Maximal operator for multilinear Calderón–Zygmund singular integral operators on weighted Hardy spaces

J. Math. Anal. Appl. 373 (2011) 384–392

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Maximal operator for multilinear Calderón–Zygmund singular integral operators on weighted Hardy spaces Wenjuan Li a , Qingying Xue a,b,∗,1 , Kôzô Yabuta c,2 a

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, People’s Republic of China c Research Center for Mathematical Sciences, Kwansei Gakuin University, Gakuen 2-1, Sanda 669-1337, Japan b

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 18 March 2010 Available online 5 August 2010 Submitted by Richard M. Aron

In this paper, the maximal operator associated with multilinear Calderón–Zygmund singular integral operators will be studied by using an improved Coltlar’s inequality. Moreover, weighted norm inequalities and some estimates on weighted Hardy spaces are obtained for this maximal operator. © 2010 Elsevier Inc. All rights reserved.

Keywords: Multiple weights Weighted norm inequalities Multilinear Calderón–Zygmund operators Weighted Hardy spaces Coltlar’s inequality

1. Introduction The study of multilinear singular integral operators of Calderón–Zygmund type continues to attract many researchers’ interests, such as [5,10,7,11,2,1,13]. Many results obtained parallel to the linear theory of classical Calderón–Zygmund operators but new interesting phenomena have been observed. See also [9] and the references therein for a detailed description of previous work in the subject. One aspect of the theory that still is being developed is the one related to the study of maximal operators associated to multilinear singular integrals and appropriate versions of multilinear weighted norm inequalities. So we first recall the definition of multilinear Calderón–Zygmund operators as well as the corresponding maximal operators. Definition 1.1 (Multilinear Calderón–Zygmund operators). Let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values in the space of tempered distributions













T : S Rn × · · · × S Rn −→ S  Rn .

*

Corresponding author at: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China. Current address: Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA. E-mail addresses: [email protected] (W. Li), [email protected] (Q. Xue). 1 The author was supported partly by NSFC (Grant No. 10701010), NSFC (Key program Grant No. 10931001), CPDRFSFP (Grant No. 200902070), Beijing Natural Science Foundation (Grant No. 1102023). 2 The author was supported partly by Grant-in-Aid for Scientific Research (C) No. 20540195, Japan Society for the Promotion of Science. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.07.057

©

2010 Elsevier Inc. All rights reserved.

W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

385

Following [5], we say that T is an m-linear Calderón–Zygmund operator if, for some 1  q j < ∞, it extends to a bounded multilinear operator from L q1 × · · · × L qm to L q , where 1q = q1 + · · · + q1 , and if there exists a function K , defined off the m

1

diagonal x = y 1 = · · · = ym in (Rn )m+1 , satisfying



T ( f 1 , . . . , f m )(x) =

K (x, y 1 , . . . , ym ) f 1 ( y 1 ) · · · f m ( ym ) dy 1 · · · dym

(Rn )m

for all x ∈ /

m

j =1 supp

f j;

  A  K ( y 0 , y 1 , . . . , ym )   ; m ( k,l=0 | yk − yl |)mn

(1.1)

A | y j − y j |ε     K ( y 0 , . . . , y j , . . . , ym ) − K y 0 , . . . , y  , . . . , ym    , m j ( k,l=0 | yk − yl |)mn+ε

(1.2)

and

for some

ε > 0 and all 0  j  m, whenever | y j − y j |  12 max0km | y j − yk |.

