arXiv:1303.4480v1 [math.CA] 19 Mar 2013
Multilinear singular and fractional integral
operators on weighted Morrey spaces
Hua Wang
Department of Mathematics,
Zhejiang University, Hangzhou 310027, P. R. China
E-mail address: wanghua@pku.edu.cn.
Wentan Yi
Department of Applied Mathematics,
Zhengzhou Information Science and Technology Institute,
Zhengzhou 450002, P. R. China
E-mail address: nlwt89@sina.com.
Abstract
In this paper, we will study the boundedness properties of multilinear Calderón–Zygmund operators and multilinear fractional integrals on
products of weighted Morrey spaces with multiple weights.
MSC(2010): 42B20; 42B35
Keywords: Multilinear Calderón–Zygmund operators; multilinear fractional integrals; weighted Morrey spaces; multiple weights
1
Introduction and main results
Multinear Calderón–Zygmund theory is a natural generalization of the linear
case. The initial work on the class of multilinear Calderón–Zygmund operators
was done by Coifman and Meyer in [4], and was later systematically studied by
Grafakos and Torres in [12–14]. Let Rn be the n-dimensional Euclidean space
and (Rn )m = Rn × · · ·× Rn be the m-fold product space (m ∈ N). We denote by
S (Rn ) the space of all Schwartz functions on Rn and by S ′ (Rn ) its dual space,
the set of all tempered distributions on Rn . Let m ≥ 2 and T be an m-linear
operator initially defined on the m-fold product of Schwartz spaces and taking
values into the space of tempered distributions,
T : S (Rn ) × · · · × S (Rn ) → S ′ (Rn ).
Following [12], for given f~ = (f1 , . . . , fm ), we say that T is an m-linear Calderón–
Zygmund
operator if for some q1 , . . . , qm ∈ [1, ∞) and q ∈ (0, ∞) with 1/q =
Pm
n
q1
1/q
k , it extends to a bounded multilinear operator from L (R ) × · · · ×
k=1
1
Lqm (Rn ) into Lq (Rn ), and if there exists a kernel function K(x, y1 , . . . , ym ) in
the class m-CZK(A, ε), defined away from the diagonal x = y1 = · · · = ym in
(Rn )m+1 such that
Z
T (f~)(x) = T (f1 , . . . , fm )(x) =
K(x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym ) dy1 · · · dym ,
(Rn )m
(1.1)
whenever f1 , . . . , fm ∈ S (Rn ) and x ∈
/ ∩m
k=1 supp fk . We say that K(x, y1 , . . . , ym )
is a kernel in the class m-CZK(A, ε), if it satisfies the size condition
A
K(x, y1 , . . . , ym ) ≤
,
(|x − y1 | + · · · + |x − ym |)mn
for some A > 0 and all (x, y1 , . . . , ym ) ∈ (Rn )m+1 with x 6= yk for some 1 ≤ k ≤
m. Moreover, for some ε > 0, it satisfies the regularity condition that
A · |x − x′ |ε
K(x, y1 , . . . , ym ) − K(x′ , y1 , . . . , ym ) ≤
(|x − y1 | + · · · + |x − ym |)mn+ε
whenever |x − x′ | ≤ 12 max1≤k≤m |x − yk |, and also that for each fixed k with
1 ≤ k ≤ m,
A · |yk − yk′ |ε
K(x, y1 , . . . , yk , . . . , ym )−K(x, y1 , . . . , y ′ , . . . , ym ) ≤
k
(|x − y1 | + · · · + |x − ym |)mn+ε
whenever |yk − yk′ | ≤ 21 max1≤i≤m |x − yi |. In recent years, many authors have
been interested in studying the boundedness of these operators on function
spaces, see e.g.[11, 15, 22, 23]. In 2009, the weighted strong and weak type estimates of multilinear Calderón–Zygmund singular integral operators were established in [21] by Lerner et al. New more refined multilinear maximal function
was defined and used in [21] to characterize the class of multiple AP~ weights.
Theorem A ([21]). Let m ≥ 2 and T be an m-linear Calderón–Zygmund
opPm
erator. If p1 , . . . , pm ∈ (1, ∞) and p ∈ (0, ∞) with 1/p =
k=1 1/pk , and
w
~ = (w1 , . . . , wm ) satisfy the AP~ condition, then there exists a constant C > 0
independent of f~ = (f1 , . . . , fm ) such that
T (f~) p
L (ν
w
~)
where νw~ =
Qm
i=1
p/pi
wi
≤C
m
Y
fi p
,
L i (w )
i
i=1
.
Theorem B ([21]). Let m ≥ 2 and T be an m-linear Calderón–Zygmund
operator.
P If p1 , . . . , pm ∈ [1, ∞), min{p1 , . . . , pm } = 1 and p ∈ (0, ∞) with
1/p = m
~ = (w1 , . . . , wm ) satisfy the AP~ condition, then there
k=1 1/pk , and w
exists a constant C > 0 independent of f~ = (f1 , . . . , fm ) such that
T (f~) p
W L (ν
w
~)
where νw~ =
Qm
i=1
p/pi
wi
≤C
m
Y
fi Lpi (wi )
i=1
.
2
,
Let m ≥ 2 and 0 < α < mn. For given f~ = (f1 , . . . , fm ), the m-linear
fractional integral operator is defined by
Z
f1 (y1 ) · · · fm (ym )
Iα (f~)(x) = Iα (f1 , . . . , fm )(x) =
dy1 · · · dym .
mn−α
(Rn )m |(x − y1 , . . . , x − ym )|
(1.2)
For the boundedness properties of multilinear fractional integrals on various
function spaces, we refer the reader to [10, 16–19, 29, 30]. In 2009, Moen [24]
considered the weighted norm inequalities for multilinear fractional integral operators and constructed the class of multiple AP~ ,q weights (see also [2]).
Theorem C ([2,24]). Let m ≥ 2, 0 < α < mn P
and Iα be an m-linear fractional
m
integral operator. If p1 , . . . , pm ∈ (1, ∞), 1/p = k=1 1/pk and 1/q = 1/p−α/n,
and w
~ = (w1 , . . . , wm ) satisfy the AP~ ,q condition, then there exists a constant
C > 0 independent of f~ = (f1 , . . . , fm ) such that
Iα (f~) q
L ((ν
w
~)
where νw~ =
Qm
i=1
q)
≤C
m
Y
fi p pi ,
L i (w )
i
i=1
wi .
