Weighted estimates with general weights for multilinear Calderón-Zygmund operators

Weighted estimates with general weights for multilinear Calderón-Zygmund operators

Acta Mathematica Scientia 2012,32B(4):1529–1544 http://actams.wipm.ac.cn WEIGHTED ESTIMATES WITH GENERAL WEIGHTS FOR MULTILINEAR ´ CALDERON-ZYGMUND O...

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Acta Mathematica Scientia 2012,32B(4):1529–1544 http://actams.wipm.ac.cn

WEIGHTED ESTIMATES WITH GENERAL WEIGHTS FOR MULTILINEAR ´ CALDERON-ZYGMUND OPERATORS∗



Hu Guoen (

)

Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou 450002, China E-mail: [email protected]

Abstract In this paper, some weighted estimates with general weights are established for the m-linear Calder´ on-Zygmund operator and the corresponding maximal operator. It m  is proved that, if p1 , · · · , pm ∈ [1, ∞] and p ∈ (0, ∞) with 1/p = 1/pk , then for any k=1

weight w, integer  with 1 ≤  ≤ m, these operators are bounded from Lp1 (Rn , MB w) × · · · × Lp (Rn , MB w) × Lp+1 (Rn , M w) × · · · × Lpm (Rn , M w) to Lp,∞ (Rn , w) or Lp (Rn , w), where B is a Young function and MB is the maximal operator associated with B. Key words multilinear Calder´ on-Zygmund operator; maximal operator; weighted norm inequality; Calder´ on-Zygmund decomposition 2010 MR Subject Classification

1

42B20

Introduction

During the last several years, considerable attention was paid to the boundedness of multilinear singular integral operators on function spaces. Let m ≥ 1, K(x; y1 , · · · , ym ) be a locally integrable function defined away from the diagonal {x = y1 = y2 = · · · = ym } in (Rn )m+1 , A > 0 and γ ∈ (0, 1] be two constants. We say that K is a kernel in m-CZK(A, γ) if it satisfies the size condition that, for all (x, y1 , · · · , ym ) ∈ (Rn )m+1 with x = yj for some 1 ≤ j ≤ m, |K(x; y1 , · · · , ym )| ≤

A mn (|x − y1 | + · · · + |x − ym |)

(1.1)

and satisfies the regularity condition that |K(x; y1 , · · · , ym ) − K(x ; y1 , · · · , ym )| ≤

A|x − x |γ (|x − y1 | + · · · + |x − ym |)mn+γ

(1.2)

whenever max |x − yk | ≥ 2|x − x |, and also that, for each fixed k with 1 ≤ k ≤ m, 1≤k≤m

|K(x; y1 , · · · , yk , · · · , ym ) − K(x; y1 , · · · , yk , · · · , ym )| ≤ ∗ Received

A|yk − yk |γ

mn+γ

(|x − y1 | + · · · + |x − ym |)

(1.3)

August 2, 2010; revised April 15, 2011. The research was supported by the NSFC (10971228).

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whenever max |x − yj | ≥ 2|yk − yk |. An operator T defined on m-fold product of Schwartz 1≤j≤m

spaces and taking values in the space of tempered distributions, is said to be an m-linear Calder´ on-Zygmund operator with kernel K if (a) T is m-linear, m  (b) for some q1 , · · · , qm ∈ [1, ∞] and some q ∈ (0, ∞) with 1/q = 1/qk , T can be k=1

extended to be a bounded operator from Lq1 (Rn ) × Lq2 (Rn ) × · · · × Lqm (Rn ) to Lq (Rn ), m  supp fk , (c) for f1 , · · · , fm ∈ L2 (Rn ) with compact supports, and a.e. x ∈ k=1

 T (f1 , · · · , fm )(x) =

(Rn )m

K(x; y1 , · · · , ym )

m 

fk (yk )dy1 · · · dym ,

(1.4)

k=1

and K is in m-CZK(A, γ) for some constants A and γ. It is obvious that, when m = 1, this operator is just the classical Calder´on-Zygmund operator. For the case of m ≥ 2, this operator has intimate connections with operator theory and partial differential equations, and was considered first by Coifman and Meyer [2], and then by many authors. In their remarkable work [7], Grafakos and Torres considered the mapping properties of T on the space of type Lp1 (Rn ) × · · · × Lpm (Rn ) with 1 ≤ p1 , · · · , pm < ∞, and established a T 1 type theorem for the operator T . Moreover, Grafakos and Torres [8] established the weighted estimates with Ap weights for T and the corresponding maximal operator, defined by T ∗ (f1 , · · · , fm )(x) = sup |T (f1 , · · · , fm )(x)|

(1.5)

>0

with

 T (f1 , · · · , fm )(x) =

 1≤j≤m

|x−yj |2 >2

K(x; y1 , · · · , ym )

m 

fk (yk )dy1 · · · dym .

k=1

Lerner et al. [12] introduced a new maximal operator and established some interesting weighted on-Zygmund norm inequalities with mixed Ap wights. For other works about multilinear Calder´ operators, we mention the papers [6], [10], [13] and the references therein. Recently, Hu [9] considered the weighted estimates with general weights for the multilinear Calder´ on-Zygmund p n operator. For a weight (nonnegative and locally integrable function) w, let L (R , w) denotes the usual weighted Lp space with weight w, and Lp,∞ (Rn , w) denotes the weighted weak Lp space with respect to the weight w, that is, Lp,∞ (Rn , w) = {f : f Lp,∞ (Rn ,w) < ∞}, where (and in the sequal)  1/p . f Lp,∞(Rn ,w) = sup λ w({x ∈ Rn : |f (x)| > λ}) λ>0

For δ ∈ R, p ∈ (0, ∞) and a suitable function f , set     |f (x)| p |f (x)|  1 dx ≤ 1 . logδ e + f Lp(log L)δ ,E = inf λ > 0 : |E| E λ λ

