WEIGHTED ENDPOINT ESTIMATES FOR MULTILINEAR LITTLEWOOD-PALEY
OPERATORS
LIU LANZHE
Abstract. In this paper, we prove weighted endpoint estimates for multilinear Littlewood-Paley operators.
1.
Introduction and Results
Let ψ be a fixed function on Rn which satisfies the following properties:
R
(1) ψ(x)dx = 0,
(2) |ψ(x)| ≤ C(1 + |x|)−(n+1) ,
(3) |ψ(x + y) − ψ(x)| ≤ C|y|(1 + |x|)−(n+2) when 2|y| < |x|;
Let m be a positive integer and A be a function on Rn . The multilinear Littlewood-Paley operator is defined by
"Z Z
#1/2
nµ
t
A
A
2 dydt
gµ (f )(x) =
|Ft (f )(x, y)| n+1
, µ > 1,
n+1
t + |x − y|
t
R+
where
FtA (f )(x, y) =
Z
Rn
Rm+1 (A; x, z)
ψt (y − z)f (z)dz,
|x − z|m
Received June 11, 2003.
2000 Mathematics Subject Classification. Primary 42B20, 42B25.
Key words and phrases. Multilinear operator, Littlewood-Paley operator, Hardy space, BMO space.
Supported by the NNSF (Grant: 10271071).
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Rm+1 (A; x, y) = A(x) −
X 1
Dα A(y)(x − y)α ,
α!
|α|≤m
and ψt (x) = t
−n
ψ(x/t) for t > 0. We denote by Ft (f )(y) = f ∗ ψt (y). We also define
!1/2
nµ
Z Z
t
dydt
gµ (f )(x) =
|Ft (f )(y)|2 n+1
,
n+1
t + |x − y|
t
R+
which is the Littlewood-Paley operator
(see [17]).
R R
1/2
2
n+1
Let H be the Hilbert space H = h : ||h|| =
<∞
. Then for each fixed x ∈ Rn ,
n+1 |h(t)| dydt/t
R
+
FtA (f )(x, y) may be viewed as a mapping from (0, +∞) to H, and it is clear that
nµ/2
t
gµA (f )(x) = FtA (f )(x, y) ,
t + |x − y|
nµ/2
t
gµ (f )(x) = Ft (f )(y) .
t + |x − y|
We also consider the variant of gµA , which is defined by
g̃µA (f )(x) =
"Z Z
n+1
R+
t
t + |x − y|
nµ
dydt
|F̃tA (f )(x, y)|2 n+1
t
#1/2
,
µ > 1,
where
F̃tA (f )(x, y) =
Z
Rn
Qm+1 (A; x, z)
ψt (y − z)f (z)dz
|x − z|m
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and
Qm+1 (A; x, z) = Rm (A; x, z) −
X
Dα A(x)(x − z)α .
|α|=m
Note that when m = 0, gµA is just the commutator of Littlewood-Paley operator (see [1], [14], [15]). It is well
known that multilinear operators, as an extension of commutators, are of great interest in harmonic analysis and
have been widely studied by many authors (see [4] – [8], [12], [13]). In [11], [16], the endpoint boundedness
properties of commutators generated by the Calderon-Zygmund operator and BMO functions are obtained. The
main purpose of this paper is to study the weighted endpoint boundedness of the multilinear Littlewood-Paley
operators. Throughout this paper, M (f ) will denote the Hardy-Littlewood maximal function of f , Q will denote
n
a cube of Rn Rwith side parallel to the
Q and any locally
R axes. For a cube
R integral function f on R , we denote
−1
#
−1
that f (Q) = Q f (x)dx, fQ = |Q|
f (x)dx and f (x) = sup |Q|
|f (y) − fQ |dy. Moreover, for a weight
Q
Q
x∈Q
functions w ∈ A1 (see [10]), f is said to belong BMO(w) if f # ∈ L∞ (w) and define ||f ||BMO(w) = ||f # ||L∞ (w) ,
if w = 1, we denote that BMO(Rn ) = BMO(w). Also, we give the concepts of atomic and weighted H 1 space.
1
A
on Q, ||a||L∞ (w) ≤ w(Q)−1 and
R function a is called a H (w) atom if there exists a cube Q such that a is supported
a(x)dx = 0. It is well known that, for w ∈ A1 , the weighted Hardy space H 1 (w) has the atomic decomposition
characterization (see [2]).
We shall prove the following theorems in Section 3.
Theorem 1. Let Dα A ∈ BMO(Rn ) for |α| = m and w ∈ A1 . Then gµA is bounded from L∞ (w) to BMO(w).
Theorem 2. Let Dα A ∈ BMO(Rn ) for |α| = m and w ∈ A1 . Then g̃µA is bounded from H 1 (w) to L1 (w).
