 m | y l y k | ) m n
(2)
and
| K ( y 0 , , y j , , y m ) K ( y 0 , , y j , , y m ) | A | y j y j | ϵ ( l , k = 0 m | y l y k | ) m n + ϵ
(3)

for some ϵ > 0 and all 0 j m , whenever | y j y j | 1 2 max 0 k m | y j y k | .

We also use some notation following [10]. Given a locally integrable vector function b = ( b 1 , , b m ) ( BMO ) m , the commutator of b and the m-linear Calderón-Zygmund operator T, denoted here by T Σ b , was introduced by Pérez and Torres in [9] and is defined via
T Σ b ( f ) = j = 1 m T b j j ( f ) ,
where
T b j j ( f ) = b j T ( f ) T ( f 1 , , b j f j , , f N ) .
The iterated commutator T Π b is defined by
T Π b ( f ) = [ b 1 , , [ b m 1 , [ b m , T ] m ] m 1 ] 1 ( f ) .
To clarify the notations, if T is associated in the usual way with a Calderón-Zygmund kernel K, then at a formal level
T Σ b ( f ) ( x ) = ( R n ) m j = 1 m ( b j ( x ) b j ( y j ) ) K ( x , y 1 , , y m ) f 1 ( y 1 ) f m ( y m ) d y 1 d y m
and
T Π b ( f ) ( x ) = ( R n ) m j = 1 m ( b j ( x ) b j ( y j ) ) K ( x , y 1 , , y m ) f 1 ( y 1 ) f m ( y m ) d y 1 d y m .
It was shown in [2] that if 1 q = 1 q 1 + + 1 q m , then an m-linear Calderón-Zygmund operator T maps from L q 1 × × L q m to L q , when 1 < q j < for all j = 1 , , m ; and from L q 1 × × L q m to L q , , when 1 q j < for all j = 1 , , m , and min 1 j m q j = 1 . The weighted strong and weak L q boundedness of T is also true for weights in the class A P which will be introduced in next section (see Corollary 3.9 [7]). It was proved in [9] that T Σ b is bounded from L q 1 × × L q m to L q for all indices satisfying 1 q = 1 q 1 + + 1 q m with q > 1 and q j > 1 , j = 1 , , m . The result was extended in [7] to all q > 1 / m . In fact, the authors obtained the weighted L q -version bounds as follows, for all ω A P :
T Σ b ( f ) L q ( ν ω ) C b BMO m j = 1 m f j L q j ( ω j ) .
As may be expected from the situation in the linear case, T Σ b is not bounded from L 1 × × L 1 to L 1 , . However, a sharp weak-type estimate in a very general sense was obtained in [7], that is, for all ω A ( 1 , , 1 ) ,
ν ω { x R n : | T Σ b ( f ) ( x ) | > t m } C j = 1 m ( R n Φ ( | f j ( x ) | t ) ω j ( x ) d x ) 1 / m ,

where Φ ( t ) = t ( 1 + log + t ) . When m = 1 , the above endpoint estimate was obtained in [11]. The same as for T Σ b , the strong type bound and the endpoint estimate for T Π b were also established by Pérez et al. in [10].

The weighted Morrey spaces L p , k ( w ) was introduced by Komori and Shirai [6]. Moreover, they showed that some classical integral operators and corresponding commutators are bounded in weighted Morrey spaces. Some other authors have been interested in this space for sublinear operators, see [1214]. In [15], Ye proved two results similar to Pérez and Trujillo-González [11] for the multilinear commutators of the normal Calderón-Zygmund operators on weighted Morrey spaces. Wang and Yi [16] considered the multilinear Calderón-Zygmund operators on weighted Morrey spaces and obtained some results similar to weighted Lebesgue spaces.

We will prove the following strong type bound for T Π b on weighted Morrey spaces.

Theorem 1.1 Let T be an m-linear Calderón-Zygmund operator; ω A P ( A ) m with
1 p = 1 p 1 + + 1 p m
and 1 < p j < , j = 1 , , m ; and b BMO m . Then, for any 0 < k < 1 , there exists a constant C such that
T Π b ( f ) L p , k ( ν ω ) C j = 1 m b j BMO j = 1 m f j L p j , k ( ω j ) .