Theorem D ([2, 24]). Let m ≥ 2, 0 < α < mn and Iα be an m-linear fractional
Pm integral operator. If p1 , . . . , pm ∈ [1, ∞), min{p1 , . . . , pm } = 1, 1/p =
~ = (w1 , . . . , wm ) satisfy the AP~ ,q conk=1 1/pk and 1/q = 1/p − α/n, and w
dition, then there exists a constant C > 0 independent of f~ = (f1 , . . . , fm ) such
that
m
Y
fi p pi ,
Iα (f~) q
q ≤ C
L
W L ((νw
~) )
i=1
where νw~ =
Qm
i=1
i (w
i
)
wi .
On the other hand, the classical Morrey spaces Lp,λ were originally introduced by Morrey in [25] to study the local behavior of solutions to second
order elliptic partial differential equations. For the boundedness of the Hardy–
Littlewood maximal operator, the fractional integral operator and the Calderón–
Zygmund singular integral operator on these spaces, we refer the reader to
[1,3,28]. For the properties and applications of classical Morrey spaces, one can
see [6–8] and the references therein.
In 2009, Komori and Shirai [20] first defined the weighted Morrey spaces
Lp,κ (w) which could be viewed as an extension of weighted Lebesgue spaces, and
studied the boundedness of the above classical operators in Harmonic Analysis
on these weighted spaces. Recently, in [31–38], we have established the continuity properties of some other operators and their commutators on the weighted
Morrey spaces Lp,κ (w).
The main purpose of this paper is to establish the boundedness properties of
multilinear Calderón–Zygmund operators and multilinear fractional integrals on
products of weighted Morrey spaces with multiple weights. We now formulate
our main results as follows.
3
Theorem 1.1. Let m ≥ 2 and T be an m-linear Calderón–Zygmund
operPm
1/p
,
ator. If p1 , . . . , pm ∈ (1, ∞) and p ∈ (0, ∞) with 1/p =
k and
k=1
w
~ = (w1 , . . . , wm ) ∈ AP~ with w1 , . . . , wm ∈ A∞ , then for any 0 < κ < 1,
there exists a constant C > 0 independent of f~ = (f1 , . . . , fm ) such that
T (f~) p,κ
L
(ν
w
~)
where νw~ =
Qm
i=1
p/pi
wi
≤C
m
Y
fi p ,κ
,
L i (wi )
i=1
.
Theorem 1.2. Let m ≥ 2 and T be an m-linear Calderón–Zygmund operator.PIf p1 , . . . , pm ∈ [1, ∞), min{p1 , . . . , pm } = 1 and p ∈ (0, ∞) with
1/p = m
~ = (w1 , . . . , wm ) ∈ AP~ with w1 , . . . , wm ∈ A∞ , then for
k=1 1/pk , and w
any 0 < κ < 1, there exists a constant C > 0 independent of f~ = (f1 , . . . , fm )
such that
m
Y
fi p ,κ
T (f~) p,κ
,
≤C
WL
L
(νw
~)
i
(wi )
i=1
where νw~ =
Qm
i=1
p/pi
wi
.
Theorem 1.3. Let m ≥ 2, 0 < α < mn and Iα
an m-linear fractional
Pbe
m
integral operator. P
If p1 , . . . , pm ∈ (1, ∞), 1/p =
k=1 1/pk , 1/qk = 1/pk −
m
α/mn and 1/q = k=1 1/qk = 1/p − α/n, and w
~ = (w1 , . . . , wm ) ∈ AP~ ,q with
q1
qm
w1 , . . . , wm ∈ A∞ , then for any 0 < κ < p/q, there exists a constant C > 0
independent of f~ = (f1 , . . . , fm ) such that
Iα (f~) q,κq/p
L
((ν
w
~)
where νw~ =
Qm
i=1
q)
≤C
m
Y
fi p ,κp q/pq pi qi ,
i (w
L i i
,w )
i=1
wi .
i
i
Theorem 1.4. Let m ≥ 2, 0 < α < mn and Iα be an m-linear fractional
Pm integral operator. If p1 , . . . , pm ∈ [1,
∞),
min{p
,
.
.
.
,
p
}
=
1,
1/p
=
1
m
k=1 1/pk ,
Pm
1/qk = 1/pk −α/mn and 1/q = k=1 1/qk = 1/p−α/n, and w
~ = (w1 , . . . , wm ) ∈
qm
AP~ ,q with w1q1 , . . . , wm
∈ A∞ , then for any 0 < κ < p/q, there exists a constant
C > 0 independent of f~ = (f1 , . . . , fm ) such that
Iα (f~) q,κq/p
WL
((ν
w
~)
where νw~ =
2
Qm
i=1
wi .
q)
≤C
m
Y
fi p ,κp q/pq pi qi ,
i
i (w
L i
,w )
i=1
i
i
Notations and definitions
The classical Ap weight theory was first introduced by Muckenhoupt in the study
of weighted Lp boundedness of Hardy–Littlewood maximal functions in [26]. A
4
weight w is a nonnegative, locally integrable function on Rn , B = B(x0 , rB )
denotes the ball with the center x0 and radius rB . For 1 < p < ∞, a weight
function w is said to belong to Ap , if there is a constant C > 0 such that for
every ball B ⊆ Rn ,
p−1
Z
Z
1
1
−1/(p−1)
w(x) dx
w(x)
dx
≤ C,
(2.1)
|B| B
|B| B
where |B| denotes the Lebesgue measure of B. For the case p = 1, w ∈ A1 , if
there is a constant C > 0 such that for every ball B ⊆ Rn ,
Z
1
w(x) dx ≤ C · ess inf w(x).
(2.2)
x∈B
|B| B
A weight function w ∈ A∞ if it satisfies the Ap condition for some 1 < p <
∞. We also need another weight class Ap,q introduced by Muckenhoupt and
Wheeden in [27]. A weight function w belongs to Ap,q for 1 < p < q < ∞ if
there is a constant C > 0 such that for every ball B ⊆ Rn ,
1/q 1/p′
Z
Z
1
1
q
−p′
w(x) dx
w(x)
dx
≤ C.
