Open Access

Weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels

Journal of Inequalities and Applications20142014:371

https://doi.org/10.1186/1029-242X-2014-371

Received: 9 February 2014

Accepted: 29 August 2014

Published: 25 September 2014

Abstract

In this paper, we prove the weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels.

Keywords

multilinear commutator singular integral operator BMO

1 Introduction

Let b BMO ( R n ) and T be the Calderón-Zygmund operator, the commutator [ b , T ] generated by b and T is defined by [ b , T ] ( f ) ( x ) = b ( x ) T ( f ) ( x ) T ( b f ) ( x ) . A classical result of Coifman et al. (see [1]) proved that the commutator [ b , T ] is bounded on L p ( R n ) ( 1 < p < ). In [2, 3], the boundedness properties of the commutators for the extreme values of p are obtained. In this paper, we will introduce the multilinear commutator of singular integral operators with non-smooth kernels and prove the weighted boundedness properties of the operator for the extreme cases.

First let us introduce some notations (see [312]). In this paper, Q will denote a cube of R n with sides parallel to the axes. For a cube Q and a function b, let b Q = | Q | 1 Q b ( x ) d x and b ( Q ) = Q b ( x ) d x , the sharp function of b is defined by
b # ( x ) = sup Q x 1 | Q | Q | b ( y ) b Q | d y .
It is well known that (see [6])
b # ( x ) = sup Q x inf c C 1 | Q | Q | b ( y ) c | d y .
Moreover, for a weight function ω (that is, a non-negative locally integrable function), b is said to belong to BMO ( ω ) if b # L ( ω ) and define b BMO ( ω ) = b # L ( ω ) , if ω = 1 , we denote BMO ( ω ) = BMO ( R n ) . It is well known that (see [11])
b b 2 k Q BMO C k b BMO .
The A p weight is defined by (see [6])
A p = { 0 < ω L loc 1 ( R n ) : sup Q ( 1 | Q | Q ω ( x ) d x ) ( 1 | Q | Q ω ( x ) 1 / ( p 1 ) d x ) p 1 < } , 1 < p <
and
A 1 = { 0 < ω L loc 1 ( R n ) : sup Q x 1 | Q | Q ω ( y ) d y c ω ( x ) , a.e. } .
Definition 1 A family of operators D t , t > 0 , is said to be an ‘approximation to the identity’ if, for every t > 0 , D t can be represented by the kernel a t ( x , y ) in the following sense:
D t ( f ) ( x ) = R n a t ( x , y ) f ( y ) d y
for every f L p ( R n ) with p 1 , and a t ( x , y ) satisfies
| a t ( x , y ) | h t ( x , y ) = C t n / 2 s ( | x y | 2 / t ) ,
where s is a positive, bounded, and decreasing function satisfying
lim r r n + ϵ s ( r 2 ) = 0

for some ϵ > 0 .

Definition 2 A linear operator T is called the singular integral operators with non-smooth kernels if T is bounded on L 2 ( R n ) and associated with a kernel K ( x , y ) such that
T ( f ) ( x ) = R n K ( x , y ) f ( y ) d y
for every continuous function f with compact support, and for almost all x not in the support of f.
  1. (1)
    There exists an ‘approximation to the identity’ { B t , t > 0 } such that T B t has associated kernel k t ( x , y ) and there exist c 1 , c 2 > 0 so that
    | x y | > c 1 t 1 / 2 | K ( x , y ) k t ( x , y ) | d x c 2 for all  y R n .
     
  2. (2)
    There exists an ‘approximation to the identity’ { A t , t > 0 } such that A t T has associated kernel K t ( x , y ) which satisfies
    | K t ( x , y ) | c 4 t n / 2 if  | x y | c 3 t 1 / 2
    and
    | K ( x , y ) K t ( x , y ) | c 4 t δ / 2 | x y | n δ if  | x y | c 3 t 1 / 2

    for some c 3 , c 4 > 0 , δ > 0 .

     
Given some locally integrable functions b j ( j = 1 , , m ). The multilinear operator associated to T is defined by
T b ( f ) ( x ) = R n [ j = 1 m ( b j ( x ) b j ( y ) ) ] K ( x , y ) f ( y ) d y .
Definition 3 Given the ‘approximations to the identity’ { A t , t > 0 } and a weight function ω.
  1. (1)
    The weighted BMO space associated with { A t , t > 0 } is defined by
    BMO A ( ω ) = { f L loc 1 ( R n ) : f BMO A ( ω ) < } ,
    where
    f BMO A ( ω ) = sup Q 1 ω ( Q ) Q | f ( x ) A t Q ( f ) ( x ) | ω ( x ) d x ,

    t Q = l ( Q ) 2 and l ( Q ) denotes the side length of Q.

     
  2. (2)
    The weighted central BMO space associated with { A t , t > 0 } is defined by
    CMO A ( ω ) = { f L loc 1 ( R n ) : f CMO A ( ω ) < } ,
    where
    f CMO ( ω ) = sup r > 1 1 ω ( Q ( 0 , r ) ) Q | f ( x ) A t Q f ( x ) | ω ( x ) d x ,

    and t Q = r 2 .

