Open Access

Weighted boundedness of a multilinear operator associated to a singular integral operator with general kernels

Journal of Inequalities and Applications20142014:188

https://doi.org/10.1186/1029-242X-2014-188

Received: 24 December 2013

Accepted: 25 April 2014

Published: 13 May 2014

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with general kernels. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.

MSC:42B20, 42B25.

Keywords

multilinear operator singular integral operator sharp maximal function weighted BMO weighted Lipschitz function

1 Introduction and preliminaries

As the development of singular integral operators (see [13]), their commutators and multilinear operators have been well studied. In [46], the authors prove that the commutators generated by singular integral operators and BMO functions are bounded on L p ( R n ) for 1 < p < . Chanillo (see [7]) proves a similar result when singular integral operators are replaced by fractional integral operators. In [8, 9], the boundedness for the commutators generated by singular integral operators and Lipschitz functions on Triebel-Lizorkin and L p ( R n ) ( 1 < p < ) spaces is obtained. In [10, 11], the boundedness for the commutators generated by singular integral operators and weighted BMO and Lipschitz functions on L p ( R n ) ( 1 < p < ) spaces is obtained. In [12], some singular integral operators with general kernel are introduced, and the boundedness for the operators and their commutators generated by BMO and Lipschitz functions is obtained (see [8, 12, 13]). Motivated by these, in this paper, we study the multilinear operator generated by the singular integral operator with general kernel and the weighted Lipschitz and BMO functions.

First, let us introduce some notations. Throughout this paper, Q denotes a cube of R n with sides parallel to the axes. For any locally integrable function f, the sharp maximal function of f is defined by
M # ( f ) ( x ) = sup Q x 1 | Q | Q | f ( y ) f Q | d y ,
where, and in what follows, f Q = | Q | 1 Q f ( x ) d x . It is well known that (see [1, 2])
M # ( f ) ( x ) sup Q x inf c C 1 | Q | Q | f ( y ) c | d y .
Let
M ( f ) ( x ) = sup Q x 1 | Q | Q | f ( y ) | d y .

For η > 0 , let M η # ( f ) ( x ) = M # ( | f | η ) 1 / η ( x ) and M η ( f ) ( x ) = M ( | f | η ) 1 / η ( x ) .

For 0 < η < n , 1 p < and the non-negative weight function w, set
M η , p , w ( f ) ( x ) = sup Q x ( 1 w ( Q ) 1 p η / n Q | f ( y ) | p w ( y ) d y ) 1 / p .

We write M η , p , w ( f ) = M p , w ( f ) if η = 0 .

The A p weight is defined by (see [1]), for 1 < p < ,
A p = { 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 < }
and
A 1 = { w L loc p ( R n ) : M ( w ) ( x ) C w ( x ) , a.e. } .
Given a non-negative weight function w, for 1 p < , the weighted Lebesgue space L p ( R n , w ) is the space of functions f such that
f L p ( w ) = ( R n | f ( x ) | p w ( x ) d x ) 1 / p < .
For 0 < β < 1 and the non-negative weight function w, the weighted Lipschitz space Lip β ( w ) is the space of functions b such that
b Lip β ( w ) = sup Q 1 w ( Q ) β / n ( 1 w ( Q ) Q | b ( y ) b Q | p w ( x ) 1 p d y ) 1 / p < ,
and the weighted BMO space BMO ( w ) is the space of functions b such that
b BMO ( w ) = sup Q ( 1 w ( Q ) Q | b ( y ) b Q | p w ( x ) 1 p d y ) 1 / p < .
Remark (1) It has been known that (see [11, 14]), for b Lip β ( w ) , w A 1 and x Q ,
| b Q b 2 k Q | C k b Lip β ( w ) w ( x ) w ( 2 k Q ) β / n .
  1. (2)
    It has been known that (see [2, 14]), for b BMO ( w ) , w A 1 and x Q ,
    | b Q b 2 k Q | C k b BMO ( w ) w ( x ) .
     
  2. (3)

    Let b Lip β ( w ) or b BMO ( w ) and w A 1 . By [14], we know that spaces Lip β ( w ) or BMO ( w ) coincide and the norms b Lip β ( w ) or b BMO ( w ) are equivalent with respect to different values 1 p < .

     
Definition 1 Let φ be a positive, increasing function on R + , and there exists a constant D > 0 such that
φ ( 2 t ) D φ ( t ) for  t 0 .
Let w be a non-negative weight function on R n and f be a locally integrable function on  R n . Set, for 1 p < ,
f L p , φ ( w ) = sup x R n , d > 0 ( 1 φ ( d ) Q ( x , d ) | f ( y ) | p w ( y ) d y ) 1 / p ,
where Q ( x , d ) = { y R n : | x y | < d } . The generalized weighted Morrey space is defined by
L p , φ ( R n , w ) = { f L loc 1 ( R n ) : f L p , φ ( w ) < } .

If φ ( d ) = d δ , δ > 0 , then L p , φ ( R n , w ) = L p , δ ( R n , w ) , which is the classical Morrey space (see [15, 16]). If φ ( d ) = 1 , then L p , φ ( R n , w ) = L p ( R n , w ) , which is the weighted Lebesgue space (see [1]).

