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Weighted boundedness of multilinear singular integral operator with general kernels for the extreme cases

Abstract

We prove the weighted boundedness properties for the multilinear operator associated to the singular integral operator with general kernels for the extreme cases.

MSC:42B20, 42B25.

1 Introduction and preliminaries

As for the development of singular integral operators, their commutators and multilinear operators have been well studied (see [19]). Let T be the Calderón-Zygmund singular integral operator and bBMO( R n ), a classical result of Coifman et al. (see [5]) stated that the commutator [b,T](f)=T(bf)bT(f) is bounded on L p ( R n ) for 1<p<. In [10], the authors obtain the boundedness properties of the commutators for the extreme values of p (that is, p=1 and p=). Note that [b,T] is not bounded for the end point boundedness. The purpose of this paper is to introduce some multilinear operator associated to the singular integral operator with general kernels (see [11]) and prove the weighted boundedness properties of the multilinear operators for the extreme cases.

First, let us introduce some preliminaries (see [6, 9]). Throughout this paper, Q will denote a cube of R n with sides parallel to the axes. For a locally integrable functions b and a weight function w (that is, a non-negative locally integrable function), let w(Q)= Q w(x)dx, w Q = | Q | 1 Q w(x)dx, the weighted sharp function of b is defined by

b # (x)= sup Q x 1 w ( Q ) Q | b ( y ) b Q | w(y)dy.

We say that b belongs to BMO(w) if b # belongs to L (w), and we define b B M O ( w ) = b # L ( w ) . If w=1, we denote BMO(w)=BMO( R n ). It has been known that (see [9])

b b 2 k Q B M O Ck b B M O .

We also define the central BMO space by CMO( R n ), which is the space of those functions f L loc ( R n ) such that

f C M O = sup r > 1 | Q ( 0 , r ) | 1 Q | f ( x ) f Q | dx<.

It is well known that (see [6, 9])

f C M O sup r > 1 inf c C | Q ( 0 , r ) | 1 Q | f ( x ) c | dx.

Definition 1 Let 1<p< and w be a non-negative weight functions on R n . We shall call B p (w) the space of those functions f on R n such that

f B p ( w ) = sup r > 1 [ w ( Q ( 0 , r ) ) ] 1 / p f χ Q ( 0 , r ) L p ( w ) <.

The A p weight is defined by (see [6])

A p = { 0 < w L loc 1 ( R n ) : sup Q ( 1 | Q | Q w ( x ) d x ) ( 1 | Q | Q w ( x ) 1 / ( p 1 ) d x ) p 1 < } , 1 < p < ,

and

A 1 = { 0 < w L loc 1 ( R n ) : sup Q x 1 | Q | Q w ( y ) d y C w ( x ) , a.e. } .

2 Theorems

In this paper, we will study the following multilinear singular integral operator (see [11]).

Definition 2 Let T:S S be a linear operator such that T is bounded on L 2 ( R n ) and has a kernel K, that is, there exists a locally integrable function K(x,y) on R n × R n {(x,y) R n × R n :x=y} such that

T(f)(x)= R n K(x,y)f(y)dy

for every bounded and compactly supported function f, where K satisfies

| K ( x , y ) | C | x y | n , 2 | y z | < | x y | ( | K ( x , y ) K ( x , z ) | + | K ( y , x ) K ( z , x ) | ) d x C ,

and there is a sequence of positive constant numbers { C k } such that for any k1,

( 2 k | z y | | x y | < 2 k + 1 | z y | ( | K ( x , y ) K ( x , z ) | + | K ( y , x ) K ( z , x ) | ) q d y ) 1 / q C k ( 2 k | z y | ) n / q ,

where 1< q <2 and 1/q+1/ q =1.

Let m j be the positive integers (j=1,,l), m 1 ++ m l =m and b j be the functions on R n (j=1,,l). Set, for 1jl,

R m j + 1 ( b j ;x,y)= b j (x) | α | m j 1 α ! D α b j (y) ( x y ) α .

The multilinear operator associated to T is defined by

T b (f)(x)= R n j = 1 l R m j + 1 ( b j ; x , y ) | x y | m K(x,y)f(y)dy.

Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 2 (see [7, 8]). Also note that when m=0, T b is just a multilinear commutator of T and b (see [13]). It is well known that a multilinear operator, as a non-trivial extension of a commutator, is of great interest in harmonic analysis and has been widely studied by many authors (see [13]). In this paper, we will study the weighted boundedness properties of the multilinear operators T b for the extreme cases (see [1215]).

We shall prove the following theorems in Section 3.

