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Boundedness for multilinear commutator of Marcinkiewicz operator with variable kernels on Hardy and Herz-Hardy spaces

Abstract

In this paper, the ( H b p , L p )- and (H K ˙ q , b α , p , K ˙ q α , p )-type boundedness for the multilinear commutator related to the Marcinkiewicz operator with variable kernels is obtained.

MSC:42B20, 42B25.

1 Introduction and definitions

Let T be the Calderón-Zygmund operator and bBMO( R n ). The commutator [b,T] generated by T and b is defined by

[b,T](f)(x)=b(x)T(f)(x)T(bf)(x).

A classical result of Coifman, Rochberg and Weiss (see [1, 2]) proved that the commutator [b,T] is bounded on L p ( R n ) (1<p<). However, it was observed that the [b,T] is not bounded, in general, from H p ( R n ) to L p ( R n ). But if H p ( R n ) is replaced by a suitable atomic space H b P ( R n ), then [b,T] maps continuously H b P ( R n ) into L p ( R n ) (see [3]). In addition, we easily know that H b p ( R n ) H p ( R n ). In recent years, the theory of Herz-type Hardy spaces have been developed (see [47]). The main purpose of this paper is to consider the continuity of multilinear commutators related to Marcinkiewicz operators with variable kernels and BMO( R n ) functions on certain Hardy and Herz-Hardy spaces. Let us first introduce some definitions (see [314]).

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 .

Definition 1 Let b i (i=1,,m) be a locally integrable function and 0<p1. A bounded measurable function a on R n is said to be a (p, b ) atom if

  1. (1)

    suppaB=B( x 0 ,r),

  2. (2)

    a L | B | 1 / p ,

  3. (3)

    B a(y)dy= B a(y) l σ b l (y)dy=0 for any σ C j m , 1jm.

A temperate distribution f is said to belong to H b p ( R n ) if, in the Schwartz distribution sense, it can be written as

f(x)= j = 1 λ j a j (x),

where every a j is (p, b ) atom, λC and j = 1 | λ | p <. Moreover, f H b p ( R n ) ( j = 1 | λ j | p ) 1 / p .

Given a set E R n , the characteristic function of E is defined by χ E . Let B k ={x R n :|x| 2 k } and C k = B k B k 1 and χ k = χ B k , kZ.

Definition 2 Let 0<p, q<, αR. For kZ, set B k ={x R n :|x| 2 k } and C k = B k B k 1 . Denote by χ k the characteristic function of C k and by χ 0 the characteristic function of B 0 .

  1. (1)

    The homogeneous Herz space is defined by

    K ˙ q α , p ( R n ) = { f L loc q ( R n { 0 } ) : f K ˙ q α , p < } ,

where

f K ˙ q α , p = [ k = 2 k α p f χ k L q p ] 1 / p .
  1. (2)

    The nonhomogeneous Herz space is defined by

    K q α , p ( R n ) = { f L loc q ( R n ) : f K q α , p < } ,

where

f K q α , p = [ k = 1 2 k α p f χ k L q p + f χ 0 L q p ] 1 / p .

Definition 3 Let αR, 1<q<, 0<α<n(11/q), b i BMO( R n ), 1im. A function a on R n is called a central (α,q, b )-atom (or a central (α,q, b )-atom of restrict type) if

  1. (1)

    suppaB=B(0,r) (or for some r1),

  2. (2)

    a L q | B | α / n ,

  3. (3)

    B a(x)dx= B a(x) l σ b l (x)dx=0 for any σ C j m , 1jm.

A temperate distribution f is said to belong to H K ˙ q , b α , p ( R n ) (or H K q , b α , p ( R n )) if it can be written as f= j = λ j a j (or f= j = 0 λ j a j ), in the Schwartz distribution sense, where a j is a central (α,q, b )-atom (or a central (α,q, b )-atom of restrict type) supported on B(0, 2 j ) and | λ j | p < (or j = 0 | λ j | p <). Moreover, f H K ˙ q , b α , p (or  f H K q , b α , p ) ( j | λ j | p ) 1 / p .