The following endpoint estimate will also be proved.

Theorem 1.2 Let T be an m-linear Calderón-Zygmund operator; 0 < k < 1 , ω A ( 1 , , 1 ) ( A ) m , and b BMO m . Then, for any λ > 0 and cube Q, there exists a constant C such that
1 ν ω ( Q ) k ν ω { x Q : | T Π b ( f ) ( x ) | > λ } C j = 1 m [ f j / λ L Φ ( m ) , k ( ω j ) ] 1 / m ,

where, Φ ( t ) = t ( 1 + log + t ) and f L Φ ( m ) , k ( ω ) = Φ ( m ) ( | f | ) L 1 , k ( ω ) .

Remark 1.1 Here we remark that the above estimate is also valid for T Σ b .

2 Some definitions and results

In this section, we introduce some definitions and results used later.

Definition 2.1 ( A p weights)

A weight ω is a nonnegative, locally integrable function on  R n . Let 1 < p < , a weight function ω is said to belong to the class A p , if there is a constant C such that for any cube Q,
( 1 | Q | Q ω ( x ) d x ) ( 1 | Q | Q ω ( x ) 1 p d x ) p 1 C ,
and to the class A 1 , if there is a constant C such that for any cube Q,
1 | Q | Q ω ( x ) d x C inf x Q ω ( x ) .

We denote A = p > 1 A p .

Definition 2.2 (Multiple weights)

For m exponents p 1 , , p m [ 1 , ) , we often write p for the number given by p = j = 1 m p j and denote by P the vector ( p 1 , , p m ) . A multiple weight ω = ( ω 1 , , ω m ) is said to satisfy the A P condition if for
ν ω = j = 1 m ω p / p j ,
we have
sup Q ( 1 | Q | Q ν ω ( x ) d x ) 1 / p j = 1 m ( 1 | Q | Q ω j ( x ) 1 p j d x ) 1 / p j < ,
when p j = 1 , ( 1 | Q | Q ω j ( x ) 1 p j d x ) 1 / p j is understood as ( inf x ω ( x ) ) 1 . As remarked in [7], j = 1 m A p j is strictly contained in A P , moreover, in general ω A P does not imply ω j L loc 1 for any j, but instead
ω A P { ( ν ω ) p A m p , ω j 1 p j A m p j , j = 1 , , m ,

where the condition ω j 1 p j A m p j in the case p j = 1 is understood as ω j 1 / m A 1 .

Definition 2.3 (Weighted Morrey spaces)

Let 0 < p < , 0 < k < 1 , and ω be a weight function on R n . The weighted Morrey space is defined by
L p , k ( ω ) = { f L loc p : f L p , k ( ω ) < } ,
where
f L p , k ( ω ) = sup Q ( 1 ω ( Q ) k Q | f ( x ) | p ω ( x ) ) 1 / p .
The weighted weak Morrey space is defined by
W L p , k ( ω ) = { f  measurable : f W L p , k ( ω ) < } ,
where
f W L p , k ( ω ) = sup Q inf λ > 0 λ ω ( Q ) k / p ω ( { x Q : | f | ( x ) > λ } ) 1 / p .

Definition 2.4 (Maximal function)

For Φ ( t ) = t ( 1 + log + t ) and a cube Q in R n we will consider the average f Φ , Q of a function f given by the Luxemburg norm
f Φ , Q = inf { λ > 0 : 1 | Q | Q Φ ( | f ( x ) | λ ) d x 1 } ,
and the corresponding maximal is naturally defined by
M Φ f ( x ) = sup Q x f Φ , Q ,
and the multilinear maximal operator M Φ , Q is given by
M Φ ( f ) ( x ) = sup Q x j = 1 m f j Φ , Q .
The following pointwise equivalence is very useful:
M Φ f ( x ) M 2 f ( x ) ,

where M is the Hardy-Littlewood maximal function. We refer reader to [7, 10] and their references for details.

We say that a weight ω satisfies the doubling condition, simply denoted ω Δ 2 , if there is a constant C > 0 such that ω ( 2 Q ) C ω ( Q ) holds for any cube Q. If ω A p with 1 p < , we know that ω ( λ Q ) λ n p [ ω ] A p ω ( Q ) for all λ > 1 ; then ω Δ 2 .