Theorem 1 Let T be the singular integral operator as Definition  2, and we have the sequence { k m C k } l 1 , w A 1 and D α b j BMO( R n ) for all α with |α|= m j and j=1,,l. Then T b is bounded from L (w) to BMO(w).

Theorem 2 Let T be the singular integral operator as Definition  2, and we have the sequence { k m C k } l 1 , 1<p<, w A 1 , and D α b j BMO( R n ) for all α with |α|= m j and j=1,,l. Then T b is bounded from B p (w) to CMO(w).

3 Proofs of theorems

We begin with two preliminaries lemmas.

Lemma 1 (see [3])

Let b be a function on R n and D α b L q ( R n ) for |α|=m and some q>n. Then

| R m ( b ; x , y ) | C | x y | m | α | = m ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) | D α b ( z ) | q d z ) 1 / q ,

where Q ˜ (x,y) is the cube centered at x and having side length 5 n |xy|.

Lemma 2 (see [11])

Let T be the singular integral operator as Definition  2, and we have the sequence { C k } l 1 . Then T is bounded on L p ( R n ,w) for w A p with 1<p<.

Proof of Theorem 1 It is only for us to prove that there exists a constant C Q such that

1 w ( Q ) Q | T b ( f ) ( x ) C Q | w(x)dxC f L ( w )

holds for any cube Q. Without loss of generality, we may assume l=2. Fix a cube Q=Q( x 0 ,d). Let Q ˜ =5 n Q and b ˜ j (x)= b j (x) | α | = m 1 α ! ( D α b j ) Q ˜ x α , then R m ( b j ;x,y)= R m ( b ˜ j ;x,y) and D α b ˜ j = D α b j ( D α b j ) Q ˜ for |α|= m j . We write, for f 1 =f χ Q ˜ and f 2 =f χ R n Q ˜ ,

T b ( f ) ( x ) = R n j = 1 2 R m j + 1 ( b ˜ j ; x , y ) | x y | m K ( x , y ) f ( y ) d y = R n j = 1 2 R m j ( b ˜ j ; x , y ) | x y | m K ( x , y ) f 1 ( y ) d y | α 1 | = m 1 1 α 1 ! R n R m 2 ( b ˜ 2 ; x , y ) ( x y ) α 1 D α 1 b ˜ 1 ( y ) | x y | m K ( x , y ) f 1 ( y ) d y | α 2 | = m 2 1 α 2 ! R n R m 1 ( b ˜ 1 ; x , y ) ( x y ) α 2 D α 2 b ˜ 2 ( y ) | x y | m K ( x , y ) f 1 ( y ) d y + | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! R n ( x y ) α 1 + α 2 D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) | x y | m K ( x , y ) f 1 ( y ) d y + R n j = 1 2 R m j + 1 ( b ˜ j ; x , y ) | x y | m K ( x , y ) f 2 ( y ) d y = T ( j = 1 2 R m j ( b ˜ j ; x , ) | x | m f 1 ) T ( | α 1 | = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ) ( x ) α 1 D α 1 b ˜ 1 | x | m f 1 ) T ( | α 2 | = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ) ( x ) α 2 D α 2 b ˜ 2 | x | m f 1 ) + T ( | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! ( x ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 | x | m f 1 ) + T ( j = 1 2 R m j + 1 ( b ˜ j ; x , ) | x | m f 2 ) ,

then

| T b ( f ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | | T ( j = 1 2 R m j ( b ˜ j ; x , ) | x | m f 1 ) | + | T ( | α 1 | = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ) ( x ) α 1 D α 1 b ˜ 1 | x | m f 1 ) | + | T ( | α 2 | = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ) ( x ) α 2 D α 2 b ˜ 2 | x | m f 1 ) | + | T ( | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! ( x ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 | x | m f 1 ) | + | T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | = I 1 ( x ) + I 2 ( x ) + I 3 ( x ) + I 4 ( x ) + I 5 ( x )

and

1 w ( Q ) Q | T b ( f ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | w ( x ) d x 1 w ( Q ) Q I 1 ( x ) w ( x ) d x + 1 w ( Q ) Q I 2 ( x ) w ( x ) d x + 1 w ( Q ) Q I 3 ( x ) w ( x ) d x + 1 w ( Q ) Q I 4 ( x ) w ( x ) d x + 1 w ( Q ) Q I 5 ( x ) w ( x ) d x = I 1 + I 2 + I 3 + I 4 + I 5 .