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The maximal operator MLp(log L)δ is defined by MLp (log L)δ f (x) = sup f Lp(log L)δ ,Q , Qx

where the supremum is taken over all cubes containing x. For simplicity, we denote by ML log L the operator ML(log L)1 . Hu [9] proved the following weighted estimate for the multilinear Calder´ on-Zygmund operators. Theorem 1.1 Let m ≥ 1, T be an m-linear Calder´ on-Zygmund operator. m  (i) If p1 ∈ [1, ∞), p2 · · · , pm ∈ [1, ∞] and p ∈ (0, ∞) with 1/p = 1/pk , then for any k=1

δ > 0, there exists a positive constant C such that, for all weight w and bounded functions f1 , · · · , fm with compact supports, T (f1 , · · · , fm )Lp,∞ (Rn ,w) ≤ Cf1 Lp1 (Rn ,ML(log L)p1 −1+δ w)

m 

fk Lpk (Rn ,Mw) ,

k=2

where and in the sequel, when pk = ∞,  · Lpk (Rn ,u) is replaced by  · L∞ (Rn ) , when j0 ≥ j, j−1

will disappear. Furthermore, if p1 ∈ (1, ∞) and p2 , · · · , pm ∈ (1, ∞], then the term k=j0

T (f1, · · · , fm )Lp (Rn ,w) ≤ Cf1 Lp1 (Rn ,ML(log L)p1 −1+δ w)

m 

fk Lpk (Rn ,Mw) .

k=2

(ii) If p1 , · · · , pm ∈ (1, ∞] and p ∈ (0, ∞) with 1/p =

m 

1/pk , then for any δ > 0, there

k=1

exists a positive constant C such that, for all weight w and bounded functions f1 , · · · , fm with compact supports, T (f1 , · · · , fm )Lp (Rn ,w) ≤ C

m 

fk Lpk (Rn ,ML(log L)p∗ −1+δ w) ,

k=1

where and in the sequel, p∗ = max{1, p}. Our first purpose in this paper is to improve Theorem 1.1. We will establish some new weighted estimates with general weights for the multilinear Calder´ on-Zygmund operator T . The new result here shows that, for the multilinear Calder´ on-Zygmund operator, the singularity can be viewed concentrated on some special variables. Our result for the multilinear operator T can be stated as follows. Theorem 1.2 Let m ≥ 2 and  be any integer such that 1 ≤  ≤ m, T be an mlinear Calder´ on-Zygmund operator, p1 ∈ [1, ∞), p2 , · · · , pm ∈ [1, ∞] and p ∈ (0, ∞) with m   1/p = 1/pk . Let r ∈ (0, ∞) with 1/r = 1/pk (note that rm = p). Then for any 1≤k≤

k=1

δ > 0, there exists a positive constant C such that for all weight w and bounded functions f1 , · · · , fm with compact supports, T (f1, · · · , fm )Lp,∞ (Rn ,w) ≤ Cf1 Lp1 (Rn ,M

r∗ −1+δ w) L(log L) 

×

m  k=+1

fk Lpk (Rn ,Mw)

 

fk Lpk (Rn ,ML(log L)δ w)

k=2

(1.6)

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with r∗ = max{1, r }. Furthermore, if p1 ∈ (1, ∞), p2 , · · · , pm ∈ (1, ∞], then  

T (f1 , · · · , fm )Lp (Rn ,w) ≤ Cf1 Lp1 (Rn ,M

r∗ −1+δ w) L(log L) 

×

m 

fk Lpk (Rn ,ML(log L)δ w)

k=2

fk Lpk (Rn ,Mw) .

(1.7)

k=+1

Remark 1 It is obvious that, for two special cases: the case  = 1, and the case  = m and r ≤ 1, inequality (1.6) follows from Theorem 1.1 directly. However, for different choices of , our conclusions lead to different weighted estimates which have independent interests. For example, for m = 4 and p1 = p2 = p3 = p4 = 6, we get from (1.7) that T (f1, · · · , f4 )L3/2 (Rn ,w) ≤ Cf1 L6 (Rn ,ML(log L)5+δ w)

4 

fk L6 (Rn ,Mw) ,

k=2

T (f1, · · · , f4 )L3/2 (Rn ,w) ≤ Cf1 L6 (Rn ,ML(log L)2+δ w) ×f2 L6 (Rn ,ML(log L)δ w)

4 

fk L6 (Rn ,Mw) ,

k=3

T (f1, · · · , f4 )L3/2 (Rn ,w) ≤ Cf1 L6 (Rn ,ML(log L)1+δ w) ×

3 

fk L6 (Rn ,ML(log L)δ w) f4 L6 (Rn ,Mw) ,

k=2

and T (f1 , · · · , f4 )L3/2 (Rn ,w) ≤ Cf1 L6 (Rn ,ML(log L)1/2+δ w)

4 

fk L6 (Rn ,ML(log L)δ w) ,

k=2

if we choose  = 1, 2, 3, 4 respectively. Our second purpose in this paper is to establish weighted estimates similar to Theorem 1.2 for the maximal operator T ∗ . We have that Theorem 1.3 Let m ≥ 2 be an integer, T be an m-linear Calder´ on-Zygmund operator ∗ and T be the corresponding maximal operator, p1 , · · · , pm ∈ [1, ∞] and p ∈ (0, ∞) with m   1/p = 1/pk . Let  be an integer such that 1 ≤  ≤ m and r ∈ (0, ∞) with 1/r = 1/pk . 1≤k≤

k=1

(i) If min pk > 1 and r ≤ 1 or min pk = 1 and r ∈ (0, 1), then for any δ > 0, there 1≤k≤

1≤k≤

exists a positive constant C such that, for all weight w, all bounded functions f1 , · · · , fm with compact supports, T ∗ (f1 , · · · , fm )Lp,∞ (Rn ,w) ≤ C

 

fk Lpk (Rn ,ML log L w)

k=1

m 

fk Lpk (Rn ,Mw) .

