Multilinear commutators of Calderón–Zygmund operators on Hardy-type spaces with non-doubling measures

J. Math. Anal. Appl. 317 (2006) 228–244 www.elsevier.com/locate/jmaa

Multilinear commutators of Calderón–Zygmund operators on Hardy-type spaces with non-doubling measures ✩ Yan Meng, Dachun Yang ∗ School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China Received 7 May 2004 Available online 5 January 2006 Submitted by William F. Ames

Abstract Under the assumption that µ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón–Zygmund operators or fractional integrals with RBMO(µ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(µ) functions, of the Hardy space H 1 (µ) of Tolsa.  2005 Elsevier Inc. All rights reserved. Keywords: Multilinear commutator; Calderón–Zygmund operator; Fractional integral; Hardy-type space; RBMO(µ); Non-doubling measure

1. Introduction We will work on the d-dimensional Euclidean space Rd with a non-negative Radon measure µ which only satisfies the following growth condition: there exists a constant C0 > 0 such that
 µ B(x, r)
C0 r n

✩

This project was supported by NNSF (No. 10271015) and RFDP (No. 20020027004) of China.

* Corresponding author.

E-mail addresses: [email protected] (Y. Meng), [email protected] (D. Yang). 0022-247X/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.12.003

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

229

for all x ∈ Rd and r > 0, where B(x, r) = {y ∈ Rd : |y − x| < r}, n is a fixed number and 0 < n
d. Since the measure µ is not necessary to satisfy the doubling condition that there exists a constant C > 0 such that µ(B(x, 2r))
Cµ(B(x, r)) for all x ∈ supp(µ) and r > 0, we call such Rd a non-homogeneous space. It is well known that the doubling condition is a key assumption in the analysis on spaces of homogeneous type. However, in recent years, a lot of papers focus on the study of function spaces and the boundedness of Calderón–Zygmund operators in non-homogeneous spaces and indicate that many classical results still hold in nonhomogeneous spaces; see [4–7,10–13] and their references. The analysis on non-homogeneous spaces was proved to play a striking role in solving the long open Painlevé’s problem by Tolsa in [14]; see also [15] for more background. The main purpose of this paper is to establish the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón–Zygmund operators or fractional integrals with RBMO(µ) functions. Before stating our results, we first recall some necessary notation and definitions. Let K be a function on Rd × Rd \ {(x, y): x = y} satisfying that for x = y,   K(x, y)
C|x − y|−n , (1.1) and for |x − y|  2|x − x  |,  δ     K(x, y) − K(x  , y) + K(y, x) − K(y, x  )
C |x − x | , |x − y|n+δ

(1.2)

where δ ∈ (0, 1] and C > 0 is a constant. The Calderón–Zygmund operator associated to the above kernel K and the measure µ is formally defined by  (1.3) T (f )(x) = K(x, y)f (y) dµ(y). Rd

This integral may be not convergent for many functions. Thus we consider the truncated operators T for  > 0 defined by  K(x, y)f (y) dµ(y). T (f )(x) = |x−y|>

We say that T is bounded on L2 (µ) if the operators T are bounded on L2 (µ) uniformly on  > 0. By a cube Q ⊂ Rd we mean a closed cube whose sides parallel to the axes and we denote its side length by l(Q). Let α and βd be positive constants such that α > 1 and βd > α n . For a cube Q, we say that Q is (α, β)-doubling if µ(αQ)
βµ(Q), where αQ denotes the cube concentric with Q and having side length αl(Q). In what follows, for definiteness, if α and β are not specified, by a doubling cube we mean a (2, 2d+1 )-doubling cube. Especially, for any given  the smallest doubling cube in the family {2k Q}k
0 . For two cubes cube Q, we denote by Q Q1 ⊂ Q2 , set NQ1 ,Q2

KQ1 ,Q2 = 1 +

 µ(2k Q1 ) , l(2k Q1 )n k=1

where NQ1 ,Q2 is the first positive integer k such that l(2k Q1 )  l(Q2 ); see [10] for some basic properties of KQ1 ,Q2 .

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Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

Definition 1.1. Let ρ > 1 be some fixed constant. We say that a function b ∈ L1loc (µ) belongs to the space RBMO(µ) if there is a constant B > 0 such that    1 b(x) − m (b) dµ(x)
B < ∞, (1.4) sup Q Q µ(ρQ) Q

and if Q1 ⊂ Q2 are doubling cubes,   mQ (b) − mQ (b)
BKQ ,Q , 1 2 1 2

(1.5)

where the supremum is taken over all cubes centered at some point of supp µ and mQ  (b) is the  mean value of b on Q, namely,  1 (b) = b(x) dµ(x). mQ   µ(Q)  Q

