Boundedness of Hyper Singular Parametric Marcinkiewicz Integrals with Variable Kernels

(1)

http://dx.doi.org/10.4236/am.2013.411A3005

Boundedness of Hyper-Singular Parametric

Marcinkiewicz Integrals with Variable Kernels

Qiquan Fang1*, Xianliang Shi2

1

Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, China

2

College of Mathematics and Computer Science, Hunan Normal University, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China),

Hunan Normal University, Changsha, China Email: *[email protected]

Received September 13, 2013; revised October 13, 2013; accepted October 20, 2013

Copyright © 2013 Qiquan Fang, Xianliang Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this article, we consider the boundedness of , ,

 

__b on Hardy type space Hbp

 

Rn . Where 0 <<n ,

  

1 2 2

, , 0 , , 2 1 2

d , t

t

f x F f x

t

 

  

_ _{ }  _ _{ } _







b b

  





_{ }

_{   }

, ,

1

,

d

m

t x y t n i

i

x x y

.

i

F f x

,

i



b x b y

 _  _f y

x y



   



 







b

 



y



1, 2, ,



,

 

,1 , 0, 1 0 1 m

n

m i i i i

b b b b _ R i m  _   

     



 

b

Keywords: Hyper-Singular Marcinkiewicz Integral; Variable Kernel; Multilinear Commutator; Hardy Type Space

1. Introduction

A function 



x z,

 

n r

L R L

defined on is said to be-

long to , if it satisfies the following

three conditions:

n n

 

n1

S 



, ,

R R

1) 



x,z



 



x z for any ,x zRn and any

> 0

 ;

2)

   





 





1

0,

: sup , d

n r n

n n

L R L S

r r

S y R

z y z z



 

 





 



  



_

  <

3)



_Sn1



x z, 

  

d z 0, for any

n

xR .

In [1], the authors considered the hyper-singular parametric Marcinkiewicz integral with variable kernel as follows:

  

1 2 2

, 0 , 2 1 2

d , t

t

f x F f x

t

 

  

_ _{ }  _ _{ } _







where

  



  

,

d .

t x y t n

x x y

F f x f y y

x y



   

 







When  0, we set ,0

 

_ _, which is the parametric Marcinkiewicz integral with variable kernels considered in [2].

For  > 0, the homogenous Lipschitz space

 

n

R 

 _{is the space of function} _f _{such that}  1

_{ }

, , 0

, sup

n

h x h R h

f x f

h 



 

 



  



where kh denotes -th difference operator (see [3]). k

In 2006, Lu and Xu studied the boundedness of the

commutator of ,

m b

_ in [4]. They proved that:

Theorem A [4]. Suppose



n1 for

S

 

  



 

1

0 < , , 0 <

2

n

b _ R .

m



      _If n ₁

p

n  

and 1 1

q p n



  , then ,

m b

 maps m_,

 

p n b s

H R

continuously into Lq

 

Rn . Here , m b

_ is defined as

(2)

follows:

  

1 2 2

, 0 , , 3

d ,

m m

b b t

t

f x F f x

t

_ _{ }  _





 

 (1)

where

  



      

, , 1 d .

m m

b t x y t n

x y

F f x b x b y f y y

x y

   

 

 _  _





Let b



b b1, 2,,bm



,

1 i m, i 0,

 

n , i _i

b _ R

    m

i

 

i1

, ,



, 0   1.

    In this article, we

mainly consider the commutator b defined by

  

1 2 2

, , 0 , , 2 1 2

d ,

t

f x F f x

t

 

  

_ _{ }  _ _{ }





b b

 

 (2)

where

  





_{ }

_{   }

, ,

1

,

d .

t

m

i i

n x y t

i

F f x

x x y

b x b y f y y

x y



 

  



 

 _  _







b

Given any positive integer , for all , we

denote by

m 1 j m

m j

C

 



 2 ,

the family of all finite subsets

of of dif-

ferent elements. For any

 

1 , ,

 

j

   





1, 2,,m



j

m j

C

  , we associate the com-

plementary sequence   given by  



1, 2,,m



\ , (see [5]).