Lemma 2.1 ([6])

Suppose ω Δ 2 , then there exists a constant D > 1 such that
ω ( 2 Q ) D ω ( Q )

for any cube.

Lemma 2.2 ([16])

If ω j A , then for any cube Q, we have
Q j = 1 m ω j θ j ( x ) d x j = 1 m ( Q ω j ( x ) d x [ ω j ] ) θ j ,

where j = 1 m θ j = 1 , 0 θ j 1 .

Lemma 2.3 ([17])

Suppose ω A , then b BMO ( ω ) b BMO . Here
BMO ( ω ) = { b : b BMO ( ω ) = sup Q 1 ω ( Q ) Q | b ( x ) b Q , ω | ω ( x ) d x < } ,

and b Q , ω = 1 ω ( Q ) Q b ( x ) ω ( x ) d x .

From the fact | b 2 j Q b Q | C j b BMO and Lemma 2.3, we deduce that | b 2 j Q , ω b Q , ω | C j b BMO . The following lemma is the multilinear version of the Fefferman-Stein type inequality.

Lemma 2.4 (Theorem 3.12 [7])

Assume that ω i is a weight in A 1 for all i = 1 , , m , and set ν = ( i = 1 m ω i ) 1 / m . Then
j = 1 m M ( f j ) L p , ( ν ) j = 1 m f j L 1 ( M ω j ) .

Lemma 2.5 (Proposition 3.13 [7])

Let 1 p = 1 p 1 + + 1 p m . If 1 p j , j = 1 , , m , then
M ( f ) L p , ( ν ω ) j = 1 m f j L p j ( M ω j ) .

Lemma 2.6 (Theorem 3.2 [10])

Let p > 0 and let ω be a weight in A . Suppose that b BMO m . Then there exist C ω (independent of b) and C ω , b such that
R n | T Π b ( f ) ( x ) | ω ( x ) d x C ω j = 1 m b j BMO R n M Φ ( f ) ( x ) p ω ( x ) d x
and
sup t > 0 1 Φ ( m ) ( 1 t ) ω ( { y R n : | T Π b ( f ) ( y ) | > t m } ) C ω , b sup t > 0 1 Φ ( m ) ( 1 t ) ω ( { y R n : | M Φ ( f ) ( y ) | > t m } )

for all f = ( f 1 , , f m ) bounded with compact support.

Lemma 2.7 (Theorem 4.1 [10])

Let ω A ( 1 , , 1 ) . Then there exists a constant C such that
ν ω ( { x R n : | M L log L ( f ) ( x ) | > t m } ) C j = 1 m ( R n Φ ( m ) ( | f j ( x ) | t ) ω j ( x ) d x ) 1 / m .

By the above two inequalities, Pérez and Trujillo-González obtained the following results.

Lemma 2.8 (Theorem 1.1 [10])

Let T be an m-linear Calderón-Zygmund operator; ω A P with
1 p = 1 p 1 + + 1 p m
and 1 < p j < , j = 1 , , m ; and b BMO m . Then there exists a constant C such that
T Π b ( f ) L p ( ν ω ) C j = 1 m b j BMO j = 1 m f j L p j ( ω j ) .

Lemma 2.9 (Theorem 1.2 [10])

Let T be an m-linear Calderón-Zygmund operator; ω A ( 1 , , 1 ) , and b BMO m . Then, for any λ > 0 and cube Q, there exists a constant C such that
ν ω { x R n : | T Π b ( f ) ( x ) | > λ } C j = 1 m ( R n Φ ( m ) ( | f j ( x ) | t ) ω j ( x ) d x ) 1 / m ,

where Φ ( t ) = t ( 1 + log + t ) and.

3 Proofs of theorems

We only present the case m = 2 for simplicity, but, as the reader will immediately notice, a complicated notation and a similar procedure can be followed to obtain the general case. Our arguments will be standard.