(2.3)
|B| B
|B| B
When p = 1, w is in the class A1,q with 1 < q < ∞ if there is a constant C > 0
such that for every ball B ⊆ Rn ,
1/q Z
1
1
q
≤ C.
(2.4)
w(x) dx
ess sup
|B| B
x∈B w(x)
Now let us recall the definitions of multiple weights. For m exponents
~
p1 , . . . , pm , we will write P~ for the vector
Pm P = (p1 , . . . , pm ). Let p1 , . . . , pm ∈
~ = (w1 , . . . , wm ), set
[1, ∞) and p ∈ (0, ∞) with 1/p = k=1 1/pk . Given w
Qm
p/pi
~ satisfies the AP~ condition if it satisfies
νw~ = i=1 wi . We say that w
sup
B
1
|B|
1/p Y
1/p′i
Z
m 1
1−p′i
νw~ (x) dx
wi (x)
dx
< ∞.
|B| B
B
i=1
Z
(2.5)
1/p′i
−1
R
1
1−p′i
When pi = 1, |B|
dx
is understood as inf x∈B wi (x)
.
B wi (x)
Pm
Let p1Q
, . . . , pm ∈ [1, ∞), 1/p = k=1 1/pk and q > 0. Given w
~ = (w1 , . . . , wm ),
m
~ satisfies the AP~ ,q condition if it satisfies
set νw~ = i=1 wi . We say that w
sup
B
1
|B|
1/p′i
1/q Y
Z
m ′
1
νw~ (x)q dx
wi (x)−pi dx
< ∞.
|B| B
B
i=1
Z
(2.6)
1/p′i
−1
R
′
1
w (x)−pi dx
is understood as inf x∈B wi (x)
.
When pi = 1, |B|
B i
Given a ball B and λ > 0, λB denotes the ball with the same center as
B whose radius is λ times that of B. For a given weight function w and a
measurable set E, we also denote the Lebesgue Rmeasure of E by |E| and the
weighted measure of E by w(E), where w(E) = E w(x) dx.
5
Lemma 2.1 ([9]). Let w ∈ Ap with 1 ≤ p < ∞. Then, for any ball B, there
exists an absolute constant C > 0 such that
w(2B) ≤ C w(B).
Lemma 2.2 ([5]). Let w ∈ A∞ . Then for all balls B ⊆ Rn , the following
reverse Jensen inequality holds.
Z
Z
1
w(x) dx ≤ C|B| · exp
log w(x) dx .
|B| B
B
Lemma 2.3 ([9]). Let w ∈ A∞ . Then for all balls B and all measurable subsets
E of B, there exists δ > 0 such that
δ
w(E)
|E|
.
≤C
w(B)
|B|
Pm
Lemma 2.4 ([21]). Let p1 , . . . , pm ∈ [1, ∞) and 1/p = k=1 1/pk . Then w
~ =
(w1 , . . . , wm ) ∈ AP~ if and only if
(
νw~ ∈ Amp ,
1−p′i
wi
where νw~ =
p/pi
and
i=1 wi
1/m
∈ A1 .
wi
Qm
understood as
∈ Amp′i ,
i = 1, . . . , m,
1−p′i
the condition wi
∈ Amp′i in the case pi = 1 is
Lemma 2.5 ([2, 24]). Let 0 < α < mn, p1 , . . . , pm ∈ [1, ∞), 1/p =
and 1/q = 1/p − α/n. Then w
~ = (w1 , . . . , wm ) ∈ AP~ ,q if and only if
(
where νw~ =
Qm
i=1
Pm
k=1
1/pk
(νw~ )q ∈ Amq ,
−p′i
wi
∈ Amp′i ,
i = 1, . . . , m,
wi .
Given a weight function w on Rn , for 0 < p < ∞, the weighted Lebesgue
space Lp (w) defined as the set of all functions f such that
f p
=
L (w)
Z
1/p
|f (x)| w(x) dx
< ∞.
p
Rn
(2.7)
We also denote by W Lp (w) the weighted weak space consisting of all measurable
functions f such that
1/p
f p
< ∞.
(2.8)
= sup λ · w x ∈ Rn : |f (x)| > λ
W L (w)
λ>0
In 2009, Komori and Shirai [20] first defined the weighted Morrey spaces
Lp,κ (w) for 1 ≤ p < ∞. In order to deal with the multilinear case m ≥ 2, we
shall define Lp,κ (w) for all 0 < p < ∞.
6
Definition 2.6. Let 0 < p < ∞, 0 < κ < 1 and w be a weight function on Rn .
Then the weighted Morrey space is defined by
Lp,κ (w) = f ∈ Lploc (w) : f Lp,κ (w) < ∞ ,
where
f p,κ
= sup
L
(w)
B
1
w(B)κ
Z
1/p
|f (x)| w(x) dx
p
B
and the supremum is taken over all balls B in Rn .
Definition 2.7. Let 0 < p < ∞, 0 < κ < 1 and w be a weight function on Rn .
Then the weighted weak Morrey space is defined by
W Lp,κ (w) = f measurable : f W Lp,κ (w) < ∞ ,
where
1/p
1
λ · w x ∈ B : |f (x)| > λ
.
κ/p
w(B)
λ>0
f p,κ
= sup sup
WL
(w)
B
Furthermore, in order to deal with the fractional order case, we need to
consider the weighted Morrey spaces with two weights.
Definition 2.8. Let 0 < p < ∞ and 0 < κ < 1. Then for two weights u and v,
the weighted Morrey space is defined by
Lp,κ (u, v) = f ∈ Lploc (u) : f Lp,κ (u,v) < ∞ ,
where
f p,κ
= sup
L
(u,v)
B
1
v(B)κ
Z
B
1/p
|f (x)|p u(x) dx
.
Throughout this article, we will use C to denote a positive constant, which
is independent of the main parameters and not necessarily the same at each
occurrence. Moreover, we will denote the conjugate exponent of p > 1 by
p′ = p/(p − 1).
3
Proofs of Theorems 1.1 and 1.2
Before proving the main theorems of this section, we need to establish the
following lemma.