(1.8)

k=+1

(ii) If r > 1 and p1 ∈ (1, ∞), then for any δ > 0, there exists a positive constant C such that, for all weight w, bounded functions f1 , · · · , fm with compact supports, T ∗ (f1 , · · · , fm )Lp,∞ (Rn ,w) ≤ Cf1 Lp1 (Rn ,M

L(log L)r +δ

×

  k=2

w)

fk Lpk (Rn ,ML(log L)1+δ w)

m  k=+1

fk Lpk (Rn ,Mw) . (1.9)

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(iii) If r > 1, sk ∈ (1, pk ] when pk < ∞ and sk = ∞ when pk = ∞ (1 ≤ k ≤ ), such that 1/sk = 1, then for any δ > 0, there exists a positive constant C such that, for all weight

1≤k≤

w, bounded functions f1 , · · · , fm with compact supports, 

T ∗(f1 , · · · , fm )Lp,∞ (Rn ,w) ≤ C

fk Lpk (Rn ,M

w) L(log L)2pk /sk −1+δ

1≤k≤,pk =sk

×



m 

fk Lpk (Rn ,ML log L w)

1≤k≤,pk =sk

fk Lpk (Rn ,Mw) .

k=+1

(1.10) Moreover, if min pk > 1, then the “norm”  · Lp,∞ (Rn ,w) in the conclusions of Theorem 1.2 1≤k≤m

can be replaced by  · Lp (Rn ,w) . We now make some conventions. Throughout this paper, we always denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. For a measurable set E, χE denotes the characteristic function of E. Given λ > 0 and a cube Q, λQ denotes the cube with the same center as Q and whose side length is λ times that of Q. For a fixed p with p ∈ [1, ∞), p denotes the dual exponent of p, namely, p = p/(p − 1).

2

Proof of Theorem 1.2

We begin with some preliminary lemmas. Lemma 2.1 Let m ≥ 2, T be an m-linear Calder´ on-Zygmund operator with kernel K in m-CZK(A, γ) for some A, γ > 0. Then, for all positive integer  with 1 ≤  < m and all bounded functions f+1 , · · · , fm with compact supports, the operator Tf+1 ,···,fm defined by Tf+1 ,···,fm (f1 , · · · , f )(x) = T (f1 , · · · , fm )(x) is a Calder´on-Zygmund operator with kernel in -CZK(A

m

k=+1

fk L∞ (Rn ) , γ).

This lemma is a combination of Lemma 3 and Theorem 2 in [7]. Lemma 2.2 Let m ≥ 2 and  be an integer such that 1 ≤  ≤ m, T be an m-linear Calder´ on-Zygmund operator, p1 ∈ [1, ∞), p2 , · · · , p ∈ [1, ∞] and p ∈ (0, ∞) with 1/p =   1/pk . Then for any δ > 0, there exists a positive constant C, such that for all weight w and k=1

all bounded functions f1 , · · · , fm with compact supports, T (f1 , · · · , fm )Lp,∞ (Rn ,w) ≤ Cf1 Lp1 (Rn ,ML(log L)p∗ −1+δ w) ×

  k=2

fk Lpk (Rn ,ML(log L)δ w)

m 

fk L∞ (Rn ) .

k=+1

Proof It is obvious for the case of p ≤ 1, Lemma 2.2 is an easy consequence of (ii) of Theorem 1.1. Thus, we only consider the case of p > 1. Without loss of generality, we may assume that p2 , · · · , p ∈ (1, ∞). Let r1 , · · · , r ∈ (1, ∞). By Lemma 2.1 and the weighted estimate with general weights for the classical Calder´on-Zygmund operator (see [14]), we know

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that, for any fixed σ > 0, there exists a positive constant C depending only on σ and r1 such that, for any weight w and bounded functions f1 , · · · , fm , T (f1 , · · · , fm )Lr1 (Rn ,w) ≤ Cf1 Lr1 (Rn ,ML(log L)r1 −1+σ w)

m 

fk L∞ (Rn ) .

k=2

Similarly, we have that T (f1 , · · · , fm )Lr2 (Rn ,w) ≤ Cf2 Lr2 (Rn ,ML(log L)r2 −1+σ w)



fk L∞ (Rn ) .

1≤k≤m,k=2

Applying the multilinear Riesz-Throin interpolation theorem (see [5, p.72]) to the last inequalities leads to that, for any s1 , s2 ∈ (1, ∞), s ∈ (1, ∞) with 1/s = 1/s1 + 1/s2 , T (f1 , · · · , fm )Ls (Rn ,w) ≤ C

2 

m 

fk Lsk (Rn ,M

L(log L)rk −1+σ

w)

k=1

fk L∞ (Rn ) ,

(2.1)

k=3

where t1 /r1 = 1/s1 , t2 /r2 = 1/s2 for some t1 , t2 ∈ [0, 1] with t1 + t2 = 1. By (2.1) and the trivial  T (f1 , · · · , fm )Lr3 (Rn ,w) ≤ Cf3 Lr3 (Rn ,ML(log L)r3 −1+σ w) fk L∞ (Rn ) , 1≤k≤m,k=3

another application of the multilinear Riesz-Throin interpolation theorem shows that, for s1 , s2 , s3 ∈ (1, ∞), T (f1 , · · · , fm )Ls (Rn ,w) ≤ C

3 

m 

fk Lsk (Rn ,M

L(log L)rk −1+σ

w)

k=1

fk L∞ (Rn ) ,

(2.2)

k=4

where t1 /r1 = 1/s1 , t2 /r2 = 1/s2 and t3 /r3 = 1/s3 , for some t1 , t2 , t3 ∈ [0, 1] and t1 +t2 +t3 = 1. Repeating the interpolation procedure as above for (2.2)  times, we finally get that, when  p1 , · · · , pm ∈ (1, ∞), p ∈ (1, ∞) such that 1/p = 1/pk , 1≤k≤

T (f1 , · · · , fm )Lp (Rn ,w) 

 

m 

fk Lpk (Rn ,M

fk L∞ (Rn ) ,

w) L(log L)rk −1+σ

k=1

where tk /rk = 1/pk , tk ∈ (0, 1) for k = 1, · · · , , and

 1≤k≤

(2.3)

k=+1

tk = 1.