As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [1720]).

In this paper, we study some singular integral operators as follows (see [12]).

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 . Moreover, let m be a positive integer and b be a function on R n . Set
R m + 1 ( b ; x , y ) = b ( x ) | α | m 1 α ! D α b ( y ) ( x y ) α .
The multilinear operator related to the operator T is defined by
T b ( f ) ( x ) = R n R m + 1 ( b ; x , y ) | x y | m K ( x , y ) f ( y ) d y .

Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 1 (see [13, 5, 6]) and that the commutator [ b , T ] ( f ) = b T ( f ) T ( b f ) is a particular operator of the multilinear operator T b if m = 0 . The multilinear operator T b is a non-trivial generalization of the commutator. It is well known that commutators and multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [2123]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator T b . As application, we obtain the weighted L p -norm inequality and Morrey space boundedness for the multilinear operator T b .

2 Theorems and lemmas

We shall prove the following theorems.

Theorem 1 Let T be a singular integral operator as in Definition  2, the sequence { k C k } l 1 , w A 1 , 0 < η < 1 , q < r < and D α b BMO ( w ) for all α with | α | = m . Then there exists a constant C > 0 such that, for any f C 0 ( R n ) and x ˜ R n ,
M η # ( T b ( f ) ) ( x ˜ ) C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .
Theorem 2 Let T be a singular integral operator as in Definition  2, the sequence { k C k } l 1 , w A 1 , 0 < η < 1 , q < r < , 0 < β < 1 and D α b Lip β ( w ) for all α with | α | = m . Then there exists a constant C > 0 such that, for any f C 0 ( R n ) and x ˜ R n ,
M η # ( T b ( f ) ) ( x ˜ ) C | α | = m D α b Lip β ( w ) w ( x ˜ ) M β , r , w ( f ) ( x ˜ ) .

Theorem 3 Let T be a singular integral operator as in Definition  2, the sequence { k C k } l 1 , w A 1 , q < u < and D α b BMO ( w ) for all α with | α | = m . Then T b is bounded from L u ( R n , w ) to L u ( R n , w 1 u ) .

Theorem 4 Let T be a singular integral operator as in Definition  2, the sequence { k C k } l 1 , w A 1 , q < u < , 0 < D < 2 n and D α b BMO ( w ) for all α with | α | = m . Then T b is bounded from L u , φ ( R n , w ) to L u , φ ( R n , w 1 u ) .

Theorem 5 Let T be a singular integral operator as in Definition  2, the sequence { k C k } l 1 , w A 1 , 0 < β < 1 , q < u < n / β , 1 / v = 1 / u β / n and D α b Lip β ( w ) for all α with | α | = m . Then T b is bounded from L u ( R n , w ) to L v ( R n , w 1 v ) .

Theorem 6 Let T be a singular integral operator as in Definition  2, the sequence { k C k } l 1 , w A 1 , 0 < β < 1 , 0 < D < 2 n , q < u < n / β , 1 / v = 1 / u β / n and D α b Lip β ( w ) for all α with | α | = m . Then T b is bounded from L u , φ ( R n , w ) to L v , φ ( R n , w 1 v ) .

To prove the theorems, we need the following lemmas.

Lemma 1 (see [[1], p.485])

Let 0 < u < v < and for any function f 0 , we define that, for 1 / r = 1 / u 1 / v ,
f W L v = sup λ > 0 λ | { x R n : f ( x ) > λ } | 1 / v , N u , v ( f ) = sup Q f χ Q L u / χ Q L r ,
where the sup is taken for all measurable sets Q with 0 < | Q | < . Then
f W L v N u , v ( f ) ( v / ( v u ) ) 1 / u f W L v .

Lemma 2 (see [12])

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

Lemma 3 (see [1, 7])

Let 0 η < n , 1 s < u < n / η , 1 / v = 1 / u η / n and w A 1 . Then
M η , s , w ( f ) L v ( w ) C f L u ( w ) .

Lemma 4 (see [1])

Let 0 < p , η < and w 1 r < A r . Then, for any smooth function f for which the left-hand side is finite,
R n M η ( f ) ( x ) p w ( x ) d x C R n M η # ( f ) ( x ) p w ( x ) d x .

Lemma 5 (see [17, 20])

Let 0 < p < , 0 < η < , 0 < D < 2 n and w A 1 . Then, for any smooth function f for which the left-hand side is finite,
M η ( f ) L p , φ ( w ) C M η # ( f ) L p , φ ( w ) .

Lemma 6 (see [17, 20])

Let 0 η < n , 0 < D < 2 n , 1 s < u < n / η , 1 / v = 1 / u η / n and w A 1 . Then
M η , s , w ( f ) L v , φ ( w ) C f L u , φ ( w ) .