As in the linear theory, a certain amount of extra smoothness is required for these operators to have such boundedness properties. We will assume that K ( y 0 , y 1 , . . . , ym ) satisfies the following estimates

 α0  Aα ∂ y · · · ∂ yαm K ( y 0 , y 1 , . . . , ym )   , m 0 mn+|α | ( m | y k − yl |) k,l=0

(1.3)

for all |α |  N, where α = (α0 , . . . , αm ) is an ordered set of m-tuples of nonnegative integers, |α | = |α0 | + · · · + |αm |, where |α j | is the order of each multiindex α j , and N is a large integer to be determined later. In this article we study maximal multilinear singular integral operator defined by





T ∗ ( f )(x) = sup T δ ( f 1 , . . . , f m )(x), δ>0

where T δ are the smooth truncations of T given by



T δ ( f 1 , . . . , f m )(x) =

K δ (x, y 1 , . . . , ym ) f 1 ( y 1 ) · · · f m ( ym ) d y .

(Rn )m



Here, d y = dy 1 · · · dym , K δ (x, y 1 , . . . , ym ) = φ( |x − y 1 |2 + · · · + |x − ym |2 /2δ) K (x, y 1 , . . . , ym ) and φ(x) is a smooth function on Rn , which vanishes if |x|  1/4 and is equal to 1 if |x| > 1/2. In [8], the authors studied the following operator T˜∗ given by

T˜∗ ( f )(x) = T˜∗ ( f 1 , . . . , f m )(x) = sup



δ>0 |x− y 1 |2 +···+|x− ym |2 >δ 2

K (x, y 1 , . . . , ym ) f 1 ( y 1 ) · · · f m ( ym ) d y .

Throughout this paper we will let W be the norm of T in the mapping T : L 1 × · · · × L 1 → L 1/m,∞ . We list some results for T˜∗ and T ∗ as follows:

η > 0, there exists a constant C η = C η (n, m) < ∞ such that for all f in any product of L qi (Rn ) spaces, with 1  q i < ∞, the following inequality holds for all x in Rn

Theorem A. (See [8].) Let T be an m-linear Calderón–Zygmund operator. Then, for all

 m

η  (1/η)   T˜∗ ( f )(x)  C η M  T ( f ) (x) + (A + W ) M f i (x) ,

(1.4)

i =1

where M denotes the Hardy–Littlewood maximal function with respect to balls on Rn . Corollary B. (See [8].) Let T be an m-linear Calderón–Zygmund operator. Then, for all exponent 1 < q i  ∞, q < ∞, and q satisfying 1 = q1 + · · · + q1 , we have q 1

m

T˜∗ ( f )(x)

Lq

 C(A + W )

m

i =1

 f i  L qi .

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W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

+ · · · + q1m , and ω ∈ A q1 ∩ · · · ∩ A qm . Let T be an m-linear Calderón–Zygmund operator. Then there exists a constant C q,n < ∞ for all f = ( f 1 , . . . , f m ) satisfying Theorem C. (See [8].) Let 1  q i < ∞, and q be such that

T˜∗ ( f )

q

Lω

 C n,q ( A + W )

m

i =1

1 q

=

1 q1

 f i  L qi . ω

Theorem D. (See [6].) Let 1 < q1 , . . . , qm , q < ∞ be fixed indices satisfying

1 q1

1

+ ··· +

qm

=

1 q

and let 0 < p 1 , . . . , pm , p  1 be real numbers satisfying

1 p1

1

+ ··· +

=

pm

1 p

.

Suppose that K satisfies (1.3) with N = [n(1/ p − 1)]. Let T be related to K and assume that T admits an extension that maps L q1 × · · · × L qm into L q with norm B. Then T ∗ extends to a bounded operator from H p 1 × · · · × H pm into L p , and satisfies the norm estimate  T ∗  H p1 ×···× H pm → L p  C ( A + B ) for some constant C = C (n, p i , qi ). Recently, the theory of weighted multilinear Calderón–Zygmund singular integral operators was established in [11] by Lerner, Ombrosi, Pérez, Torres, Trujillo-González and the multiple weights A p were constructed. This together with the results for multiple weights A (p ,q) adapted to multilinear fractional integral operators [1,13] answered an open problem in [9]. That is, the existence of multiple weights theory for multilinear Calderón–Zygmund operators and multilinear fractional integral operators. Meanwhile, a new more refined multiple maximal function M

 m

  1  f i ( y i ) dy i | Q | Q x

M( f )(x) = sup

i =1

Q

was used in [11] to characterize the class of A p and to obtain some weighted estimates for the multilinear Calderón– Zygmund singular integral operators. So let us recall the definition of A p weights. For m-exponents p 1 , . . . , pm , we will often write p for the number given by

