Skip to main content

Inequalities and boundedness for commutators related to integral operator with general kernel

Abstract

In this paper, we establish the sharp maximal function inequalities for the commutators related to some integral operator with general kernel and the BMO and Lipschitz functions. As an application, we obtain the boundedness of the commutators on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

MSC:42B20, 42B25.

1 Introduction and preliminaries

As the development of singular integral operators (see [13]), their commutators have been well studied (see [46]). In [57], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on L p ( R n ) for 1<p<. Chanillo (see [8]) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [911], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and L p ( R n ) (1<p<) spaces are 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 are obtained (see [12, 13]). The purpose of this paper is to prove the sharp maximal function inequalities for the commutator associated with some integral operator with general kernel and the BMO and Lipschitz functions. As an application, we obtain the boundedness of the commutator on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

First, let us introduce some notations. Throughout this paper, Q will denote 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 | dy;

here, and in what follows, f Q = | Q | 1 Q f(x)dx. It is well known that (see [1, 2])

M # (f)(x) sup Q x inf c C 1 | Q | Q | f ( y ) c | dy.

We say that f belongs to BMO( R n ) if M # (f) belongs to L ( R n ) and define f B M O = M # ( f ) L . It has been known that (see [14])

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

Let

M(f)(x)= sup Q x 1 | Q | Q | f ( y ) | dy.

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

For 0<η<1 and 1r<, set

M η , r (f)(x)= sup Q x ( 1 | Q | 1 r η / n Q | f ( y ) | r d y ) 1 / r .

The A p weight is defined by (see [1])

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 < } , 1 < p < ,

and

A 1 = { w L loc p ( R n ) : M ( w ) ( x ) C w ( x ) ,  a.e. } .

For β>0 and p>1, let F ˙ p β , ( R n ) be the weighted homogeneous Triebel-Lizorkin space (see [11]).

For β>0, the Lipschitz space Lip β ( R n ) is the space of functions f such that

f Lip β = sup x , y R n x y | f ( x ) f ( y ) | | x y | β <.

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

Definition 1 Let F t (x,y) be defined on R n × R n ×[0,+) and b be a locally integrable function on R n , set

F t (f)(x)= R n F t (x,y)f(y)dy

and

F t b (f)(x)= R n ( b ( x ) b ( y ) ) F t (x,y)f(y)dy

for every bounded and compactly supported function f.

Let H be the Banach space H={h:h<}. For each fixed x R n , we view F t (f)(x) and F t b (f)(x) as the mappings from [0,+) to H. Set

T(f)(x)= F t ( f ) ( x ) ,

which T is bounded on L 2 ( R n ). The commutator related to F t b is defined by

T b (f)(x)= F t b ( f ) ( x )

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

2 | y z | < | x y | ( F t ( x , y ) F t ( x , z ) + F t ( y , x ) F t ( z , x ) ) dxC

and

( 2 k | z y | | x y | < 2 k + 1 | z y | ( F t ( x , y ) F t ( x , z ) + F t ( y , x ) F t ( 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.

Definition 2 Let φ be a positive, increasing function on R + and there exists a constant D>0 such that

φ(2t)Dφ(t)for t0.

Let f be a locally integrable function on R n . Set, for 0η<n and 1p<n/η,

f L p , η , φ = sup x R n , d > 0 ( 1 φ ( d ) 1 p η / n Q ( x , d ) | f ( y ) | p d y ) 1 / p ,

where Q(x,d)={y R n :|xy|<d}. The generalized fractional Morrey space is defined by

L p , η , φ ( R n ) = { f L loc 1 ( R n ) : f L p , η , φ < } .

We write L p , η , φ ( R n )= L p , φ ( R n ) if η=0, which is the generalized Morrey space. If φ(d)= d δ , δ>0, then L p , φ ( R n )= L p , δ ( R n ), which is the classical Morrey spaces (see [14, 15]). If φ(d)=1, then L p , φ ( R n ,w)= L p ( R n ), which is the Lebesgue spaces.

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 [7, 1619]).

It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [5, 6]). In [6], Pérez and Trujillo-Gonzalez prove a sharp estimate for the multilinear commutator. The main purpose of this paper is to prove the sharp maximal inequalities for the commutator. As the application, we obtain the L p -norm inequality, Morrey and Triebel-Lizorkin spaces boundedness for the commutator.