The minimal constant B in (1.4) and (1.5) is defined to be the RBMO(µ) norm of b and is denoted by b∗ . The space RBMO(µ) was introduced by Tolsa [11] and he proved there that the definition of RBMO(µ) is independent of the choices of numbers ρ. In the sequel, we will choose ρ = 2. Let T be the Calderón–Zygmund operator defined by (1.3). In what follows, we will always assume that T is bounded on L2 (µ). We now fix a T, which is a weak limit as  → 0 of some subsequence of the uniformly L2 (µ) bounded operators T on  > 0; see [11, p. 141]. It is easy to deduce that T is still bounded on L2 (µ); moreover, for f ∈ L2 (µ) whose support is not all of Rd ,   T (f )(x) = K(x, y)f (y) dµ(y) Rd

with the same K as in T , which satisfies (1.1) and (1.2). For such a T, m ∈ N and bi ∈ RBMO(µ), i = 1, 2, . . . , m, we formally define the multilinear commutator Tb by    (1.6) Tb (f )(x) = bm , bm−1 , . . . , [b1 , T] · · · (f )(x). A such type of multilinear commutators when µ is the d-dimensional Lebesgue measure was first introduced by Pérez and Trujillo-González in [9]. When µ is a non-doubling measure, it was proved in [10] for m = 1 and in [3] for m > 1 that if T is bounded on L2 (µ) and bi ∈ RBMO(µ) for i = 1, . . . , m, then Tb is bounded on Lp (µ) for 1 < p < ∞. But Tb is not bounded from H 1 (Rd ) to L1 (Rd ) even when µ is the d-dimensional Lebesgue measure and m = 1; see [8]. In this paper, motivated by [8], we will first prove that for any m ∈ N, Tb is bounded from some subspace of H 1 (µ) associated with b into L1 (µ), in analogy with the result established by Pérez in [8] with µ being the d-dimensional Lebesgue measure. In the sequel, for 1
i
m, we denote by Cim the family of all finite subsets σ = {σ (1), . . . , σ (i)} of {1, 2, . . . , m} with i different elements. For any σ ∈ Cim , the complementary sequence σ  is given by σ  = {1, 2, . . . , m} \ σ . Let b = (b1 , b2 , . . . , bm ) be a finite family of locally integrable functions. For all 1
i
m and σ = {σ (1), . . . , σ (i)} ∈ Cim , we define       b(x) − b(y) σ = bσ (1) (x) − bσ (1) (y) · · · bσ (i) (x) − bσ (i) (y) ,       b(x) − mQ (b) σ = bσ (1) (x) − mQ (bσ (1) ) · · · bσ (i) (x) − mQ (bσ (i) ) ,

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

and

231

      mR (b) − mQ (b) σ = mR (bσ (1) ) − mQ (bσ (1) ) · · · mR (bσ (i) ) − mQ (bσ (i) ) ,

where Q and R are cubes in Rd and x, y ∈ Rd . With this notation, we write b σ ∗ = bσ (1) ∗ · · · bσ (i) ∗ . If σ = {1, . . . , m}, we simply write ∗ = b1 ∗ · · · bm ∗ . b Definition 1.2. Let ρ > 1, 1 < p
∞ and γ , τ ∈ N. Suppose bi ∈ RBMO(µ) for i = 1, 2, . . . , τ . p, τ, γ )-atomic block if A function h ∈ L1loc (µ) is called a (b, (a) (b) (c) (d)

there exists some cube R such that supp(h) ⊂ R; h(y) dµ(y) = 0; Rd τ h(y)b σ (y) dµ(y) = 0 for all 1
i
τ and σ ∈ Ci ; Rd for j = 1, 2, there are functions aj supported on cube Qj ⊂ R and numbers λj ∈ R such that h = λ1 a1 + λ2 a2 , and  1/p−1 −γ aj Lp (µ)
µ(ρQj ) KQj ,R .

Then we denote |h|H 1,p

(µ) b,τ,γ

= |λ1 | + |λ2 |. 1,p (µ) b,τ,γ

We say that f ∈ H f=

∞ 

p, τ, γ )-atomic blocks {hk }k∈N such that if there are (b,

hk

k=1

with

∞

k=1 |hk |H 1,p (µ) b,τ,γ

f H 1,p

b,τ,γ

1,p (µ) b,τ,γ

< ∞. The H

 = inf |bk |H 1,p (µ) k

b,τ,γ

norm of f is defined by

, (µ)

p, τ, γ )-atomic where the infimum is taken over all the possible decompositions of f in (b, blocks. Remark 1.1. By an argument similar to that in [10], we easily see that the above definition is 1,p independent of the chosen constant ρ > 1. If τ = 0, the space H (µ) is just the atomic Hardy b,τ,γ space introduced by Tolsa in [10] when γ = 1 and when γ > 1 by the authors in [2], which was proved in [2,10] to be the Hardy space H 1 (µ) of Tolsa in [13] with equivalent norms; see also 1,p Definition 1.4 below. However, it is still open if the spaces H (µ) are equivalent for any fixed b,τ,γ τ ∈ N and different γ ∈ N and 1 < p
∞. But we have the following obvious properties that for any τ ∈ N, 1 < p
∞ and γ1 , γ2 ∈ N with 1
γ1 < γ2 , 1,p 1,p (µ) ⊂ H (µ) ⊂ H 1 (µ), b,τ,γ2 b,τ,γ1