Now, let us estimate I 1 , I 2 , I 3 , I 4 , and I 5 , respectively. First, for xQ and y Q ˜ , by Lemma 1, we get

R m ( b ˜ j ;x,y)C | x y | m | α j | = m D α j b j B M O ,

thus, by the L p (w)-boundedness of T for 1<p< (Lemma 2) and Hölder’s inequality, we obtain

I 1 C j = 1 2 ( | α j | = m j D α j b j B M O ) 1 w ( Q ) Q | T ( f 1 ) ( x ) | w ( x ) d x C j = 1 2 ( | α j | = m j D α j b j B M O ) ( 1 w ( Q ) R n | T ( f 1 ) ( x ) | p w ( x ) d x ) 1 / p C j = 1 2 ( | α j | = m j D α j b j B M O ) ( 1 w ( Q ) R n | f 1 ( x ) | p w ( x ) d x ) 1 / p C j = 1 2 ( | α j | = m j D α j b j B M O ) ( w ( Q ˜ ) w ( Q ) ) 1 / p f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

For I 2 , since w A 1 , w satisfies the reverse of Hölder’s inequality:

( 1 | Q | Q w ( x ) p 0 d x ) 1 / p 0 C | Q | Q w(x)dx

for all cube Q and some 1< p 0 < (see [13]), thus, by the L p -boundedness of T for p>1, we get

I 2 C | α 2 | = m 2 D α 2 b 2 B M O | α 1 | = m 1 1 w ( Q ) Q | T ( D α 1 b ˜ 1 f 1 ) ( x ) | w ( x ) d x C | α 2 | = m 2 D α 2 b 2 B M O | α 1 | = m 1 ( 1 w ( Q ) R n | T ( D α 1 b ˜ 1 f 1 ) ( x ) | p w ( x ) d x ) 1 / p C | α 2 | = m 2 D α 2 b 2 B M O | α 1 | = m 1 ( 1 w ( Q ) R n | D α 1 b ˜ 1 ( x ) f 1 ( x ) | p w ( x ) d x ) 1 / p C | α 2 | = m 2 D α 2 b 2 B M O | α 1 | = m 1 ( 1 | Q | Q ˜ | D α 1 b 1 ( x ) ( D α 1 b 1 ) Q ˜ | p p 0 d x ) 1 / p p 0 × w ( Q ) 1 / p | Q | 1 / p ( 1 | Q ˜ | Q ˜ w ( x ) p 0 d x ) 1 / p p 0 f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) w ( Q ) 1 / p | Q | 1 / p ( 1 | Q ˜ | Q ˜ w ( x ) d x ) 1 / p f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

For I 3 , similar to the proof of I 2 , we get

I 3 C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

Similarly, for I 4 , choose 1< r 1 , r 2 < such that 1/ r 1 +1/ r 2 +1/ p 0 =1, we obtain, by Hölder’s inequality and the reverse of Hölder’s inequality,

I 4 C | α 1 | = m 1 , | α 2 | = m 2 1 w ( Q ) Q | T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) ( x ) | w ( x ) d x C | α 1 | = m 1 , | α 2 | = m 2 ( 1 w ( Q ) R n | T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) ( x ) | p w ( x ) d x ) 1 / p C | α 1 | = m 1 , | α 2 | = m 2 w ( Q ) 1 / p ( R n | D α 1 b ˜ 1 ( x ) D α 2 b ˜ 2 ( x ) f 1 ( x ) | p w ( x ) d x ) 1 / p C | α 1 | = m 1 , | α 2 | = m 2 ( 1 | Q ˜ | Q ˜ | D α 1 b ˜ 1 ( x ) | p r 1 d x ) 1 / p r 1 ( 1 | Q ˜ | Q ˜ | D α 2 b ˜ 2 ( x ) | p r 2 d x ) 1 / p r 2 × w ( Q ) 1 / p | Q | 1 / p ( 1 | Q ˜ | Q ˜ w ( x ) p 0 d x ) 1 / p p 0 f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

For I 5 , we write

T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( x 0 ) = R n ( K ( x , y ) | x y | m K ( x 0 , y ) | x 0 y | m ) j = 1 2 R m j ( b ˜ j ; x , y ) f 2 ( y ) d y + R n ( R m 1 ( b ˜ 1 ; x , y ) R m 1 ( b ˜ 1 ; x 0 , y ) ) R m 2 ( b ˜ 2 ; x , y ) | x 0 y | m K ( x 0 , y ) f 2 ( y ) d y + R n ( R m 2 ( b ˜ 2 ; x , y ) R m 2 ( b ˜ 2 ; x 0 , y ) ) R m 1 ( b ˜ 1 ; x 0 , y ) | x 0 y | m K ( x 0 , y ) f 2 ( y ) d y | α 1 | = m 1 1 α 1 ! R n [ R m 2 ( b ˜ 2 ; x , y ) ( x y ) α 1 | x y | m K ( x , y ) R m 2 ( b ˜ 2 ; x 0 , y ) ( x 0 y ) α 1 | x 0 y | m K ( x 0 , y ) ] × D α 1 b ˜ 1 ( y ) f 2 ( y ) d y | α 2 | = m 2 1 α 2 ! R n [ R m 1 ( b ˜ 1 ; x , y ) ( x y ) α 2 | x y | m K ( x , y ) R m 1 ( b ˜ 1 ; x 0 , y ) ( x 0 y ) α 2 | x 0 y | m K ( x 0 , y ) ] × D α 2 b ˜ 2 ( y ) f 2 ( y ) d y + | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! R n [ ( x y ) α 1 + α 2 | x y | m K ( x , y ) ( x 0 y ) α 1 + α 2 | x 0 y | m K ( x 0 , y ) ] × D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) f 2 ( y ) d y = I 5 ( 1 ) ( x ) + I 5 ( 2 ) ( x ) + I 5 ( 3 ) ( x ) + I 5 ( 4 ) ( x ) + I 5 ( 5 ) ( x ) + I 5 ( 6 ) ( x ) .