2 Theorems

We shall prove the following theorems.

Theorem 1 Let T be the integral operator as Definition 1, the sequence { C k } l 1 , 0<β<1, q s< and b Lip β ( R n ). Then there exists a constant C>0 such that, for any f C 0 ( R n ) and x ˜ R n ,

M # ( T b ( f ) ) ( x ˜ )C b Lip β ( M β , s ( f ) ( x ˜ ) + M β , s ( T ( f ) ) ( x ˜ ) ) .

Theorem 2 Let T be the integral operator as Definition 1, the sequence { 2 k β C k } l 1 , 0<β<1, q s< and b Lip β ( R n ). Then there exists a constant C>0 such that, for any f C 0 ( R n ) and x ˜ R n ,

sup Q x ˜ inf c R n 1 | Q | 1 + β / n Q | T b (f)(x)c|dxC b Lip β ( M s ( f ) ( x ˜ ) + M s ( T ( f ) ) ( x ˜ ) ) .

Theorem 3 Let T be the integral operator as Definition 1, the sequence {k C k } l 1 , q s< and bBMO( R n ). Then there exists a constant C>0 such that, for any f C 0 ( R n ) and x ˜ R n ,

M # ( T b ( f ) ) ( x ˜ )C b B M O ( M s ( f ) ( x ˜ ) + M s ( T ( f ) ) ( x ˜ ) ) .

Theorem 4 Let T be the integral operator as Definition 1, the sequence { C k } l 1 , 0<β<min(1,n/ q ), q <p<n/β, 1/r=1/pβ/n and b Lip β ( R n ). Then T b is bounded from L p ( R n ) to L r ( R n ).

Theorem 5 Let T be the integral operator as Definition 1, the sequence { C k } l 1 , 0<D< 2 n , 0<β<min(1,n/ q ), q <p<n/β, 1/r=1/pβ/n and b Lip β ( R n ). Then T b is bounded from L p , β , φ ( R n ) to L r , φ ( R n ).

Theorem 6 Let T be the integral operator as Definition 1, the sequence { 2 k β C k } l 1 , 0<β<min(1,n/ q ), q <p<n/β, 1/r=1/pβ/n and b Lip β ( R n ). Then T b is bounded from L p ( R n ) to F ˙ r β , ( R n ).

Theorem 7 Let T be the integral operator as Definition 1, the sequence {k C k } l 1 and bBMO( R n ). Then T b is bounded on L p ( R n ) for q p<.

3 Proofs of theorems

To prove the theorems, we need the following lemma.

Lemma 1 (see [12])

Let T be the integral operator as Definition 1, the sequence { C k } l 1 . Then T is bounded on L p ( R n ) for 1<p<.

Lemma 2 (see [11])

For 0<β<1 and 1<p<, we have

f F ˙ p β , sup Q 1 | Q | 1 + β / n Q | f ( x ) f Q | d x L p sup Q inf c 1 | Q | 1 + β / n Q | f ( x ) c | d x L p .

Lemma 3 (see [1])

Let 0<p< and w 1 r < A r . Then, for any smooth function f for which the left-hand side is finite,

R n M(f) ( x ) p w(x)dxC R n M # (f) ( x ) p w(x)dx.

Lemma 4 (see [8])

Suppose that 0<η<n, 1<s<p<n/η and 1/q=1/pη/n. Then

M η , s ( f ) L q C f L p .

Lemma 5 Let 1<p<, 0<D< 2 n . Then, for any smooth function f for which the left-hand side is finite,

M ( f ) L p , φ C M # ( f ) L p , φ .