For any  





   

1 , 2 ,,

 

j



Cmj

     



1, 2, ,



, we will

denote  b b  b j

  .



b   1 2

and the product

j

b_ b_ b_ b_ When C₀m, we have   ,

by definition, we have . Similarly, when ,

we have

1

b_  m

m

C

      and b_{ } 1. With this notation, if

 1  j ,

 

     we write

 

 1    

1 .

j j

b b

  

    __   __

b

When 1

m i i

 





, we write

 

 i m

i i b

 _

  



  __

b

i

1 .

Definition 1.1. Let 0 <p1, b be defined as above

such that

 

,1 , 0 < < 1, ₁

i

m n

i i i

b  _ R  i m 



_ 

n

.

A function a x

 

on R is called a -atom if







p q b, ,



1) supp_a__B _x₀,_r _:_



_x__Rn: xx₀ r



, for some

0

n

x R and r> 0;

2)





1 1

0, q p

q

a  B x r 

3) _n

   

d for any

Ra x b  x x







, 1, , \

m j

C m

   

0 <

and j1, 2,, .m

1

Definition 1.2. Let , we say that a distribu-

tion

p

f on Rn belongs to Hbp

 

Rn if and only if f

can be written as i i

i

f a

 





in the distributional sense,

where each ai is a



p q, ,b



-atom and

. p i i



 

 



Moreover,

1

inf

p

p p i H

i

f 

 

 

 

 

 __ _ _

 

 

 



b

with the infimum taken over all the above decomposi- tions of f as above

Definition 1.3. A function

 

   

1



, n r n 1

x z L R L S  r



   

,

r

L

is said to satisfy the 

-Dini condition, if

 

1 1

0 d < , 0 1,

r



 

 

    



(3)

where  r

 

r

denotes the integral modulus of continu- ity of order of  defined by

 









 

1 1

, 0 ,

1

,

sup sup

, d .

n

n n

r _S

x R y S y z

r r

x z y

x z z z

 

  

 

 

  

    



 

 _ 





  

  _







We will denote simply Lr-Dini condition for Lr,- Dini condition when  0.

2. Main Theorem

Now let us formulate our main results as follows.

Theorem 2.1.Suppose that , ,

 

_b is the commutator

(2), and let

1 1

0 1,1 p n ,

q p n .

   

 



      



 

   

1

, n r n

x z L R L S 

   , then , ,

 

b is bounded

from Lp

 

Rn into Lq

 

Rn . That is,

 

, , Lq Rn C  f LpRn . 



_b  b _

Theorem 2.2. Suppose that , ,

 

__b is the commutator

(2), and let n < p 1,1 1

n q p n



.

  



  

  If

 

x z,

 satisfies the following two conditions: 1) 

 

x z, satisfies Lr,-Dini condition (3); 2) there exists

(3)

> max , , 1 2

n n n

r

n n n

n n

p p

   

 

 

 _{ } 

      

 

 

such

that 

 

x z, L

  

Rn L Sr n1 then









 

1

0 0

< 2

2 < 4

, ,

d

d . r r

n n

K x K n

n _r

r

y K y K

x x x y x x x

x

x y x

y CK

K

 





   

 



 



 _ _ _ _ _ 

 _ 

 

 

 

   

 





, ,

 

__b is boun-

ded from Hbp

 

Rn



q n

L R

into



. That is _{where the constant} _{is independent of} _and

.

> 0

C K

y

 

_{ }

, , f Lq Rn C  f HpRn . 