H

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Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

and for any τ, γ ∈ N and 1 < p1 < p2
∞, 1,p2 1,p (µ) ⊂ H 1 (µ) ⊂ H 1 (µ). b,τ,γ b,τ,γ

H 1,∞ (µ) ⊂ H b,τ,γ

Here is one of the main results of this paper. Theorem 1.1. Let 1 < p
∞, m ∈ N and bi ∈ RBMO(µ) for i = 1, 2, . . . , m. Let T and Tb be as in (1.3) and (1.6), respectively. Suppose that T is bounded on L2 (µ). Then the multilinear 1,p commutator Tb is bounded from H (µ) into L1 (µ). b,m,m+1

Remark 1.2. Let us consider the multilinear commutator of the fractional integral operator, Iα;b , defined by  m   Iα;b (f )(x) = bi (x) − bi (y) Rd

i=1

f (y) dµ(y), |x − y|n−α

(1.7)

where m ∈ N, bi ∈ RBMO(µ) for i = 1, 2, . . . , m and 0 < α < n. By an argument similar to the proof of Theorem 1.1, we can prove the following result for Iα;b and we omit the details by similarity. Theorem 1.2. Let 0 < α < n, 1 < p
∞, m ∈ N, bi ∈ RBMO(µ) for i = 1, 2, . . . , m, and Iα;b 1,p be as in (1.7). Then Iα;b is bounded from H (µ) into Ln/(n−α) (µ). b,m,m+1

In [1], the authors proved that T is bounded on the Hardy space H 1 (µ) if T∗ (1) = 0. Moti1,p (µ) into H 1 (µ) with the vated by this, we now consider the boundedness of Tb from H b,m,m+2 assumption that T ∗ (1) = 0. Here, by T ∗ (1) = 0, we mean that for any bounded function h with b b compact support satisfying (b) and (c) of Definition 1.2,  Tb (h)(x) dµ(x) = 0. (1.8) Rd 1,p (µ) b,m,m+2

We point out that for a such function h, it is easy to see that h ∈ H

and therefore,

Tb (h) ∈ by Theorem 1.1. Also, if Tb (h) ∈ then Tb (h) should satisfy (1.8) by the definition of the Hardy space H 1 (µ); see [10,13] or Definition 1.4 below. Thus, in some sense, condition (1.8) is also necessary. We remark that this result is new even when µ is the d-dimensional Lebesgue measure. L1 (µ)

H 1 (µ),

Definition 1.3. Given f ∈ L1loc (µ), we set      MΦ f (x) = sup  f ϕ dµ, ϕ∼x

Rd

where the notation ϕ ∼ x means that ϕ ∈ L1 (µ) ∩ C 1 (Rd ) and satisfies

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

233

(i) ϕL1 (µ)
1, (ii) 0
ϕ(y)
|y − x|−n for all y ∈ Rd , and (iii) |∇ϕ(y)|
|y − x|−(n+1) for all y ∈ Rd , where ∇ = (∂/∂x1 , . . . , ∂/∂xd ). Based on Theorem 1.2 of Tolsa in [13], we define the Hardy space H 1 (µ) as follows. Definition 1.4. The Hardy space H 1 (µ) is the set of all functions f ∈ L1 (µ) satisfying that 1 1 Rd f dµ = 0 and MΦ f ∈ L (µ). Moreover, we define the norm of f ∈ H (µ) by f H 1 (µ) = f L1 (µ) + MΦ f L1 (µ) . Another main result of this paper is as follows. Theorem 1.3. Let 1 < p
∞, m ∈ N and bi ∈ RBMO(µ) for i = 1, 2, . . . , m. Let T and Tb be as in (1.3) and (1.6), respectively. Suppose that T is bounded on L2 (µ) and T ∗ (1) = 0 as in b 1,p (1.8). Then Tb is bounded from H (µ) into H 1 (µ). b,m,m+2

