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

Abstract

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

1 Introduction

Let bBMO( 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(bf)(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)dx and b(Q)= Q b(x)dx, the sharp function of b is defined by

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

It is well known that (see [6])

b # (x)= sup Q x inf c C 1 | Q | Q |b(y)c|dy.

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 Ck 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)dy

for every f L p ( R n ) with p1, 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)dy

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)|dx 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 |xy| c 3 t 1 / 2

    and

    |K(x,y) K t (x,y)| c 4 t δ / 2 | x y | n δ if |xy| 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)dy.

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)dx,

    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)dx,

    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 1jm, 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 bBMO( R n ). Then

  1. (a)

    for every f L ( R n ), 1p<, 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), 1r<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 xQ and y 2 k + 1 Q 2 k Q, we have |xy| 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)dx

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>ru (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 1jm. 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)dxC 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)dy, 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)dy R n ( b 1 ( y ) ( b 1 ) Q ) K t (x,y)f(y)dy.

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)dxC 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)dy, 1jm, 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 |xy|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)dxC 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 1jm. 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)dxC 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)|dy, 1jm, 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 |xy|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)dxC b BMO f B p ( ω ) .

This completes the proof of Theorem 2. □

References

  1. 1.

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

  2. 2.

    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.

  3. 3.

    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

  4. 4.

    Deng DG, Yan LX: Commutators of singular integral operators with non-smooth kernels. Acta Math. Sci. 2005, 25: 137-144.

  5. 5.

    Duong XT, McIntosh A: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 1999, 15: 233-265.

  6. 6.

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

  7. 7.

    Liu LZ: Sharp function boundedness for vector-valued multilinear singular integral operators with non-smooth kernels. J. Contemp. Math. Anal. 2010, 45: 185-196. 10.3103/S1068362310040011

  8. 8.

    Liu LZ: Multilinear singular integral operators on Triebel-Lizorkin and Lebesgue spaces. Bull. Malays. Math. Soc. 2012, 35: 1075-1086.