Lemma 7 (see [22])

Let b be a function on R n and D α A L u ( R n ) for all α with | α | = m and any u > n . Then
| R m ( b ; x , y ) | C | x y | m | α | = m ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) | D α b ( z ) | u d z ) 1 / u ,

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

3 Proofs of theorems

Proof of Theorem 1 It suffices to prove for f C 0 ( R n ) and some constant C 0 that the following inequality holds:
( 1 | Q | Q | T b ( f ) ( x ) C 0 | η d x ) 1 / η C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .
Fix a cube Q = Q ( x 0 , d ) and x ˜ Q . Let Q ˜ = 5 n Q and b ˜ ( x ) = b ( x ) | α | = m 1 α ! ( D α b ) Q ˜ x α , then R m ( b ; x , y ) = R m ( b ˜ ; x , y ) and D α b ˜ = D α b ( D α b ) Q ˜ for | α | = m . We write, for f 1 = f χ Q ˜ and f 2 = f χ R n Q ˜ ,
T b ( f ) ( x ) = R n R m ( b ˜ ; x , y ) | x y | m K ( x , y ) f 1 ( y ) d y | α | = m 1 α ! R n ( x y ) α D α b ˜ ( y ) | x y | m K ( x , y ) f 1 ( y ) d y + R n R m + 1 ( b ˜ ; x , y ) | x y | m K ( x , y ) f 2 ( y ) d y = T ( R m ( b ˜ ; x , ) | x | m f 1 ) T ( | α | = m 1 α ! ( x ) α D α b ˜ | x | m f 1 ) + T b ˜ ( f 2 ) ( x ) ,
then
( 1 | Q | Q | T b ( f ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | η d x ) 1 / η C ( 1 | Q | Q | T ( R m ( b ˜ ; x , ) | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T ( | α | = m ( x ) α D α b ˜ | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | η d x ) 1 / η = I 1 + I 2 + I 3 .
For I 1 , noting that 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 [1]). We take u = r p 0 / ( r + p 0 1 ) in Lemma 7 and have 1 < u < r and p 0 = u ( r 1 ) / ( r u ) . Then by Lemma 7 and Hölder’s inequality, we get
| R m ( b ˜ ; x , y ) | C | x y | m | α | = m ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) | D α b ˜ ( z ) | u d z ) 1 / u C | x y | m | α | = m | Q ˜ | 1 / u ( Q ˜ ( x , y ) | D α b ˜ ( z ) | u w ( z ) u ( 1 r ) / r w ( z ) u ( r 1 ) / r d z ) 1 / u C | x y | m | α | = m | Q ˜ | 1 / u ( Q ˜ ( x , y ) | D α b ˜ ( z ) | r w ( z ) 1 r d z ) 1 / r × ( Q ˜ ( x , y ) w ( z ) u ( r 1 ) / ( r u ) d z ) ( r u ) / r u C | x y | m | α | = m | Q ˜ | 1 / u D α b BMO ( w ) w ( Q ˜ ) 1 / r | Q ˜ | ( r u ) / r u × ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) w ( z ) p 0 d z ) ( r u ) / r u C | x y | m | α | = m D α b BMO ( w ) | Q ˜ | 1 / u w ( Q ˜ ) 1 / r | Q ˜ | 1 / u 1 / r ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) w ( z ) d z ) ( r 1 ) / r C | x y | m | α | = m D α b BMO ( w ) | Q ˜ | 1 / u w ( Q ˜ ) 1 / r | Q ˜ | 1 / u 1 / r w ( Q ˜ ) 1 1 / r | Q ˜ | 1 / r 1 C | x y | m | α | = m D α b BMO ( w ) w ( Q ˜ ) | Q ˜ | C | x y | m | α | = m D α b BMO ( w ) w ( x ˜ ) .
Thus, by the L s -boundedness of T (see Lemma 2) for 1 < s < r and w A 1 A r / s , we obtain
I 1 C | Q | Q | T ( R m ( b ˜ ; x , ) | x | m f 1 ) | d x C | α | = m D α b BMO ( w ) w ( x ˜ ) ( 1 | Q | R n | T ( f 1 ) ( x ) | s d x ) 1 / s C | α | = m D α b BMO ( w ) w ( x ˜ ) | Q | 1 / s ( R n | f 1 ( x ) | s d x ) 1 / s C | α | = m D α b BMO ( w ) w ( x ˜ ) | Q | 1 / s ( Q ˜ | f ( x ) | s w ( x ) s / r w ( x ) s / r d x ) 1 / s C | α | = m D α b BMO ( w ) w ( x ˜ ) | Q | 1 / s ( Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r ( Q ˜ w ( x ) s / ( r s ) d x ) ( r s ) / r s C | α | = m D α b BMO ( w ) w ( x ˜ ) | Q | 1 / s w ( Q ˜ ) 1 / r ( 1 w ( Q ˜ ) Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r × ( 1 | Q ˜ | Q ˜ w ( x ) s / ( r s ) d x ) ( r s ) / r s ( 1 | Q ˜ | Q ˜ w ( x ) d x ) 1 / r | Q ˜ | 1 / s w ( Q ˜ ) 1 / r C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .
For I 2 , by the weak ( L 1 , L 1 ) boundedness of T (see Lemma 2) and Kolmogoro’s inequality (see Lemma 1), we obtain
I 2 C | α | = m ( 1 | Q | Q | T ( D α b f ˜ 1 ) ( x ) | η d x ) 1 / η C | α | = m | Q | 1 / η 1 | Q | 1 / η T ( D α b f ˜ 1 ) χ Q L η χ Q L η / ( 1 η ) C | α | = m 1 | Q | T ( D α b f ˜ 1 ) W L 1 C | α | = m 1 | Q | R n | D α b ˜ ( x ) f 1 ( x ) | d x C | α | = m 1 | Q | Q ˜ | D α b ( x ) ( D α b ) Q ˜ | w ( x ) 1 / r | f ( x ) | w ( x ) 1 / r d x C | α | = m 1 | Q | ( Q ˜ | ( D α b ( x ) ( D α b ) Q ˜ ) | r w ( x ) 1 r d x ) 1 / r ( Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r C | α | = m 1 | Q | D α b BMO ( w ) w ( Q ˜ ) 1 / r w ( Q ˜ ) 1 / r ( 1 w ( Q ˜ ) Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r C | α | = m D α b BMO ( w ) w ( Q ˜ ) | Q ˜ | M r , w ( f ) ( x ˜ ) C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .
For I 3 , noting that | x y | | x 0 y | for x Q and y R n Q , we write
| T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | R n | R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | | K ( x , y ) | | x y | m | f 2 ( y ) | d y + R n | K ( x , y ) | x y | m K ( x 0 , y ) | x 0 y | m | | R m ( b ˜ ; x 0 , y ) | | f 2 ( y ) | d y + | α | = m 1 α ! R n | K ( x , y ) ( x y ) α | x y | m K ( x 0 , y ) ( x 0 y ) α | x 0 y | m | | D α b ˜ ( y ) | | f 2 ( y ) | d y = I 3 ( 1 ) ( x ) + I 3 ( 2 ) ( x ) + I 3 ( 3 ) ( x ) .
For I 3 ( 1 ) ( x ) , by the formula (see [22])
R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) = | γ | < m 1 γ ! R m | γ | ( D γ b ˜ ; x , x 0 ) ( x y ) γ
and Lemma 7, we have, similar to the proof of I 1 ,
| R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | C | γ | < m | α | = m | x x 0 | m | γ | | x y | | γ | D α b BMO ( w ) w ( x ˜ )
and
| R m ( b ˜ ; x 0 , y ) | C | α | = m | x x 0 | m D α b BMO ( w ) w ( x ˜ ) .
Thus, by w A 1 A r , we get
I 3 ( 1 ) ( x ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ | R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | | K ( x , y ) | | x y | m | f ( y ) | d y C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 | f ( y ) | d y C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 d ( 2 k d ) n + 1 2 k Q ˜ | f ( y ) | w ( y ) 1 / r w ( y ) 1 / r d y C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 d ( 2 k d ) n + 1 ( 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r × ( 2 k Q ˜ w ( y ) 1 / ( r 1 ) d y ) ( r 1 ) / r C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 d ( 2 k d ) n + 1 w ( 2 k Q ˜ ) 1 / r × ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) 1 / ( r 1 ) d y ) ( r 1 ) / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r | 2 k Q ˜ | w ( 2 k Q ˜ ) 1 / r C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) k = 1 2 k C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .
For I 3 ( 2 ) ( x ) , we take 1 < p < such that 1 / p + 1 / q + 1 / r = 1 . Recalling r > q and w A 1 A r / p + 1 , we get
I 3 ( 2 ) ( x ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | | R m ( b ˜ ; x 0 , y ) | | x y | m | f ( y ) | w ( y ) 1 / r w ( y ) 1 / r d y + 2 k + 1 Q ˜ 2 k Q ˜ | 1 | x y | m 1 | x 0 y | m | | K ( x 0 , y ) | | R m ( b ˜ ; x 0 , y ) | | f ( y ) | d y C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 0 ( 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 ˜ | f ( y ) | r w ( y ) d y ) 1 / r ( 2 k + 1 Q ˜ 2 k Q ˜ w ( y ) p / r d y ) 1 / p + C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 0 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 | f ( y ) | w ( y ) 1 / r w ( y ) 1 / r d y C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 C k | 2 k Q | 1 / q w ( 2 k Q ˜ ) 1 / r × ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) p / r d y ) 1 / p × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r | 2 k Q ˜ | 1 / r + 1 / p w ( 2 k Q ˜ ) 1 / r + C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 d ( 2 k d ) n + 1 × ( 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r ( 2 k Q ˜ w ( y ) 1 / ( r 1 ) d y ) ( r 1 ) / r C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 ( C k + 2 k ) ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .
Similarly, we have, for r < p 1 < , 1 < s 1 , s 2 < with 1 / p 1 + 1 / q + 1 / r + 1 / s 1 = 1 and 1 / q + 1 / r + 1 / s 2 = 1 ,
I 3 ( 3 ) ( x ) | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | | ( x y ) α | | x y | m × | D α b ( y ) ( D α b ) 2 k + 1 Q ˜ | w ( y ) ( 1 p 1 ) / p 1 | f ( y ) | w ( y ) 1 / r w ( y ) ( p 1 1 ) / p 1 1 / r d y + | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | | ( x y ) α | | x y | m × | ( D α b ) 2 k + 1 Q ˜ ( D α b ) Q ˜ | | f ( y ) | w ( y ) 1 / r w ( y ) 1 / r d y + | α | = m 2 k + 1 Q ˜ 2 k Q ˜ | ( x y ) α | x y | m ( x 0 y ) α | x 0 y | m | | K ( x 0 , y ) | | f ( y ) | | D α b ˜ ( y ) | d y C | α | = m k = 0 ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | q d y ) 1 / q ( 2 k + 1 Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r × ( 2 k + 1 Q ˜ | D α b ( y ) ( D α b ) 2 k + 1 Q ˜ | p 1 w ( y ) 1 p 1 d y ) 1 / p 1 × ( 2 k + 1 Q ˜ w ( y ) ( 1 / r 1 / p 1 ) s 1 d y ) 1 / s 1 + C | α | = m k = 0 ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | q d y ) 1 / q × ( 2 k + 1 Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r × | ( D α b ) 2 k + 1 Q ˜ ( D α b ) Q ˜ | ( 2 k + 1 Q ˜ w ( y ) s 2 / r d y ) 1 / s 2 + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 × | f ( y ) | | D α b ( y ) ( D α b ) 2 k + 1 Q ˜ | w ( y ) 1 / r w ( y ) 1 / r d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 × | f ( y ) | | ( D α b ) 2 k + 1 Q ˜ ( D α b ) Q ˜ | w ( y ) 1 / r w ( y ) 1 / r d y C | α | = m k = 1 C k | 2 k Q | 1 / q D α b BMO ( w ) w ( 2 k Q ˜ ) 1 / p 1 w ( 2 k Q ˜ ) 1 / r × ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r 1 / p 1 ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) ( 1 / r 1 / p 1 ) s 1 d y ) 1 / s 1 × | 2 k Q ˜ | 1 / s 1 + 1 / r 1 / p 1 w ( 2 k Q ˜ ) 1 / r + 1 / p 1 + C | α | = m k = 1 C k | 2 k Q | 1 / q k D α b BMO ( w ) w ( x ˜ ) w ( 2 k Q ˜ ) 1 / r × ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) s 2 / r d y ) 1 / s 2 | 2 k Q ˜ | 1 / s 2 + 1 / r w ( 2 k Q ˜ ) 1 / r + C | α | = m k = 1 d ( 2 k d ) n + 1 ( 2 k Q ˜ | D α b ( y ) ( D α b ) 2 k Q ˜ | r w ( y ) 1 r d y ) 1 / r × ( 2 k + 1 Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r + C | α | = m k = 1 d ( 2 k d ) n + 1 k D α b BMO ( w ) w ( x ˜ ) ( 2 k Q ˜ w ( y ) r / r d y ) 1 / r × ( 2 k + 1 Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r | α | = m D α b BMO ( w ) k = 1 C k w ( 2 k Q ˜ ) | 2 k Q ˜ | ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r + | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 k C k ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r + C | α | = m D α b BMO ( w ) k = 1 2 k w ( 2 k Q ˜ ) | 2 k Q ˜ | ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r + C | α | = m D α b BMO ( w ) w ( x ˜ ) k = 1 k 2 k ( 1 w ( 2 k Q ˜ ) 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .
Thus
I 3 C | α | = m D α b BMO ( w ) w ( x ˜ ) M r , w ( f ) ( x ˜ ) .