Lemma 3.1. Let m ≥ 2, p1 , . . . , pm ∈ [1, ∞) and p ∈ (0, ∞) with 1/p =
Pm
Qm
p/pi
, then for any
~ =
k=1 1/pk . Assume that w1 , . . . , wm ∈ A∞ and νw
i=1 wi
ball B, there exists a constant C > 0 such that
m Z
Y
i=1
B
p/pi
Z
νw~ (x) dx.
≤C
wi (x) dx
B
7
Proof. Since w1 , . . . , wm ∈ A∞ , then by using Lemma 2.2, we have
p/pi
p/pi
Z
m Y
1
|B| · exp
≤C
wi (x) dx
log wi (x) dx
|B| B
B
i=1
Z
m Y
1
p/pi
p/pi
· exp
=C
|B|
dx
log wi (x)
|B| B
i=1
!
Z
m
X
Pm
1
p/p
i
= C · |B| i=1
· exp
log wi (x)p/pi dx .
|B| B
i=1
m Z
Y
i=1
Qm
Pm
Note that i=1 p/pi = 1 and νw~ (x) = i=1 wi (x)p/pi . Then by Jensen inequality, we obtain
m Z
Y
i=1
p/pi
Z
1
log νw~ (x) dx
≤ C · |B| · exp
wi (x) dx
|B| B
B
Z
≤C
νw~ (x) dx.
B
We are done.
Proof of Theorem 1.1. For any ball B = B(x0 , rB ) ⊆ Rn and let fi = fi0 + fi∞ ,
where fi0 = fi χ2B , i = 1, . . . , m and χ2B denotes the characteristic function of
2B. Then we write
m
Y
fi (yi ) =
m Y
i=1
i=1
=
fi0 (yi ) + fi∞ (yi )
X
αm
f1α1 (y1 ) · · · fm
(ym )
α1 ,...,αm ∈{0,∞}
=
m
Y
i=1
fi0 (yi ) +
X′
αm
f1α1 (y1 ) · · · fm
(ym ),
P′
where each term of
contains at least one αi 6= 0. Since T is an m-linear
operator, then we have
1
νw~ (B)κ/p
Z
B
1/p
T (f1 , . . . , fm )(x)p νw~ (x) dx
1/p
p
1
0
T (f10 , . . . , fm
νw~ (x) dx
)(x)
νw~ (B)κ/p
B
Z
1/p
X′
p
1
αm
T (f α1 , . . . , fm
νw~ (x) dx
+
)(x)
1
νw~ (B)κ/p
B
X′
I α1 ,...,αm .
=I 0 +
≤
Z
8
In view of Lemma 2.4, we have that νw~ ∈ Amp . Applying Theorem A, Lemma
3.1 and Lemma 2.1, we get
1/pi
m Z
Y
1
0
pi
I ≤C·
|fi (x)| wi (x) dx
νw~ (B)κ/p i=1 2B
Qm
m
κ/pi
Y
i=1 wi (2B)
fi p ,κ
·
≤C
L i (wi )
κ/p
νw~ (B)
i=1
m
Y
νw~ (2B)κ/p
fi p ,κ
·
≤C
i
L
(wi )
νw~ (B)κ/p
i=1
≤C
m
Y
fi p ,κ
.
L i (w )
i
i=1
For the other terms, let us first consider the case when α1 = · · · = αm = ∞.
By the size condition, for any x ∈ B, we obtain
Z
|f1 (y1 ) · · · fm (ym )|
∞
T (f1∞ , . . . , fm
)(x) ≤ C
dy1 · · · dym
(|x
−
y1 | + · · · + |x − ym |)mn
n
m
m
(R ) \(2B)
∞ Z
X
|f1 (y1 ) · · · fm (ym )|
dy1 · · · dym
≤C
(|x
−
y1 | + · · · + |x − ym |)mn
j+1
m
j
m
B) \(2 B)
j=1 (2
∞ Y
m Z
X
|fi (yi )|
≤C
dyi
n
j+1 B\2j B |x − yi |
j=1 i=1 2
Z
∞ Y
m
X
1
fi (yi ) dyi ,
(3.1)
≤C
j+1
|2 B| 2j+1 B
j=1 i=1
where we have used the notation E m = E × · · · × E. Furthermore, by using
Hölder’s inequality, the multiple AP~ condition and Lemma 3.1, we deduce that
∞ Y
m
X
∞
T (f1∞ , . . . , fm
)(x) ≤ C
j=1 i=1
1
|2j+1 B|
Z
2j+1 B
1
fi (yi )pi wi (yi ) dyi
Pm
1/pi Z
′
wi (yi )1−pi dyi
2j+1 B
1
m + i=1 (1− p ) Y
κ/pi i
|2j+1 B| p
1
j+1
fi p ,κ
2
B
w
·
L i (wi ) i
|2j+1 B|m
νw~ (2j+1 B)1/p
i=1
j=1
∞ Qm
m
j+1
X
Y
B)κ/pi
i=1 wi (2
fi Lpi ,κ (wi ) ·
≤C
νw~ (2j+1 B)1/p
j=1
i=1
≤C
≤C
∞
X
∞
m
X
Y
(κ−1)/p
fi p ,κ
νw~ 2j+1 B
.
·
L i (wi )
j=1
i=1
Since νw~ ∈ Amp ⊂ A∞ , then it follows directly from Lemma 2.3 that
δ
νw~ (B)
|B|
.