     
Definition 4 Let 1 < p < and ω be a weighted function on R n . We shall call B p ( ω ) the space of those functions f on R n , such that
f B p ( ω ) = sup r > 1 [ ω ( Q ( 0 , r ) ) ] 1 / p f χ Q ( 0 , r ) L p ( ω ) < .

For b j BMO ( R n ) ( j = 1 , , m ), set b BMO = j = 1 m b j BMO . Given a positive integer m and 1 j m , we denote by C j m the family of all finite subsets σ = { σ ( 1 ) , , σ ( j ) } of { 1 , , m } of j different elements. For σ C j m , set σ c = { 1 , , m } σ . For b = ( b 1 , , b m ) and σ = { σ ( 1 ) , , σ ( j ) } C j m , set b σ = ( b σ ( 1 ) , , b σ ( j ) ) , b σ = b σ ( 1 ) b σ ( j ) and b σ BMO = b σ ( 1 ) BMO b σ ( j ) BMO .

2 Theorems and proofs

We begin with some preliminaries lemmas.

Lemma 1 ([5, 7])

Let ω A 1 , 1 < p , and T be the singular integral operators with non-smooth kernels. Then T is boundedness on L p ( w ) .

Lemma 2 Let ω A 1 , { A t , t > 0 } be anapproximation to the identityand b BMO ( R n ) . Then
  1. (a)
    for every f L ( R n ) , 1 p < , and any cube Q,
    ( 1 | Q | Q | A t Q ( ( b b Q ) f ) ( y ) | p d y ) 1 / p C b BMO f L ;
     
  2. (b)
    for every f B p ( ω ) , 1 r < p < , and any cube Q,
    ( 1 ω ( Q ) Q | A t Q ( ( b b Q ) f ) ( y ) | r ω ( y ) d y ) 1 / r C b BMO f B p ( ω ) ,
     

where t Q = l ( Q ) 2 and l ( Q ) denotes the side length of Q.

Proof (a) Write
( 1 | Q | Q | A t Q ( ( b b Q ) f ) ( y ) | p d y ) 1 / p ( 1 | Q | Q R n h t Q ( x , y ) p | ( b ( y ) b Q ) f ( y ) | p d y d x ) 1 / p ( 1 | Q | Q 2 Q h t Q ( x , y ) p | ( b ( y ) b Q ) f ( y ) | p d y d x ) 1 / p + ( k = 1 1 | Q | Q 2 k + 1 Q 2 k Q h t Q ( x , y ) p | ( b ( y ) b Q ) f ( y ) | p d y d x ) 1 / p = I 1 + I 2 .
We have, by Hölder’s inequality,
I 1 ( C | Q | | 2 Q | Q 2 Q | ( b ( y ) b Q ) f ( y ) | p d y d x ) 1 / p C f L ( 1 | 2 Q | 2 Q | b ( y ) b Q | p d y ) 1 / p C b BMO f L .
For I 2 , for x Q and y 2 k + 1 Q 2 k Q , we have | x y | 2 k 1 t Q and h t Q ( x , y ) C s ( 2 2 ( k 1 ) ) | Q | . Thus
I 2 C k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( 1 | 2 k + 1 Q | 2 k + 1 Q | ( b ( y ) b Q ) f ( y ) | p d y ) 1 / p C f L k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( 1 | 2 k + 1 Q | 2 k + 1 Q | b ( y ) b 2 k + 1 Q | p d y ) 1 / p + C f L k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) | b Q b 2 k + 1 Q | C f L k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( k + 1 ) b BMO C b BMO f L ,
where the last inequality follows from
k = 2 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( k + 1 ) C k = 2 k 2 ( k 1 ) ϵ <
for some ϵ > 0 .
  1. (b)
    Write
    ( 1 ω ( Q ) Q | A t Q ( ( b b Q ) f ) ( y ) | r ω ( y ) d y ) 1 / r ( 1 ω ( Q ) Q R n h t Q ( x , y ) r | ( b ( y ) b Q ) f ( y ) | r ω ( y ) d y d x ) 1 / r ( 1 ω ( Q ) Q 2 Q h t Q ( x , y ) p | ( b ( y ) b Q ) f ( y ) | r ω ( y ) d y d x ) 1 / r + ( k = 1 1 ω ( Q ) Q 2 k + 1 Q 2 k Q h t Q ( x , y ) r | ( b ( y ) b Q ) f ( y ) | r ω ( y ) d y d x ) 1 / r = I + II .
     