In what follows, C always denotes a positive constant that is independent of the main parameters involved but whose value may differ from line to line. For any index p ∈ [1, ∞], we denote by p  its conjugate index, namely, 1/p + 1/p  = 1. 2. Proof of Theorem 1.1 We begin with some necessary lemmas. Lemma 2.1. Let 1 < p < ∞, m ∈ N and bi ∈ RBMO(µ) for i = 1, 2, . . . , m. Let T and Tb be as in (1.3) and (1.6), respectively. Suppose that T is bounded on L2 (µ). Then Tb is bounded on ∗ , where C > 0 is a constant. Lp (µ) with the norm no more than Cb Lemma 2.1 was proved by Tolsa for the case m = 1 in [10] and by the authors for the cases m > 1 in [3]. The following Lemma 2.2 is a simple corollary of the John–Nirenberg inequality with non-doubling measure; see [10]. Lemma 2.2. Let m ∈ N, bi ∈ RBMO(µ) for i = 1, 2, . . . , m, ρ > 1 and 1
p < ∞. Then there exists a constant C > 0 such that for any cube Q, 1/p   m m

p  1 bi (x) − m (bi ) dµ(x)
C bi ∗ . Q µ(ρQ) Q i=1

i=1

p, m, m+1)Proof of Theorem 1.1. By a standard argument, it suffices to verify that for any (b, atomic block h as in Definition 1.2 with ρ = 4, Tb (h) is in L1 (µ) with norm no more than C|h|H 1,p (µ) , where C > 0 is a constant independent of h. Let all the notation be the same as b,m,m+1

in Definition 1.2. By our choices, aj , j = 1, 2, now satisfies the following size condition: aj Lp (µ)
µ(4Qj )1/p−1 [KQj ,R ]−(m+1) .

(2.1)

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Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

Write 

  T (h)(x) dµ(x) =



b

Rd



  T (h)(x) dµ(x) +

  T (h)(x) dµ(x) = M + N.

b

b

Rd \2R

2R

To estimate M, we further decompose M


2 



2    T (aj )(x) dµ(x) + |λj |

|λj |

j =1



  T (aj )(x) dµ(x) = M1 + M2 .

b

j =1

2Qj

b

2Rj \2Qj

From the Hölder inequality, Lemma 2.1 and (2.1), it follows that M1


2  j =1

2     1/p    |λj | Tb (aj ) Lp (µ) µ(2Qj )
Cb∗ |λj |aj Lp (µ) µ(2Qj )1/p j =1

∗
Cb

2 

|λj |.

j =1

Let Nj,1 = N2Qj ,2Rj for j = 1, 2. With the aid of the formula that for x, y ∈ Rd , m m 

       bi (x) − bi (y) = b(x) − mQ j (b) σ mQ j (b) − b(y) σ  ,

(2.2)

i=0 σ ∈Cim

i=1

where if i = 0, then σ  = {1, 2, . . . , m} and σ = ∅, by (1.1), the Hölder inequality, Lemma 2.2 and (2.1), we easily obtain  m  Nj,1  2         M2
bi (x) − bi (y) K(x, y)aj (y) dµ(y) dµ(x) |λj |    j =1


C

k=1

2 

|λj |

j =1

 Qj 2 

×



Nj,1 m   

    b(x) − m  (b)  Qj

σ

i=0 σ ∈Cim k=1 k+1 2 Qj \2k Qj

|λj |

j =1



Qj i=1

    m  (b) − b(y)   |aj (y)| dµ(y) dµ(x) Qj σ |x − y|n

×


C

2k+1 Qj \2k Qj

1/p   1/p m          aj (y)p dµ(y)  m  (b) − b(y)  p dµ(y) Qj σ i=0 σ ∈Cim Q j

Nj,1 i    l=0 η∈C i k=1 l

 ×

2k+1 Qj

Qj

1 l(2k Qj )n

  b(x) − m

k+1 Q 2
j

      (b) η m
(b) − mQ j (b) η dµ(x) k+1 2

Qj

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

∗
Cb

2 

|λj |aj Lp (µ) µ(2Qj )1/p



j =1

∗
Cb

2 

235

Nj,1 m    i µ(2k+2 Qj )  K
k+1 k n Qj ,2 Qj l(2 Qj ) m i=0 σ ∈Ci k=1



|λj |aj Lp (µ) µ(2Qj )1/p [KQj ,R ]m+1

j =1

∗
Cb

2 

|λj |,

j =1

where we have used the fact that for any 1
k
Nj,1 , K 

k+1 Q Qj ,2
j


CKQj ,R ; see [10].