Proof of Theorem 1.1 For any cube Q, we split f j into f j 0 + f j , where f j 0 = f j χ 2 Q and f j = f j f j 0 , j = 1 , 2 . Then we only need to verify the following inequalities:
I = ( 1 ν ω ( Q ) k Q | T Π b ( f 1 0 , f 2 0 ) ( x ) | p ν ω ( x ) d x ) 1 / p C j = 1 2 b j BMO j = 1 2 f j L p j , k ( ω j ) , II = ( 1 ν ω ( Q ) k Q | T Π b ( f 1 0 , f 2 ) ( x ) | p ν ω ( x ) d x ) 1 / p C j = 1 2 b j BMO j = 1 2 f j L p j , k ( ω j ) , III = ( 1 ν ω ( Q ) k Q | T Π b ( f 1 , f 2 0 ) ( x ) | p ν ω ( x ) d x ) 1 / p C j = 1 2 b j BMO j = 1 2 f j L p j , k ( ω j ) , IV = ( 1 ν ω ( Q ) k Q | T Π b ( f 1 , f 2 ) ( x ) | p ν ω ( x ) d x ) 1 / p C j = 1 2 b j BMO j = 1 2 f j L p j , k ( ω j ) .
From Lemma 2.8 and Lemma 2.2, we get
I C 1 ν ω ( Q ) k / p j = 1 2 b j BMO ( R n | f j 0 ( x ) | p j ω j ( x ) d x ) 1 / p j C 1 ν ω ( Q ) k / p j = 1 2 [ b j BMO ω j ( 2 Q ) k / p j f j L p j , k ( ω j ) ] C j = 1 2 [ b j BMO f j L p j , k ( ω j ) ] .
Since II and III are symmetric we only estimate II. Taking λ j = ( b j ) Q , ω j , the operator T Π b can be divided into four parts:
T Π b ( f 1 0 , f 2 ) ( x ) = ( b 1 ( x ) λ 1 ) ( b 2 ( x ) λ 2 ) T ( f 1 0 , f 2 ) ( x ) ( b 1 ( x ) λ 1 ) T ( f 1 0 , ( b 2 λ 2 ) f 2 ) ( x ) ( b 2 ( x ) λ 2 ) T ( ( b 1 λ 1 ) f 1 0 , f 2 ) ( x ) + T ( ( b 1 λ 1 ) f 1 0 , ( b 2 λ 2 ) f 2 ) ( x ) = II 1 + II 2 + II 3 + II 4 .
Using the size condition (2) of K, Definition 2.2, and Lemma 2.2, we deduce that for any x Q ,
| T ( f 1 0 , f 2 ) ( x ) | C 2 Q R n 2 Q | f 1 ( y 1 ) f 2 ( y 2 ) | ( | x y 1 | + | x y 2 | ) 2 n d y 2 d y 1 C 2 Q | f 1 ( y 1 ) | d y 1 l = 1 1 | 2 l Q | 2 2 l + 1 Q 2 l Q | f 2 ( y 2 ) | d y 2 C l = 1 j = 1 2 1 | 2 l + 1 Q | 2 l + 1 Q | f j ( y j ) | d y j C l = 1 j = 1 2 1 | 2 l + 1 Q | ( 2 l + 1 Q | f j ( y j ) | p j ω j ( y j ) d y j ) 1 / p j × ( 2 l + 1 Q ω j ( y j ) 1 p j d y j ) 1 / p j C l = 1 1 | 2 l + 1 Q | 2 | 2 l + 1 Q | 1 p + 1 p 1 + 1 p 2 ν ω ( 2 l + 1 Q ) j = 1 2 f j L p j , k ( ω j ) ω j ( 2 l + 1 Q ) k / p j C j = 1 2 f j L p j , k ( ω j ) l = 1 ν ω ( 2 l + 1 Q ) ( k 1 ) / p .
Taking the above estimate together with Hölder’s inequality and Lemma 2.3, we have
( 1 ν ω ( Q ) k Q | II 1 | p ν ω ( x ) d x ) 1 / p 1 ν ω ( Q ) k / p ( Q | ( b 1 ( x ) λ 1 ) ( b 2 ( x ) λ 2 ) | p ν ω ( x ) d x ) 1 / p × j = 1 2 f j L p j , k l = 1 ν ω ( 2 l + 1 Q ) ( k 1 ) / p ν ω ( Q ) 1 / p ν ω ( Q ) k / p j = 1 2 ( 1 ν ω ( Q ) Q | ( b j ( x ) λ 1 ) | 2 p ν ω ( x ) d x ) 1 / 2 p × j = 1 2 f j L p j , k l = 1 ν ω ( 2 l + 1 Q ) ( k 1 ) / p j = 1 2 b j BMO f j L p j , k ( ω j ) ,
where the last inequality is obtained by the property of A : there is a constant δ > 0 such that
ν ω ( Q ) ν ω ( 2 l + 1 Q ) C ( | Q | | 2 l + 1 Q | ) δ .
For II 2 , from the size condition (2) of K, the A P condition, Lemma 2.2, and Lemma 2.3, it follows that