Proof For any cube Q=Q( x 0 ,d) in R n , we know M( χ Q ) A 1 for any cube Q=Q(x,d) by [20]. Noticing that M( χ Q )1 and M( χ Q )(x) d n / ( | x x 0 | d ) n if x Q c , by Lemma 3, we have, for f L p , φ ( R n ),

Q M ( f ) ( x ) p d x = R n M ( f ) ( x ) p χ Q ( x ) d x R n M ( f ) ( x ) p M ( χ Q ) ( x ) d x C R n M # ( f ) ( x ) p M ( χ Q ) ( x ) d x = C ( Q M # ( f ) ( x ) p M ( χ Q ) ( x ) d x + k = 0 2 k + 1 Q 2 k Q M # ( f ) ( x ) p M ( χ Q ) ( x ) d x ) C ( Q M # ( f ) ( x ) p d x + k = 0 2 k + 1 Q 2 k Q M # ( f ) ( x ) p | Q | | 2 k + 1 Q | d x ) C ( Q M # ( f ) ( x ) p d x + k = 0 2 k + 1 Q M # ( f ) ( x ) p 2 k n d y ) C M # ( f ) L p , φ p k = 0 2 k n φ ( 2 k + 1 d ) C M # ( f ) L p , φ p k = 0 ( 2 n D ) k φ ( d ) C M # ( f ) L p , φ p φ ( d ) ,

thus

( 1 φ ( d ) Q M ( f ) ( x ) p d x ) 1 / p C ( 1 φ ( d ) Q M # ( f ) ( x ) p d x ) 1 / p

and

M ( f ) L p , φ C M # ( f ) L p , φ .

This finishes the proof. □

Lemma 6 Let T be the integral operator as Definition 1, 0η<n, 0<D< 2 n and 1p<. Then

T ( f ) L p , η , φ C f L p , η , φ .

Lemma 7 Let 0<D< 2 n , 0<η<n, 1s<p<n/η and 1/q=1/pη/n. Then

M η , s ( f ) L q , φ C f L p , η , φ .

The proofs of the two lemmas are similar to that of Lemma 5 by Lemmas 1 and 4, we omit the details.

Proof of Theorem 1 It suffices to prove for f C 0 ( R n ) and some constant C 0 , the following inequality holds:

1 | Q | Q | T b (f)(x) C 0 |dxC b Lip β ( M β , s ( f ) ( x ˜ ) + M β , s ( T ( f ) ) ( x ˜ ) ) .

Fix a cube Q=Q( x 0 ,d) and x ˜ Q. Write, for f 1 =f χ 2 Q and f 2 =f χ ( 2 Q ) c ,

F t b (f)(x)= ( b ( x ) b 2 Q ) F t (f)(x) F t ( ( b b 2 Q ) f 1 ) (x) F t ( ( b b 2 Q ) f 2 ) (x).

Then

1 | Q | Q F t b ( f ) ( x ) F t ( ( b 2 Q b ) f 2 ) ( x 0 ) d x 1 | Q | Q ( b ( x ) b 2 Q ) F t ( f ) ( x ) d x + 1 | Q | Q F t ( ( b b 2 Q ) f 1 ) ( x ) d x + 1 | Q | Q F t ( ( b b 2 Q ) f 2 ) ( x ) F t ( ( b b 2 Q ) f 2 ) ( x 0 ) d x = I 1 + I 2 + I 3 .

For I 1 , by Hölder’s inequality and Lemma 2, we obtain

I 1 C | Q | sup x 2 Q | b ( x ) b 2 Q | | Q | 1 1 / s ( Q | T ( f ) ( x ) | s d x ) 1 / s C | Q | 1 / s b Lip β | 2 Q | β / n | 2 Q | 1 / s β / n ( 1 | Q | 1 s β / n Q | T ( f ) ( x ) | s d x ) 1 / s C b Lip β M β , s ( T ( f ) ) ( x ˜ ) .

For I 2 , by the boundedness of T, we get

I 2 ( 1 | Q | R n | T ( ( b b 2 Q ) f 1 ) ( x ) | s d x ) 1 / s C ( 1 | Q | R n | ( b ( x ) b 2 Q ) f 1 ( x ) | s d x ) 1 / s C | Q | 1 / s | 2 Q | β / n | | 2 Q | 1 / s β / n ( 1 | 2 Q | 1 s β / n 2 Q | f ( x ) | s d x ) 1 / s C b Lip β M β , s ( f ) ( x ˜ ) .