_  _

b

b b

lemma 2.2. [7] Let 0 <<n, 1 < p< n,1 1

q p n



  

and T _, be defined as

  





Remark Obviously, , ,

 

__b is the commutator of the

operator ,

 

_ in [1]. At the same time, we change the

course of the statement in [4]. , x _Rn f y

 

d .y

x y

 

 







, n

x x y

T f If there exists

In order to prove our Theorems, we need several

pre-liminary lemmas. r> p, such that

 

   

1

, n r n ,

x z L R L S 

   then

Lemma 2.1. [6] Let 0<n







,  r1

and suppose

If there exists a

 

  

n r n1

x z L R L S

   .

,

T  is bounded from Lp

 

Rn into Lq

 

Rn . That is

 

_{ }

_{   }

1

_{ }

, _Lq _Rn _L _Rn _{L S}r n _Lp_Rn .

T__ f C   _  f

constant 0 < < 1 2

R such that y <RK, then for any

3. Proofs

0

n

x R ,

3.1. Proof of Theorem 2.1.

Applying the Minkowski’ inequality, we can get

  





_{ }

_{   }





_{ }

_{   }

 

1 2 2

, , 0 , 2 1 2

1

2 2

2 1 2 0

1

1 2 2 1 2 1

1 ,

d

, d

d

, _d

d

sup

n

n i n

t

m

i i

n x y t

i

m

i i

n

R x y

i

m

i i

R i x y R

x y

t

f x F f x

t

x x y t

b x b y f y y

t

x y

x x y _t

b x b y f y y

t

x y

b x b y

C

x y

 

  

 

  



   



 

  _



   



 



 

 _ _

 _ _ _ _ 

 _ _ 

_ _  _ _

 _ 

 _ _ 

 

  _ _

  _ _

 



 







 







b b





 

 





 

,

 

, 1

d

,

d .

n n n

n

R R R

x x y

x y f y

x y x y

x x y

C f y y T f

x y

 



 

   

 

   

 

 

 

 

 

 





 

b b

y

x 

3.2. Proof of Theorem 2.2. By Lemma 2.2 , we have

 

_{ }

 

_{ }

   

1

 

, ,

,

.

q n

n _q _n

n r n n p n

L R

R _L _R

L R L S R L R

f

C T f

C f



 



 



 



  

 



 



b

Noting that r> n

n  , we can choose l1 such that 1

< < n

r l

 



 . It is easy to see that . Next , we

choose such that

1

l

>

r

2

l

2 1

1 1

.

l l n

 

  It follows from

(4)

Theorem 2.1 that __{, ,}_b_ is bounded from _Ll1

 

_Rn

into _Ll2

 

_Rn . That is

 

1

 

2

, ,f Ll Rn C  Rn f Ll Rn .

_b  b_





(4)

By the atomic decomposition theory on Hardy type space, it suffices to prove that there is a constant

such that for all -atom the following holds

> 0

C



p l b, ,1

 

, , _q _n _Rn .

L R

a C

 



b  b 

Without loss of generality we may assume that



0



suppa B B x d, . We write



0





8 ,8 :

B nBB x nd B x d₀, 



. We split

 

, ,a _Lq _Rn

__b into two parts as follows:

 



 



 

1 1

, , q n , , d , , d 1 2

q q

C

B B

L R

a a x x a x x

  

  

        

 



b b b : I I .

We can easily see that l2>q. By (4) and the size condition of atom a, we have

 



2



2

 

2 2 ₁ 2 1

1

1 1 1 1 1 1 1 1

1 , , d n l n n .

l _l

q l q l q l l p

R L R R

B

I a x x B C B a C B B C

 

 

 _  _  _  

   



_

_b  b _  b _ _Rn





 b _

Next we estimate I2. Let us consider , , a x

 

 

_b :

 





 

   





_{ }

_{   }

0

1 2 2

2

, , 0 2 1 2

1

1 2 2

2 1 2 2

1 21 22

, d

d

, d

d

: .

m x x d

i i

n x y t

i

m

i i

n x x d x y t

i

x x y t

a x b x b y a y y

t

x y

x x y t

b x b y a y y

t

x y

I I



  

 

      





 

   



 _ _ _ _ 

 _ _ 

_ _  _ _

 _ 

 _ _ 

 