By Lemma 1 and the inequality (see [9])

| b Q 1 b Q 2 |Clog ( | Q 2 | / | Q 1 | ) b B M O for  Q 1 Q 2 ,

we know that, for xQ and y 2 k + 1 Q ˜ 2 k Q ˜ ,

| R m j ( b ˜ j ; x , y ) | C | x y | m j | α | = m j ( D α b j B M O + | ( D α b j ) Q ˜ ( x , y ) ( D α b j ) Q ˜ | ) C k | x y | m j | α | = m j D α b j B M O .

Note that |xy|| x 0 y| for xQ and y R n Q ˜ , we obtain, by the conditions on K,

| I 5 ( 1 ) ( x ) | R n | 1 | x y | m 1 | x 0 y | m | | K ( x , y ) | j = 1 2 | R m j ( b ˜ j ; x , y ) | | f 2 ( y ) | d y + R n | K ( x , y ) K ( x 0 , y ) | | x 0 y | m j = 1 2 | R m j ( b ˜ j ; x , y ) | | f 2 ( y ) | d y k = 0 2 k + 1 Q ˜ 2 k Q ˜ | 1 | x y | m 1 | x 0 y | m | | K ( x , y ) | j = 1 2 | R m j ( b ˜ j ; x , y ) | | f ( y ) | d y + k = 0 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | | x 0 y | m j = 1 2 | R m j ( b ˜ j ; x , y ) | | f ( y ) | d y C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 0 k 2 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 | f ( y ) | d y + C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 0 k 2 ( 2 k + 1 Q ˜ 2 k Q ˜ | f ( y ) | q d y ) 1 / q × ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | q d y ) 1 / q C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 1 k 2 ( 2 k + C k ) f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

For I 5 ( 2 ) (x), by the formula (see [3]):

R m j ( b ˜ j ;x,y) R m j ( b ˜ j ; x 0 ,y)= | β | < m 1 β ! R m | β | ( D β b ˜ j ; x , x 0 ) ( x y ) β

and Lemma 1, we have

| R m j ( b ˜ j ; x , y ) R m j ( b ˜ j ; x 0 , y ) | C | β | < m j | α | = m j | x x 0 | m j | β | | x y | | β | D α b j B M O ,

thus

| I 5 ( 2 ) ( x ) | C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ k | x x 0 | | x 0 y | n + 1 | f ( y ) | d y C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 1 k 2 k f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

Similarly,

| I 5 ( 3 ) ( x ) | C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

For I 5 ( 4 ) (x), similar to the proof of I 5 ( 1 ) (x) and I 5 ( 2 ) (x), we get

| I 5 ( 4 ) ( x ) | C | α 1 | = m 1 R n | R m 2 ( b ˜ 2 ; x , y ) R m 2 ( b ˜ 2 ; x 0 , y ) | | ( x 0 y ) α 1 K ( x 0 , y ) | | x 0 y | m × | D α 1 b ˜ 1 ( y ) | | f 2 ( y ) | d y + C | α 1 | = m 1 R n | ( x y ) α 1 | x y | m ( x 0 y ) α 1 | x 0 y | m | | K ( x , y ) | | R m 2 ( b ˜ 2 ; x , y ) | × | D α 1 b ˜ 1 ( y ) | | f 2 ( y ) | d y + C | α 1 | = m 1 R n | K ( x , y ) K ( x 0 , y ) | | ( x 0 y ) α 1 | x 0 y | m | | R m 2 ( b ˜ 2 ; x , y ) | × | D α 1 b ˜ 1 ( y ) | | f 2 ( y ) | d y C | α 2 | = m 2 D α 2 b 2 B M O k = 1 k 2 k | α 1 | = m 1 ( 1 | 2 k Q ˜ | 2 k Q ˜ | D α 1 b ˜ 1 ( y ) | d y ) f L ( w ) + C | α 2 | = m 2 D α 2 b 2 B M O | α 1 | = m 1 k = 0 k ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | q d y ) 1 / q × ( 2 k + 1 Q ˜ 2 k Q ˜ | D α 1 b ˜ 1 ( y ) | q d y ) 1 / q f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 1 k 2 ( 2 k + C k ) f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