≤C
νw~ (2j+1 B)
|2j+1 B|
9
(3.2)
1/p′i
Hence
∞
I ∞,...,∞ ≤ νw~ (B)(1−κ)/p T (f1∞ , . . . , fm
)(x)
∞
m
X
Y
νw~ (B)(1−κ)/p
fi p ,κ
·
≤C
L i (wi )
ν ~ (2j+1 B)(1−κ)/p
j=1 w
i=1
δ(1−κ)/p
∞ m
X
Y
|B|
fi p ,κ
·
≤C
L i (wi )
|2j+1 B|
j=1
i=1
≤C
m
Y
fi p ,κ
,
L i (wi )
i=1
where the last inequality holds since 0 < κ < 1 and δ > 0. We now consider
the case where exactly ℓ of the αi are ∞ for some 1 ≤ ℓ < m. We only give
the arguments for one of these cases. The rest are similar and can easily be
obtained from the arguments below by permuting the indices. Using the size
condition again, we deduce that for any x ∈ B,
Z
Z
|f1 (y1 ) · · · fm (ym )|
T (f ∞ , . . . , f ∞ , f 0 , . . . , f 0 )(x) ≤ C
dy1 · · · dym
ℓ+1
m
ℓ
1
mn
(Rn )ℓ \(2B)ℓ (2B)m−ℓ (|x − y1 | + · · · + |x − ym |)
m Z
Y
fi (yi ) dyi
≤C
×
i=ℓ+1
∞
X
1
|2j+1 B|m
j=1
≤C
≤C
2B
m Z
Y
i=ℓ+1 2B
∞ Y
m
X
Z
(2j+1 B)ℓ \(2j B)ℓ
∞
X
fi (yi ) dyi ×
1
|2j+1 B|
j=1 i=1
j=1
Z
2j+1 B
f1 (y1 ) · · · fℓ (yℓ ) dy1 · · · dyℓ
ℓ Z
Y
1
fi (yi ) dyi
|2j+1 B|m i=1 2j+1 B\2j B
fi (yi ) dyi , (3.3)
and we arrived at the expression considered in the previous case. So for any
x ∈ B, we also have
m
Y
0
0
fi T (f1∞ , . . . , fℓ∞ , fℓ+1
, . . . , fm )(x) ≤ C
Lpi ,κ (wi )
i=1
·
∞
X
j=1
νw~ 2j+1 B
(κ−1)/p
.
Therefore, by the inequality (3.2) and the above pointwise inequality, we have
0
0
, . . . , fm
)(x)
I α1 ,...,αm ≤ νw~ (B)(1−κ)/p T (f1∞ , . . . , fℓ∞ , fℓ+1
∞
m
X
Y
νw~ (B)(1−κ)/p
fi p ,κ
·
≤C
L i (wi )
ν ~ (2j+1 B)(1−κ)/p
j=1 w
i=1
10
≤C
≤C
∞ m
X
Y
fi p ,κ
·
L i (wi )
i=1
m
Y
i=1
j=1
|B|
|2j+1 B|
δ(1−κ)/p
fi p ,κ
.
L i (w )
i
Combining the above estimates and then taking the supremum over all balls
B ⊆ Rn , we complete the proof of Theorem 1.1.
Proof of Theorem 1.2. For any ball B = B(x0 , rB ) ⊆ Rn and decompose fi =
fi0 + fi∞ , where fi0 = fi χ2B , i = 1, . . . , m. Then for any given λ > 0, we can
write
1/p
νw~ x ∈ B : T (f1 , . . . , fm ) > λ
1/p
1/p X′
αm 0 ) > λ/2m
+
νw~ x ∈ B : T (f1α1 , . . . , fm
) > λ/2m
≤νw~ x ∈ B : T (f10 , . . . , fm
X′
I∗α1 ,...,αm ,
=I∗0 +
P′
where each term of
contains at least one αi 6= 0. By Lemma 2.4 again, we
know that νw~ ∈ Amp with 1 ≤ mp < ∞. Applying Theorem B, Lemma 3.1 and
Lemma 2.1, we have
1/pi
m Z
CY
I∗0 ≤
|fi (x)|pi wi (x) dx
λ i=1 2B
Qm
m
C · i=1 wi (2B)κ/pi Y fi p ,κ
≤
L i (wi )
λ
i=1
≤
≤
m
C · νw~ (2B)κ/p Y fi p ,κ
L i (wi )
λ
i=1
m
C · νw~ (B)κ/p Y fi p ,κ
.
L i (wi )
λ
i=1
In the proof of Theorem 1.1, we have already showed the following pointwise
estimate (see (3.1) and (3.3)).
Z
∞ Y
m
X
1
αm
≤C
T (f α1 , . . . , fm
fi (yi ) dyi .
)(x)
(3.4)
1
j+1
|2 B| 2j+1 B
j=1 i=1
Without loss of generality, we may assume that p1 = · · · = pℓ = min{p1 , . . . , pm } =
1, and pℓ+1 , . . . , pm > 1. Using Hölder’s inequality, the multiple AP~ condition
and Lemma 3.1, we obtain
Z
Z
m
∞ Y
ℓ
Y
X
1
1
T (f α1 , . . . , f αm )(x) ≤ C
fi (yi ) dyi ×
fi (yi ) dyi
m
1
j+1
j+1
|2 B| 2j+1 B
|2 B| 2j+1 B
j=1 i=1
i=ℓ+1
≤C
∞ Y
ℓ
X
j=1 i=1
1
|2j+1 B|
11
Z
2j+1 B
fi (yi )wi (yi ) dyi
inf
yi ∈2j+1 B
−1
wi (yi )
×
m
Y
i=ℓ+1
m
Y
≤C
i=1
1
|2j+1 B|
Z
2j+1 B
fi (yi )pi wi (yi ) dyi
1/pi Z
′
wi (yi )1−pi dyi
2j+1 B
1/p′i
∞
X
(κ−1)/p
fi p ,κ
νw~ 2j+1 B
.
L i (w )
i
j=1
Observe that νw~ ∈ Amp with 1 ≤ mp < ∞. Thus, it follows from the inequality
(3.2) that for any x ∈ B,
m
Y
αm
fi p ,κ
T (f α1 , . . . , fm
=C
·
)(x)
1
L i (wi )
∞
X
1
νw~ (B)(1−κ)/p
(1−κ)/p
νw~ (B)
ν ~ (2j+1 B)(1−κ)/p
i=1
j=1 w
δ(1−κ)/p
∞ m
X
Y
|B|
1
fi p ,κ
·
≤C
L i (wi ) ν (B)(1−κ)/p
|2j+1 B|
w
~
j=1
i=1
≤C
m
Y
fi p ,κ
·
L i (wi )
1
.
νw~ (B)(1−κ)/p
i=1
(3.5)
αm
If x ∈ B : T (f1α1 , . . . , fm
)(x) > λ/2m = Ø, then the inequality
m
C · νw~ (B)κ/p Y fi p ,κ
L i (wi )
λ
i=1
αm
holds trivially. Now if instead we suppose that x ∈ B : T (f1α1 , . . . , fm
)(x) >
λ/2m 6= Ø, then by the pointwise inequality (3.5), we have
I∗α1 ,...,αm ≤
λ<C
m
Y
fi Lpi ,κ (wi )
i=1
which is equivalent to
νw~ (B)1/p ≤
Therefore
·
1
,
νw~ (B)(1−κ)/p
m
C · νw~ (B)κ/p Y fi p ,κ
.