For I, since ω A 1 , ω satisfies the reverse of Hölder’s inequality
( 1 | Q | Q ω ( x ) q d x ) 1 / q C | Q | Q ω ( x ) d x
for some 1 < q < , and ω ( Q 2 ) | Q 2 | | Q 1 | ω ( Q 1 ) C for all cubes Q 1 , Q 2 with Q 1 Q 2 , ω A p / r u for 1 < u , v < with u v = q and p > r u (see [6]). We have, by Hölder’s inequality,
I ( C ω ( Q ) | Q | Q 2 Q | ( b ( y ) b Q ) f ( y ) | r ω ( y ) d y d x ) 1 / r I C ( 1 ω ( Q ) 2 Q | ( b ( y ) b Q ) f ( y ) | r ω ( y ) d y ) 1 / r I C [ | 2 Q | ω ( Q ) ( 1 | 2 Q | 2 Q | b ( y ) b Q | r u ω ( y ) u d y ) 1 / u ( 1 | 2 Q | 2 Q | f ( y ) | r u d y ) 1 / u ] 1 / r I C ( | 2 Q | ω ( Q ) ) 1 / r ( 1 | 2 Q | 2 Q | b ( y ) b Q | r u v d y ) 1 / r u v ( 1 | 2 Q | 2 Q ω ( y ) u v d y ) 1 / r u v I × ( 1 | 2 Q | 2 Q | f ( y ) | r u d y ) 1 / r u I C b BMO ( | 2 Q | ω ( 2 Q ) ) 1 / r ( ω ( 2 Q ) | 2 Q | ) 1 / r ( 1 | 2 Q | 2 Q | f ( y ) | r u ω ( y ) r u p ω ( y ) r u p d y ) 1 / r u I C b BMO ( 1 | 2 Q | 2 Q ( | f ( y ) | r u ω ( y ) r u p ) p r u d y ) 1 / p ( 1 | 2 Q | 2 Q ω ( y ) r u p p p r u d y ) ( p r u ) / p r u I C b BMO ( 1 | 2 Q | ) 1 / p f χ 2 Q L p ( ω ) ( 1 | 2 Q | 2 Q ω ( y ) d y ) 1 / p I × [ ( 1 | 2 Q | 2 Q ω ( y ) d y ) ( 1 | 2 Q | 2 Q ω ( y ) 1 p r u 1 d y ) p r u 1 ] 1 / p I C b BMO ω ( 2 Q ) 1 / p f χ 2 Q L p ( ω ) I C b BMO f B p ( ω ) ; II C ( | Q | ω ( Q ) ) 1 / r k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( 1 | 2 k + 1 Q | 2 k + 1 Q | ( b ( y ) b Q ) f ( y ) | r ω ( y ) d y ) 1 / r II C ( | Q | ω ( Q ) ) 1 / r k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( 1 | 2 k + 1 Q | 2 k + 1 Q | b ( y ) b 2 k + 1 Q | r u ω ( y ) u d y ) 1 / r u II × ( 1 | 2 k + 1 Q | 2 k + 1 Q | f ( y ) | r u d y ) 1 / r u II C ( | Q | ω ( Q ) ) 1 / r k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( 1 | 2 k + 1 Q | 2 k + 1 Q | b ( y ) b 2 k + 1 Q | r u v d y ) 1 / r u v II × ( 1 | 2 k + 1 Q | 2 k + 1 Q ω ( y ) u v d y ) 1 / r u v ( 1 | 2 k + 1 Q | 2 k + 1 Q f ( y ) r u d y ) 1 / r u II C b BMO k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( k + 1 ) ( | Q | ω ( Q ) ω ( 2 k + 1 Q ) | 2 k + 1 Q | ) 1 / r II × ( 1 | 2 k + 1 Q | 2 k + 1 Q f ( y ) r u d y ) 1 / r u II C k = 1 2 ( k 1 ) n s ( 2 2 ( k 1 ) ) ( k + 1 ) b BMO ω ( 2 k + 1 Q ) 1 / p f χ 2 k + 1 Q L p ( ω ) II C b BMO f B p ( w ) .

This completes the proof. □

Theorem 1 Let T be the singular integral operators with non-smooth kernels, ω A 1 and b = ( b 1 , , b m ) with b j BMO ( R n ) for 1 j m . Then T b is bounded from L ( ω ) to BMO A ( ω ) .

Proof It suffices to prove, for f C 0 ( R n ) , the following inequality holds:
1 ω ( Q ) Q | T b ( f ) ( x ) A t Q T b ( f ) ( x ) | ω ( x ) d x C f L ( ω ) .

We fix a cube Q = Q ( x 0 , d ) . We decompose f into f = f 1 + f 2 with f 1 = f χ Q , f 2 = f χ ( R n Q ) .