Similarly,

| I 5 ( 5 ) ( x ) | C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

For I 5 ( 6 ) (x), taking 1< r 1 , r 2 < such that 1/ r 1 +1/ r 2 =1, then

| I 5 ( 6 ) ( x ) | C | α 1 | = m 1 , | α 2 | = m 2 R n | ( x y ) α 1 + α 2 K ( x , y ) | x y | m ( x 0 y ) α 1 + α 2 K ( x 0 , y ) | x 0 y | m | × | D α 1 b ˜ 1 ( y ) | | D α 2 b ˜ 2 ( y ) | | f 2 ( y ) | d y C | α 1 | = m 1 , | α 2 | = m 2 k = 1 ( 2 k + C k ) f L ( w ) × ( 1 | 2 k Q ˜ | 2 k Q ˜ | D α 1 b ˜ 1 ( y ) | r 1 d y ) 1 / r 1 ( 1 | 2 k Q ˜ | 2 k Q ˜ | D α 2 b ˜ 2 ( y ) | r 2 d y ) 1 / r 2 C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 1 k 2 ( 2 k + C k ) f L ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

Thus

I 5 C j = 1 2 ( | α j | = m j D α j b j B M O ) f L ( w ) .

This completes the proof of Theorem 1. □

Proof of Theorem 2 It is only for us to prove that there exists a constant C Q such that

1 w ( Q ) Q | T b ( f ) ( x ) C Q | w(x)dxC f B p ( w )

holds for any cube Q=Q(0,d) with d>1. Without loss of generality, we may assume l=2. Fix a cube Q=Q(0,d) with d>1. Let Q ˜ =5 n Q and b ˜ j (x)= b j (x) | α | = m j 1 α ! ( D α b j ) Q ˜ x α , then R m j ( b j ;x,y)= R m j ( b ˜ j ;x,y) and D α b ˜ j = D α b j ( D α b j ) Q ˜ for |α|= m j . Similar to the proof of Theorem 1, we write, for f 1 =f χ Q ˜ and f 2 =f χ R n Q ˜ ,

1 w ( Q ) Q | T b ( f ) ( x ) T b ˜ ( f 2 ) ( 0 ) | w ( x ) d x 1 w ( Q ) Q | T ( j = 1 2 R m j ( b ˜ j ; x , ) | x | m f 1 ) | w ( x ) d x + 1 w ( Q ) Q | T ( | α 1 | = m 1 1 α 1 ! R m 2 ( b ˜ 2 ; x , ) ( x ) α 1 D α 1 b ˜ 1 | x | m f 1 ) | w ( x ) d x + 1 w ( Q ) Q | T ( | α 2 | = m 2 1 α 2 ! R m 1 ( b ˜ 1 ; x , ) ( x ) α 2 D α 2 b ˜ 2 | x | m f 1 ) | w ( x ) d x + 1 w ( Q ) Q | T ( | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! ( x ) α 1 + α 2 D α 1 b ˜ 1 D α 2 b ˜ 2 | x | m f 1 ) | w ( x ) d x + 1 w ( Q ) Q | T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( 0 ) | w ( x ) d x = L 1 + L 2 + L 3 + L 4 + L 5 .

Similar to the proof of Theorem 1, we get

L 1 C j = 1 2 ( | α j | = m j D α j b j B M O ) ( 1 w ( Q ) R n | T ( f 1 ) ( x ) | p w ( x ) d x ) 1 / p C j = 1 2 ( | α j | = m j D α j b j B M O ) w ( Q ˜ ) 1 / p f χ Q ˜ L p ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f B p ( w ) .