L i (wi )
λ
i=1
I∗α1 ,...,αm ≤ νw~ (B)1/p ≤
m
C · νw~ (B)κ/p Y fi p ,κ
.
L i (wi )
λ
i=1
Summing up all the above estimates and then taking the supremum over all
balls B ⊆ Rn and all λ > 0, we complete the proof of Theorem 1.2.
By using Hölder’s inequality, it is easy to check that if each wi is in Api ,
then
m
Y
Api ⊂ AP~ .
i=1
and this inclusion is strict (see [21]). Thus, as direct consequences of Theorems
1.1 and 1.2, we immediately obtain the following
12
Corollary 3.2. Let m ≥ 2 and T be an m-linear Calderón–Zygmund
operPm
1/p
,
ator. If p1 , . . . , pm ∈Q(1, ∞) and p ∈ (0, ∞) with 1/p =
k and
k=1
,
then
for
any
0
<
κ
<
1,
there
exists
a conw
~ = (w1 , . . . , wm ) ∈ m
A
i=1 pi
~
stant C > 0 independent of f = (f1 , . . . , fm ) such that
T (f~) p,κ
L
(ν
w
~)
where νw~ =
Qm
i=1
p/pi
wi
≤C
m
Y
fi p ,κ
,
L i (wi )
i=1
.
Corollary 3.3. Let m ≥ 2 and T be an m-linear Calderón–Zygmund operator.PIf p1 , . . . , pm ∈ [1, ∞), min{p1 , . . .Q
, pm } = 1 and p ∈ (0, ∞) with
m
m
~ = (w1 , . . . , wm ) ∈ i=1 Api , then for any 0 < κ < 1,
1/p = k=1 1/pk , and w
there exists a constant C > 0 independent of f~ = (f1 , . . . , fm ) such that
T (f~) p,κ
WL
(ν
w
~)
where νw~ =
4
Qm
i=1
p/pi
wi
≤C
m
Y
fi Lpi ,κ (wi )
,
i=1
.
Proofs of Theorems 1.3 and 1.4
Following along the same lines as that of Lemma 3.1, we can also show the
following result, which plays an important role in our proofs of Theorems 1.3
and 1.4.
Lemma
4.1. Let m ≥ 2, q1 , . . . , qm ∈ [1, ∞) and q Q
∈ (0, ∞) with 1/q =
Pm
m
q1
qm
~ =
i=1 wi , then for any
k=1 1/qk . Assume that w1 , . . . , wm ∈ A∞ and νw
ball B, there exists a constant C > 0 such that
q/qi
Z
m Z
Y
qi
νw~ (x)q dx.
≤C
wi (x) dx
i=1
B
B
Proof of Theorem 1.3. Arguing as in the proof of Theorem 1.1, fix a ball B =
B(x0 , rB ) ⊆ Rn and decompose fi = fi0 + fi∞ , where fi0 = fi χ2B , i = 1, . . . , m.
Since Iα is an m-linear operator, then we have
Z
1/q
q
1
q
I
(f
,
.
.
.
,
f
)(x)
ν
(x)
dx
α 1
m
w
~
q
κ/p
νw
B
~ (B)
Z
1/q
q
1
0
Iα (f10 , . . . , fm
νw~ (x)q dx
≤ q
)(x)
νw~ (B)κ/p
B
Z
1/q
X′
q
1
α1
q
αm
+
I
(f
)(x)
ν
(x)
dx
,
.
.
.
,
f
α
w
~
q
m
1
κ/p
νw
B
~ (B)
X′
J α1 ,...,αm ,
=J 0 +
13
P′
where each term of
contains at least one αi 6= 0. In view of Lemma 2.5, we
can see that (νw~ )q ∈ Amq . Using Theorem C, Lemma 4.1 and Lemma 2.1, we
get
1/pi
m Z
Y
1
pi
pi
J ≤C· q
|fi (x)| wi (x) dx
νw~ (B)κ/p i=1 2B
Qm
m
Y
wqi (2B)κq/pqi
fi Lpi ,κpi q/pqi (wpi ,wqi ) · i=1 q i κ/p
≤C
i
i
νw~ (B)
i=1
κ/p
Q
m
qi
q/qi
m
Y
i=1 wi (2B)
fi p ,κp q/pq pi qi ·
=C
q
i
i (w
L i
κ/p
i ,wi )
νw
~ (B)
i=1
0
≤C
≤C
m
q
κ/p
Y
fi p ,κp q/pq pi qi · νw~q(2B)
i
i
i
L
(wi ,wi )
νw~ (B)κ/p
i=1
m
Y
fi p ,κp q/pq pi qi .
i
i (w
L i
,w )
i
i=1
i
For the other terms, let us first deal with the case when α1 = · · · = αm = ∞.
By the definition of Iα , for any x ∈ B, we obtain
Z
|f1 (y1 ) · · · fm (ym )|
Iα (f ∞ , . . . , f ∞ )(x) =
dy1 · · · dym
1
m
mn−α
(|x
−
y
n
m
m
1 | + · · · + |x − ym |)
(R ) \(2B)
∞ Z
X
|f1 (y1 ) · · · fm (ym )|
dy1 · · · dym
=
mn−α
j+1 B)m \(2j B)m (|x − y1 | + · · · + |x − ym |)
j=1 (2
m Z
∞ Y
X
|fi (yi )|
≤C
dyi
n−α/m
j+1 B\2j B |x − yi |
j=1 i=1 2
Z
∞ Y
m
X
1
fi (yi ) dyi .