When m = 1 , set ( b 1 ) Q = | Q | 1 Q b 1 ( y ) d y , we have
T b 1 ( f ) ( x ) = R n [ ( b 1 ( x ) ( b 1 ) Q ) ( b 1 ( y ) ( b 1 ) Q ) ] K ( x , y ) f ( y ) d y = ( b 1 ( x ) ( b 1 ) Q ) R n K ( x , y ) f ( y ) d y R n ( b 1 ( y ) ( b 1 ) Q ) K ( x , y ) f ( y ) d y
and
A t Q T b 1 ( f ) ( x ) = ( b 1 ( x ) ( b 1 ) Q ) R n K t ( x , y ) f ( y ) d y R n ( b 1 ( y ) ( b 1 ) Q ) K t ( x , y ) f ( y ) d y .
Then
| T b 1 ( f ) ( x ) A t Q T b 1 ( f ) ( x ) | | ( b 1 ( x ) ( b 1 ) Q ) R n K ( x , y ) f ( y ) d y | + | R n ( b 1 ( y ) ( b 1 ) Q ) K ( x , y ) f 1 ( y ) d y | + | ( b 1 ( x ) ( b 1 ) Q ) R n K t ( x , y ) f ( y ) d y | + | R n ( b 1 ( y ) ( b 1 ) Q ) K t ( x , y ) f 1 ( y ) d y | + | R n ( b 1 ( y ) ( b 1 ) Q ) ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y | = I 1 ( x ) + I 2 ( x ) + I 3 ( x ) + I 4 ( x ) + I 5 ( x ) .
For I 1 ( x ) , let 1 / p + 1 / p = 1 , 1 / q + 1 / q = 1 , by the reverse of Hölder’s inequality with 1 < q < , Lemma 1, and Hölder’s inequality, we have
1 ω ( Q ) Q | I 1 ( x ) | ω ( x ) d x C ω ( Q ) ( Q | b 1 ( x ) ( b 1 ) Q | p ω ( x ) d x ) 1 / p ( R n | T ( f ) ( x ) | p ω ( x ) χ Q ( x ) d x ) 1 / p C ω ( Q ) ( Q | b 1 ( x ) ( b 1 ) Q | p ω ( x ) d x ) 1 / p ( R n | f ( x ) | p ω ( x ) χ Q ( x ) d x ) 1 / p C ω ( Q ) ( Q | b 1 ( x ) ( b 1 ) Q | p ω ( x ) d x ) 1 / p f L ( ω ) ( Q ω ( x ) d x ) 1 / p C ω ( Q ) [ ( Q | b 1 ( x ) ( b 1 ) Q | p q d x ) 1 / q ( Q ω ( x ) q d x ) 1 / q ] 1 / p f L ( ω ) ω ( Q ) 1 / p C ω ( Q ) 1 / p 1 | Q | 1 / p b 1 BMO ( 1 | Q | Q ω ( x ) q d x ) 1 / p q f L ( ω ) C b 1 BMO f L ( ω ) .
For I 2 ( x ) , taking p > 1 , by Hölder’s inequality, we have
1 ω ( Q ) Q | I 2 ( x ) | ω ( x ) d x ( 1 ω ( Q ) R n | T ( ( b 1 ( b 1 ) Q ) f 1 ) ( x ) | p ω ( x ) χ Q ( x ) d x ) 1 / p C ω ( Q ) 1 / p ( R n | ( b 1 ( x ) ( b 1 ) Q ) f 1 ( x ) | p ω ( x ) χ Q ( x ) d x ) 1 / p C ω ( Q ) 1 / p [ ( Q | b 1 ( x ) ( b 1 ) Q | p q d x ) 1 / q ( Q | f ( x ) | p q ω ( x ) q d x ) 1 / q ] 1 / p C ω ( Q ) 1 / p ( Q | b 1 ( x ) ( b 1 ) Q | p q d x ) 1 / p q ( Q | f ( x ) | p q ω ( x ) q d x ) 1 / p q C ω ( Q ) 1 / p ( Q | b 1 ( x ) ( b 1 ) Q | p q d x ) 1 / p q ( Q ω ( x ) q d x ) 1 / p q f L ( ω ) C ω ( Q ) 1 / p | Q | 1 / p q b 1 BMO | Q | 1 / p q ( 1 | Q | Q ω ( x ) q d x ) 1 / p q f L ( ω ) C b 1 BMO ( | Q | ω ( Q ) ) 1 / p ( 1 | Q | Q ω ( x ) d x ) 1 / p f L ( ω ) C b 1 BMO f L ( ω ) .
For I 3 ( x ) and I 4 ( x ) , we get, for 1 < p 1 , p 2 < with 1 / p 1 + 1 / p 2 + 1 / q = 1 ,
1 ω ( Q ) Q | I 3 ( x ) | ω ( x ) d x C ω ( Q ) Q | b 1 ( x ) ( b 1 ) Q | | A t Q ( f ) ( x ) | ω ( x ) d x C | Q | ω ( Q ) ( 1 | Q | Q | b 1 ( x ) ( b 1 ) Q | p 1 d x ) 1 / p 1 × ( 1 | Q | Q | A t Q ( f ) ( x ) | p 2 d x ) 1 / p 2 ( 1 | Q | Q ω ( x ) q d x ) 1 / q C | Q | ω ( Q ) b 1 BMO f L ( ω ) ω ( Q ) | Q | C b 1 BMO f L ( ω ) , 1 ω ( Q ) Q | I 4 ( x ) | ω ( x ) d x 1 ω ( Q ) R n | A t Q ( ( b 1 ( b 1 ) Q ) f 1 ) ( x ) | ω ( x ) d x C | Q | ω ( Q ) ( 1 | Q | Q | A t Q ( ( b 1 ( b 1 ) Q ) f 1 ) ( x ) | q d x ) 1 / q ( 1 | Q | Q ω ( x ) q d x ) 1 / q C | Q | ω ( Q ) b 1 BMO f L ( ω ) ω ( Q ) | Q | C b 1 BMO f L ( ω ) .
