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