For L 2 , taking r,s,t>1 such that r<p, t=p p 0 /(pr), and 1/s+1/(p/r)+1/t=1, then, by the reverse of Hölder’s inequality,

L 2 C | α 2 | = m 2 D α 2 b 2 B M O | α 1 | = m 1 ( 1 w ( Q ) R n | T ( D α 1 b ˜ 1 f 1 ) ( x ) | r w ( x ) d x ) 1 / r C | α 2 | = m 2 D α 2 b 2 B M O w ( Q ) 1 / r | α 1 | = m 1 ( R n | D α 1 b ˜ 1 ( x ) f 1 ( x ) | r w ( x ) d x ) 1 / r C | α 2 | = m 2 D α 2 b 2 B M O w ( Q ) 1 / r | α 1 | = m 1 ( Q ˜ | D α b ˜ 1 ( x ) | r s d x ) 1 / r s × ( Q ˜ | f ( x ) | p w ( x ) d x ) 1 / p ( Q ˜ w ( x ) ( 1 r / p ) t d x ) 1 / r t C j = 1 2 ( | α j | = m j D α j b j B M O ) w ( Q ) 1 / r | Q | 1 / r s f χ Q ˜ L p ( w ) | Q | 1 / r t × ( 1 | Q ˜ | Q ˜ w ( x ) p 0 d x ) 1 / r t C j = 1 2 ( | α j | = m j D α j b j B M O ) w ( Q ) 1 / r | Q | 1 / r s f χ Q ˜ L p ( w ) ( 1 | Q ˜ | Q ˜ w ( x ) d x ) p 0 / r t C j = 1 2 ( | α j | = m j D α j b j B M O ) w ( Q ˜ ) 1 / p f χ Q ˜ L p ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f B p ( w ) , L 3 C j = 1 2 ( | α j | = m j D α j b j B M O ) f B p ( w ) .

For L 4 , taking r, s 1 , s 2 ,t>1 such that r<p, t=p p 0 /(pr), and 1/ s 1 +1/ s 2 +1/(p/r)+1/t=1, then, by the reverse of Hölder’s inequality,

L 4 C | α 1 | = m 1 , | α 2 | = m 2 ( 1 w ( Q ) R n | T ( D α 1 b ˜ 1 D α 2 b ˜ 2 f 1 ) ( x ) | r w ( x ) d x ) 1 / r C w ( Q ) 1 / r | α 1 | = m 1 , | α 2 | = m 2 ( R n | D α 1 b ˜ 1 ( x ) D α 2 b ˜ 2 ( x ) f 1 ( x ) | r w ( x ) d x ) 1 / r C w ( Q ) 1 / r | α 1 | = m 1 , | α 2 | = m 2 ( Q ˜ | D α b ˜ 1 ( x ) | r s 1 d x ) 1 / r s 1 ( Q ˜ | D α b ˜ 2 ( x ) | r s 2 d x ) 1 / r s 2 × ( Q ˜ | f ( x ) | p w ( x ) d x ) 1 / p ( Q ˜ w ( x ) ( 1 r / p ) t d x ) 1 / r t C j = 1 2 ( | α j | = m j D α j b j B M O ) w ( Q ) 1 / r | Q | 1 / r s 1 + 1 / r s 2 + 1 / r t f χ Q ˜ L p ( w ) × ( 1 | Q ˜ | Q ˜ w ( x ) d x ) p 0 / r t C j = 1 2 ( | α j | = m j D α j b j B M O ) w ( Q ˜ ) 1 / p f χ Q ˜ L p ( w ) C j = 1 2 ( | α j | = m j D α j b j B M O ) f B p ( w ) .

For L 5 , similar to the proof of the proof of I 5 in Theorem 1, we have

T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( 0 ) = R n ( K ( x , y ) | x y | m K ( 0 , y ) | y | m ) j = 1 2 R m j ( b ˜ j ; x , y ) f 2 ( y ) d y + R n ( R m 1 ( b ˜ 1 ; x , y ) R m 1 ( b ˜ 1 ; 0 , y ) ) R m 2 ( b ˜ 2 ; x , y ) | y | m K ( 0 , y ) f 2 ( y ) d y + R n ( R m 2 ( b ˜ 2 ; x , y ) R m 2 ( b ˜ 2 ; 0 , y ) ) R m 1 ( b ˜ 1 ; x 0 , y ) | y | m K ( 0 , y ) f 2 ( y ) d y | α 1 | = m 1 1 α 1 ! R n [ R m 2 ( b ˜ 2 ; x , y ) ( x y ) α 1 | x y | m K ( x , y ) R m 2 ( b ˜ 2 ; 0 , y ) ( y ) α 1 | y | m K ( 0 , y ) ] × D α 1 b ˜ 1 ( y ) f 2 ( y ) d y | α 2 | = m 2 1 α 2 ! R n [ R m 1 ( b ˜ 1 ; x , y ) ( x y ) α 2 | x y | m K ( x , y ) R m 1 ( b ˜ 1 ; 0 , y ) ( y ) α 2 | y | m K ( 0 , y ) ] × D α 2 b ˜ 2 ( y ) f 2 ( y ) d y + | α 1 | = m 1 , | α 2 | = m 2 1 α 1 ! α 2 ! R n [ ( x y ) α 1 + α 2 | x y | m K ( x , y ) ( y ) α 1 + α 2 | y | m K ( 0 , y ) ] × D α 1 b ˜ 1 ( y ) D α 2 b ˜ 2 ( y ) f 2 ( y ) d y = L 5 ( 1 ) ( x ) + L 5 ( 2 ) ( x ) + L 5 ( 3 ) ( x ) + L 5 ( 4 ) ( x ) + L 5 ( 5 ) ( x ) + L 5 ( 6 ) ( x ) .