 = ( p 1 , . . . , p m ). p

1 p

=

1 p1

+ ··· +

1 pm

 for the vector , and p

 = (ω1 , . . . , ωm ), set Definition 1.2 (Multiple A p weights). (See [11].) Let 1  p 1 , . . . , pm < ∞. Given ω

νω =

m

p/ p j

ωj

.

j =1

 satisfies the A p condition if We say that ω

sup Q

1

|Q |



m Q

When p j = 1, ( | Q1 |

ωi

p pi

 1p

i =1



1

i =1



Q

   m

p 1  ωi 1− p i i < ∞. |Q |

ω i 1− p i )

1 p i

(1.5)

Q

is understood as (inf Q

ω j )−1 .

In particular, when m = 1, we note that A p will be degenerated to the classical A p weight. Moreover, if m = 1 and p i = 1, then this class of weights coincides with the classical A 1 weights. It is well known that if ω ∈ A p for 1 < p < ∞, then ω ∈ A r for all r > p and ω ∈ A q for some 1 < q < p. We thus use qω := inf{q > 1: ω ∈ A q } to denote the critical index of ω . We will refer to (1.5) as the multilinear A p condition. We list some results in [11] as follows:

 = (ω1 , . . . , ωm ) and 1  p i < ∞. Then ω  ∈ A p if and only if Theorem E. (See [11].) Let ω





ωi1− p i ∈ Amp i  i = 1, . . . , m, νω ∈ Amp 1− p i 

where the condition ωi

∈ Amp i  in the case p i = 1 is understood as ω1/m ∈ A 1 .

W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

387 1 p

Theorem F. (See [11].) Let T be an m-linear Calderón–Zygmund operator, satisfying (1.1), (1.2),

ω satisfy the A p condition. Then M( f )(x)

p

Lν ω

C

m

i =1

p

Lν ω

C

m

i =1

1 p1

+ ··· +

1 pm

 f i L pi .

, 1 < p i < ∞ and

(1.6)

ωi

Theorem G. (See [11].) Let T be an m-linear Calderón–Zygmund operator, satisfying (1.1), (1.2), condition, and 1 < p i < ∞. Then

T ( f )(x)

=

1 p

 f i L pi .

=

1 p1

+ ··· +

1 pm

 satisfy the A p ,ω

(1.7)

ωi

On the other hand, in 1989, Strömberg and Torchinsky in [14] defined weighted Hardy spaces and obtained some boundedness for Calderón–Zygmund operators. We will use Garcia-Cuerva’s atomic decomposition theory [3] for weighted Hardy spaces. We characterize weighted Hardy spaces in terms of atoms in the following way. Definition 1.3 (Weighted Hardy spaces). Assume that ω ∈ A q with critical index qω . Let [·] be the greatest integer function. For s ∈ Z satisfying s  [n(qω / p − 1)], a real-valued function a(x) is called ( p , q, s)-atom centered at x0 with respect to ω (or ω − ( p , q, s)-atom centered at x0 ) if q

(a) a ∈ L ω (Rn ) and is supported in a cube Q centered at x0 , 1

(b) a L q  ω( Q ) q (c)



− 1p

ω

,

α Rn a(x)x dx = 0 for every multi-index

α with |α |  s.