Now, let δ > 0, p1 , · · · , p ∈ (1, ∞), p ∈ (1, ∞) such that 1/p =

 1≤k≤

1/pk . We can choose

σ ∈ (0, δ) such that 1 + δ − σ < p. Noting that 1/p ∈ (1/(p + δ − σ), 1/(1 + δ − σ)), we can choose t1 , t2 ∈ (0, 1) such that t1 + ( − 1)t2 = 1,

( − 1)t2 1 t1 + = . p+δ−σ 1+δ−σ p

Setting r1 = p + δ − σ, r2 = · · · = r = 1 + δ − σ in (2.3), we complete the proof of Lemma 2.2. 2 Proof of Theorem 1.2 Let f1 , · · · , fm be bounded functions with compact supports and f1 Lp1 (Rn ,M

r∗ −1+δ w) L(log L) 

= · · · = f Lp (Rn ,M

r∗ −1+δ w) L(log L) 

= f+1 Lp+1 (Rn ,Mw) = · · · = fm Lpm (Rn ,Mw) = 1.

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For each fixed λ > 0 and each k with  + 1 ≤ k ≤ m, applying the Calder´ on-Zygmund decomposition to |fk |pk at level λp , we obtain sequences of cubes {Qjk }j with disjoint interiors, such that (i) for any fixed j and k with  + 1 ≤ k ≤ m,  1 λp < j |fk (y)|pk dy ≤ 2n λp . |Qk | Qjk (ii) |fk (x)| ≤ Cλp/pk , a.e. x ∈ Rn \ Set Ω = 4nQjk and

j

Qjk .

+1≤k≤m j

gk (x) =

bk (x) =

⎧  ⎪ VQj (fk )χQj (x), ⎨ fk (x)χRn \ Qj (x) + j

⎪ ⎩ f (x), k

k

k

j

k

if pk = ∞,

⎧  j  ⎪ fk (x) − VQj (fk ) χQj (x) = bk (x), ⎨ ⎪ ⎩0

if pk < ∞,

k

j

k

if pk < ∞,

j

if pk = ∞,

where and in the sequel, for a function f and a measurable set E with |E| < ∞, VE (f ) denotes  the mean value of f on E, namely, VE (f ) = |E|−1 E f (y)dy. Recall that gk L∞ (Rn ) ≤ Cλp/pk . Thus, by Lemma 2.2, w({x ∈ Rn : |T (f1 , · · · , f , g+1 , · · · , gm )(x)| > λ}) ≤ Cλ−r ≤ Cλ−r

 

m 

fk rLpk (Rn ,M

k=1 m 

r∗ −1+δ w) L(log L) 

gk rL∞ (Rn )

k=+1

λr p/pk

k=+1 −p

≤ Cλ

.

Note that w(Ω) ≤ C

 +1≤k≤m

 w(4nQj ) k

j

|4nQjk |

|Qjk | ≤ Cλ−p .

Our proof for (1.6) is now reduced to prove that w({x ∈ Rn \Ω : |T (f1 , · · · , f , v+1 , · · · , vm )(x)| > λ}) ≤ Cλ−p ,

(2.4)

where vk ∈ {gk , bk } for k with  + 1 ≤ k ≤ m, and at least one vk = bk . We only prove (2.4) for the case vm = bm . The argument for other cases are similar and j will be omitted. For each fixed j, denote by ym and l(Qjm ) the center and side length of Qjm respectively. By the vanishing moment of bjm and the regularity condition (1.3), we see that, for x ∈ Rn \Ω,    j γ |y − ym |    m K(x; y1 , · · · , ym )bjm (ym )dym  ≤ C |bj (ym )|dy. (2.5)  |x − yk |)nm+γ m Rn Rn ( 1≤k≤m

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Note that, for any r > 0 and any σ > 0,  1 |f (y)|dy ≤ Cr−σ M f (x). (|x − y| + r)n+σ n R

(2.6)

It then follows from (2.5) and (2.6) that, when x ∈ Rn \Ω,   

(Rn )m

 ≤C

Rn  

≤C

K(x; y1 , · · · , ym )

 

fk (yk )

k=1

 (Rn )m−1

M fk (x)

k=1

  vk (yk )bjm (ym )dy1 · · · dym 

m−1  k=+1

m−1  j γ   |y − ym |  m |f (y )| |vk (yk )||bjm (ym )|dy1 · · · dym k k nm+γ ( |x − yk |) k=1

1≤k≤m

m−1 



M vk (x)

Rn

k=+1

k=+1

j γ |ym − ym | |bj (ym )|dym . |x − ym |n+γ m

This, in turn, implies that, for x ∈ Rn \Ω, |T (f1 , · · · , f , v+1 , · · · , vm−1 , bm )(x)| ≤ C

 

m−1 

M fk (x)

k=1

M vk (x)Mm (x),

k=+1

where Mm (x) is the Marcinkiewicz function defined by Mm (x) =

 {l(Qjm )}γ bjm L1 (Rn ) j

j n+γ (l(Qjm ) + |x − ym |)

.

Note that bjm L1 (Rn ) ≤ bjm Lpm (Rn ) |Qjm |1−1/pm ≤ C|Qjm |λp/pm and w({x ∈ R : Mm (x) > Cλ n

p/pm

}) ≤ C

 Rn

j

≤C

 j

≤ Cλ−p

{l(Qjm )}n+γ w(x)dx j + |x − ym |)n+γ

(l(Qjm )

|Qjm | inf M w(y) y∈Qjm



≤ Cλ−p .