Now we turn to estimate N. Denote the center of R by xR . The propositions (b) and (c) of Definition 1.2, (2.2), (1.2), the Hölder inequality, Lemma 2.2 and (2.1) lead to     m       bi (x) − bi (y) K(x, y) − K(x, xR ) h(y) dµ(y) dµ(x) N=    R i=1

Rd \2R

m   ∞        b(y) − m (b)  h(y)
C R σ i=0 σ ∈Cim k=1 R



× 2k+1 R\2k R


C

2 

|λj |

j =1

δ     b(x) − m (b)  |x − xR | dµ(x) dµ(y) R σ |x − y|n+δ

 m−i         b(y) − m  (b) m  (b) − m (b)  Qj Qj R η η

m   i=0 σ ∈Cim

l=0 η∈C m−i Q j l

      × aj (y) dµ(y)  ×

i   ∞  s=0 θ∈Csi k=1



l(R)δ l(2k Qj )n+δ

  b(x) − m

×

      (b) θ m
(b) − mR(b) θ  dµ(x)
k+1 k+1

2

R

R

2

2k+1 R


C

2 

p

|λj |aj L (µ)

j =1

×

m−i 



l=0 η∈C m−i l

∗
Cb

2  j =1



m  

b σ ∗

∞  µ(2k+2 R)l(R)δ 

i=0 σ ∈Cim

k=1

l(2k Qj )n+δ

K
k+1 R,2

i

R

1/p     p   b(y) − m  (b) m  (b) − m (b)   dµ(y) Qj Qj R η η

Qj 

|λj |aj Lp (µ) µ(2Qj )1/p [KQj ,R ]m

236

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

∗
Cb

2 

|λj |.

j =1

Combining the estimates for M and N, we have finished the proof of Theorem 1.1.

2

3. Proof of Theorem 1.3 In order to prove Theorem 1.3, we first recall the following basic fact in [1]. Lemma 3.1. Let MΦ be as in Definition 1.3 and 1 < p < ∞. Then MΦ is bounded on Lp (µ). Proof of Theorem 1.3. Just as in the proof of Theorem 1.1, we only need to verify that for any p, m, m + 2)-atomic block h as in Definition 1.2 with ρ = 4, T (h) is in H 1 (µ) with norm (b, b no more than C|h|H 1,p (µ) , where C > 0 is a constant independent of h. By our choices, for b,m,m+2

j = 1, 2, aj satisfies the following size condition that aj Lp (µ)
µ(4Qj )1/p−1 [KQj ,R ]−(m+2) .

(3.1)

T ∗ (1) = 0, b

Theorem 1.1 and Definition 1.4, we deduce that the proof By the assumption that of Theorem 1.3 can be reduced to proving that    MΦ T (h)  1
Cb ∗ |h| 1,p . (3.2) b L (µ) H (µ) b,m,m+2

Write        MΦ T (h)  1 = MΦ T (h) (x) dµ(x) + b b L (µ)

  MΦ Tb (h) (x) dµ(x) = I + II.

Rd \4R

4R

Noting that MΦ is sublinear, we can control I by         I
MΦ Tb (h) χ8R (x) dµ(x) + MΦ Tb (h) χRd \8R (x) dµ(x) = I1 + I2 . 4R

4R

From the fact that for j = 1, 2, Qj ⊂ R, it follows that for any z ∈ Qj and any y ∈ 2k+1 R \ 2k R, k  3, |y − z|  l(2k−2 R). By this fact, (ii) of Definition 1.3, (2.2), (1.1), Lemma 2.2 and (3.1), we obtain       T (h)(y)ϕ(y) dµ(y) dµ(x) I2
sup b ϕ∼x




4R

Rd \8R

2 

∞  

|λj |

j =1

k=3 4R



 ∞        bi (y) − bi (z) K(y, z)aj (z) dµ(z)   

i=1 2k+1 R\2k R Qj

1 dµ(y) dµ(x) |x − y|n 2 m   ∞    |λj |
C ×

j =1



i=0 σ ∈Cim k=3 4R k+1 2 R\2k R

|[b(y) − mQ j (b)]σ | |x − y|n

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

×

 |[m (b) − b(z)]  ||a (z)| j j σ Q |y − z|n

Qj


C

2  j =1

dµ(z) dµ(y) dµ(x)

 1/p m   ∞    p  µ(4R) aj Lp (µ)   mQ |λj | dµ(z) j (b) − b(z) σ  l(2k R)n l(2k R)n m i=0 σ ∈Ci k=3



Qj

  b(y) − m

×

   dµ(y)

(b) + m
(b) − mR(b) + mR(b) − mQ j (b) σ k+1 R 2
2k+1 R

2k+1 R

∗
Cb

2 

|λj |aj Lp (µ) µ(2Qj )1/p



j =1 m   ∞  i  i  µ(4R)µ(2k+2 R)  K
× + KQ j ,R  k R)n l(2k R)n R,2k+1 R l(2 m i=0 σ ∈Ci k=3

∗
Cb

2 

|λj |.

j =1

To estimate I1 , we write I1


2 

 4Qj

2 

 |λj |

j =1

+

   MΦ Tb (aj ) χ8R (x) dµ(x)

|λj |

j =1

+

2 

   MΦ Tb (aj ) χ2Qj (x) dµ(x)

4R\4Qj

 |λj |

j =1

   MΦ Tb (aj ) χ8R\2Qj (x) dµ(x)

4R\4Qj

= I11 + I12 + I13 . The Hölder inequality, Lemma 3.1, Lemma 2.1 and (3.1) lead to I11


2  j =1


C

237

    |λj |µ(4Qj )1/p MΦ Tb (aj ) χ8R Lp (µ)

2  j =1

  |λj |µ(4Qj )1/p Tb (aj )Lp (µ)

∗
Cb

2 

|λj |aj Lp (µ) µ(4Qj )1/p

j =1

∗
Cb

2  j =1

|λj |.