∞ =  f ∞ . When q = ∞, L ∞ ω will be taken to mean L and  f  L ∞ ω p

Theorem H. (See [3].) Let w ∈ A q , 0 < p  1  q  ∞, and p =  q. For each f ∈ H ω (Rn ), there exist a sequence ai of   ω − ( p , q, [n(q w / p ) − 1])-atoms and a sequence λi of real numbers with |λi | p  C  f  p p such that f = λi ai both in the Hω

p

sense of distributions and in the H ω norm. At present, combining the above, we have obtained the boundedness of multilinear Calderón–Zygmund operators on weighted Hardy spaces in [12], so it is natural to ask the following interesting question. Are there any weighted results for maximal operators for multilinear singular integral operators on weighted Hardy spaces? Our result is as follows: Theorem 1.1. Let 1 < q1 , . . . , qm , q < ∞, 0 < p 1 , . . . , pm , p  1, satisfying

1 q1

+ ··· +

1 qm

=

1 q

and

1 p1

+ ··· +

1 pm

=

1 p

.

Let T be an m-linear Calderón–Zygmund operators such that K satisfies (1.1), (1.2), (1.3) with N = max1i m {[n((qi )ω / p i − 1)], [(qi / p i − 1)mn]}. We have the following results: (i) If ω ∈ A q1 ∩ · · · ∩ A qm , then

T ∗ ( f )(x)

p

Lω

C

m

i =1

 f i  H pi . ω

(1.8)

(ii) If for each i, ωi ∈ A 1 , then

T ∗ ( f )(x)

p

Lν ω

C

m

i =1

 f i  H pi . ωi

(1.9)

388

W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

2. Proof of Theorem 1.1 We will use the following facts as in [12]: (a) Let q  1 and ω ∈ A q . Suppose a is an ω -( p , ∞, s)-atom. Then a is an ω -( p , q, s)-atom. (b) Let ω ∈ A p , p  1, then for any cube Q and λ > 1, ω(λ Q )  C λnp ω( Q ), where C does not dependent on Q nor on λ (see [4] for details). We now turn our attention to Theorem 1.1. p p We prove this theorem using the atomic decomposition of H ω spaces. Since finite sums of atoms are dense in H ω , we will work with such sums and obtain estimates independent of the number of terms in each sum. · · · × L 2m into L 2 by Lemma 2.2 below when w i = 1, we can assume each f i , 1  i  m, Since T ∗ is bounded from L 2m × pi as a finite sum of H ω -atoms, f i = k λi ,k ai ,k , where ai ,k are ( p i , ∞, s)-atom, this means they are supported in cubes Q i ,k and |ai ,k |  ω( Q )−1/ p i . However, by the above fact (a), we know that ai ,k satisfies 1

− p1

ai ,k Lqi  ω( Q ) qi ω  ai ,k (x)xα dx = 0,

(2.1)

,

i

 

 |α |  s, s  n (qi )ω / p i − 1 .

(2.2)

Q i ,k

√

Denote by c i ,k and | Q i ,k | the center and the side length of Q i ,k , and let Q˜ i ,k = 8 n Q i ,k , employing multilinearity we write

T ∗ ( f )(x) = T ∗ ( f 1 , . . . , f m )(x) 



···

k1



λ1,k1 · · · λm,km T ∗ (a1,k1 , . . . , am,km )(x).

(2.3)

km

For x ∈ Rn , we split the right side of (2.3) into two terms I 1 (x) + I 2 (x), where

I 1 (x) =



···

k1

and

I 2 (x) =



  |λ1,k1 | · · · |λm,km | T ∗ (a1,k1 , . . . , am,km )(x)χ Q˜ 1,k

km



···

k1



1

  |λ1,k1 | · · · |λm,km | T ∗ (a1,k1 , . . . , am,km )(x)χ Q˜ c

1,k1

km

∩···∩ Q˜ m,km

. c ∪···∪ Q˜ m ,km

Now, let us begin to discuss I 1 (x). For fixed k1 , . . . , km , assume that Q˜ 1,k1 ∩ · · · ∩ Q˜ m,km = ∅, otherwise, there is nothing need to be proved. Suppose that Q i ∗ ,ki∗ , i ∗ ∈ 1, 2, . . . , m, has the smallest size among all these cubes. We take a cube G k1 ,...,km such that

