Open Access

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

Journal of Inequalities and Applications20142014:341

https://doi.org/10.1186/1029-242X-2014-341

Received: 1 June 2014

Accepted: 21 August 2014

Published: 3 September 2014

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.

Keywords

multilinear operator singular integral operator BMO space

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 b B M O ( R n ) , a classical result of Coifman et al. (see [5]) stated that the commutator [ b , T ] ( f ) = T ( b f ) b T ( 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 ) d x , w Q = | Q | 1 Q w ( x ) d x , the weighted sharp function of b is defined by
b # ( x ) = sup Q x 1 w ( Q ) Q | b ( y ) b Q | w ( y ) d y .
We say that b belongs to B M O ( w ) if b # belongs to L ( w ) , and we define b B M O ( w ) = b # L ( w ) . If w = 1 , we denote B M O ( w ) = B M O ( R n ) . It has been known that (see [9])
b b 2 k Q B M O C k b B M O .
We also define the central B M O space by C M O ( 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 | d x < .
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 | d x .
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 ) d y
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 k 1 ,
( 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 1 j l ,
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 ) d y .

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 B M O ( R n ) for all α with | α | = m j and j = 1 , , l . Then T b is bounded from L ( w ) to B M O ( 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 B M O ( R n ) for all α with | α | = m j and j = 1 , , l . Then T b is bounded from B p ( w ) to C M O ( 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 | x y | .

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 ) d x C 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 x Q 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 ) d x
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 | C log ( | Q 2 | / | Q 1 | ) b B M O for  Q 1 Q 2 ,
we know that, for x Q 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 | x y | | x 0 y | for x Q 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 ) d x C 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 / ( p r ) , 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 / ( p r ) , 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. □

Declarations

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).

Authors’ Affiliations

(1)
College of Science, Guangxi University of Science and Technology

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© Zhang and Guo; licensee Springer. 2014

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