Theorem 3. Let Dα A ∈ BMO(Rn ) for |α| = m and w ∈ A1 . Then gµA is bounded from H 1 (w) to weak L1 (w).
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Theorem 4. Let Dα A ∈ BMO(Rn ) for |α| = m and w ∈ A1 .
(i) If for any H 1 (w)-atom a supported on certain cube Q and u ∈ 3Q \ 2Q, there is
nµ/2X
Z
α Z
1
(x
−
u)
t
α
w(x)dx ≤ C,
ψ
(y
−
z)D
A(z)a(z)dz
t
m
α! |x − u|
(4Q)c t + |x − y|
Q
|α|=m
gµA
1
then
is bounded from H (w) to L1 (w);
(ii) If for any cube Q and u ∈ 3Q \ 2Q, there is
nµ/2 X
Z Z
α
1
1
t
(u − z)
α
α
w(x)dx
(D
A(x)
−
(D
A)
)
·
ψ
(y
−
z)f
(z)dz
Q
t
m
w(Q) Q t + |x − y|
α!
c |u − z|
(4Q)
|α|=m
≤ C||f ||L∞ (w) ,
then g̃µA is bounded from L∞ (w) to BMO(w).
Remark. In general, gµA is not bounded from H 1 (w) to L1 (w).
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2.
Some Lemmas
We begin with two preliminary lemmas.
Lemma 1. (see [7].) Let A be a function on Rn and Dα A ∈ Lq (Rn ) for |α| = m and some q > n. Then
!1/q
Z
X
1
m
α
q
|Rm (A : x, y)| ≤ C|x − y|
|D A(z)| dz
,
|Q̃(x, y)| Q̃(x,y)
|α|=m
√
where Q̃(x, y) is the cube centered at x and having side length 5 n|x − y|.
Lemma 2. Let w ∈ A1 , 1 < p < ∞ 1 < r ≤ ∞, 1/q = 1/p + 1/r and Dα A ∈ BMO(Rn ) for |α| = m. Then
gµA is bounded from Lp (w) to Lq (w), that is
X
||gµA (f )||Lq (w) ≤ C
||Dα A||BMO ||f ||Lp (w) .
|α|=m
Proof. By Minkowski inequality and the condition of ψ, we have
gµA (f )(x)
!1/2
nµ
Z
Z
|f (z)||Rm+1 (A; x, z)|
t
dydt
≤
|ψt (y − z)|2
dz
n+1
|x − z|m
t + |x − y|
t1+n
Rn
R+
Z ∞ Z
nµ
1/2
Z
|f (z)||Rm+1 (A; x, z)|
t−2n
t
dydt
≤ C
·
dz
2n+2
|x − z|m
t + |x − y|
t1+n
Rn
0
Rn (1 + |y − z|/t)
Z ∞ nµ
1/2
Z
Z t
dy
|f (z)||Rm+1 (A; x, z)|
−n
t
·
tdt
dz,
≤ C
|x − z|m
t + |x − y|
(t + |y − z|)2n+2
0
Rn
Rn
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noting that
Z
t−n
Rn
t
t + |x − y|
nµ
dy
≤ CM
(t + |y − z|)2n+2
and
Z
0
∞
1
(t + | · −z|)2n+2
(x) ≤ C
1
(t + |x − z|)2n+2
tdt
= C|x − z|−2n ,
(t + |x − z|)2n+2
we obtain
gµA (f )(x)
Z
≤ C
Rn
Z
= C
Rn
thus, the lemma follows from [8] [9].
|f (z)|
|Rm+1 (A; x, z)|
|x − z|m
|f (z)||Rm+1 (A; x, z)|
dz,
|x − z|m+n
Z
0
∞
tdt
(t + |x − z|)2n+2
1/2
dz
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3.
Proof of Theorems
Proof of Theorem 1. It is only to prove that there exists a constant CQ such that
Z
1
|g A (f )(x) − CQ |w(x)dx ≤ C||f ||L∞ (w)
w(Q) Q µ
P
√
holds for any cube Q. Fix a cube Q = Q(x0 , l). Let Q̃ = 5 nQ and Ã(x) = A(x) −
1
α
α
α! (D A)Q̃ x ,
|α|=m
= FtA (f1 ) + FtA (f2 )
then Rm (A; x, y) = Rm (Ã; x, y) and Dα Ã = Dα A − (Dα A)Q̃ for |α| = m. We write FtA (f )
for f1 = f χQ̃ and f2 = f χRn \Q̃ , then
Z
1
|g A (f )(x) − gµA (f2 )(x0 )|w(x)dx
w(Q) Q µ
Z 1
t
t
nµ/2 A
nµ/2 A
=
||(
)
Ft (f )(x, y)|| − ||(
)
Ft (f2 )(x0 , y)|| w(x)dx
w(Q) Q t + |x − y|
t + |x0 − y|
nµ/2
Z
Z 1
1
t
A
≤
g (f1 )(x)w(x)dx +
FtA (f2 )(x, y)
w(Q) Q µ
w(Q) Q t + |x − y|
nµ/2
t
FtA (f2 )(x0 , y) w(x)dx
−
t + |x0 − y|
:= I(x) + II(x).