˜

˜

Q˜ 1,k1 ∩ · · · ∩ Q˜ m,km ⊂ G k1 ,...,km ⊂ G˜ k1 ,...,km ⊂ Q˜ 1,k1 ∩ · · · ∩ Q˜ m,km and

ω(G k1 ,...,km )  C ω( Q i∗ ,ki∗ ). By using the Hölder’s inequality, Lemma 2.2 when w i = w, and (2.1), we have



1

ω(G k1 ,...,km ) 

   T ∗ (a1,k , . . . , am,k )(x)ω(x) dx m 1

G k1 ,...,km

1

ω(G k1 ,...,km )







ω(G k1 ,...,km )1/q T ∗ (a1,k1 , . . . , am,km )(x) Lqω

 C ω(G k1 ,...,km )−1/q 1

 C ω(G k1 ,...,km )− q

m

i =1 m

ai ,ki Lqi ω

1

ω( Q i,ki ) qi

− p1

i

i =1

C =C

m

i =1 m

i =1

− q1

ω(G k1 ,...,km )

m

i

1

ω( Q i∗ ,ki∗ ) qi

− p1

i

i =1

− p1

ω( Q i∗ ,ki∗ )

i

.

(∗)

W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

389

In order to get the estimate for I 1 (x), we need the following lemma: Lemma 2.1. Let 0 < p  1. Then there is a constant C = C ( p ) such that for all finite collections of cubes { Q k }m in Rn and all k=1 nonnegative functions gk ∈ L ω with supp gk ⊂ Q k we have

m  gk k =1

L p (ω )

m  1   C gk (x)ω(x) dxχ Q˜ k ω( Q k ) k =1

Qk

(2.4)

. L p (ω )

p

The proof of this lemma can be easily obtained by substituting L p norm by L ω and | Q1 | k in [6]. Using the above lemma and the fact (b), we have

 I 1 L ωp

m

1   −  C ··· |λ1,k1 | · · · |λm,km | ω( Q i∗ ,ki∗ ) pi χG˜ k ,...,k m 1 k1

i =1

km

m   − p1  C |λi ,ki |ω( Q i ∗ ,ki∗ ) i χG˜ k ,...,k m p 1 i =1 ki Lω m   1 − p1  C |λi ,ki |ω( Q i ∗ ,ki∗ ) i ω(x) pi χG˜ k ,...,k m 1 i =1

C

i =1

=C

ki

m 

 Qk

gk (x) dx by ω(1Q ) k

 Qk

gk (x)ω dx

p

Lω

Lp

 p1 i ˜ pi − 1 ˜ |λi ,ki | ω( Q i ∗ ,ki∗ ) ω( Q i ∗ ,ki∗ )

ki

m 

i =1

|λi ,ki |

pi

 p1

i

(2.5)

.

ki

Secondly, we consider the estimate of I 2 (x). Let A be a nonempty subset of {1, . . . , m}, and we denote the cardinality of A by | A |, then 1  | A |  m. Let A c = {1, . . . , m} \ A. If A = {1, . . . , m}, we define



Q˜ ic,k

i∈ A

 ∩

i





Q˜ i ,ki

=

i∈ Ac



Q˜ ic,k ,

i∈ A

i

then we have c Q˜ 1c,k1 ∪ · · · ∪ Q˜ m ,km =



We set E A = (

i∈ A

A ⊂{1,...,m}



Q˜ ic,k ) ∩ ( i





i∈ Ac

i∈ A

Q˜ ic,k

 i

∩



Q˜ i ,ki

 .

i∈ Ac

Q˜ i ,ki ). For fixed A, we assume that the side length of the cube Q i ∗ ,ki∗ , i ∗ ∈ A, is the

smallest among the side lengths of the cube Q i ,ki , i ∈ A. Let P cNi∗,k (x, y 1 , . . . , ym ) be the Nth order Taylor polynomial of i∗