For I 3 , recalling that s> q , we have

I 3 1 | Q | Q ( 2 Q ) c | b ( y ) b 2 Q | | f ( y ) | F t ( x , y ) F t ( x 0 , y ) d y d x 1 | Q | Q k = 1 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) | b ( y ) b 2 k + 1 Q | | f ( y ) | d y d x + 1 | Q | Q k = 1 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) | b 2 k + 1 Q b 2 Q | | f ( y ) | d y d x C | Q | Q k = 1 ( 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) q d y ) 1 / q × sup y 2 k + 1 Q | b ( y ) b 2 k + 1 Q | ( 2 k + 1 Q | f ( y ) | q d y ) 1 / q d x + C | Q | Q k = 1 | b 2 k + 1 Q b 2 Q | ( 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) q d y ) 1 / q × ( 2 k + 1 Q | f ( y ) | q d y ) 1 / q d x C k = 1 C k ( 2 k d ) n / q | 2 k + 1 Q | β / n b Lip β | 2 k + 1 Q | 1 / q 1 / s | 2 k + 1 Q | 1 / s β / n × ( 1 | 2 k + 1 Q | 1 s β / n 2 k + 1 Q | f ( y ) | s d y ) 1 / s + C k = 1 b Lip β | 2 k Q | β / n C k ( 2 k d ) n / q | 2 k + 1 Q | 1 / q 1 / s | 2 k + 1 Q | 1 / s β / n × ( 1 | 2 k + 1 Q | 1 s β / n 2 k + 1 Q | f ( y ) | s d y ) 1 / s C b Lip β M β , s ( f ) ( x ˜ ) k = 1 C k C b Lip β M β , s ( f ) ( x ˜ ) .

This completes the proof of Theorem 1. □

Proof of Theorem 2 It suffices to prove for f C 0 ( R n ) and some constant C 0 , the following inequality holds:

1 | Q | 1 + β / n Q | T b (f)(x) C 0 |dxC b Lip β ( M s ( f ) ( x ˜ ) + M s ( T ( f ) ) ( x ˜ ) ) .

Fix a cube Q=Q( x 0 ,d) and x ˜ Q. Write, for f 1 =f χ 2 Q and f 2 =f χ ( 2 Q ) c ,

F t b (f)(x)= ( b ( x ) b 2 Q ) F t (f)(x) F t ( ( b b 2 Q ) f 1 ) (x) F t ( ( b b 2 Q ) f 2 ) (x).

Then

1 | Q | 1 + β / n Q F t b ( f ) ( x ) F t ( ( b 2 Q b ) f 2 ) ( x 0 ) d x 1 | Q | 1 + β / n Q ( b ( x ) b 2 Q ) F t ( f ) ( x ) d x + 1 | Q | 1 + β / n Q F t ( ( b b 2 Q ) f 1 ) ( x ) d x + 1 | Q | 1 + β / n Q F t ( ( b b 2 Q ) f 2 ) ( x ) F t ( ( b b 2 Q ) f 2 ) ( x 0 ) d x = I I 1 + I I 2 + I I 3 .

By using the same argument as in the proof of Theorem 1, we get

I I 1 C | Q | 1 + β / n sup x 2 Q | b ( x ) b 2 Q | | Q | 1 1 / s ( Q | T ( f ) ( x ) | s d x ) 1 / s C b Lip β | 2 Q | β / n | Q | 1 / s | Q | 1 / s β / n ( 1 | Q | Q | T ( f ) ( x ) | s d x ) 1 / s C b Lip β M s ( T ( f ) ) ( x ˜ ) , I I 2 1 | Q | 1 + β / n | Q | 1 1 / s ( R n | T ( ( b b 2 Q ) f 1 ) ( x ) | s d x ) 1 / s C | Q | 1 + β / n | Q | 1 1 / s ( R n | ( b ( x ) b 2 Q ) f 1 ( x ) | s d x ) 1 / s C | Q | 1 + β / n | Q | 1 1 / s | 2 Q | β / n | 2 Q | 1 / s ( 1 | 2 Q | 2 Q | f ( x ) | s d x ) 1 / s C b Lip β M s ( f ) ( x ˜ ) , I I 3 1 | Q | 1 + β / n Q ( 2 Q ) c | b ( y ) b 2 Q | | f ( y ) | F t ( x , y ) F t ( x 0 , y ) d y d x 1 | Q | 1 + β / n Q k = 1 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) | b ( y ) b 2 k + 1 Q | | f ( y ) | d y d x + 1 | Q | 1 + β / n Q k = 1 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) | b 2 k + 1 Q b 2 Q | | f ( y ) | d y d x C | Q | 1 + β / n Q k = 1 ( 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) q d y ) 1 / q × sup y 2 k + 1 Q | b ( y ) b 2 k + 1 Q | ( 2 k + 1 Q | f ( y ) | q d y ) 1 / q d x + C | Q | 1 + β / n Q k = 1 | b 2 k + 1 Q b 2 Q | ( 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) q d y ) 1 / q × ( 2 k + 1 Q | f ( y ) | q d y ) 1 / q d x C | Q | β / n k = 1 C k ( 2 k d ) n / q | 2 k + 1 Q | β / n b Lip β | 2 k + 1 Q | 1 / q × ( 1 | 2 k + 1 Q | 2 k + 1 Q | f ( y ) | s d y ) 1 / s + C | Q | β / n k = 1 b Lip β | 2 k Q | β / n C k ( 2 k d ) n / q | 2 k + 1 Q | 1 / q × ( 1 | 2 k + 1 Q | 2 k + 1 Q | f ( x ) | s d x ) 1 / s C b Lip β M s ( f ) ( x ˜ ) k = 1 2 k β C k C b Lip β M s ( f ) ( x ˜ ) .