For L 5 ( 1 ) (x), taking 1<r< such that 1/p+1/q+1/r=1, by w A 1 A p / r + 1 , we get

| L 5 ( 1 ) ( x ) | C k = 0 2 k + 1 Q ˜ 2 k Q ˜ | 1 | x y | m 1 | y | m | | K ( x , y ) | j = 1 2 | R m j ( b ˜ j ; x , y ) | | f ( y ) | d y + k = 0 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | | y | m × j = 1 2 | R m j ( b ˜ j ; x , y ) | | f ( y ) | w ( y ) 1 / p w ( y ) 1 / p d y C j = 1 2 ( | α | = m j D α b j B M O ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ k 2 d ( 2 k d ) n + 1 | f ( y ) | d y + C j = 1 2 ( | α | = m j D α b j B M O ) k = 0 k 2 ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | q d y ) 1 / q × ( 2 k + 1 Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p ( 2 k + 1 Q ˜ w ( y ) r / p d y ) 1 / r C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 1 k 2 2 k w ( 2 k Q ˜ ) 1 / p ( 2 k Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / p ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) 1 / ( p 1 ) d y ) ( p 1 ) / p + C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 1 k 2 C k w ( 2 k Q ˜ ) 1 / p ( 2 k Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / p ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) r / p d y ) 1 / r C j = 1 2 ( | α j | = m j D α j b j B M O ) f B p ( w ) .

Similarly, we get, for 1< r 1 , r 2 , r 3 , r 4 ,s< with 1/p+1/ r 1 +1/s=1, 1/p+1/q+1/ r 2 +1/s=1, and 1/p+1/q+1/ r 3 +1/ r 4 +1/s=1,

| L 5 ( 2 ) ( x ) + L 5 ( 3 ) ( x ) + L 5 ( 4 ) ( x ) + L 5 ( 5 ) ( x ) + L 5 ( 6 ) ( x ) | C ( | α | = m 2 D α b 2 B M O ) | α | = m 1 k = 0 2 k + 1 Q ˜ 2 k Q ˜ k d ( 2 k d ) n + 1 | D α 1 b ˜ 1 ( y ) | | f ( y ) | d y + C ( | α | = m 1 D α b 1 B M O ) | α | = m 2 k = 0 2 k + 1 Q ˜ 2 k Q ˜ k d ( 2 k d ) n + 1 | D α 2 b ˜ 2 ( y ) | | f ( y ) | d y + C ( | α | = m 2 D α b 2 B M O ) | α | = m 1 k = 0 k 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | × | D α 1 b ˜ 1 ( y ) | | f ( y ) | d y + C ( | α | = m 1 D α b 1 B M O ) | α | = m 2 k = 0 k 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | × | D α 2 b ˜ 2 ( y ) | | f ( y ) | d y + C | α 1 | = m 1 , | α 2 | = m 2 k = 0 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | | D α 1 b ˜ 1 ( y ) | | D α 2 b ˜ 2 ( y ) | | f ( y ) | d y C ( | α | = m 2 D α b 2 B M O ) | α | = m 1 k = 0 k ( 2 k + 1 Q ˜ | D α 1 b ˜ 1 ( y ) | r 1 d y ) 1 / r 1 × ( 2 k + 1 Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p ( 2 k + 1 Q ˜ w ( y ) s / p d y ) 1 / s + C ( | α | = m 1 D α b 1 B M O ) | α | = m 2 k = 0 k ( 2 k + 1 Q ˜ | D α 2 b ˜ 2 ( y ) | r 1 d y ) 1 / r 1 × ( 2 k + 1 Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p ( 2 k + 1 Q ˜ w ( y ) s / p d y ) 1 / s + C ( | α | = m 2 D α b 2 B M O ) | α | = m 1 k = 0 k ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | q d y ) 1 / q × ( 2 k + 1 Q ˜ | D α 1 b ˜ 1 ( y ) | r 2 d y ) 1 / r 2 ( 2 k + 1 Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p ( 2 k + 1 Q ˜ w ( y ) s / p d y ) 1 / s + C ( | α | = m 1 D α b 1 B M O ) | α | = m 2 k = 0 k ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | q d y ) 1 / q × ( 2 k + 1 Q ˜ | D α 2 b ˜ 2 ( y ) | r 2 d y ) 1 / r 2 ( 2 k + 1 Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p ( 2 k + 1 Q ˜ w ( y ) s / p d y ) 1 / s + C | α 1 | = m 1 , | α 2 | = m 2 k = 0 ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( 0 , y ) | q d y ) 1 / q ( 2 k + 1 Q ˜ w ( y ) s / p d y ) 1 / s × ( 2 k + 1 Q ˜ | D α 1 b ˜ 1 ( y ) | r 3 d y ) 1 / r 3 ( 2 k + 1 Q ˜ | D α 2 b ˜ 2 ( y ) | r 4 d y ) 1 / r 4 × ( 2 k + 1 Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p C j = 1 2 ( | α j | = m j D α j b j B M O ) k = 1 k 2 ( 2 k + C k ) w ( 2 k Q ˜ ) 1 / p ( 2 k Q ˜ | f ( y ) | p w ( y ) d y ) 1 / p C j = 1 2 ( | α j | = m j D α j b j B M O ) f B p ( w ) .