K (x, y 1 , . . . , ym ) about the variable y i ∗ at the c i ∗,ki∗ . Since ai ∗,ki∗ has zero vanishing moments up to N, and observe that the kernels K δ satisfy (1.3) uniformly in δ > 0. By (1.3) we get

   T ∗ (a1,k , . . . , am,k )(x) m 1     m 

    N  sup ai ,ki ( y i ) ai ∗,ki∗ ( y i ∗ ) K δ (x, y 1 , . . . , ym ) − P c i∗,k (x, y 1 , . . . , ym ) d y  i∗  δ>0  ∗ (Rn )m−1

 C (Rn )m−1

i =1,i =i

Rn

 m

    ai ,k ( y i ) ai ∗,k ∗ ( y i ∗ )| y i ∗ − c i ∗,k ∗ | N +1 |x − ξ | + i i i

i =1,i =i ∗

where ξ is between y i ∗ and c i ∗,ki∗ .

Rn

m  j =1, j =i ∗

−mn− N −1 |x − y j |

d y ,

390

W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

As argument in [12], when N = max1i m [n((qi )ω / p i − 1)], [(qi / p i − 1)mn], we have

 I 2 L ωp  C

m 

i =1

|λi ,ki | p i

 p1

i

(2.6)

.

ki

In conclusion, summing the estimates (2.5) for I 1 and (2.6) for I 2 , we can take limit and obtain

T ∗ ( f )(x)

p

Lω

C

m

i =1

 f i  H pi . ωi

We complete the proof of Theorem 1.1 for case (i). Now we turn to prove case (ii). Procedure is similar as in proof of case (i), we only show the differences. To prove I 1 (x), we must introduce another lemma about weighted norm inequality with multiple weights for maximal operators. Lemma 2.2. Let T be an m-linear Calderón–Zygmund operator, satisfying (1.1), (1.2), condition, and 1 < p i < ∞. Then

T ∗ ( f )(x)

p

Lν ω

C

m

i =1

1 p

=

1 p1

+ ··· +

1 pm

 satisfy the A p , and ω

 f i L pi .

(2.7)

ωi

We will postpone its proof until the last section. We take a cube G k1 ,...,km such that for each i, ωi (G k1 ,...,km )  C ωi ( Q i ∗ ,ki∗ ). Thus by Lemma 2.2, we have



1

νω (G k1 ,...,km )

   T ∗ (a1,k , . . . , am,k )(x)νω (x) dx m 1

G k1 ,...,km

 C νω (G k1 ,...,km )−1/q

m

i =1

m

C

− p1

ω( Q i∗ ,ki∗ )

i

ai ,ki Lqi

ωi

.

i =1

By Lemma 2.1, then

 I 1 L νp

ω

m  

− p1  C ··· |λ1,k1 | · · · |λm,km | ωi ( Q i∗ ,ki∗ ) i χG˜ k ,...,k m 1 k1

C

 m 

1 −1 ∗ ,k ∗ ) p i ωi (x) p i χ ˜ |λ | ω ( Q i i , k i i G k1 ,...,km i i =1

C

i =1

km

ki

m 

i =1

|λi ,ki | p i

p

Lν ω

L pi

 p1

i

.

ki

3. Proof of Lemma 2.2 In order to prove Lemma 2.2, we must give an improved multiple Coltlar’s inequality associated to Theorem A. For integrity, we give its proof with some modifications in [8] as follows.

 m

η  (1/η)      M T ( f ) (x) + (A + W ) M f i (x) .

T ∗ ( f )(x)  C η

(3.1)

i =1

Now, we prove (3.1) firstly. We will denote by S δ (x) the cube {y : sup1 j m |x − y j |  δ}, and denote U δ = {y ∈ S 2δ }. It is clear that it is enough to prove for η arbitrary small, so we only discuss for 0 < η < 1/m. Fix x in Rn . Then we have

 

sup δ>0

Uδ

 

K δ (x, y 1 , . . . , ym ) f 1 ( y 1 ) · · · f m ( ym ) d y   C A M( f )(x).