Definition 4 Let Ω be homogeneous of degree zero on R n such that ω r (δ) is defined

ω r (δ)= sup | ρ | < δ ( S n 1 | Ω ( ρ x ´ ) Ω ( x ´ ) | r d σ ( x ´ ) ) 1 / r ,

where |ρ|= sup x ´ S n 1 |ρ x ´ x ´ |.

If 0 1 ω r ( δ ) δ dδ<, we say Ω(x) satisfied L r -Dini condition.

Definition 5 Let 0<ϵ<n, 0<γ1 and Ω be homogeneous of degree zero on R n such that S n 1 Ω( x )dσ( x )=0. Assume that Ω Lip γ ( S n 1 ), that is, there exists a constant M>0 such that for any x,y S n 1 , |Ω(x)Ω(y)|M | x y | γ . The Marcinkiewicz multilinear commutator is defined by

μ ϵ b ˜ (f)(x)= ( 0 | F t b ˜ ( f ) ( x ) | 2 d t t 3 ) 1 / 2 ,

where

F t b ˜ (f)(x)= | x y | t Ω ( x y ) | x y | n 1 ϵ [ j = 1 m ( b j ( x ) b j ( y ) ) ] f(y)dy.

Set

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

we also define that

μ ϵ (f)(x)= ( 0 | F t ( f ) ( x ) | 2 d t t 3 ) 1 / 2 ,

which is the Marcinkiewicz operator (see [9, 13, 14]).

2 Theorems and proofs

We begin with three preliminary lemmas.

Lemma 1 (see [12])

Let 1<r<, b j BMO( R n ) for j=1,,k and kN. Then we have

1 | Q | Q j = 1 k | b j ( y ) ( b j ) Q | dyC j = 1 k b j BMO ( R n )

and

( 1 | Q | Q j = 1 k | b j ( y ) ( b j ) Q | r d y ) 1 / r C j = 1 k b j BMO ( R n ) .

Lemma 2 (see [14])

Let 0<ϵ<n, 1<s<n/ϵ and 1/r=1/sϵ/n. Then μ ϵ b is bounded from L s ( R n ) to L r ( R n ).

Lemma 3 (see [15])

Let 0<μ<n, Ω(x,z) L ( R n ) satisfy L r ( S n 1 ) (r1) conditions, that is, there exists a constant 0< a 0 <1/2 such that for | y 0 |< a 0 R,

( R < | x | < 2 R | Ω ( x , x y ) | x y | n μ Ω ( x , x ) | x y | n μ | r d x ) 1 / r C R n / r ( n μ ) ( | y | R + | y | / 2 R | y | / R ω r ( δ ) δ d δ ) .

Theorem 1 Let 0<ϵ<n, n/(n+1/2ϵ)<q1, 1/q=1/pϵ/n, b =( b 1 ,, b m ), b i BMO, 1im. Then μ ϵ b is bounded from H b p ( R n ) to L q ( R n ).

Proof It suffices to show that there exists a constant C>0 such that for every (p, b ) atom a,

μ ϵ b ( a ) L q C.

Let a be a (p, b ) atom supported on a ball B=B( x 0 ,2d). We write

R n | μ ϵ b ( a ) ( x ) | q dx= | x x 0 | 2 d | μ ϵ b ( a ) ( x ) | q dx+ | x x 0 | > 2 d | μ ϵ b ( a ) ( x ) | q dx=I+II.

For I, taking r,s>1 with q<s<n/ϵ and 1/r=1/sϵ/n, by Hölder’s inequality and the ( L s , L r )-boundedness of μ ϵ b , we get

IC μ ϵ b ( a ) L r q | B ( x 0 , 2 d ) | 1 q / r C a L s q | B | 1 q / r C | B | q / p + q / s + 1 q / r C.