| T ( f 1 0 , ( b 2 λ 2 ) f 2 ) ( x ) | C 2 Q | f 1 ( y 1 ) | d y 1 l = 1 1 | 2 l Q | 2 2 l + 1 Q 2 l Q | ( b 2 ( y 2 ) λ 2 ) f 2 ( y 2 ) | d y 2 C l = 1 1 | 2 l + 1 Q | 2 ( 2 l + 1 Q | f 1 ( y 1 ) | p 1 ω j ( y 1 ) d y 1 ) 1 / p 1 ( 2 l + 1 Q ω 1 ( y 1 ) 1 p 1 d y j ) 1 / p 1 × ( 2 l + 1 Q | f 2 ( y 2 ) | p 2 ω 2 ( y 2 ) d y 2 ) 1 / p 2 × ( 2 l + 1 Q | b 2 ( y 2 ) λ 2 | p 2 ω 2 ( y 2 ) p 2 / p 2 d y 2 ) 1 / p 2 C l = 1 l j = 1 2 1 | 2 l + 1 Q | ( 2 l + 1 Q | f j ( y j ) | p j ω j ( y j ) d y j ) 1 / p j ( 2 l + 1 Q ω j ( y j ) 1 p j d y j ) 1 / p j C j = 1 2 f j L p j , k ( ω j ) l = 1 l ν ω ( 2 l + 1 Q ) ( k 1 ) / p .
The third inequality can be deduced by the fact that
( 1 ω ( 2 j + 1 Q ) 2 l + 1 Q | b ( y ) b Q , ω | p ω ( y ) d y ) 1 / p C l b BMO ( ω ) .
Hölder’s inequality and Lemma 2.3 tell us
( 1 ν ω ( Q ) k Q | II 2 | p ν ω ( x ) d x ) 1 / p C 1 ν ω ( Q ) k / p ( Q | ( b 1 ( x ) λ 1 ) | p ν ω ( x ) d x ) 1 / p j = 1 2 f j L p j , k l = 1 l ν ω ( 2 l + 1 Q ) ( k 1 ) / p C ν ω ( Q ) 1 / p ν ω ( Q ) k / p j = 1 2 f j L p j , k l = 1 l ν ω ( 2 l + 1 Q ) ( k 1 ) / p C j = 1 2 b j BMO f j L p j , k ( ω j ) .
Similarly, we get
| T ( f 1 0 , ( b 2 λ 2 ) f 2 ) ( x ) | C l = 1 1 | 2 l + 1 Q | 2 ( 2 l + 1 Q | f 1 ( y 1 ) | p 1 ω j ( y 1 ) d y 1 ) 1 / p 1 × ( 2 l + 1 Q | b 1 ( y 1 ) λ 1 | p 1 ω 1 ( y 1 ) 1 p 1 d y j ) 1 / p 1 × ( 2 l + 1 Q | f 2 ( y 2 ) | p 2 ω 2 ( y 2 ) d y 2 ) 1 / p 2 ( 2 l + 1 Q ω 2 ( y 2 ) p 2 / p 2 d y 2 ) 1 / p 2 C j = 1 2 f j L p j , k ( ω j ) l = 1 l ν ω ( 2 l + 1 Q ) ( k 1 ) / p ,
and so
( 1 ν ω ( Q ) k Q | II 3 | p ν ω ( x ) d x ) 1 / p C j = 1 2 b j BMO f j L p j , k ( ω j ) .
The term II 4 is estimated in a similar way and we deduce
| T ( ( b 1 λ 1 ) f 1 0 , ( b 2 λ 2 ) f 2 ) ( x ) | C l = 1 1 | 2 l + 1 Q | 2 j = 1 2 ( 2 l + 1 Q | f j ( y j ) | p j ω j ( y j ) d y j ) 1 / p j × ( 2 l + 1 Q | b j ( y j ) λ j | p j ω j ( y j ) p j / p j d y j ) 1 / p j C j = 1 2 f j L p j , k ( ω j ) l = 1 l 2 ν ω ( 2 l + 1 Q ) ( k 1 ) / p .
So,
( 1 ν ω ( Q ) k Q | II 4 | p ν ω ( x ) d x ) 1 / p C j = 1 2 b j BMO f j L p j , k ( ω j ) .
Finally, we still split T Π b ( f 1 , f 2 ) ( x ) into four terms:
T Π b ( f 1 , f 2 ) ( x ) = ( b 1 ( x ) λ 1 ) ( b 2 ( x ) λ 2 ) T ( f 1 , f 2 ) ( x ) ( b 1 ( x ) λ 1 ) T ( f 1 , ( b 2 λ 2 ) f 2 ) ( x ) ( b 2 ( x ) λ 2 ) T ( ( b 1 λ 1 ) f 1 , f 2 + T ( ( b 1 λ 1 ) f 1 , ( b 2 λ 2 ) f 2 ) ( x ) ) ( x ) = IV 1 + IV 2 + IV 3 + IV 4 .
Because each term of IV j is completely analogous to II j , j = 1 , 2 , 3 , 4 with a small difference, we only estimate IV 1 :
| T ( f 1 , f 2 ) ( x ) | C ( R n ) 2 ( 2 Q ) 2 | f 1 ( y 1 ) f 2 ( y 2 ) | ( | x y 1 | + | x y 2 | ) 2 n d y 2 d y 1 C l = 1 ( 2 l + 1 Q ) 2 ( 2 l Q ) 2 | f 1 ( y 1 ) f 2 ( y 2 ) | ( | x y 1 | + | x y 2 | ) 2 n d y 2 d y 1 C l = 1 1 | 2 l + 1 Q | 2 ( 2 l + 1 Q ) 2 j = 1 2 | f j ( y j ) | d y j C j = 1 2 f j L p j , k ( ω j ) l = 1 ν ω ( 2 l + 1 Q ) ( k 1 ) / p .
Hence,
( 1 ν ω ( Q ) k Q | IV 1 | p ν ω ( x ) d x ) 1 / p C j = 1 2 b j BMO f j L p j , k ( ω j ) .