238

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

For j = 1, 2, denote NQj ,4R simply by Nj,2 . By (ii) of Definition 1.3, the Hölder inequality, Lemma 2.1 and (3.1), we have I12


2 

|λj |

j =1


C

2 



k=2

2k+1 Qj \2k Qj

|λj |

j =1


C

2  j =1

     sup  Tb (aj )(y)ϕ(y) dµ(y) dµ(x)

Nj,2 

ϕ∼x

2Qj

Nj,2 



k=2

2k+1 Qj \2k Qj

1 dµ(x) l(2k Qj )n



  T (aj )(y) dµ(y) b

2Qj

Nj,2   µ(2k+1 Qj )  T (aj ) p µ(2Qj )1/p |λj | b L (µ) k n l(2 Qj )

∗
Cb

k=2

2 



|λj |aj Lp (µ) µ(2Qj )1/p KQj ,R

j =1

∗
Cb

2 

|λj |.

j =1

For I13 , we further decompose it into I13


2 

|λj |

j =1

+

+

Nj,2 



k=2

2k+1 Qj \2k Qj

   MΦ Tb (aj )χ2k+2 Qj \2k−1 Qj (x) dµ(x)

Nj,2 



j =1

k=2

2k+1 Qj \2k Qj

2 

Nj,2 



k=2

2k+1 Qj \2k Qj

2 

|λj |

|λj |

j =1

   MΦ Tb (aj )χmax{2k+2 Qj ,8R}\2k+2 Qj (x) dµ(x)    MΦ Tb (aj )χ2k−1 Qj \2Qj (x) dµ(x)

= E + F + G. The Hölder inequality, Lemma 3.1, (2.2), (1.1), Lemma 2.2 and (3.1) tell us that E


2 

Nj,2

|λj |

j =1


C

2 


 1/p     MΦ T (aj )χ k+2 µ 2k+1 Qj 2 Qj \2k−1 Qj Lp (µ) b k=2

 
1/p k+1 |λj | µ 2 Qj

j =1


C

2  j =1

|λj |

Nj,2



k=2

2k+2 Qj \2k−1 Qj

  T (aj )p dµ(y) b

Nj,2  m    1/p
µ 2k+1 Qj i=0 σ ∈Cim k=2

1/p



2k+2 Qj \2k−1 Qj

    b(y) − m  (b) p Qj

σ

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

239

 p 1/p      ×  (b) − b(z) K(y, z)a (z) dµ(z) dµ(y) mQ j j  σ Qj


C

2 

|λj |

j =1

Nj,2   m        µ(2k+1 Qj )1/p  m  (b) − b(z)  aj (z) dµ(z) Q σ j l(2k Qj )n m i=0 σ ∈Ci k=2

 

  b(y) − m

×

(b) + m
k+2 Q 2
2k+2 Qj j

2k+2 Q


C

Qj

2 

j

|λj |

j =1

m  

b σ ∗

i=0 σ ∈Cim



1/p  p  (b) − mQ dµ(y) j (b) σ

Nj,2  i µ(2k+3 Qj )  K
k+2 k n Qj ,2 Qj l(2 Qj ) k=2

1/p   1/p       aj (z)p dµ(z)  m  (b) − b(z)  p dµ(z) Qj σ

× Qj

∗
Cb

Qj 2 



|λj |aj Lp (µ) µ(2Qj )1/p [KQj ,R ]m+1

j =1

∗
Cb

2 

|λj |.

j =1

By (ii) of Definition 1.3, (2.2), (1.1), the Hölder inequality, Lemma 2.2 and (3.1), we easily see G