This completes the proof of Theorem 2. □

Proof of Theorem 3 It suffices to prove for f C 0 ( R n ) and some constant C 0 , the following inequality holds:

1 | Q | Q | T b (f)(x) C 0 |dxC b B M O ( M s ( f ) ( x ˜ ) + M s ( T ( f ) ) ( x ˜ ) ) .

Fix a cube Q=Q( x 0 ,d) and x ˜ Q. Write, for f 1 =f χ 2 Q and f 2 =f χ ( 2 Q ) c ,

F t b (f)(x)= ( b ( x ) b 2 Q ) F t (f)(x) F t ( ( b b 2 Q ) f 1 ) (x) F t ( ( b b 2 Q ) f 2 ) (x).

Then

1 | Q | Q F t b ( f ) ( x ) F t ( ( b 2 Q b ) f 2 ) ( x 0 ) d x 1 | Q | Q ( b ( x ) b 2 Q ) F t ( f ) ( x ) d x + 1 | Q | Q F t ( ( b b 2 Q ) f 1 ) ( x ) d x + 1 | Q | Q F t ( ( b b 2 Q ) f 2 ) ( x ) F t ( ( b b 2 Q ) f 2 ) ( x 0 ) d x = I I I 1 + I I I 2 + I I I 3 .

For II I 1 , by Hölder’s inequality, we get

I I I 1 ( 1 | Q | Q | b ( x ) b 2 Q | s d x ) 1 / s ( 1 | Q | Q | T ( f ) ( x ) | s d x ) 1 / s C b B M O M s ( T ( f ) ) ( x ˜ ) .

For II I 2 , choose 1<r<s, by Hölder’s inequality and the boundedness of T, we obtain

I I I 2 ( 1 | Q | R n | T ( ( b b 2 Q ) f 1 ) ( x ) | r d x ) 1 / r C | Q | 1 / r ( R n | ( b b 2 Q ) f 1 ( x ) | r d x ) 1 / r C | Q | 1 / r ( 2 Q | b ( x ) b 2 Q | s r / ( s r ) d x ) ( s r ) / s r ( 2 Q | f ( x ) | s d x ) 1 / s C b B M O ( 1 | 2 Q | 2 Q | f ( x ) | s d x ) 1 / s C b B M O M s ( f ) ( x ˜ ) .

For II I 3 , recalling that s> q , taking 1<p<, 1<r<s with 1/p+1/q+1/r=1, by Hölder’s inequality, we obtain

I I I 3 1 | Q | Q ( 2 Q ) c | b ( y ) b 2 Q | | f ( y ) | F t ( x , y ) F t ( x 0 , y ) d y d x 1 | Q | Q k = 1 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) | b ( y ) b 2 Q | | f ( y ) | d y d x 1 | Q | Q k = 1 ( 2 k d | y x 0 | < 2 k + 1 d F t ( x , y ) F t ( x 0 , y ) q d y ) 1 / q × ( 2 k + 1 Q | b ( x ) b 2 Q | p d x ) 1 / p ( 2 k + 1 Q | f ( y ) | r d y ) 1 / r d x C k = 1 k ( 2 k d ) n / q ( 2 k d | y x 0 | < 2 k + 1 d K ( x , y ) K ( x 0 , y ) q d y ) 1 / q × b B M O ( 1 | 2 k + 1 Q | 2 k + 1 Q | f ( y ) | s d y ) 1 / s C b B M O M s ( f ) ( x ˜ ) k = 1 k C k C b B M O M s ( f ) ( x ˜ ) .