Rn

|fm (x)|pm M w(x)dx (2.7)

Let w∗ (x) = w(x)χRn \Ω (x). As it was pointed out in [4, p.159] that, for any cube Qjk , sup M w∗ (y) ≤ C inf M w∗ (y).

y∈Qjk

y∈Qjk

This, via a trivial computation involving the H¨ older inequality, leads to that, for k with  + 1 ≤ k ≤ m,   pk pk gk Lpk (Rn ,Mw∗ ) ≤ C |f (x)| M w(x)dx + {VQj (|fk |)}pk |Qjk | inf j M w∗ (y) k Rn \

≤ C +C

j

 j

≤C

Qjk

Qjk

j

k

y∈Qk

|fk (x)|pk M w(x)dx (2.8)

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and bk pLkpk (Rn ,Mw∗ ) ≤ C

 j

Qjk

|fk (x)|pk M w(x)dx ≤ C.

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(2.9)

An argument involving the inequalities (2.7), (2.8), (2.9) and the estimates that M hL1,∞ (Rn ,w) ≤ ChL1 (Rn ,Mw) , M hLs(Rn ,w) ≤ ChLs (Rn ,Mw) , 1 < s < ∞

(2.10)

gives us that w({x ∈ Rn \Ω : |T (f1 , · · · , f , v+1 , · · · , vm−1 , bm )(x)| > λ}) ≤

 

w({x ∈ Rn : M fk (x) > λp/pk })

k=1 m−1 

+

w({x ∈ Rn \Ω : M vk (x) > λp/pk }) + w({x ∈ Rn : Mm (x) > Cλp/pm })

k=+1 −p

≤ Cλ

,

and then establishes (2.4). We now prove (1.7). For each fixed p1 , · · · , pm ∈ (1, ∞) and fixed δ > 0, we can choose p11 , · · · , pm+1,1 ; p12 , · · · , pm+1,2 ; · · · ; p1m , · · · , pm+1,m and δ1 , · · · , δm+1 ∈ (0, δ), such that (a) pjk ∈ (1, ∞) for any 1 ≤ k ≤ m and 1 ≤ j ≤ m + 1, (1/p1 , · · · , 1/pm , 1/p) is in the open convex hull of the points (1/p11 , · · · , 1/p1m , 1/p1 ), · · · , (1/pm+1,1 , · · · , 1/pm+1,m, 1/pm+1 ), m  where 1/pj = 1/pjk ; k=1

∗ + δj < r∗ + δ, (b) for a fixed  with 1 ≤  < m, and each fixed j with 1 ≤ j ≤ m + 1, rj   1/pjk . where rj ∈ (0, ∞) with 1/rj = k=1

We know from (1.6) that, for any j with 1 ≤ j ≤ m + 1, T (f1 , · · · , fm )Lpj ,∞ (Rn ,w) ≤ C

 

fk Lpjk (Rn ,M

r∗ −1+δj L(log L) j

k=1

≤C

 

fk Lpjk (Rn ,M

m 

r∗ −1+δ w) L(log L) 

k=1

fk Lpjk (Rn ,Mw)

w) k=+1 m 

fk Lpjk (Rn ,Mw) .

k=+1

Inequality (1.7) then follows from the multilinear Marcinkiewicz interpolation (see [5, 72–73]) and the last inequality immediately. 2

3

Proof of Theorem 1.3

We begin with some lemmas which will be used in the proof of Theorem 1.3 and have independent interest.  Lemma 3.1 Let s1 , · · · , sm ∈ [1, ∞], s ∈ (0, ∞) such that 1/s = 1/sk , R be an 1≤k≤m

m-linear operator from S(Rn ) × · · · × S(Rn ) to S  (Rn ). Suppose that there exists nonnegative constants σ1 , · · · , σm and C such that for any weight u1 , · · · , um , R(f1 , · · · , fm )Ls (Rn ,uS ) ≤ C

m  k=1

fk Lsk (Rn ,ML(log L)σk uk ) ,

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where and in the sequel, for weights u1 , · · · , um and s1 , · · · , sm ∈ ([1, ∞), uS =

m

k=1

p/pk

uk

.

Then for pk ∈ [sk , ∞) when sk < ∞ or pk = ∞ when sk = ∞, δk > σk pk /sk , 1 ≤ k ≤ m,  with 1/p = 1/pk ∈ (0, 1), there exists a positive constant C such that for any weight w1 , · · · , wm ,

1≤k≤m

R(f1 , · · · , fm )Lp (Rn ,w P ) ≤ C

 1≤k≤m,pk =sk



×

fk Lpk (Rn ,ML(log L)σk wk ) fk Lpk (Rn ,M

L(log L)pk /sk −1+δk

1≤k≤m,pk >sk

wk ) .

Proof Obviously, it suffices to consider the case max pk < ∞. We will employ some 1≤k≤m

ideas from [3]. For fixed p1 ∈ [s1 , ∞), · · · , pm ∈ [sm , ∞) and p ∈ (0, ∞) such that 1/p =  1/pk , set r = p/s and rk = pk /sk . For fixed w1 , · · · , wm and nonnegative function h, 1≤k≤m

note that p/p1

m  s/sk    1/r  1/r wk k w  P k hr /rk .

p/pm · · · wm h=

w1

k=1

A standard duality argument now tells us that  s R(f1 , · · · , fm )Lp (Rn ,w  ) = sup P

h Lr (Rn ,w 

≤C  1/rk

Note that, for pk = sk , w  P

t log−σk (2 + t) ≤



 P

≤1 )

Rn

m 

sup h Lr (Rn ,w 

|R(f1 , · · · , fm )|s h(x)w  P (x)dx

 P

≤1 k=1 )

fk s

   .  1/r  1/r Lsk Rn ,ML(log L)σk wk k w   k hr /rk P



hr /rk ≡ 1. Let ηk = δk − rk σk . Write t1/rk

log

σk +(rk −1+ηk )/rk



(2 + t)

× t1/rk log(rk −1+ηk )/rk (2 + t).