2 

|λj |

j =1




2 

|λj |

j =1


C

2 

Nj,2 



k=2

2k+1 Qj \2k Qj

Nj,2 



k=2

2k+1 Qj \2k Qj

|λj |

j =1



   T (aj )(y)ϕ(y) dµ(y) dµ(x) b



sup

Nj,2 m   

ϕ∼x

2k−1 Qj \2Qj



k−2  l=1

2l+1 Qj \2l Qj



k−2 

|Tb (aj )(y)| dµ(y) dµ(x) |y − x|n 

|[b(y) − mQ j (b)]σ | |y − x|n

i=0 σ ∈Cim k=2 k+1 l=1 l+1 2 Qj \2k Qj 2 Qj \2l Qj

        mQ × j (b) − b(z) σ  K(y, z)aj (z) dµ(z) dµ(y) dµ(x) Qj


C

2  j =1

Nj,2 k−2 m    µ(2k+1 Q)  1 |λj | k n l l(2 Qj ) l(2 Qj )n m

 × Qj

i=0 σ ∈Ci k=2

l=1



    b(y) − m  (b)  dµ(y) Qj σ

2l+1 Qj

1/p   1/p       aj (z)p dµ(z)  m  (b) − b(z)  p dµ(z) Qj σ Qj

240

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

∗
Cb

2 

|λj |

j =1

∗
Cb

2 

Nj,2 k−2 m    i µ(2k+1 Q)  µ(2l+2 Qj )  K
l+1 k n l n Qj ,2 Qj l(2 Qj ) l(2 Qj ) m i=0 σ ∈Ci k=2

l=1



|λj |aj Lp (µ) µ(2Qj )1/p [KQj ,R ]m+2

j =1

∗
Cb

2 

|λj |.

j =1

An argument similar to the estimate for G can also give us a desired estimate for F. The estimates for E, F and G lead us a desired estimate for I13 . Combining the estimates for I11 , I12 , I13 and I2 yield  I=

2    ∗ ∗ |h| 1,p MΦ Tb (h) (x) dµ(x)
Cb |λj | = Cb H

(µ) b,m,m+2

j =1

4R

.

(3.3)

Now we turn to the estimate for II. Invoking that T ∗ (1) = 0, we obtain b

 II = Rd \4R




Rd \4R

      sup  Tb (h)(y) ϕ(y) − ϕ(xR ) dµ(y) dµ(x) ϕ∼x Rd

       sup  Tb (h)(y) ϕ(y) − ϕ(xR ) dµ(y) dµ(x) ϕ∼x 2R



+ Rd \4R

   sup  ϕ∼x

    Tb (h)(y) ϕ(y) − ϕ(xR ) dµ(y) dµ(x)

Rd \2R

= II1 + II2 . Note that for any z ∈ 2R, x ∈ 2k+1 R \ 2k R, k  2, |x − z|  l(2k−2 R). This together with (iii) of Definition 1.3 and the mean value theorem leads to   l(R) ϕ(y) − ϕ(xR )
C (3.4) l(2k−2 R)n+1 for y ∈ 2R. By (3.4), (2.2), (1.1), the Hölder inequality, Lemmas 2.1 and 2.2, and (3.1), we have II1


2 

|λj |

j =1

+

2 



k=2

2k+1 R\2k R

|λj |

j =1


C

2  j =1



∞ 

|λj |



sup ϕ∼x

∞ 



k=2

2k+1 R\2k R

2R\2Qj

 sup ϕ∼x

m   ∞ 

    T (aj )(y)ϕ(y) − ϕ(xR ) dµ(y) dµ(x) b     T (aj )(y)ϕ(y) − ϕ(xR ) dµ(y) dµ(x) b

2Qj



i=0 σ ∈Cim k=2 k+1 2 R\2k R

Nj,2 −1  l(R) k n+1 l(2 R) l=1



    b(y) − m  (b)  Qj

2l+1 Qj \2l Qj

σ

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244



    m  (b) − b(z)   |aj (z)| dµ(z) dµ(y) dµ(x) Qj σ |y − z|n

× Qj

+C

2 

|λj |

j =1


C

2 

|λj |

j =1

 l=1





k=2

2k+1 R\2k R

   l(R)   T (aj ) χ2Q  1 dµ(x) j L (µ) b k n+1 l(2 R)

m   ∞  µ(2k+1 R)l(R) l(2k R)n+1 m i=0 σ ∈Ci k=2

Nj,2 −1

×

∞ 



1 l l(2 Qj )n

    b(y) − m  (b)  dµ(y) Qj σ

2l+1 Qj

    aj (z) m  (b) − b(z)   dµ(z) Qj σ

× Qj

+C

2 

|λj |

j =1

∗
Cb

∞     µ(2k+1 R)l(R)   T (aj ) χ2Q  p µ(2Qj )1/p j L (µ) b k n+1 l(2 R) k=2

2 

∞  µ(2k+1 R)l(R)

|λj |aj Lp (µ) µ(2Qj )1/p

j =1

k=2

l(2k R)n+1

Nj,2 −1

 µ(2l+2 Qj )  m K
Qj ,2l+1 Qj l(2l Qj )n

×

l=1

+C

2 

|λj |aj Lp (µ) µ(2Qj )1/p



j =1

∗
Cb

2 



|λj |aj Lp (µ) µ(2Qj )1/p [KQj ,R ]m+1

j =1

∗
Cb

2 

|λj |.