For II, denoting λ=( λ 1 ,, λ m ) with λ i = ( b i ) B , 1im, where ( b i ) B = | B ( x 0 , 2 d ) | 1 × B ( x 0 , 2 d ) b i (x)dx, by Hölder’s inequality and the vanishing moment of a, we get

II [ 0 | x x 0 | + 2 d | | x y | < t j = 1 m ( b j ( x ) b j ( y ) ) a ( y ) Ω ( x , x y ) | x y | n 1 ϵ d y | 2 d t t 3 ] 1 / 2 + [ | x x 0 | + 2 d | | x y | < t j = 1 m ( b j ( x ) b j ( y ) ) a ( y ) Ω ( x , x y ) | x y | n 1 ϵ d y | 2 d t t 3 ] 1 / 2 = II 1 + II 2 .

Note that |xy||x x 0 ||x x 0 |+2d for |x x 0 |>2d, yB. For 1/t+1/r=1, we have

II 1 C R n ( | x y | | x x 0 | + 2 d d t t 3 ) 1 / 2 j = 1 m | b j ( x ) b j ( y ) | | a ( y ) | | Ω ( x , x y ) | | x y | n 1 ϵ d y C R n | 1 | x y | 2 1 ( | x x 0 | + 2 d ) 2 | 1 / 2 j = 1 m | b j ( x ) b j ( y ) | | a ( y ) | | Ω ( x , x y ) | | x y | n 1 ϵ d y C R n j = 1 m | b j ( x ) b j ( y ) | | a ( y ) | | Ω ( x , x y ) | | x y | n 1 ε | y x 0 | 1 / 2 | x x 0 | 3 / 2 d y C R n j = 1 m | b j ( x ) b j ( y ) | | a ( y ) | | Ω ( x , x y ) | | x x 0 | n + 1 / 2 ϵ | y x 0 | 1 / 2 d y C j = 1 m 1 σ C j m 1 | x x 0 | n + 1 / 2 ϵ ( B | ( b ( y ) λ ) σ c | | a ( y ) | | Ω ( x , x y ) | | y x 0 | 1 / 2 d y ) × | ( b ( x ) λ ) σ | C j = 1 m 1 σ C j m 1 | x x 0 | n + 1 / 2 ϵ ( B ( | ( b ( y ) λ ) σ c | | a ( y ) | | y x 0 | 1 / 2 ) t d y ) 1 / t × ( B ( | Ω ( x , x y ) | ) r d y ) 1 / r | ( b ( x ) λ ) σ | C j = 1 m 1 σ C j m d n ( 1 1 / p + 1 / t + 1 / r + 1 / 2 n ) | x x 0 | n + 1 / 2 ϵ b σ c BMO Ω L × L r | ( b ( x ) λ ) σ | C j = 1 m 1 σ C j m d n ( 1 / p + 1 / t + 1 / r + 1 / 2 n ) | x x 0 | n + 1 / 2 ϵ b σ c BMO | ( b ( x ) λ ) σ | ,

so we have

For II 2 , we can obtain

II 2 = R n ( | x x 0 | + 2 d d t t 3 ) 1 / 2 | j = 1 m ( b j ( x ) b j ( y ) ) a ( y ) Ω ( x , x y ) | x y | n 1 ϵ d y | C | R n j = 1 m ( b j ( x ) b j ( y ) ) a ( y ) | Ω ( x , x y ) | x y | n 1 ϵ Ω ( x , x ) | x | n 1 ϵ | d y | 1 | x x 0 | + 2 d C R n j = 1 m | b j ( x ) b j ( y ) | | a ( y ) | | Ω ( x , x y ) | x y | n ϵ Ω ( x , x ) | x | n ϵ | d y C j = 1 m 1 σ C j m | ( b ( x ) λ ) σ | B | ( b ( y ) λ ) σ c | | a ( y ) | | Ω ( x , x y ) | x y | n ϵ Ω ( x , x ) | x | n ϵ | d y .