These complete the proof of Theorem 1. □

Proof of Theorem 2 It suffices to prove for f C 0 ( R n ) and some constant C 0 that the following inequality holds:
( 1 | Q | Q | T b ( f ) ( x ) C 0 | η d x ) 1 / η C | α | = m D α b Lip β ( w ) w ( x ˜ ) M β , r , w ( f ) ( x ˜ ) .
Fix a cube Q = Q ( x 0 , d ) and x ˜ Q . Similar to the proof of Theorem 1, we have, for f 1 = f χ Q ˜ and f 2 = f χ R n Q ˜ ,
( 1 | Q | Q | T b ( f ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | η d x ) 1 / η C ( 1 | Q | Q | T ( R m ( b ˜ ; x , ) | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T ( | α | = m ( x ) α D α b ˜ | x | m f 1 ) | η d x ) 1 / η + C ( 1 | Q | Q | T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | η d x ) 1 / η = J 1 + J 2 + J 3 .
For J 1 and J 2 , by using the same argument as in the proof of Theorem 1, we get
| R m ( b ˜ ; x , y ) | C | x y | m | α | = m | Q ˜ | 1 / q ( Q ˜ ( x , y ) | D α b ˜ ( z ) | q w ( z ) q ( 1 r ) / r w ( z ) q ( r 1 ) / r d z ) 1 / q C | x y | m | α | = m | Q ˜ | 1 / q ( Q ˜ ( x , y ) | D α b ˜ ( z ) | r w ( z ) 1 r d z ) 1 / r × ( Q ˜ ( x , y ) w ( z ) q ( r 1 ) / ( r q ) d z ) ( r q ) / r q C | x y | m | α | = m | Q ˜ | 1 / q D α b Lip β ( w ) w ( Q ˜ ) β / n + 1 / r | Q ˜ | ( r q ) / r q × ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) w ( z ) p 0 d z ) ( r q ) / r q C | x y | m | α | = m D α b Lip β ( w ) | Q ˜ | 1 / q w ( Q ˜ ) β / n + 1 / r | Q ˜ | 1 / q 1 / r × ( 1 | Q ˜ ( x , y ) | Q ˜ ( x , y ) w ( z ) d z ) ( r 1 ) / r C | x y | m | α | = m D α b Lip β ( w ) | Q ˜ | 1 / q w ( Q ˜ ) β / n + 1 / r | Q ˜ | 1 / q 1 / r w ( Q ˜ ) 1 1 / r | Q ˜ | 1 / r 1 C | x y | m | α | = m D α b Lip β ( w ) w ( Q ˜ ) β / n + 1 | Q ˜ | C | x y | m | α | = m D α b Lip β ( w ) w ( Q ˜ ) β / n w ( x ˜ ) .
Thus
J 1 C | α | = m D α b Lip β ( w ) w ( Q ˜ ) β / n w ( x ˜ ) | Q | 1 / s ( R n | f 1 ( x ) | s d x ) 1 / s J 1 C | α | = m D α b Lip β ( w ) w ( Q ˜ ) β / n w ( x ˜ ) | Q | 1 / s J 1 × ( Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r ( Q ˜ w ( x ) s / ( r s ) d x ) ( r s ) / r s J 1 C | α | = m D α b Lip β ( w ) w ( x ˜ ) | Q ˜ | 1 / s w ( Q ˜ ) 1 / r ( 1 w ( Q ˜ ) 1 r β / n Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r J 1 × ( 1 | Q ˜ | Q ˜ w ( x ) s / ( r s ) d x ) ( r s ) / r s ( 1 | Q ˜ | Q ˜ w ( x ) d x ) 1 / r | Q ˜ | 1 / s w ( Q ˜ ) 1 / r J 1 C | α | = m D α b Lip β ( w ) w ( x ˜ ) M β , r , w ( f ) ( x ˜ ) , J 2 C | α | = m 1 | Q | Q ˜ | D α b ( x ) ( D α b ) Q ˜ | w ( x ) 1 / r | f ( x ) | w ( x ) 1 / r d x J 2 C | α | = m 1 | Q | ( Q ˜ | ( D α b ( x ) ( D α b ) Q ˜ ) | r w ( x ) 1 r d x ) 1 / r ( Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r J 2 C | α | = m 1 | Q | D α b Lip β ( w ) w ( Q ˜ ) β / n + 1 / r w ( Q ˜ ) 1 / r β / n ( 1 w ( Q ˜ ) 1 r β / n Q ˜ | f ( x ) | r w ( x ) d x ) 1 / r J 2 C | α | = m D α b Lip β ( w ) w ( Q ˜ ) | Q ˜ | M β , r , w ( f ) ( x ˜ ) J 2 C | α | = m D α b Lip β ( w ) w ( x ˜ ) M β , r , w ( f ) ( x ˜ ) .
For J 3 , we have
| R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | C | γ | < m | α | = m | x x 0 | m | γ | | x y | | γ | D α b Lip β ( w ) w ( x ˜ ) w ( 2 k Q ˜ )
and
| R m ( b ˜ ; x , y ) | C | α | = m | x x 0 | m D α b Lip β ( w ) w ( x ˜ ) w ( 2 k Q ˜ ) .