This via the generalization of H¨older inequality (see [16, p.64]) states that ML(log L)σk (h1 h2 )(x) ≤ MLrk (log L)rk −1+rk σk +ηk (h1 )(x)MLrk (log L)−1−(rk −1)ηk (h2 )(x). Thus, for each k with 1 ≤ k ≤ m and pk > sk , and each h with hLr (Rn ,w  ) ≤ 1, P       1/r 1/r |fk (x)|sk ML(log L)σk wk k w  P k hr /rk (x)dx Rn   rk 1/rk 1/r ≤C |fk (x)|pk MLrk (log L)rk −1+rk σk +ηk (wk k )(x) dx Rn    rk 1/rk   1/r  MLrk (log L)−1−(rk −1)ηk (w ×   k hr /rk )(x) dx P Rn   rk 1/rk 1/r ≤C |fk (x)|pk MLrk (log L)rk −1+rk σk +ηk (wk k )(x) dx , Rn

where, in the last inequality, we have invoked the fact that the operator MLrk (log L)−1−(rk −1)ηk 

is bounded on Lrk (Rn ) (see [15, Theorem 1.7]). On the other hand, it is easy to verify that  1/rk 1/r MLrk (log L)rk −1+rk σk +ηk (wk k )(x) ≤ C ML(log L)rk −1+δk wk (x) .

´ G.E. Hu: WEIGHTED ESTIMATES FOR CALDERON-ZYGMUND OPERATORS

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1539

Combining the above estimates yields the desired result. 2 Lemma 3.2 Let m ≥ 2, T be an m-linear Calder´ on-Zygmund operator, p1 , · · · , pm ∈  (1, ∞] and p ∈ (0, ∞) with 1/p = 1/pk . 1≤k≤m

(a) If p ∈ (0, 1], then there exists a positive constant C such that for all weights w1 , · · · , wm T ∗ (f1 , · · · , fm )Lp (Rn ,w P ) ≤ C

m 

fk Lpk (Rn ,ML log L wk ) ,

(3.1)

k=1 m

p/p  P = wk k . where and in the sequel, for weights w1 , · · · , wm and P = (p1 , · · · , pm ), we set w k=1

(b) If p ∈ (1, ∞), then, for any δ > 0, there exists a positive constant C such that for all weights w1 , · · · , wm , T ∗(f1 , · · · , fm )Lp (Rn ,w P ) ≤ C

m 

fk Lpk (Rn ,ML(log L)p+δ wk ) .

(3.2)

k=1



(c) If p ∈ (1, ∞), sk ∈ (1, pk ] when pk < ∞ or sk = ∞ when pk = ∞, such that 1/sk = 1, then for any δ > 0, there exists a positive constant C such that, for all weights

1≤k≤ w1 , · · · , wm ,

T ∗ (f1 , · · · , fm )Lp (Rn ,w P ) ≤ C



fk Lpk (Rn ,M

wk ) L(log L)2pk /sk −1+δ

1≤k≤m,sk =pk



×

fk Lpk (Rn ,ML log L wk ) .

(3.3)

1≤k≤m,sk =pk

Proof Set U (x) = {(y1 , · · · , ym ) ∈ (Rn )m : max |x − yj | > }, 1≤j≤m

and define an operator T∗ by T∗ (f1 , · · · , fm )(x) = sup |T (f1 , · · · , fm )(x)|,

(3.4)

>0

where T (f1 , · · · , fm )(x) =

 U (x)

K(x, y1 , · · · , ym )

m 

fk (yk )dy1 , · · · , dym .

(3.5)

k=1

We know from [8] that m     ∗  ∗ T (f , · · · , f )(x) − T (f , · · · , f )(x) ≤ C M fk (x).   1 m 1 m

(3.6)

k=1

Let s ∈ (0, 1/2) and M0,s be the sharp maximal operator defined by

f (x) = sup inf inf{λ > 0 : |{x ∈ Q : |f (x) − c| > λ}| < s|Q|}. M0,s Qx c∈C

This operator was introduced by Str¨ omberg [17]. As it is well known, for any fixed δ ∈ (0, 1],

f (x) ≤ Cs−δ Mδ (f )(x), M0,s

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where Mδ is the sharp maximal operator defined by  1  1/δ Mδ h(x) = sup inf |h(y) − c|δ dy . Qx c∈C |Q| Q It was shown in [11, Theorem 3.2] that, for δ ∈ (0, 1/m), the maximal operator T∗ also enjoys the sharp function estimate Mδ (T∗ (f1 , · · · , fm ))(x) ≤ C

m 

M fk (x).

k=1

On the other hand, a trivial computaion gives us that, when p ∈ (0, 1],  p



M0,s (|h|p )(x) ≤ M0,s h(x) . This, via a clever idea of Lerner [11], leads to that, for any p ∈ (0, 1] and weight u,      p

∗  T (f1 , · · · , fm )(x) u(x)dx ≤ C (T∗ (f1 , · · · , fm ))p (x)M u(x)dx M0,s Rn

Rn

 ≤C

m 

Rn

p M fk (x) M u(x)dx,

k=1

and so, by (3.6),  Rn



 p T ∗ (f1 , · · · , fm )(x) u(x)dx ≤ C

m 

Rn

p M fk (x) M u(x)dx.