≤C
(4.1)
j+1
1−α/mn
|2 B|
2j+1 B
j=1 i=1
Moreover, by using Hölder’s inequality, the multiple AP~ ,q condition and Lemma
4.1, we deduce that
m
∞ Y
X
∞
Iα (f1∞ , . . . , fm
)(x) ≤ C
Z
1/pi
1
fi (yi )pi wi (yi )pi dyi
|2j+1 B|1−α/mn
2j+1 B
j=1 i=1
1/p′i
Z
′
×
wi (yi )−pi dyi
2j+1 B
≤C
∞
X
j=1
1
+
Pm
(1− p1 )
i=1
1
|2j+1 B| q
·
q
|2j+1 B|m−α/n
νw~ (2j+1 B)1/q
14
i
×
m Y
fi p ,κp q/pq pi qi wqi 2j+1 B κq/pqi
i
i (w
L i
,w ) i
i
i=1
i
∞
m
X
Y
fi p ,κp q/pq pi qi ·
≤C
i (w ,w )
L i i
≤C
i=1
m
Y
i=1
i
i
fi p ,κp q/pq pi qi ·
i (w ,w )
L i i
i
i
j=1
∞
X
" Qm
qi j+1
B)q/qi
i=1 wi (2
q
j+1 B)1/q
νw
~ (2
q
j+1
B
νw
~ 2
j=1
κ/p−1/q
κ/p #
.
Since (νw~ )q ∈ Amq ⊂ A∞ , then it follows immediately from Lemma 2.3 that
q
νw
~ (B)
≤C
q
j+1 B)
νw
(2
~
|B|
|2j+1 B|
δ′
.
(4.2)
Hence
q
1/q−κ/p ∞
J ∞,...,∞ ≤ νw
Iα (f1∞ , . . . , fm
)(x)
~ (B)
∞
m
q
1/q−κ/p
X
Y
νw
~ (B)
fi p ,κp q/pq pi qi ·
≤C
q
i
i (w
L i
,w
)
i
i
ν ~ (2j+1 B)1/q−κ/p
j=1 w
i=1
δ′ (1/q−κ/p)
∞ m
X
Y
|B|
fi p ,κp q/pq pi qi ·
≤C
i
i (w
L i
i ,wi )
|2j+1 B|
j=1
i=1
≤C
m
Y
fi p ,κp q/pq pi qi ,
i
i (w
L i
,w )
i
i=1
i
where in the last inequality we have used the fact that 0 < κ < p/q and δ ′ > 0.
We now consider the case where exactly ℓ of the αi are ∞ for some 1 ≤ ℓ < m.
We only give the arguments for one of these cases. The rest are similar and can
easily be obtained from the arguments below by permuting the indices. Using
the definition of Iα again, we can see that for any x ∈ B,
Z
Z
|f1 (y1 ) · · · fm (ym )|
0
0
Iα (f1∞ , . . . , fℓ∞ , fℓ+1
dy1 · · · dym
, . . . , fm )(x) =
mn−α
(|x
−
y
n
ℓ
ℓ
m−ℓ
1 | + · · · + |x − ym |)
(R ) \(2B)
(2B)
m Z
Y
fi (yi ) dyi
≤C
×
i=ℓ+1
∞
X
j=1
≤C
≤C
2B
1
j+1
|2 B|m−α/n
m Z
Y
i=ℓ+1 2B
∞ Y
m
X
j=1 i=1
15
Z
(2j+1 B)ℓ \(2j B)ℓ
∞
X
fi (yi ) dyi ×
j=1
1
|2j+1 B|1−α/mn
Z
f1 (y1 ) · · · fℓ (yℓ ) dy1 · · · dyℓ
ℓ Z
Y
1
fi (yi ) dyi
j+1
m−α/n
|2 B|
j+1 B\2j B
i=1 2
2j+1 B
fi (yi ) dyi ,
(4.3)
and we arrived at the expression considered in the previous case. Thus, for any
x ∈ B, we also have
∞
m
X
Y
κ/p−1/q
q
j+1
fi p ,κp q/pq pi qi ·
Iα (f ∞ , . . . , f ∞ , f 0 , . . . , f 0 )(x) ≤ C
B
.
νw
1
ℓ
ℓ+1
m
~ 2
i
i (w
L i
,w )
i
i=1
i
j=1
Therefore, by the inequality (4.2) and the above pointwise inequality, we obtain
0
0
, . . . , fm
)(x)
J α1 ,...,αm ≤ ν q (B)1/q−κ/p Iα (f1∞ , . . . , fℓ∞ , fℓ+1
w
~
∞
m
X
Y
fi p ,κp q/pq pi qi ·
i
i (w
L i
,w )
q
1/q−κ/p
νw
~ (B)
q
j+1
i
i
ν ~ (2 B)1/q−κ/p
j=1 w
i=1
δ′ (1/q−κ/p)
∞ m
X
Y
|B|
fi p ,κp q/pq pi qi ·
≤C
i
i (w
L i
i ,wi )
|2j+1 B|
j=1
i=1
≤C
≤C
m
Y
fi p ,κp q/pq pi qi .
i
i (w
L i
,w )
i=1
i
i
Summarizing the estimates derived above and then taking the supremum over
all balls B ⊆ Rn , we finish the proof of Theorem 1.3.
Proof of Theorem 1.4. As before, fix a ball B = B(x0 , rB ) ⊆ Rn and split fi
into fi = fi0 + fi∞ , where fi0 = fi χ2B , i = 1, . . . , m. Then for each fixed λ > 0,
we can write
1/q
q x ∈ B : Iα (f1 , . . . , fm ) > λ
νw
~
1/q
1/q X′ q q αm 0 ) > λ/2m
+
νw~ x ∈ B : Iα (f1α1 , . . . , fm
x ∈ B : Iα (f10 , . . . , fm
) > λ/2m
≤νw
~
X′
J∗α1 ,...,αm ,
=J∗0 +
P′
where each term of
contains at least one αi 6= 0. By Lemma 2.5 again, we
know that (νw~ )q ∈ Amq with 1 < mq < ∞. Using Theorem D, Lemma 4.1 and
Lemma 2.1, we have
1/pi
m Z
CY
|fi (x)|pi wi (x)pi dx
λ i=1 2B
Qm
m
C · i=1 wiqi (2B)κq/pqi Y fi p ,κp q/pq pi qi
≤
i
i (w
L i
i ,wi )
λ
i=1
J∗0 ≤
≤
≤
m
q
κ/p Y
C · νw
~ (2B)
fi p ,κp q/pq pi qi
i
i (w
L i
i ,wi )
λ
i=1
m
q
κ/p Y
C · νw
~ (B)
fi p ,κp q/pq pi qi .
i (w
L i i
i ,wi )
λ
i=1
16
In the proof of Theorem 1.3, we have already proved the following pointwise
estimate (see (4.1) and (4.3)).