Thus, for 1 < p < with 1 / p + 1 / q + 1 / r = 1 and r < p 1 < , 1 < s 1 , s 2 < with 1 / p 1 + 1 / q + 1 / r + 1 / s 1 = 1 and 1 / q + 1 / r + 1 / s 2 = 1 , we obtain
| T b ˜ ( f 2 ) ( x ) T b ˜ ( f 2 ) ( x 0 ) | R n | R m ( b ˜ ; x , y ) R m ( b ˜ ; x 0 , y ) | | K ( x , y ) | | x y | m | f 2 ( y ) | d y + R n | K ( x , y ) | x y | m K ( x 0 , y ) | x 0 y | m | | R m ( b ˜ ; x 0 , y ) | | f 2 ( y ) | d y + | α | = m 1 α ! R n | K ( x , y ) ( x y ) α | x y | m K ( x 0 , y ) ( x 0 y ) α | x 0 y | m | | D α b ˜ ( y ) | | f 2 ( y ) | d y C | α | = m D α b Lip β ( w ) w ( x ˜ ) k = 0 w ( 2 k + 1 Q ˜ ) β / n 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 | f ( y ) | d y + C | α | = m D α b Lip β ( w ) w ( x ˜ ) k = 0 w ( 2 k + 1 Q ˜ ) β / n ( 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 ˜ | f ( y ) | r w ( y ) d y ) 1 / r ( 2 k + 1 Q ˜ 2 k Q ˜ w ( y ) p / r d y ) 1 / p + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 | D α b ( y ) ( D α b ) 2 k + 1 Q ˜ | | f ( y ) | d y + C | α | = m k = 0 2 k + 1 Q ˜ 2 k Q ˜ | x x 0 | | x 0 y | n + 1 | ( D α b ) 2 k + 1 Q ˜ ( D α b ) Q ˜ | | f ( y ) | d y + C | α | = m k = 0 ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | q d y ) 1 / q ( 2 k + 1 Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r × ( 2 k + 1 Q ˜ | D α b ( y ) ( D α b ) 2 k + 1 Q ˜ | p 1 w ( y ) 1 p 1 d y ) 1 / p 1 ( 2 k + 1 Q ˜ w ( y ) ( 1 / r 1 / p 1 ) s 1 d y ) 1 / s 1 + C | α | = m k = 0 ( 2 k + 1 Q ˜ 2 k Q ˜ | K ( x , y ) K ( x 0 , y ) | q d y ) 1 / q ( 2 k + 1 Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r × | ( D α b ) 2 k + 1 Q ˜ ( D α b ) Q ˜ | ( 2 k + 1 Q ˜ w ( y ) s 2 / r d y ) 1 / s 2 C | α | = m D α b Lip β ( w ) w ( x ˜ ) k = 1 k d ( 2 k d ) n + 1 w ( 2 k Q ˜ ) β / n ( 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) 1 / ( r 1 ) d y ) ( r 1 ) / r ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r | 2 k Q ˜ | w ( 2 k Q ˜ ) 1 / r + C | α | = m D α b Lip β ( w ) w ( x ˜ ) k = 1 C k | 2 k Q | 1 / q w ( 2 k Q ˜ ) β / n ( 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) p / r d y ) 1 / p ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r | 2 k Q ˜ | 1 / p + 1 / r w ( 2 k Q ˜ ) 1 / r + C | α | = m k = 1 d ( 2 k d ) n + 1 ( 2 k Q ˜ | D α b ( y ) ( D α b ) 2 k Q ˜ | r w ( y ) 1 r d y ) 1 / r × ( 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r + C | α | = m D α b Lip β ( w ) w ( x ˜ ) k = 1 C k | 2 k Q | 1 / q w ( 2 k Q ˜ ) β / n ( 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) ( 1 / r 1 / p 1 ) s 1 d y ) 1 / s 1 ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r 1 / p 1 × | 2 k Q ˜ | 1 / s 1 + 1 / r 1 / p 1 w ( 2 k Q ˜ ) 1 / r + 1 / p 1 + C | α | = m D α b Lip β ( w ) w ( x ˜ ) k = 1 k C k | 2 k Q | 1 / q w ( 2 k Q ˜ ) β / n ( 2 k Q ˜ | f ( y ) | r w ( y ) d y ) 1 / r × ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) d y ) 1 / r ( 1 | 2 k Q ˜ | 2 k Q ˜ w ( y ) s 2 / r d y ) 1 / s 2 | 2 k Q ˜ | 1 / s 2 + 1 / r w ( 2 k Q ˜ ) 1 / r C | α | = m D α b Lip β ( w ) w ( x ˜ ) k = 1 k ( C k + 2 k ) × ( 1 w ( 2 k Q ˜ ) 1 r β / n 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r + C | α | = m D α b Lip β ( w ) k = 1 k ( C k + 2 k ) w ( 2 k Q ˜ ) | 2 k Q ˜ | × ( 1 w ( 2 k Q ˜ ) 1 r β / n 2 k Q ˜ | f ( y ) | r w ( y ) d x ) 1 / r C | α | = m D α b Lip β ( w ) w ( x ˜ ) M β , r , w ( f ) ( x ˜ ) .