For I 5 ( x ) , we have
I 5 ( x ) = | R n ( b 1 ( y ) ( b 1 ) Q ) ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y | C k = 0 2 k + 1 Q 2 k Q | b 1 ( y ) ( b 1 ) Q | | f ( y ) | d δ | x 0 y | n + δ d y C k = 1 d δ ( 2 k 1 d ) n + δ | 2 k Q | ( 1 | 2 k Q | 2 k Q | f ( y ) | p d y ) 1 / p × ( 1 | 2 k Q | 2 k Q | b 1 ( y ) ( b 1 ) Q | p d y ) 1 / p C k = 1 k m 2 k δ b 1 BMO f L ( ω ) C b 1 BMO f L ( ω ) ,
so
1 ω ( Q ) Q | I 5 ( x ) | ω ( x ) d x C b 1 BMO f L ( ω ) .
When m > 1 , set b Q = ( ( b 1 ) Q , , ( b m ) Q ) R n , where ( b j ) Q = | Q | 1 Q b j ( y ) d y , 1 j m , we have
T b ( f ) ( x ) = j = 1 m ( b j ( x ) ( b j ) Q ) R n K ( x , y ) f ( y ) d y + j = 1 m 1 σ C j m ( 1 ) m j ( b ( x ) ( b ) Q ) σ R n ( b ( y ) ( b ) Q ) σ c K ( x , y ) f ( y ) d y + ( 1 ) m R n j = 1 m ( b j ( y ) ( b j ) Q ) K ( x , y ) f ( y ) d y
and
A t Q T b ( f ) ( x ) = j = 1 m ( b j ( x ) ( b j ) Q ) R n K t ( x , y ) f ( y ) d y + j = 1 m 1 σ C j m ( 1 ) m j ( b ( x ) ( b ) Q ) σ R n ( b ( y ) ( b ) Q ) σ c K t ( x , y ) f ( y ) d y + ( 1 ) m R n j = 1 m ( b j ( y ) ( b j ) Q ) K t ( x , y ) f ( y ) d y ,
then
| T b ( f ) ( x ) A t Q T b ( f ) ( x ) | | j = 1 m ( b j ( x ) ( b j ) Q ) R n K ( x , y ) f ( y ) d y | + | j = 1 m 1 σ C j m ( b ( x ) ( b ) Q ) σ R n ( b ( y ) ( b ) Q ) σ c K ( x , y ) f ( y ) d y | + | R n j = 1 m ( b j ( y ) ( b j ) Q ) K ( x , y ) f 1 ( y ) d y | + | j = 1 m ( b j ( x ) ( b j ) Q ) R n K t ( x , y ) f ( y ) d y | + | j = 1 m 1 σ C j m ( b ( x ) ( b ) Q ) σ R n ( b ( y ) ( b ) Q ) σ c K t ( x , y ) f ( y ) d y | + | R n j = 1 m ( b j ( y ) ( b j ) Q ) K t ( x , y ) f 1 ( y ) d y | + | R n j = 1 m ( b j ( y ) ( b j ) Q ) ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y | = J 1 ( x ) + J 2 ( x ) + J 3 ( x ) + J 4 ( x ) + J 5 ( x ) + J 6 ( x ) + J 7 ( x ) .
For J 1 ( x ) , same as m = 1 , for some 1 < q < , let 1 / q 1 + 1 / q 2 + + 1 / q m + 1 / q = 1 , 1 / p + 1 / p = 1 , by Hölder’s inequality, and the reverse of Hölder’s inequality, we get
1 ω ( Q ) Q | J 1 ( x ) | ω ( x ) d x C ω ( Q ) ( Q | ( b 1 ( x ) ( b 1 ) Q ) ( b m ( x ) ( b m ) Q ) | p ω ( x ) d x ) 1 / p × ( Q | T ( f ) ( x ) | p ω ( x ) d x ) 1 / p C ω ( Q ) ( Q | b 1 ( x ) ( b 1 ) Q | p | b m ( x ) ( b m ) Q | p ω ( x ) d x ) 1 / p × f L ( ω ) ( Q ω ( x ) d x ) 1 / p C ω ( Q ) f L ( ω ) ω ( Q ) 1 / p j = 1 m [ ( Q | b j ( x ) ( b j ) Q | p q j d x ) 1 / q j ( Q ω ( x ) q d x ) 1 / q ] 1 / p C b BMO f L ( ω ) ω ( Q ) 1 / p + 1 / p 1 | Q | 1 / p ( 1 / q 1 + + 1 / q m + 1 / q 1 ) C b BMO f L ( ω ) .
For J 2 ( x ) , by Hölder’s inequality and the reverse of Hölder’s inequality, we have