j =1

We further estimate II2 by    ∞   II2 = sup  k=2




ϕ∼x

2k+1 R\2k R

∞ 



k=2

2k+1 R\2k R

+

    Tb (h)(y) ϕ(y) − ϕ(xR ) dµ(y) dµ(x)

Rd \2R

   MΦ Tb (h)χ2k+2 R\2k−1 R (x) dµ(x)

∞ 



k=2

2k+1 R\2k R





sup

   T h(y)ϕ(xR ) dµ(y) dµ(x) b

ϕ∼x 2k+2 R\2k−1 R

241

242

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

+

+



∞ 



k=2

2k+1 R\2k R

    T (h)(y) ϕ(y) + ϕ(xR ) dµ(y) dµ(x)



sup

b

ϕ∼x

∞ 



k=2

2k+1 R\2k R

Rd \2k+2 R





sup

    T (h)(y) ϕ(y) + ϕ(xR ) dµ(y) dµ(x) b

ϕ∼x 2k−1 R\2R

= II21 + II22 + II23 + II24 . From the Hölder inequality, Lemma 3.1, (b) and (c) of Definition 1.2, (2.2), (1.2), Lemma 2.2 and (3.1), we can deduce that II21


∞    
1/p  MΦ T (h)χ k+2 k−1  p µ 2k+1 R 2 R\2 R L (µ) b k=2 ∞ 


C


1/p µ 2k+1 R

k=2



 m      bi (y) − bi (z) K(y, z) − K(y, xR )  

 2k+2 R\2k−1 R

p 1/p   × h(z) dµ(z) dµ(y) 

m   ∞ 
1/p
C µ 2k+1 R

R i=1



i=0 σ ∈Cim k=2



    b(y) − m (b) p R

σ

2k+2 R\2k−1 R

1/p  p       mR(b) − b(z) σ  K(y, z) − K(y, xR ) h(z) dµ(z) dµ(y) × 
C

R m 

   ∞   µ(2k+1 R)1/p l(R)δ

i=0 σ ∈Cim k=2



l(2k R)n+δ

     m (b) − b(z)  h(z) dµ(z) R σ

×

1/p     b(y) − m (b) p dµ(y) R σ

2k+2 R

R


C

2 

|λj |

j =1



×

m   i=0 σ ∈Cim

∞  i µ(2k+3 R)l(R)δ  K
k+2 R k n+δ R, 2 l(2 R) k=2

1/p   1/p       aj (z)p dµ(z)  m (b) − b(z)  p dµ(z) R σ

Qj

∗
Cb

b σ ∗

Qj 2 



|λj |aj Lp (µ) µ(2Qj )1/p [KQj ,R ]m

j =1

∗
Cb

2 

|λj |.

j =1

An argument similar to the estimate for II21 can also give us a desired estimate for II22 .

Y. Meng, D. Yang / J. Math. Anal. Appl. 317 (2006) 228–244

243

Finally, we estimate II23 . By (b) and (c) of Definition 1.2, (ii) of Definition 1.3, (2.2), (1.2), the Hölder inequality, Lemma 2.2 and (3.1), we obtain    ∞ ∞ m       K(y, z) − K(y, xR ) bi (y) − bi (z) II23
k=2

2k+1 R\2k R

l=k+2

     × h(z) dµ(z)

i=1

2l+1 R\2l R R



1 1 dµ(y) dµ(x) + n |y − x| |xR − x|n  2 m   ∞  ∞   µ(2k+1 R) l(R)δ
C |λj | l(2k R)n l(2l R)n+δ m j =1



×

i=0 σ ∈Ci k=2 l=k+2

    b(y) − m (b)  dµ(y) R σ

2l+1 R

     m (b) − b(z)  aj (z) dµ(z) R σ

Qj

∗
Cb

2 

|λj |aj Lp (µ) µ(2Qj )1/p

j =1

×



m  

[KQj R ]m−i

i=0 σ ∈Cim

∞  ∞  i µ(2k+1 R) µ(2l+2 R)l(R)δ  K  l+1 R k n l n+δ R, 2 l(2 R) l(2 R) k=2 l=k+2

∗
Cb

2 

|λj |.

j =1

An argument similar to the estimate for II23 can also give us a desired estimate for II24 . Combining the estimates for II21 , II22 , II23 and II24 , we obtain a desired estimate for II2 . The estimates for II1 and II2 tell us that    II = MΦ Tb (h) (x) dµ(x)
C|b|H 1,p (µ) . (3.5) Rd \4R

b,m,m+2

The estimates (3.3) and (3.5) lead to (3.2) and this completes the proof of our theorem.

2

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