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Commutators for multilinear singular integrals on weighted Morrey spaces

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.

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 mN, 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 0jm, 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)dxC 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(2Q)Cω(Q) holds for any cube Q. If ω A p with 1p<, we know that ω(λQ) 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(2Q)Dω(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)dx 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)dx.

From the fact | b 2 j Q b Q |Cj b BMO and Lemma 2.3, we deduce that | b 2 j Q , ω b Q , ω |Cj 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)dx C ω j = 1 m b j BMO R n M Φ ( f ) ( x ) p ω(x)dx

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 xQ,

| 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 Cl 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 tΦ( 1 t )>1, we have

VIC 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 NN. 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 1jN. Assume pj>n/sj and w j A p j s j / n for 1jN. 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

suppΨ { ξ R N n : 1 / 2 | ξ | 2 } , k Z Ψ ( ξ / 2 k ) =1for 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.

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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.

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Wang, S., Jiang, Y. Commutators for multilinear singular integrals on weighted Morrey spaces. J Inequal Appl 2014, 109 (2014). https://doi.org/10.1186/1029-242X-2014-109

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Keywords

  • multilinear singular integrals
  • multiple weights
  • commutators