This completes the proof of Theorem 2. □

Proof of Theorem 3 Choose 1 < r < u in Theorem 1 and notice w 1 u A 1 , then we have, by Lemmas 3 and 4,
T b ( f ) L u ( w 1 u ) M η ( T b ( f ) ) L u ( w 1 u ) C M η # ( T b ( f ) ) L u ( w 1 u ) C | α | = m D α b BMO ( w ) w M r , w ( f ) L u ( w 1 u ) = C | α | = m D α b BMO ( w ) M r , w ( f ) L u ( w ) C | α | = m D α b BMO ( w ) f L u ( w ) .

This completes the proof of Theorem 3. □

Proof of Theorem 4 Choose 1 < r < u in Theorem 1 and notice w 1 u A 1 , then we have, by Lemmas 5 and 6,
T b ( f ) L u , φ ( w 1 u ) M η ( T b ( f ) ) L u , φ ( w 1 u ) C M η # ( T b ( f ) ) L u , φ ( w 1 u ) C | α | = m D α b BMO ( w ) w M r , w ( f ) L u , φ ( w 1 u ) = C | α | = m D α b BMO ( w ) M r , w ( f ) L u , φ ( w ) C | α | = m D α b BMO ( w ) f L u , φ ( w ) .

This completes the proof of Theorem 4. □

Proof of Theorem 5 Choose 1 < r < u in Theorem 2 and notice w 1 v A 1 , then we have, by Lemmas 3 and 4,
T b ( f ) L v ( w 1 v ) M η ( T b ( f ) ) L v ( w 1 v ) C M η # ( T b ( f ) ) L v ( w 1 v ) C | α | = m D α b Lip β ( w ) w M β , r , w ( f ) L v ( w 1 v ) = C | α | = m D α b Lip β ( w ) M β , r , w ( f ) L v ( w ) C | α | = m D α b Lip β ( w ) f L u ( w ) .

This completes the proof of Theorem 5. □

Proof of Theorem 6 Choose 1 < r < u in Theorem 2 and notice w 1 v A 1 , then we have, by Lemmas 5 and 6,
T b ( f ) L v , φ ( w 1 v ) M η ( T b ( f ) ) L v , φ ( w 1 v ) C M η # ( T b ( f ) ) L v , φ ( w 1 v ) C | α | = m D α b Lip β ( w ) w M β , r , w ( f ) L v , φ ( w 1 v ) = C | α | = m D α b Lip β ( w ) M β , r , w ( f ) L v , φ ( w ) C | α | = m D α b Lip β ( w ) f L u , φ ( w ) .

This completes the proof of Theorem 6. □

Declarations

Authors’ Affiliations

(1)
Changsha Commerce and Tourism College

References

  1. Garcia-Cuerva J, Rubio de Francia JL North-Holland Mathematics Studies 16. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.Google Scholar
  2. Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.MATHGoogle Scholar
  3. Torchinsky A Pure and Applied Math. 123. In Real Variable Methods in Harmonic Analysis. Academic Press, New York; 1986.Google Scholar
  4. Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611–635. 10.2307/1970954MathSciNetView ArticleMATHGoogle Scholar
  5. Pérez C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal. 1995, 128: 163–185. 10.1006/jfan.1995.1027MathSciNetView ArticleMATHGoogle Scholar
  6. Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174View ArticleMathSciNetMATHGoogle Scholar
  7. Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002MathSciNetView ArticleMATHGoogle Scholar
  8. Lin Y: Sharp maximal function estimates for Calderón-Zygmund type operators and commutators. Acta Math. Sci., Ser. A 2011, 31: 206–215.MathSciNetMATHGoogle Scholar
  9. Paluszynski M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J. 1995, 44: 1–17.MathSciNetView ArticleMATHGoogle Scholar
  10. Bloom S: A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 1985, 292: 103–122. 10.1090/S0002-9947-1985-0805955-5View ArticleMathSciNetMATHGoogle Scholar
  11. Hu B, Gu J: Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz spaces. J. Math. Anal. Appl. 2008, 340: 598–605. 10.1016/j.jmaa.2007.08.034MathSciNetView ArticleMATHGoogle Scholar
  12. Chang DC, Li JF, Xiao J: Weighted scale estimates for Calderón-Zygmund type operators. Contemp. Math. 2007, 446: 61–70.MathSciNetView ArticleMATHGoogle Scholar
  13. Liu LZ: Sharp maximal function estimates and boundedness for commutators associated with general integral operator. Filomat 2011,25(4):137–151. 10.2298/FIL1104137LMathSciNetView ArticleMATHGoogle Scholar
  14. Garcia-Cuerva J:Weighted H p spaces. Diss. Math. 1979, 162: 1–63.MathSciNetMATHGoogle Scholar
  15. Peetre J:On convolution operators leaving L p , λ -spaces invariant. Ann. Mat. Pura Appl. 1966, 72: 295–304. 10.1007/BF02414340MathSciNetView ArticleMATHGoogle Scholar
  16. Peetre J:On the theory of L p , λ -spaces. J. Funct. Anal. 1969, 4: 71–87. 10.1016/0022-1236(69)90022-6MathSciNetView ArticleGoogle Scholar
  17. Di Fazio G, Ragusa MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A 1991, 5: 323–332.MathSciNetMATHGoogle Scholar
  18. Di Fazio G, Ragusa MA: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 1993, 112: 241–256. 10.1006/jfan.1993.1032MathSciNetView ArticleMATHGoogle Scholar
  19. Liu LZ: Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators. Acta Math. Sci., Ser. B 2005, 25: 89–94.MathSciNetMATHGoogle Scholar
  20. Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. Harmonic Analysis: Proceedings of a Conference Held in Sendai 1990, 183–189. JapanGoogle Scholar
  21. Cohen J, Gosselin J:On multilinear singular integral operators on R n . Stud. Math. 1982, 72: 199–223.MathSciNetMATHGoogle Scholar
  22. Cohen J, Gosselin J: A BMO estimate for multilinear singular integral operators. Ill. J. Math. 1986, 30: 445–465.MathSciNetMATHGoogle Scholar
  23. Ding Y, Lu SZ: Weighted boundedness for a class rough multilinear operators. Acta Math. Sin. 2001, 17: 517–526.MathSciNetView ArticleMATHGoogle Scholar

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© Feng; licensee Springer. 2014

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