1 ω ( Q ) Q | J 2 ( x ) | ω ( x ) d x j = 1 m 1 σ C j m C ω ( Q ) ( Q | ( b ( x ) b Q ) σ | p ω ( x ) d x ) 1 / p × ( Q | T ( ( b b Q ) σ c f ) ( x ) | p ω ( x ) d x ) 1 / p C j = 1 m 1 σ C j m ( 1 ω ( Q ) Q | ( b ( x ) b Q ) σ | p ω ( x ) d x ) 1 / p × ( 1 ω ( Q ) Q | T ( ( b b Q ) σ c f ) ( x ) | p ω ( x ) d x ) 1 / p C j = 1 m 1 σ C j m ω ( Q ) 1 / p [ ( Q | ( b ( x ) b Q ) σ | p q d x ) 1 / q ( Q ω ( x ) q d x ) 1 / q ] 1 / p × ω ( Q ) 1 / p ( R n | ( b ( x ) b Q ) σ c f ( x ) | p ω ( x ) χ Q ( x ) d x ) 1 / p C j = 1 m 1 σ C j m ω ( Q ) 1 / p | Q | 1 / p q + 1 / p q 1 / p ω ( Q ) 1 / p b σ BMO × ω ( Q ) 1 / p ( Q | ( b ( x ) b Q ) σ c | p q d x ) 1 / p q ( Q | f ( x ) | p q ω q ( x ) d x ) 1 / p q C j = 1 m 1 σ C j m b σ BMO b σ c BMO ( | Q | ω ( Q ) ) 1 / p × ( 1 | Q | Q ω ( x ) d x ) 1 / p f L ( ω ) C b BMO f L ( ω ) .
For J 3 ( x ) , taking p > 1 , by the L p ( ω ) -boundedness of T, we have
1 ω ( Q ) Q | J 3 ( x ) | ω ( x ) d x ( 1 ω ( Q ) R n | T ( ( b 1 ( b 1 ) Q ) ( b m ( b m ) Q ) f 1 ) ( x ) | p ω ( x ) d x ) 1 / p C ω ( Q ) 1 / p ( R n | ( b 1 ( x ) ( b 1 ) Q ) ( b m ( x ) ( b m ) Q ) f 1 ( x ) | p ω ( x ) d x ) 1 / p C ω ( Q ) 1 / p | Q | 1 / p q b BMO | Q | 1 / p q ( 1 | Q | Q ω q d x ) 1 / p q f L ( ω ) C b BMO ( | Q | ω ( Q ) ) 1 / p ( ω ( Q ) | Q | ) 1 / p f L ( ω ) C b BMO f L ( ω ) .
For J 4 ( x ) , J 5 ( x ) , and J 6 ( x ) , choose 1 < p , q j < , j = 1 , , m , such that 1 / p + 1 / q 1 + + 1 / q m + 1 / q , by Lemma 2 and similar to the proofs of J 1 ( x ) , J 2 ( x ) , and J 3 ( x ) , we get
1 ω ( Q ) Q | J 4 ( x ) | ω ( x ) d x C | Q | ω ( Q ) j = 1 m ( 1 | Q | Q | ( b j ( x ) ( b j ) Q ) | q j d x ) 1 / q j × ( 1 | Q | Q | A t Q ( f ) ( x ) | p d x ) 1 / p ( 1 | Q | Q ω ( x ) q d x ) 1 / q C b BMO f L ( ω ) , 1 ω ( Q ) Q | J 5 ( x ) | ω ( x ) d x C | Q | ω ( Q ) j = 1 m 1 σ C j m ( 1 | Q | Q | ( b ( x ) b Q ) σ | q d x ) 1 / q × ( 1 | Q | Q | A t Q ( ( b b Q ) σ c f ) ( x ) | p d x ) 1 / p ( 1 | Q | Q ω ( x ) q d x ) 1 / q C | Q | ω ( Q ) j = 1 m 1 σ C j m b σ BMO b σ c BMO f L ( ω ) ω ( Q ) | Q | C b BMO f L ( ω ) , 1 ω ( Q ) Q | J 6 ( x ) | ω ( x ) d x C | Q | ω ( Q ) ( 1 | Q | Q | A t Q ( ( b 1 ( b 1 ) Q ) ( b m ( b m ) Q ) f 1 ) ( x ) | q d x ) 1 / q × ( 1 | Q | Q ω ( x ) q d x ) 1 / q C b BMO f L ( ω ) .
For J 7 ( x ) , note that | x y | d = t 1 / 2 , taking 1 < q j < , j = 1 , , m such that 1 / q 1 + + 1 / q m + 1 / r = 1 , then
J 7 ( x ) C k = 0 2 k + 1 Q 2 k Q j = 1 m | ( b j ( y ) ( b j ) Q ) | | f ( y ) | d δ | x 0 y | n + δ d y C k = 1 d δ ( 2 k 1 d ) n + δ | 2 k Q | ( 1 | 2 k Q | 2 k Q | f ( y ) | r d y ) 1 / r × j = 1 m ( 1 | 2 k Q | 2 k Q | b j ( y ) ( b j ) Q | q j d y ) 1 / q j C k = 1 2 k δ f L ( ω ) j = 1 m ( 1 | 2 k Q | 2 k Q | b j ( y ) ( b j ) Q | q j d y ) 1 / q j C k = 1 k m 2 k δ j = 1 m b j BMO f L ( ω ) C b BMO f L ( ω ) ,
so
1 ω ( Q ) Q | J 7 ( x ) | ω ( x ) d x C b BMO f L ( ω ) .