Thus

L 5 C j = 1 2 ( α j | = m j D α j b j B M O ) f B p ( w ) .

This finishes the proof of Theorem 2. □

References

  1. Cohen J:A sharp estimate for a multilinear singular integral on R n . Indiana Univ. Math. J. 1981, 30: 693-702. 10.1512/iumj.1981.30.30053

    MathSciNet  Article  MATH  Google Scholar 

  2. Cohen J, Gosselin J:On multilinear singular integral operators on R n . Stud. Math. 1982, 72: 199-223.

    MathSciNet  MATH  Google Scholar 

  3. Cohen J, Gosselin J: A BMO estimate for multilinear singular integral operators. Ill. J. Math. 1986, 30: 445-465.

    MathSciNet  MATH  Google Scholar 

  4. Coifman R, Meyer Y Cambridge Studies in Advanced Math. 48. In Wavelets, Calderón-Zygmund and Multilinear Operators. Cambridge University Press, Cambridge; 1997.

    Google Scholar 

  5. Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611-635. 10.2307/1970954

    MathSciNet  Article  MATH  Google Scholar 

  6. Garcia-Cuerva J, Rubio de Francia JL North-Holland Mathematics Studies 116. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.

    Google Scholar 

  7. Pérez C, Pradolini G: Sharp weighted endpoint estimates for commutators of singular integral operators. Mich. Math. J. 2001, 49: 23-37. 10.1307/mmj/1008719033

    Article  MATH  Google Scholar 

  8. Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672-692. 10.1112/S0024610702003174

    Article  MathSciNet  MATH  Google Scholar 

  9. Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.

    MATH  Google Scholar 

  10. Harboure E, Segovia C, Torrea JL: Boundedness of commutators of fractional and singular integrals for the extreme values of p . Ill. J. Math. 1997, 41: 676-700.

    MathSciNet  MATH  Google Scholar 

  11. Chang DC, Li JF, Xiao J: Weighted scale estimates for Calderón-Zygmund type operators. Contemp. Math. 2007, 446: 61-70.

    MathSciNet  Article  MATH  Google Scholar 

  12. Lin Y: Sharp maximal function estimates for Calderón-Zygmund type operators and commutators. Acta Math. Sci. Ser. A Chin. Ed. 2011, 31: 206-215.

    MathSciNet  MATH  Google Scholar 

  13. Liu LZ: Weighted boundedness of multilinear operators for the extreme cases. Taiwan. J. Math. 2006, 10: 669-690.

    MathSciNet  MATH  Google Scholar 

  14. Liu LZ: Weighted Herz type spaces estimates of multilinear singular integral operators for the extreme cases. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2007, 101: 87-98.

    MathSciNet  MATH  Google Scholar 

  15. Liu LZ: Endpoint estimates for multilinear fractional singular integral operators on some Hardy spaces. Math. Notes 2010, 88: 735-752.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The work is supported by NNSFC (No. 11301097), Master Foundation of Guangxi University of Science and Technology (No. 0816208), Guangxi Education Institution Scientific Research Item (No. 2013YB170), GXNSF Grant (No. 2013GXNSFAA019001).

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The authors completed the paper. MZ carried out the ideas and methods of studies, YG participated in the design of the study and performed the statistical analysis, the sequence alignment and drafted the manuscript. The authors read and approved the final manuscript.

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Zhang, M., Guo, Y. Weighted boundedness of multilinear singular integral operator with general kernels for the extreme cases. J Inequal Appl 2014, 341 (2014). https://doi.org/10.1186/1029-242X-2014-341

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Keywords

  • multilinear operator
  • singular integral operator
  • BMO space