 _ _ _ _ 

 _ _ 

_ _  _ _

 _ 

 _ _ 

 

 









b 

 



0 0 2

x  y x x  x x  d for



0,



,

 

C

yB x d x B . By the mean value theorem, we

have





2 2

2 2 2 1 2

0 2 .

xy    xx  d    Cd xy 

Thus, by the Minkowski’s inequality for integrals,





_{ }

_{   }

 





 

_

_

 





 





0

1 2 2

21 2 1 2

1

1 2

2 2 2 2

0 1

1 ₁

2

2 ₂

0 1 2

, _d

d

, 1 1

d 2

,

d ,

n

n n

m _{x x} _d

i i

n

R x y

i

n

R B

n n

R B R B

x x y _t

I b x b y a y y

t

x y

x x y

C x y a y y

x y x y x x d

x x y _d

C a y y C d x x

x y x y



 

  



    

 

   

 

   



  



  

   

 

  _ _

  _ _

 



 

  _ _

   

 

 _    _

 

  

 







 

b

b b  x xy a

 

y d .y

Applying the Hölder inequality and the size condition of a, we have



  











 









 







 

_{ }

_{   }

1

1 1

0 0

0 1

1

1 1

0

1

1 1 1

2 2

1

2 ₁ 1 1 1 1 1

1 0

2

, d

, d d , d d

, d d l n n r n .

n B

r

r r l

r r r l r l

B B x x x y x x B

r

x x _n r r l n r r p

L R L R L S

x x S

x x y a y y

x x y y a y y x x y y a y y B

t  x z  z t a B C   x x B

 _

 

    



   

 

 

 

   _   _

 

 _ _ 

_  _    

 



(5)

So we can get

 

   

1

1 2 1 2

21 n n r n 0 .

n r n n r n p n

R L R L S

I C d x x



 

 

      

 

 b _  

> n ,

r

n   we have Noting that

 





 

   

 





 



 



 

1

1 1

1 2 1 2

21: C B

J 



_ 21 0

1 1 2

1 2 1 1 2 1 2

d

d .

n n r n

n n n

q q

n r n q

q n r n p n

C

R L R L S B

q n r n q

n r n p n n n r n p n n r p n

R d R R

I C d x x x

C d C d d C



  

 

  

 

   

  

  

 _{ } _ _

        

   

   

  





  

b

b b n b

For I22, we write

  





 

   

,

1

,

: d

m

t _{x y t} n i

i

x x y

G a x b x b y a y y

x y



 

  _

 

 _  _







b i .

  

 



   









   



 



  

 



   









   



 

, 0

0

0 1

1

0 0

0

,

1 d

,

d ,

1 d

m j

m

m j

t x y t n

j C

0

.

i i x y t n

i m

m j

n x y t j C

x x y

G a x x x y x a y y

x y

x x y

b x b x a y y

x y

x x y

m

x x y x a y y

x y



 





 





 

   

  



 _

 

   

 

    



 

  

  _

 

    



 



 



b b b b b

b b b b

So





22

I is dominated by

 



  

 



   





    





 

0

0 2 ₀

x x d j

  



 _





1

2 ₂

22 2 0 2 1 2

1

2 ₂

1

0 0 2 1 2

1 2

, d

d

, d

1 d

: .

m j

m

i i n

x x d x y t

i m

m j

n x y t C

x x y t

I b x b x a y y

t

x y

x x y t

x x y x a y y

t

x y

K K

  

 





  

  _  



 

   

  

 _ _ 

 _ _ 

_ _  _ _



 

 

 _ _ 



    _

 

 

 







b b



b b

Now let us estimate K1. By the vanishing condition of a, we have

 



  

 



  

 



  

_

_

 



  

0

1

2 2

1 2 0 0 2 1 2

1

1 2

0 0 2 2 1 2

1

i





0 0

0

0 0

d

, , d

d

, , d

1

, , d

2

, , d .