Z
∞ Y
m
X
1
αm
Iα (f α1 , . . . , fm
≤C
fi (yi ) dyi . (4.4)
)(x)
1
j+1
1−α/mn
|2 B|
2j+1 B
j=1 i=1
Without loss of generality, we may assume that p1 = · · · = pℓ = min{p1 , . . . , pm } =
1, and pℓ+1 , . . . , pm > 1. By using Hölder’s inequality, the multiple AP~ ,q condition and Lemma 4.1, we obtain
∞ Y
ℓ
X
αm
Iα (f α1 , . . . , fm
)(x) ≤ C
1
Z
1
fi (yi ) dyi
|2j+1 B|1−α/mn 2j+1 B
j=1 i=1
Z
m
Y
1
fi (yi ) dyi
×
j+1
1−α/mn
|2 B|
2j+1 B
i=ℓ+1
ℓ
∞ Y
X
−1
Z
1
fi (yi )wi (yi ) dyi
inf
w
(y
)
i i
yi ∈2j+1 B
|2j+1 B|1−α/mn 2j+1 B
j=1 i=1
1/p′i
Z
1/pi Z
m
Y
pi
′
1
−p
p
i
fi (yi ) wi (yi ) dyi
wi (yi ) i dyi
×
|2j+1 B|1−α/mn
2j+1 B
2j+1 B
i=ℓ+1
≤C
≤C
m
Y
fi i=1
p
q
Lpi ,κpi q/pqi (wi i ,wi i )
·
∞
X
q
j+1
B
νw
~ 2
j=1
κ/p−1/q
.
Note that (νw~ )q ∈ Amq with 1 < mq < ∞. Hence, it follows from the inequality
(4.2) that for any x ∈ B,
∞
q
1/q−κ/p
X
νw
1
~ (B)
q
1/q−κ/p
i
i
νw
ν q~ (2j+1 B)1/q−κ/p
~ (B)
j=1 w
i=1
δ′ (1/q−κ/p)
m
∞ Y
X
1
|B|
fi Lpi ,κpi q/pqi (wpi ,wqi ) · q
≤C
i
i
νw~ (B)1/q−κ/p j=1 |2j+1 B|
i=1
m
Y
fi p ,κp q/pq pi qi ·
Iα (f α1 , . . . , f αm )(x) = C
m
1
i
i (w
L i
,w )
≤C
m
Y
fi i=1
p
q
Lpi ,κpi q/pqi (wi i ,wi i )
·
1
.
q
1/q−κ/p
νw
(B)
~
αm
If x ∈ B : Iα (f1α1 , . . . , fm
)(x) > λ/2m = Ø, then the inequality
(4.5)
m
q
κ/p Y
C · νw
~ (B)
fi p ,κp q/pq pi qi
i
i (w
L i
i ,wi )
λ
i=1
αm
holds trivially. Now if instead we assume that x ∈ B : Iα (f1α1 , . . . , fm
)(x) >
λ/2m 6= Ø, then by the pointwise inequality (4.5), we get
J∗α1 ,...,αm ≤
λ<C
m
Y
fi p ,κp q/pq pi qi ·
i
i (w
L i
,w )
i
i=1
17
i
1
,
q
1/q−κ/p
νw
~ (B)
which in turn gives that
q
1/q
νw
~ (B)
Therefore
m
q
κ/p Y
C · νw
~ (B)
fi p ,κp q/pq pi qi .
≤
i (w ,w )
L i i
i
i
λ
i=1
m
q
κ/p Y
C · νw
~ (B)
fi p ,κp q/pq pi qi .
i
i (w
L i
i ,wi )
λ
i=1
q
1/q
≤
J∗α1 ,...,αm ≤ νw
~ (B)
Collecting all the above estimates and then taking the supremum over all balls
B ⊆ Rn and all λ > 0, we conclude the proof of Theorem 1.4.
PmBy using Hölder’s inequality, it is easy to verify that if 1 ≤ pi < qi , 1/q =
k=1 1/qk and each wi is in Api ,qi , then we have
m
Y
Api ,qi ⊂ AP~ ,q .
i=1
and this inclusion is strict (see [24]).Also recall that w ∈ Ap,q if and only if wq ∈
A1+q/p′ ⊂ A∞ (see [27]). Thus, as straightforward consequences of Theorems
1.3 and 1.4, we finally obtain the following
Corollary 4.2. Let m ≥ 2, 0 < α < mn and
PmIα be an m-linear fractional
integral operator.
If
p
,
.
.
.
,
p
∈
(1,
∞),
1/p
=
1
m
k=1 1/pk , 1/qk = 1/p
Qmk −α/mn
Pm
1/q
=
1/p
−
α/n,
and
w
~
=
(w
,
.
.
.
,
w
)
∈
and 1/q =
k
1
m
i=1 Api ,qi ,
k=1
then for any 0 < κ < p/q, there exists a constant C > 0 independent of f~ =
(f1 , . . . , fm ) such that
Iα (f~) q,κq/p
L
((ν
w
~
where νw~ =
Qm
i=1
)q )
≤C
m
Y
fi p ,κp q/pq pi qi ,
i (w
L i i
,w )
i=1
wi .
i
i
Corollary 4.3. Let m ≥ 2, 0 < α < mn and Iα be an m-linear fractional
inPm
tegral operator. If p1 , . . . , pm ∈ P
[1, ∞), min{p1 , . . . , pm } = 1, 1/p = k=1 1/pk ,
m
1/q
~ = (w1 , . . . , wm ) ∈
Qmk = 1/pk −α/mn and 1/q = k=1 1/qk = 1/p−α/n, and w
,
then
for
any
0
<
κ
<
p/q,
there
exists
a
constant
C > 0 indepenA
p
,q
i
i
i=1
dent of f~ = (f1 , . . . , fm ) such that
Iα (f~) q,κq/p
WL
((ν
w
~
where νw~ =
Qm
i=1
wi .
)q )
≤C
m
Y
fi p ,κp q/pq pi qi ,
i
i (w
L i
,w )
i=1
18
i
i
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