This completes the proof of Theorem 1. □

Theorem 2 Let 1 < p < , ω A 1 and b = ( b 1 , , b m ) with b j BMO ( R n ) for 1 j m . Then T b is bounded from B p ( ω ) to CMO A ( ω ) .

Proof It suffices to prove for f C 0 ( R n ) , the following inequality holds:
1 ω ( Q ) Q | T b ( f ) ( x ) A t Q T b ( f ) ( x ) | ω ( x ) d x C f B p ( ω )
for any cube Q = Q ( 0 , d ) with d > 1 . Fix a cube Q = Q ( 0 , d ) with d > 1 . Set f 1 = f χ Q , f 2 = f χ ( R n Q ) and b Q = ( ( b 1 ) Q , , ( b m ) Q ) R n , where ( b j ) Q = | Q | 1 Q | b j ( y ) | d y , 1 j m , we have
| T b ( f ) ( x ) A t Q T b ( f ) ( x ) | | j = 1 m ( b j ( x ) ( b j ) Q ) R n K ( x , y ) f ( y ) d y | + | j = 1 m 1 σ C j m ( b ( x ) ( b ) Q ) σ R n ( b ( y ) ( b ) Q ) σ c K ( x , y ) f ( y ) d y | + | R n j = 1 m ( b j ( y ) ( b j ) Q ) K ( x , y ) f 1 ( y ) d y | + | j = 1 m ( b j ( x ) ( b j ) Q ) R n K t ( x , y ) f ( y ) d y | + | j = 1 m 1 σ C j m ( b ( x ) ( b ) Q ) σ R n ( b ( y ) ( b ) Q ) σ c K t ( x , y ) f ( y ) d y | + | R n j = 1 m ( b j ( y ) ( b j ) Q ) K t ( x , y ) f 1 ( y ) d y | + | R n j = 1 m ( b j ( y ) ( b j ) Q ) ( K ( x , y ) K t ( x , y ) ) f 2 ( y ) d y | = L 1 ( x ) + L 2 ( x ) + L 3 ( x ) + L 4 ( x ) + L 5 ( x ) + L 6 ( x ) + L 7 ( x ) .
For L 1 ( x ) , we have
1 ω ( Q ) Q | L 1 ( x ) | ω ( x ) d x C ω ( Q ) ( Q | ( b 1 ( x ) ( b 1 ) Q ) ( b m ( x ) ( b m ) Q ) | p ω ( x ) d x ) 1 / p × ( Q | T ( f ) ( x ) | p ω ( x ) d x ) 1 / p C ω ( Q ) [ ( Q | ( b 1 ( x ) ( b 1 ) Q ) ( b m ( x ) ( b m ) Q ) | p q d x ) 1 / q ( Q ω ( x ) q d x ) 1 / q ] 1 / p × ( Q | f ( x ) | p ω ( x ) d x ) 1 / p C ω ( Q ) | Q | 1 / p q b BMO | Q | 1 / p q ( ω ( Q ) | Q | ) 1 / p f χ Q L p ( ω ) C b BMO ω ( Q ) 1 / p f χ Q L p ( ω ) C b BMO f B p ( ω ) .
For L 2 ( x ) , taking 1 < s , s < , and 1 / s + 1 / s = 1 , we have
1 ω ( Q ) Q | L 2 ( x ) | ω ( x ) d x C j = 1 m 1 σ C j m ( 1 ω ( Q ) Q | ( b ( x ) b Q ) σ | s ω ( x ) d x ) 1 / s × ( 1 ω ( Q ) Q | T ( ( b b Q ) σ c f ) ( x ) | s ω ( x ) d x ) 1 / s C j = 1 m 1 σ C j m ω ( Q ) 1 / s [ ( Q | ( b ( x ) b Q ) σ | s q d x ) 1 / q ( Q ω q d x ) 1 / q ] 1 / s × ω ( Q ) 1 / s ( Q | ( b ( x ) b Q ) σ c f ( x ) | s ω ( x ) d x ) 1 / s C j = 1 m 1 σ C j m ω ( Q ) 1 / s | Q | 1 / s q + 1 / s q 1 / s ω ( Q ) 1 / s b σ BMO × ω ( Q ) 1 / s | Q | 1 / r s b σ c BMO ( Q | f ( x ) | p ω ( x ) d x ) 1 / p ( Q ω ( x ) q d x ) ( p s ) / p q s C j = 1 m 1 σ C j m b σ BMO b σ c BMO ω ( Q ) 1 / p f χ Q L p ( ω ) C b BMO f B p ( ω ) .
For L 3 ( x ) , we have