(3.7)

k=1

We can now conclude the proof of Lemma 3.2. Note that, for weights w1 , · · · , wm , and p1 , · · · , pm ∈ (1, ∞], m  (M wk (x))p/pk . M (w  P )(x) ≤ C k=1

Recall that M (ML(log L)δ u)(x) ≤ CML(log L)δ+1 u(x)

(3.8)

(see Theorem 1.2 in [1]). (3.7), along with the H¨ older inequality and the inequality (2.10), leads to that, for p ∈ (0, 1], T ∗ (f1 , · · · , fm )Lp (Rn ,w P ) ≤ C

m 

M fk Lpk (Rn ,Mwk ) ≤ C

k=1

m 

fk Lpk (Rn ,ML log L wk ) .

k=1

On the other hand, by (3.7), the H¨ older inequality and (3.8), and the fact that M is bounded  p n 1−p p n from L (R , u ) to L (R , (ML(log L)p−1+δ u)1−p ) (see [15, Corollary 1.8] and the proof therein), we deduce that, if p ∈ (1, ∞), then for any δ > 0 and nonnegative function h with hLp (Rn ,w  )1−p ) ≤ 1, P

 Rn

T ∗ (f1 , · · · , fm )(x)h(x)dx ≤ C ≤C



m 

M fk (x)M h(x)dx

Rn k=1   m Rn

k=1

p M fk (x) ML(log L)p−1+δ w  P (x)dx

No.4

´ G.E. Hu: WEIGHTED ESTIMATES FOR CALDERON-ZYGMUND OPERATORS

 ×

(M h(x)) Rn m 

≤C

p



ML(log L)p−1+δ w  P

1−p

1541

dx

fk Lpk (Rn ,ML(log L)p+δ wk ) .

k=1

Our desired result (b) then follows directly. As for conclusion (c), it follows from Lemma 3.1 and conclusion (a) directly. 2 Proof of Theorem 1.3 To prove (i), we first consider the case of  = m. If min pk > 1, 1≤k≤m

(i) follows from (a) in Lemma 3.2 directly. Now let p1 , · · · , pm ∈ [1, ∞] and p ∈ (0, 1) with  1/p = 1/pk and min pk = 1. For a fixed λ > 0 and bounded functions f1 , · · · , fm with 1≤k≤m

1≤k≤m

compact support, and f1 Lp1 (Rn ,ML log L w) = · · · = fm Lpm (Rn ,ML log L w) = 1, applying the Calder´ on-Zygmund decomposition to |fk |pk at level λp , we then obtain sequences j of cubes {Qk }j , functions gk , bjk and bk , which are the same as in the proof of Theorem 1.2. m = Qjk , and w(x)  = w(x)χRn \Ω Setting Ω  (x), we also have k=1 j

 ≤ w(Ω)

m   k=1

w(Qjk ) ≤ Cλ−p .

j

Choose tk ∈ (pk , ∞), 1 ≤ k ≤ m, such that 1/t =

 1≤k≤m

1/tk ∈ (1, 1/p). As in the proof of

Theorem 1.2, we have gk L∞ (Rn ) ≤ Cλp/pk , gk Lpk (Rn ,ML log L w)  ≤ C. Thus, by (a) of Lemma 3.2,  : T ∗ (g1 , · · · , gm )(x) > λ}) w({x ∈ Rn \Ω ≤ λ−p T ∗ (g1 , , .., gm )pLp (Rn ,w) −t

≤ Cλ

≤ Cλ−t ≤ Cλ−t

m  k=1 m  k=1 m 

gk tLtk (Rn ,ML log L w)  (t −p )t/tk

k k gk L∞ (Rn )

m 

p t/t

gl Llpl (Rl n ,ML log L w) 

l=1

λpt(tk −pk )/(tk pk )

k=1 −p

≤ Cλ

.

By (3.6), the proof of (1.8) with  = m is now reduced to prove that  : T∗ (v1 , · · · , vm )(x) > λ}) ≤ Cλ−p , w({x ∈ Rn \Ω where vk ∈ {gk , bk } and at least one vk = bk .

(3.9)

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We now prove (3.9). For simplicity, we only consider the case vm = bm . For each fixed

> 0, write T (v1 , · · · , vm−1 , bm )(x)   m−1  = K(x; y1 , · · · , ym )bm (ym )dym vk (yk )dy1 · · · dym−1 (Rn )m−1

|x−ym |>



k=1



+ {(Rn )m−1 :

×

m−1 

max

1≤k≤m−1

|x−yk |>}





×

{(Rn )m−1 : m−1 

K(x; y1 , · · · , ym )bm (ym )dym

vk (yk )dy1 · · · dym−1

k=1



Rn

max

1≤k≤m−1

|x−yk |>}

|x−ym |>

K(x; y1 , · · · , ym )bm (ym )dym

vk (yk )dy1 · · · dym−1 .

k=1

This, in turn, implies that      T (v1 , · · · , vm−1 , bm )(x) ≤

(Rn )m−1

×

m−1 

  

 +

(Rn )m−1

×

Rn

|vk (yk )|dy1 · · · dym−1

k=1

m−1 

  K(x; y1 , · · · , ym )bm (ym )dym 

  

|x−ym |>

  K(x; y1 , · · · , ym )bm (ym )dym 

|vk (yk )|dy1 · · · dym−1

k=1

= I(x) + II(x). As the estimate for the classical maximal singular integral operator, it follows that, for x ∈  Rn \Ω,     K(x; y1 , · · · , ym )bm (ym )dym   |x−ym |>



 j

Rn



(

{l(Qjm )}γ



1≤k≤m−1

j |x − yk | + |x − ym |)nm+γ

+C /2≤|x−ym |≤2



 j

Rn

+C n

(

(



1≤k≤m

1≤k≤m−1

j |x − yk | + |x − ym |)nm+γ

1 m−1 

(

k=1

1 |bm (ym )|dym |x − yk |)nm

{l(Qjm )}γ



|x − yk | + )nm

|bjm (ym )|dym

M bm (x).

|bjm (ym )|dym

´ G.E. Hu: WEIGHTED ESTIMATES FOR CALDERON-ZYGMUND OPERATORS

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1543

 This, together with the estimate (2.6), leads to that, for x ∈ Rn \Ω, II(x) ≤ C

m−1 

  M vk (x) Mm (x) + M bm (x) .

k=1

 On the other hand, as in the proof of Theorem 1.2, we know that, for x ∈ Rn \Ω, I(x) ≤ C

m−1 

M vk (x)Mm (x).

k=1

 Therefore, for x ∈ Rn \Ω, T∗ (v1 , · · · vm−1 , bm )(x) ≤ C

m−1 

  M vk (x) Mm (x) + M bm (x) .