Thus, by Lemma 3, we have

This finishes the proof of Theorem 1. □

Theorem 2 Let 0<ϵ<n, 0<p<, 1< q 1 , q 2 <, 1/ q 1 1/ q 2 =ϵ/n, n(11/ q 1 )α<n(11/ q 1 )+1/2+ϵ and b i BMO( R n ), 1im, b =( b 1 ,, b m ). Then μ ϵ b is bounded from H K ˙ q 1 , b α , p ( R n ) to K ˙ q 2 α , p ( R n ).

Proof Let fH K ˙ q , b α , p ( R n ) and f(x)= j = λ j a j (x) be the atomic decomposition for f as in Definition 3. We write

For JJ, by the ( L q 1 , L q 2 )-boundedness of μ ϵ b , we get

JJ C k = 2 k α p ( j = k 2 | λ j | a j L q 1 ) p C k = 2 k α p ( j = k 2 | λ j | 2 j α ) p { C k = 2 k α p ( j = k 2 | λ j | p 2 j α p ) , 0 < p 1 , C k = 2 k α p ( j = k 2 | λ j | p 2 j α p / 2 ) ( j = k 2 2 j α p / 2 ) p / p , p > 1 , { C j = | λ j | p ( k = j + 2 2 ( k j ) α p ) , 0 < p 1 , C j = | λ j | p ( k = j + 2 2 ( k j ) α p / 2 ) ( k = j + 2 2 ( k j ) α p / 2 ) p / p , p > 1 , C j = | λ j | p C f H K ˙ q 1 , b α , p p .

For J, let x B k B k 1 , b j i = | B j | 1 B j b i (x)dx, 1im, b =( b j 1 ,, b j m ), we have

μ ϵ b ( a j ) ( x ) = ( 0 | | x y | < t i = 1 m ( b i ( x ) b i ( y ) ) Ω ( x , x y ) | x y | n 1 ϵ a j ( y ) d y | 2 d t t 3 ) 1 / 2 = ( 0 | x | + 2 j | | x y | < t i = 1 m ( b i ( x ) b i ( y ) ) Ω ( x , x y ) | x y | n 1 ϵ a j ( y ) d y | 2 d t t 3 ) 1 / 2 + ( | x | + 2 j | | x y | < t i = 1 m ( b i ( x ) b i ( y ) ) Ω ( x , x y ) | x y | n 1 ϵ a j ( y ) d y | 2 d t t 3 ) 1 / 2 = G + H .

For G, noting that y B j , xB(0, 2 k )B(0, 2 k 1 ), jk3, we know |xy||x||x|+ 2 j . Then, similar to the proof of Theorem 1, we obtain