1 ω ( Q ) Q | L 3 ( x ) | ω ( x ) d x C ( 1 ω ( Q ) R n | T ( ( b 1 ( b 1 ) Q ) ( b m ( b m ) Q ) f 1 ) ( x ) | s ω ( x ) d x ) 1 / s C ω ( Q ) 1 / s ( Q | ( b 1 ( x ) ( b 1 ) Q ) ( b m ( x ) ( b m ) Q ) f ( x ) | s ω ( x ) d x ) 1 / s C ω ( Q ) 1 / p b BMO f χ Q L p ( ω ) C b BMO f B p ( ω ) .
For L 4 ( x ) , L 5 ( x ) , and L 6 ( x ) , by Lemma 2, we have
1 ω ( Q ) Q | L 4 ( x ) | ω ( x ) d x C ( 1 ω ( Q ) Q | ( b 1 ( x ) ( b 1 ) Q ) ( b m ( x ) ( b m ) Q ) | s ω ( x ) d x ) 1 / s × ( 1 ω ( Q ) Q | A t Q ( f ) ( x ) | s ω ( x ) d x ) 1 / s C ( 1 ω ( Q ) ) 1 / s [ ( Q | ( b 1 ( x ) ( b 1 ) Q ) ( b m ( x ) ( b m ) Q ) | s q d x ) 1 / q × ( Q ω ( x ) q d x ) 1 / q ] 1 / s f B p ( w ) C ( 1 ω ( Q ) ) 1 / s | Q | 1 / s q b BMO | Q | 1 / s q ( ω ( Q ) | Q | ) 1 / s f B p ( w ) C b BMO f B p ( ω ) ; 1 ω ( Q ) Q | L 5 ( x ) | ω ( x ) d x C j = 1 m 1 σ C j m ( 1 ω ( Q ) Q | ( b ( x ) b Q ) σ | s ω ( x ) d x ) 1 / s × ( 1 ω ( Q ) Q | A t Q ( ( b b Q ) σ c f ) ( x ) | s ω ( x ) d x ) 1 / s C j = 1 m 1 σ C j m ω ( Q ) 1 / s [ ( Q | ( b ( x ) b Q ) σ | s q d x ) 1 / q ( Q ω q d x ) 1 / q ] 1 / s × b σ c BMO f B p ( ω ) C j = 1 m 1 σ C j m ω ( Q ) 1 / s | Q | 1 / s q + 1 / s q 1 / s ω ( Q ) 1 / s b σ BMO b σ c BMO f B p ( ω ) C b σ BMO f B p ( ω ) ; 1 ω ( Q ) Q | L 6 ( x ) | ω ( x ) d x ( 1 ω ( Q ) R n | A t Q ( ( b 1 ( b 1 ) Q ) ( b m ( b m ) Q ) f 1 ) ( x ) | s ω ( x ) d x ) 1 / s C b BMO f B p ( ω ) .
For L 7 ( x ) , note that | x y | d = t 1 / 2 , taking 1 < u < p , then
L 7 ( x ) C Q c j = 1 m | b j ( y ) ( b j ) Q | | f ( y ) | d δ | x 0 y | n + δ d y C k = 0 2 k + 1 Q 2 k Q j = 1 m | b j ( y ) ( b j ) Q | | f ( y ) | d δ | x 0 y | n + δ d y C k = 1 d δ ( 2 k 1 d ) n + δ | 2 k Q | ( 1 | 2 k Q | 2 k Q | f ( y ) | u d y ) 1 / u × ( 1 | 2 k Q | 2 k Q j = 1 m | b j ( y ) ( b j ) Q | u d y ) 1 / u C b BMO k = 1 k m 2 k δ ( 1 | 2 k Q | ) 1 / u × [ ( 2 k Q | f ( y ) | p ω ( y ) d y ) u p ( 2 k Q ω ( y ) u p u d y ) p u p ] 1 / u C b BMO k = 1 k m 2 k δ ( 1 | 2 k Q | ) 1 / u f χ 2 k Q L p ( ω ) ( ω ( 2 k Q ) | 2 k Q | ) 1 / p | 2 k Q | ( p u 1 ) 1 p × [ ( 1 | 2 k Q | 2 k Q ω ( y ) d y ) ( 1 | 2 k Q | 2 k Q ω ( y ) 1 p u 1 d y ) p u 1 ] 1 / p C b BMO k = 1 k m 2 k δ ω ( 2 k Q ) 1 / p f χ 2 k Q L p ( ω ) C b BMO f B p ( ω ) ,
so
1 ω ( Q ) Q | L 7 ( x ) | ω ( x ) d x C b BMO f B p ( ω ) .

This completes the proof of Theorem 2. □

Declarations

Acknowledgements

Project was supported by the National Natural Science Foundation of China (No. 11061003).

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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