(3.10)

k=1

We thus obtain from (2.7), (2.8), (2.9), (2.10) and (3.10) that  : T∗ (v1 , · · · , vm−1 , bm )(x) > λ}) ≤ Cλ−p , w({x ∈ Rn \Ω and then establish (3.9). We now prove inequality (1.8) for the case of 1 ≤  < m. As we have proved for (i) with  = m, we see that, when r ≤ 1 and min pk > 1, or r < 1 and min pk = 1, 1≤k≤

 

T ∗ (f1 , · · · , fm )Lr ,∞ (Rn ,w) ≤ C

1≤k≤

fk Lpk (Rn ,ML log L w)

k=1

m 

fk L∞ (Rn ) .

k=+1

This, via an argument involving the Calderon-Zygmund decomposition as in the proof of (3.9), leads to desired result for (1.8) when 1 ≤  < m. To prove the conclusion (ii), we will employ the Cotlar inequality for the multilinear Calder´ on-Zygmund operator (see [8, Theorem 1]), which states that for any σ ∈ (0, 1/m), there exists a positive constant C such that for bounded functions f1 , · · · , fm with compact supports, m    T ∗ (f1 , · · · , fm )(x) ≤ CMσ T (f1 , · · · , fm ) (x) + C M fk (x). k=1

It is well known that the operator Mσ for σ ∈ (0, 1/m) is bounded from Lp,∞ (Rn , w) to Lp,∞ (Rn , M w). This together with Lemma 2.2 implies that, for any  with 1 ≤  ≤ m, T ∗(f1 , · · · , fm )Lp,∞ (Rn ,w) ≤ Cf1 Lp1 (Rn ,M

r∗ +δ w) L(log L) 

 

×

fk Lpk (Rn ,ML(log L)1+δ w)

k=2

m 

fk L∞ (Rn ) ,

(3.11)

k=+1

provided that p1 ∈ (1, ∞) and p2 , · · · , pm ∈ (1, ∞]. Using (3.11) and employing some argument used in the proof of (1.8), we can obtain (1.9). As for inequality (1.10), note that, when r > 1, sk ∈ (1, pk ] when pk < ∞ and sk = ∞  when pk = ∞ (1 ≤ k ≤ ), such that 1/sk = 1, it follows from (3.3) that 1≤k≤



T (f1 , · · · , fm )Lr (Rn ,w) ≤ C



1≤k≤,pk =sk

×

 1≤k≤,pk =sk

fk Lpk (Rn ,M

w) L(log L)2pk /sk −1+δ

fk Lpk (Rn ,ML log L w)

m  k=+1

fk L∞ (Rn ) ,

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which, along with an argument involving the Calder´on-Zygmund decomposition as in the proof of conclusion (i), leads to (1.10). Finally, when min pk > 1, using the multilinear Marcinkiewicz interpolation theorem as 1≤k≤m

in the proof of Theorem 1.2, we see that  · Lp,∞ (Rn ,w) in (1.8), (1.9) and (1.10) can be replaced by Lp (Rn , w). We omit the details for brevity. 2 References [1] Carrozza M, Passarelli Di Napoli A. Composition of maximal operators. Publ Mat, 1996, 40: 397–409 [2] Coifman R R, Meyer Y. On commutators of singular integrals and bilinear singular integrals. Trans Amer Math Soc, 1975, 212: 315–331 [3] Cruz-Uribe SFO D, P´erez C. Two weight extrapolation via the maxiaml operator. J Funct Anal, 2000, 174: 1–17 [4] Garc´ıa-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics. Amsterdam: North-Holland Publishing Co, 1985 [5] Grafakos L. Classical Fourier Analysis. 2nd ed. New York: Springer, 2008 [6] Grafakos L. Multilinear Calder´ on-Zygmund singular integral operators: background and recent developments. To appear [7] Grafakos L, Torres R H. Multilinear Calder´ on-Zygmund theory. Adv Math, 2002, 165: 124–164 [8] Grafakos L, Torres R H. Maximal operators and weighted norm inequalities for multilinear singular integrals. Indiana Univ Math J, 2002, 51: 1261–1276 [9] Hu G. Weighted norm inequalities for the multilinear Calder´ on-Zygmund operators. Sci China Math, 2010, 53: 1863–1876 [10] Hu G, Zhu Y. Weighted norm inequalities for the commutators of multilinear singular integral operators. Acta Math Sci, 2011, 31B(3): 749–764 [11] Lerner A K. Weighted norm inequalities for the local sharp maximal function. J Fourier Anal Appl, 2004, 10: 645–674 [12] Lerner A K, Ombrosi S, P´erez C, Torres R H, Trujillo-Gonz´ alez R. New maximal functions and multiple weights for the multilinear Calder´ on-Zygmund theory. Adv Math, 2009, 220: 1222–1264 [13] Li W, Xue Q, Yabuta K. Multilinear Calderon-Zygmund operators on weighted Hardy spaces. Studia Math, 2010, 199: 1–16 [14] P´ erez C. Weighted norm inequalities for singular integral operators. J London Math Soc, 1994, 49: 296–308 [15] P´ erez C. On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp -spaces with different weights. Proc London Math Soc, 1995, 49: 135–157 [16] Rao M, Ren Z. Theory of Orlicz Spaces. New York: Marcel Dekker Inc, 1991 [17] Str¨ omberg J O. Bounded mean oscillation with Orlicz norm and duality of Hardy spaces. Indiana Univ Math J, 1979, 28: 511–544