This completes the proof of Theorem 3. □

Proof of Theorem 4 Choose q <s<p in Theorem 1, we have, by Lemmas 1, 4, and 5,

T b ( f ) L r M ( T b ( f ) ) L r C M # ( T b ( f ) ) L r C b Lip β ( M β , s ( T ( f ) ) L r + M β , s ( f ) L r ) C b Lip β ( T ( f ) L p + f L p ) C b Lip β f L p .

This completes the proof of Theorem 4. □

Proof of Theorem 5 Choose q <s<p in Theorem 1, we have, by Lemmas 5-7,

T b ( f ) L r , φ M ( T b ( f ) ) L r , φ C M # ( T b ( f ) ) L r , φ C b Lip β ( M β , s ( T ( f ) ) L r , φ + M β , s ( f ) L r , φ ) C b Lip β ( T ( f ) L p , β , φ + f L p , β , φ ) C b Lip β f L p , φ .

This completes the proof of Theorem 5. □

Proof Theorem 6 Choose q <s<p in Theorem 2. By using Lemma 3, we obtain

T b ( f ) F ˙ r β , C sup Q 1 | Q | 1 + β / n Q | T b ( f ) ( x ) T ( ( b 2 Q b ) f 2 ) ( x 0 ) | d x L r C b Lip β ( M s ( T ( f ) ) L r + M s ( f ) L r ) C b Lip β ( T ( f ) L p + f L p ) C b Lip β f L p .

This completes the proof of the theorem. □

Proof of Theorem 7 Choose q s<p in Theorem 3, we have

T b ( f ) L p M ( T b ( f ) ) L p C M # ( T b ( f ) ) L p C b B M O ( M s ( T ( f ) ) L p + M s ( f ) L p ) C b B M O ( T ( f ) L p + f L p ) C b B M O f L p .

This completes the proof of the theorem. □

4 Applications

In this section we shall apply Theorems 1-6 of the paper to some particular operators such as the Littlewood-Paley operators, Marcinkiewicz operator and Bochner-Riesz operator.

Application 1 Littlewood-Paley operators.

Fixed ε>0 and μ>(3n+2)/n. Let ψ be a fixed function which satisfies:

  1. (1)

    R n ψ(x)dx=0,

  2. (2)

    |ψ(x)|C ( 1 + | x | ) ( n + 1 ) ,

  3. (3)

    |ψ(x+y)ψ(x)|C | y | ε ( 1 + | x | ) ( n + 1 + ε ) when 2|y|<|x|.

We denote Γ(x)={(y,t) R + n + 1 :|xy|<t} and the characteristic function of Γ(x) by χ Γ ( x ) . The Littlewood-Paley commutators are defined by

g ψ b ( f ) ( x ) = ( 0 | F t b ( f ) ( x ) | 2 d t t ) 1 / 2 , S ψ b ( f ) ( x ) = [ Γ ( x ) | F t b ( f ) ( x , y ) | 2 d y d t t n + 1 ] 1 / 2 ,

and

g μ b (f)(x)= [ R + n + 1 ( t t + | x y | ) n μ | F t b ( f ) ( x , y ) | 2 d y d t t n + 1 ] 1 / 2 ,

where

F t b ( f ) ( x ) = R n ( b ( x ) b ( y ) ) ψ t ( x y ) f ( y ) d y , F t b ( f ) ( x , y ) = R n ( b ( x ) b ( z ) ) f ( z ) ψ t ( y z ) d z

and ψ t (x)= t n ψ(x/t) for t>0. Set F t (f)(y)=f ψ t (y). We also define

g ψ ( f ) ( x ) = ( 0 | F t ( f ) ( x ) | 2 d t t ) 1 / 2 , S ψ ( f ) ( x ) = ( Γ ( x ) | F t ( f ) ( y ) | 2 d y d t t n + 1 ) 1 / 2