Combining all estimates, we complete the proof of Theorem 1.1. □

We now turn to the proof of Theorem 1.2.

Proof of Theorem 1.2 By homogeneity, we may assume that λ = b 1 BMO = b 2 BMO = 1 and we only need to prove that
ν ω { x Q : | T Π b ( f 1 , f 2 ) ( x ) | > 1 } C ν ω ( Q ) k j = 1 2 ( f j L Φ ( 2 ) , k ( ω j ) ) 1 / 2 .
To prove the above inequality, we can write
ν ω { x Q : | T Π b ( f 1 , f 2 ) ( x ) | > 1 } ν ω { x Q : | T Π b ( f 1 0 , f 2 0 ) ( x ) | > 1 / 4 } + ν ω { x Q : | T Π b ( f 1 0 , f 2 ) ( x ) | > 1 / 4 } + ν ω { x Q : | T Π b ( f 1 , f 2 0 ) ( x ) | > 1 / 4 } + ν ω { x Q : | T Π b ( f 1 , f 2 ) ( x ) | > 1 / 4 } = V + VI + VII + VIII
for any cube Q. Employing Lemma 2.9 and Lemma 2.2, we have
V C j = 1 2 ( R n Φ ( m ) ( | f j ( x ) | ) ω j ( x ) d x ) 1 / 2 C j = 1 2 [ ω j ( Q ) k f j L Φ ( m ) , k ( ω j ) ] 1 / 2 C ν ω ( Q ) k j = 1 2 [ f j L Φ ( m ) , k ( ω j ) ] 1 / 2 .
From Lemma 2.6 and Lemma 2.4, we deduce that
ν ω { x Q : | T Π b ( f 1 0 , f 2 ) ( x ) | > 1 / 4 } sup t > 0 1 Φ ( m ) ( 1 t ) ν ω { x Q : | T Π b ( f 1 0 , f 2 ) ( x ) | > t 2 } C ν ω , b sup t > 0 1 Φ ( m ) ( 1 t ) ν ω ( { y Q : | M Φ ( f 1 0 , f 2 ) ( y ) | > t 2 } ) C ν ω , b sup t > 0 1 Φ ( m ) ( 1 t ) ν ω ( { y Q : | M Φ ( f 1 0 ) ( y ) M Φ ( f 2 ) ( y ) | > t 2 } ) C ν ω , b t ( R n Φ ( | f 1 0 | ) ( y ) M ( χ Q ω 1 ) ( y ) d y R n Φ ( | f 2 | ) ( y ) M ( χ Q ω 2 ) ( y ) d y ) 1 / 2 C ν ω , b t [ ω j ( Q ) k f j L Φ , k ( ω j ) ] 1 / 2 ,
where the last inequality holds by the (3.10) and (3.11) in [15]. Then from Lemma 2.2 and the fact that t Φ ( 1 t ) > 1 , we have
VI C ν ω ( Q ) [ ω j ( Q ) k f j L Φ , k ( ω j ) ] 1 / 2 .
A similar statement follows:
VII C ν ω ( Q ) [ ω j ( Q ) k f j L Φ , k ( ω j ) ] 1 / 2 ; VIII C ν ω ( Q ) [ ω j ( Q ) k f j L Φ , k ( ω j ) ] 1 / 2 .