G C B j | | x y | | x | + 2 j d t t 3 | 1 / 2 i = 1 m | b i ( x ) b i ( y ) | | Ω ( x , x y ) | | a j ( y ) | | x y | n 1 ϵ d y B j | 1 | x y | 2 1 ( | x | + 2 j ) 2 | 1 / 2 i = 1 m | b i ( x ) b i ( y ) | | Ω ( x , x y ) | | a j ( y ) | | x y | n 1 ϵ d y C 2 j ( 1 / 2 + ϵ ) B j 1 | x | n + 1 / 2 i = 1 m | b i ( x ) b i ( y ) | | Ω ( x , x y ) | | a j ( y ) | d y C 2 j ( 1 / 2 + ϵ ) | x | n + 1 / 2 i = 0 m σ C i m | ( b ( x ) b ) σ | B j | Ω ( x , x y ) | | a j ( y ) | | ( b ( y ) b ) σ c | d y C 2 j ( 1 / 2 + ϵ ) | x | n + 1 / 2 j = 1 m 1 σ C j m ( B j ( | ( b ( y ) b ) σ c | | a j ( y ) | ) t d y ) 1 / t × ( B ( | Ω ( x , x y ) | ) r d y ) 1 / r | ( b ( x ) b ) σ | C 2 j ( 1 / 2 + ϵ + n ( 1 / t + 1 / r 1 / q 1 α ) ) | x | n + 1 / 2 i = 0 m σ C i m b σ c BMO | ( b ( x ) b ) σ | , H ( | x | + 2 j | | x y | < t i = 1 m ( b i ( x ) b i ( y ) ) | Ω ( x y , x ) | x y | n 1 ϵ Ω ( x , x ) | x | n 1 ϵ | a j ( y ) d y | 2 d t t 3 ) 1 / 2 B j 1 | x | + 2 j | Ω ( x y , x ) | x y | n 1 ϵ Ω ( x , x ) | x | n 1 ϵ | i = 1 m | b i ( x ) b i ( y ) | | a j ( y ) | d y C 1 | x | + 2 j i = 0 m σ C i m | ( b ( x ) b ) σ | B j | Ω ( x y , x ) | x y | n 1 ϵ Ω ( x , x ) | x | n 1 ϵ | | a j ( y ) | | ( b ( y ) b ) σ c | d y C 1 | x | + 2 j i = 0 m σ C i m | ( b ( x ) b ) σ | ( B j | Ω ( x y , x ) | x y | n 1 ϵ Ω ( x , x ) | x | n 1 ϵ | r d y ) 1 / r × ( B j ( | a j ( y ) | | ( b ( y ) b ) σ c ) t | d y ) 1 / t C 1 | x | + 2 j i = 0 m σ C i m a L q 1 | B j | 1 / t 1 / q 1 b σ c BMO | ( b ( x ) b ) σ | ( 1 2 j + B j ω r ( δ ) δ d δ ) C 2 j n ( 1 / q 1 + 1 / t α ) | x | + 2 j i = 0 m σ C i m b σ c BMO | ( b ( x ) b ) σ | ,

thus

For the sake of simplicity, we denote

then

μ ϵ b ( a j ) χ k L q 2 C b BMO W(j,k),

we obtain

J C b BMO p k = 2 k α p ( j = k 3 | λ j | W ( j , k ) ) p { C b BMO p j = | λ j | p k = j + 3 W ( j , k ) p , 0 < p 1 , C b BMO p j = | λ j | p [ k = j + 3 W ( j , k ) p / 2 ] [ k = j + 3 W ( j , k ) p / 2 ] p / p , p > 1 C b BMO p j = | λ j | p C b BMO p f H K ˙ q 1 , b α , p p .

This completes the proof of Theorem 2. □

Remark Theorem 2 also holds for nonhomogeneous Herz-type spaces.

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Acknowledgements

This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar under Grant No. 50925727, National Natural Science Foundation of China under Grant No. 60876022, The National Defense Advanced Research Project Grant No. C1120110004, Hunan Provincial Science and Technology Foundation of China under Grant No. 2010J4 and 2011JK2023, the Key Grant Project of Chinese Ministry of Education under Grant No. 313018.

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Correspondence to Wenxin Yu.

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Authors’ contributions

In this paper, WY carried out the ( H b p , L p ) and (H K ˙ q , b α , p , K ˙ q α , p )-type boundedness for the multilinear commutator related to the Marcinkiewicz operator with variable kernels. YH, QL, JZ, XW participated in the analysis. All authors read and revised the final manuscript.

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Yu, W., He, Y., Luo, Q. et al. Boundedness for multilinear commutator of Marcinkiewicz operator with variable kernels on Hardy and Herz-Hardy spaces. J Inequal Appl 2012, 308 (2012). https://doi.org/10.1186/1029-242X-2012-308

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Keywords

  • Marcinkiewicz operator
  • multilinear commutator
  • BMO
  • Hardy space
  • Herz-Hardy space