W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

391

So it suffices to show (3.1) with T ∗ ( f )(x) replaced by





T¯∗ ( f )(x) = sup T¯ δ ( f 1 , . . . , f m )(x),

(3.2)

δ>0

where



T¯ δ ( f 1 , . . . , f m )(x) =

K δ (x, y 1 , . . . , ym ) f 1 ( y 1 ) · · · f m ( ym ) d y .

(3.3)

y ∈ / S δ (x)

Fix δ > 0 and let B (x, δ) be the ball of center x and radius δ .√ Since T is an m-linear Calderón–Zygmund operator, T ( f ) is in p L ω and hence it is finite almost everywhere. For z ∈ B (x, δ/2 m) and y ∈ / S δ (x), we get



| z − y 1 |2 + · · · + | z − y m |2

1/2

1/2 √   | x − y 1 |2 + · · · + | x − y m |2 − m|x − z| > 2δ − δ = δ.

Hence,

T¯ δ ( f )( z) = T ( f )( z) − T ( f0 )( z), where f0 = ( f 1 χ B (0,2δ) , . . . , f m χ B (0,2δ) ).

(3.4)

By (1.2), we obtain



   T¯ δ ( f )(x) − T¯ δ ( f )( z) 

A |x − z|ε

y ∈ / S δ (x)

(|x −

m

j =1 | f j ( y j )| y 1 | + · · · + |x − ym |)mn+ε

d y = I .

Now, the right side of the above inequality can be rewritten as a sum of integral for some { j 1 , . . . , jm }  {1, . . . , m} so that for y we have |x − y j |  δ if and only if j ∈ { j 1 , . . . , jl }. We denote the set for y satisfying |x − y j |  δ by Q . Then l < m and it follows that m  A |x − z|ε | f ( y )| j =1

I

y ∈ / S δ (x)









(Rn \ Q )m−l



| f j | dy j

∞  k =0

∞  k =0



j ∈{ / j 1 ,..., jl } | f j | dy j

(|x − y 1 | + · · · + |x − ym |)mn+ε

| Q |ε/n k (2 | Q |1/n )nm+ε 

| Q |ε/n (2k | Q |1/n )nm+ε

d y

|x − z|ε

| f j | dy j

j ∈{ j 1 ,..., jm } Q

CA

j



j ∈{ j 1 ,..., jm } Q



j

(|x − y 1 | + · · · + |x − ym |)mn+ε



(2k+1 Q )m





| f j | dy j

j ∈{ / j 1 ,..., jl }

(2k+1 Q )m−l

| f j | dy j

j =1

 C A M( f )(x). In [8], we can still modify the last inequality easily. Other part is the same as in [8]. Now, we begin to prove Lemma 2.2.  ∈ A p , then by Theorem E, we have νω ∈ Amp . If we choose η < m1 , mp < ηp , then νω ∈ A p . Employing Theorem F and If ω η Theorem G, we get

T ∗ ( f )

p

Lν ω

  η  (1/η)  C η M  T ( f ) (x) + ( A + W )M( f )(x) L p νω   η   (1/η)      C η M T ( f ) (x) L p + C η ( A + W ) M( f )(x) L p

νω

νω

 η  (1/η)  C η  T ( f ) (x) L p + C η ( A + W ) νω

 C η T ( f ) L p + C η ( A + W ) ν ω

 Cη( A + W )

m

f i (x) i =1

p

L ωi

.

i

Therefore, we complete the proof of Lemma 2.2.

m

f i (x)

m

f i (x)

i =1

p

L ωi

i

i =1 p

L ωi

i

392

W. Li et al. / J. Math. Anal. Appl. 373 (2011) 384–392

Acknowledgment The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions.

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