Thus we complete the proof of Theorem 1.2. □

4 A problem

Fix N N . Let m C L ( R N n { 0 } ) , for some positive integer L, satisfying the following condition:
| ξ 1 α 1 ξ N α N m ( ξ 1 , , ξ N ) | C α 1 , , α N ( | ξ 1 | + + | ξ N | ) | α |
(4)
for all | α | s and ξ R N n { 0 } , where α = ( α 1 , , α N ) and ξ = ( ξ 1 , , ξ N ) . The multilinear Fourier multiplier operator T N is defined by
T m ( f ) ( x ) = 1 ( 2 π ) N n ( R N n ) e i x ( ξ 1 + + ξ N ) m ( ξ 1 , , ξ N ) f ˆ 1 ( ξ 1 ) f ˆ N ( ξ N ) d ξ 1 d ξ N
(5)
for all f 1 , , f N S ( R n ) , where f = ( f 1 , , f N ) . If F 1 m is an integrable function, then this can also be written as
T m ( f ) ( x ) = ( R N n ) F 1 m ( x y 1 , , x y N ) f ( y 1 ) f ( y N ) d y 1 d y N .

In [18], Fujita and Tomita obtained the following theorem.

Theorem 4.1 Let 1 < p 1 , , p N < , 1 p 1 + + 1 p N = 1 p and n 2 < s j n for 1 j N . Assume p j > n / s j and w j A p j s j / n for 1 j N . If m L ( R N n ) satisfies
m k W ( s 1 , , s N ) = ( R N n ( 1 + | ξ 1 | 2 ) 1 / 2 | ( 1 + | ξ N | 2 ) 1 / 2 m ˆ ( ξ ) | 2 d ξ ) 1 / 2 < ,
then T N is bounded from L p 1 ( ω 1 ) × × L p N ( ω N ) to L p ( ν ω ) , where
m j ( ξ ) = m ( 2 j ξ 1 , , 2 j ξ N ) Ψ ( ξ 1 , ξ N ) ,
where Ψ is the Schwarz function and satisfies
supp Ψ { ξ R N n : 1 / 2 | ξ | 2 } , k Z Ψ ( ξ / 2 k ) = 1 for all  ξ R N n { 0 } .

A natural problem is whether the Lebesgue spaces L p j ( ω j ) and L p ( ν ω ) can be replaced by L p j , k ( ω ) and L p , k ( ν ω ) . It should be pointed out that the method in this paper may not be suitable to address this problem.

Declarations

Acknowledgements

The authors would like to thank the referee for some very valuable suggestions. This research was supported by NSF of China (no. 11161044, no. 11261055) and by XJUBSCX-2012004.

Authors’ Affiliations

(1)
College of Mathematics and System Sciences, Xinjiang University

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© Wang and Jiang; licensee Springer. 2014

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