and

g μ (f)(x)= ( R + n + 1 ( t t + | x y | ) n μ | F t ( f ) ( y ) | 2 d y d t t n + 1 ) 1 / 2 ,

which are the Littlewood-Paley operators (see [3]). Let H be the space

H= { h : h = ( 0 | h ( t ) | 2 d t / t ) 1 / 2 < }

or

H= { h : h = ( R + n + 1 | h ( y , t ) | 2 d y d t / t n + 1 ) 1 / 2 < } ,

then, for each fixed x R n , F t b (f)(x) and F t b (f)(x,y) may be viewed as the mapping from [0,+) to H, and it is clear that

g ψ b (f)(x)= F t b ( f ) ( x ) , g ψ (f)(x)= F t ( f ) ( x ) ,
S ψ b (f)(x)= χ Γ ( x ) F t b ( f ) ( x , y ) , S ψ (f)(x)= χ Γ ( x ) F t ( f ) ( y )

and

g μ b (f)(x)= ( t t + | x y | ) n μ / 2 F t b ( f ) ( x , y ) , g μ (f)(x)= ( t t + | x y | ) n μ / 2 F t ( f ) ( y ) .

It is easily to see that g ψ b , S ψ b , and g μ b satisfy the conditions of Theorems 1-6 (see [2123]), thus Theorems 1-6 hold for g ψ b , S ψ b , and g μ b .

Application 2 Marcinkiewicz operators.

Fixed λ>max(1,2n/(n+2)) and 0<γ1. Let Ω be homogeneous of degree zero on R n with S n 1 Ω( x )dσ( x )=0. Assume that Ω Lip γ ( S n 1 ). The Marcinkiewicz commutators are defined by

μ Ω b ( f ) ( x ) = ( 0 | F t b ( f ) ( x ) | 2 d t t 3 ) 1 / 2 , μ S b ( f ) ( x ) = [ Γ ( x ) | F t b ( f ) ( x , y ) | 2 d y d t t n + 3 ] 1 / 2 ,

and

μ λ b (f)(x)= [ R + n + 1 ( t t + | x y | ) n λ | F t b ( f ) ( x , y ) | 2 d y d t t n + 3 ] 1 / 2 ,

where

F t b (f)(x)= | x y | t ( b ( x ) b ( y ) ) Ω ( x y ) | x y | n 1 f(y)dy

and

F t b (f)(x,y)= | y z | t ( b ( x ) b ( z ) ) Ω ( y z ) | y z | n 1 f(z)dz.

Set

F t (f)(x)= | x y | t Ω ( x y ) | x y | n 1 f(y)dy.

We also define

μ Ω ( f ) ( x ) = ( 0 | F t ( f ) ( x ) | 2 d t t 3 ) 1 / 2 , μ S ( f ) ( x ) = ( Γ ( x ) | F t ( f ) ( y ) | 2 d y d t t n + 3 ) 1 / 2 ,

and

μ λ (f)(x)= ( R + n + 1 ( t t + | x y | ) n λ | F t ( f ) ( y ) | 2 d y d t t n + 3 ) 1 / 2 ,

which are the Marcinkiewicz operators (see [24]). Let H be the space

H= { h : h = ( 0 | h ( t ) | 2 d t / t 3 ) 1 / 2 < }

or

H= { h : h = ( R + n + 1 | h ( y , t ) | 2 d y d t / t n + 3 ) 1 / 2 < } .

Then it is clear that

μ Ω b ( f ) ( x ) = F t b ( f ) ( x ) , μ Ω ( f ) ( x ) = F t ( f ) ( x ) , μ S b ( f ) ( x ) = χ Γ ( x ) F t b ( f ) ( x , y ) , μ S ( f ) ( x ) = χ Γ ( x ) F t ( f ) ( y ) ,

and

μ λ b (f)(x)= ( t t + | x y | ) n λ / 2 F t b ( f ) ( x , y ) , μ λ (f)(x)= ( t t + | x y | ) n λ / 2 F t ( f ) ( y ) .

It is easy to see that μ Ω b , μ S b , and μ λ b satisfy the conditions of Theorems 1-6 (see [2124]), thus Theorems 1-6 hold for μ Ω b , μ S b , and μ λ b .