n

m

j j

x x d x y t

i m

j j _R _{x x} _d

R B

t

K b x b x D x x y a y y

t

b x b x D x x y a y y

t

C x x D x x y a y y

x x d

C x x D x x y a y y



 

  



 

   





   

 

  

 

 

_ _  _ _

 

 

  _ _

 

 

 

 









b



where



0









0



0

, ,

, , .

n n

x x x

x x y

D x x y

x y  x x 

 

 

 

1 < < 0,

n n

p  p 

(6)

Hölder’s inequality and Lemma 2.1,  





 





 



_{ }





 



 





 









1 0 1 0 1 1 1 0 0 1 0 0 1 0 0 2 2 0 0 2 2 0

, , d d

2 , ,

j j j j j q q C B q q C B B

q q q

C

B B

q q q

B d x x d

j

B d x x d

j

K

C D x x y x x a y y x

C a y D x x y x x x y

C a y d D x x y

                            _{ }  _      _{ }  _        _  _        _  _    











b b b b    





 





 

1 0 0 2 0 1 0 1 1 0 2 2 0 2 0 2 0 1 0 2 d d 2 , 1 2 2 2 1 2 2 2 j j j j j q q n n r r

j q r

B d x x d

j

y x n n n

n _r

j q r j r d

y x j B j d n n j _p y x j j B j x y

C a y d D x x y x y

C a y d d

y x

C a y d

d

, d d

d dy                          _{   } _{ }     _  _                              _{ } _  













b b b   

 

0 2 1 0 2 0 1 1 1 1 1 2 1 1 2 1 0 2 1 ₁ 1 1 0 d d

2 2 d d

1 d . j j j l n y x r d d y x n n n

j n j n n

p p r

p d y x B j d n n r p _l L R n n n n

n n p l p l

y

C d a y y

C d a B

C d C

                                                   _  _                               _  _    







b b b b    

Now we estimate K2. Applying Minkowski's inequality, the size condition of a, we obtain

 



 





 







 



 





 











 

0 0 0 1 1 ₂

2 0 0 2 2 1 2

0 1 0 0 0 0 1 0 0 0 0 , _d 1 d , ₁ d 2 sup s n m j m j i m j m m j n

R x x d

j C m n B j C m i i

x x x x

j C i i

x x y _t

K C a y x x y x y

t

x y

x x y

C a y x x y x y

x y x x d

b x b x

C x x

x x                    _ _                        _ _      _ _                

 



 



  



b b b b

 





_{ }

 





 



  

0 0 0 1 0 0 0 1 0 0 , d up , d

, d .

i n m j n m j i i n B m n R B j C m n R B j C

b y b x x x y

y x a

x y

y x

x x y

C x x y x a y y

x y

C x x d x x y a y y

                            _{ }                    



 



 



b b   y y

So we have

 

   

1

2 0

0

.

n n r n

m j

n n

m n n

n r p

r

R L R L S

j C

K C x x  d 

         _{  }       

(7)

Thus

 





 

   

 

1

1 2

1 1

0 =0

1 =0

d

.

n n r n

m j

n

m j

n q q C B

n n n

r p

R L R L S

m _n n _q q

r C

B

j C

n n_m n n

n n

r p r q

R

j C

R

K

C d

d x x

C d d

C 

 



 

 

  



 



 

 

 

  _{  } 

 

 

 



      







 

 

 _  _

 





 



 



b

x

, we have So when |xx₀|>Cd

 





 

1

22: 22 n ,

q q

C _R

B

J I C

  



_

 b 

 

2 21 22 _Rn .

I J J C

 

   b 

Combining the estimates for I1 and I2, we have

 

, ,a Lq Rn C  Rn .

b  b 

This completes the proof of Theorem 2.2.

4. Acknowledgements

The authors would like to thank anonymous rev for their comments and suggestions. The authors are par-tially supported by project 11226108, 11071065, 11171306 funded by NSF of China, project 20094306110004 fu ed by RFDP of high education of China.

REFE

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