Application 3 Bochner-Riesz operator.

Let δ>(n1)/2, B t δ ( f ˆ )(ξ)= ( 1 t 2 | ξ | 2 ) + δ f ˆ (ξ) and B t δ (z)= t n B δ (z/t) for t>0. Set

F δ , t b (f)(x)= R n ( b ( x ) b ( y ) ) B t δ (xy)f(y)dy.

The maximal Bochner-Riesz commutator is defined by

B δ , b (f)(x)= sup t > 0 | B δ , t b ( f ) ( x ) | .

We also define

B δ , (f)(x)= sup t > 0 | B t δ ( f ) ( x ) | ,

which is the maximal Bochner-Riesz operator (see [25]). Let H be the space H={h:h= sup t > 0 |h(t)|<}, then

B δ , b (f)(x)= B δ , t b ( f ) ( x ) , B δ (f)(x)= B t δ ( f ) ( x ) .

It is easy to see that B δ , b satisfies the conditions of Theorems 1-6 (see [21]), thus Theorems 1-6 hold for B δ , b .

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.

    MATH  Google 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. 10.2307/1970954

    Article  MathSciNet  MATH  Google Scholar 

  5. Pérez C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal. 1995, 128: 163–185. 10.1006/jfan.1995.1027

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. 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.1993.1032

    Article  MathSciNet  MATH  Google Scholar 

  8. Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen WG: Besov estimates for a class of multilinear singular integrals. Acta Math. Sin. 2000, 16: 613–626. 10.1007/s101140000059

    Article  MathSciNet  MATH  Google Scholar 

  10. Janson S: Mean oscillation and commutators of singular integral operators. Ark. Math. 1978, 16: 263–270. 10.1007/BF02386000

    Article  MathSciNet  MATH  Google Scholar 

  11. Paluszynski M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J. 1995, 44: 1–17.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Peetre J:On convolution operators leaving L p , λ -spaces invariant. Ann. Mat. Pura Appl. 1966, 72: 295–304. 10.1007/BF02414340

    Article  MathSciNet  MATH  Google Scholar 

  15. Peetre J:On the theory of L p , λ -spaces. J. Funct. Anal. 1969, 4: 71–87. 10.1016/0022-1236(69)90022-6

    Article  MathSciNet  Google Scholar 

  16. Chiarenza F, Frasca M: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. 1987, 7: 273–279.

    MathSciNet  MATH  Google Scholar 

  17. Di Fazio G, Ragusa MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A (7) 1991, 5: 323–332.

    MathSciNet  MATH  Google Scholar 

  18. Liu LZ: Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators. Acta Math. Sci. 2005,25(1):89–94.

    MathSciNet  MATH  Google Scholar 

  19. Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. In: Harmonic Analysis ICM-90 Satellite Conference Proceedings (Sendai, 1990), 1991, 183–189.

    Google Scholar 

  20. Coifman RR, Rochberg R: Another characterization of BMO. Proc. Am. Math. Soc. 1980, 79: 249–254. 10.1090/S0002-9939-1980-0565349-8

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu LZ: The continuity of commutators on Triebel-Lizorkin spaces. Integral Equ. Oper. Theory 2004, 49: 65–76. 10.1007/s00020-002-1186-8

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu LZ: Triebel-Lizorkin space estimates for multilinear operators of sublinear operators. Proc. Indian Acad. Sci. Math. Sci. 2003, 113: 379–393. 10.1007/BF02829632

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu LZ: Boundedness for multilinear Littlewood-Paley operators on Triebel-Lizorkin spaces. Methods Appl. Anal. 2004,10(4):603–614.

    MathSciNet  MATH  Google Scholar 

  24. Torchinsky A, Wang S: A note on the Marcinkiewicz integral. Colloq. Math. 1990,60(61):235–243.

    MathSciNet  MATH  Google Scholar 

  25. Lu SZ: Four Lectures on Real Hp Spaces. World Scientific, River Edge; 1995.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jiasheng Zeng.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zeng, J. Inequalities and boundedness for commutators related to integral operator with general kernel. J Inequal Appl 2014, 172 (2014). https://doi.org/10.1186/1029-242X-2014-172

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-172

Keywords