Now, let us estimate I and II. First, by the L∞ boundedness of gµA (Lemma 2), we gain
I ≤ ||gµA (f1 )||L∞ (w) ≤ C||f ||L∞ (w) .
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To estimate II, we write
t
t + |x − y|
FtA (f2 )(x, y)
nµ/2
t
−
FtA (f2 )(x0 , y)
t + |x0 − y|
1
−
ψt (y − z)Rm (Ã; x, z)f2 (z)dz
|x0 − z|m
nµ/2 Z t
1
t + |x − y|
|x − z|m
nµ/2 Z
t
ψt (y − z)f2 (z)
+
[Rm (Ã; x, z) − Rm (Ã; x0 , z)]dz
t + |x − y|
|x0 − z|m
"
nµ/2 nµ/2 #
Z t
t
ψt (y − z)Rm (Ã; x0 , z)f2 (z)
+
−
dz
t + |x − y|
t + |x0 − y|
|x0 − z|m
X 1 Z t
(x − z)α
(
)nµ/2
−
α!
t + |x − y|
|x − z|m
|α|=m

nµ/2
α
t
(x0 − z) 
−
ψt (y − z)Dα Ã(z)f2 (z)dz
t + |x0 − y|
|x0 − z|m
=
nµ/2
:= II1t (x) + II2t (x) + II3t (x) + II4t (x),
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Note that |x − z| ∼ |x0 − z| for x ∈ Q̃ and z ∈ Rn \ Q̃, similar to the proof of Lemma 2 and by Lemma 1,
we have
Z
1
||II1t (x)||w(x)dx
w(Q) Q
!
Z
Z
C
|x − x0 ||f (z)|
≤
|Rm (Ã; x, z)|dz w(x)dx
n+m+1
w(Q) Q
Rn \Q̃ |x − z|
!
Z
∞ Z
X
C
|x − x0 ||f (z)|
≤
|Rm (Ã; x, z)|dz w(x)dx
n+m+1
w(Q) Q
k+1 Q̃\2k Q̃ |x − z|
k=0 2
Z
∞
X
kl(2k l)m X
α
≤ C
||D A||BMO (
|f (z)|dz)
(2k l)n+m+1
2k+1 Q̃
|α|=m
k=0
≤ C
X
||Dα A||BMO ||f ||L∞ (w)
|α|=m
For
II2t (x),
∞
X
k2−k ≤ C
X
||Dα A||BMO ||f ||L∞ (w) ;
|α|=m
k=0
by the formula (see [7]):
Rm (Ã; x, z) − Rm (Ã; x0 , z) = Rm (Ã; x, x0 ) +
X
0<|β|<m
1
Rm−|β| (Dβ Ã; x0 , z)(x − x0 )β
β!
and Lemma 1, we get
|Rm (Ã; x, z) − Rm (Ã; x0 , z)| ≤ C
X
|α|=m
||Dα A||BMO (|x − x0 |m +
X
|x0 − z|m−|β| |x − x0 ||β| ),
0<|β|<m
thus, for x ∈ Q,
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||II2t (x)||
Z
≤ C
|f2 (z)|
|Rm (Ã; x, z) − Rm (Ã; x0 , z)|dz
m+n
Rn |x − z|
Z |x − x |m + P
m−|β|
X
|x − x0 ||β|
0
0<|β|<m |x0 − z|
α
≤ C
|D A||BMO
|f2 (z)|dz
|x0 − z|m+n
Rn
|α|=m
≤ C
X
|α|=m
≤ C
X
|α|=m
≤ C
X
kDα A||BMO
∞
X
k=0
klm
k
(2 l)m+n
||Dα A||BMO ||f ||L∞ (w)
∞
X
Z
|f (z)|dz
2k+1 Q̃
k2−km
k=1
||Dα A||BMO ||f ||L∞ (w) ;
|α|=m
II3t (x),
For
by the inequality: a1/2 − b1/2 ≤ (a − b)1/2 for a ≥ b > 0, we obtain, similar to the estimate of
Lemma 2 and II1 ,
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||II3t (x)||
Z
≤ C
Z
n+1
R+
Rn
Z
≤ C
Rn
Z
≤ C
Rn
!1/2
2
tnµ/2 |x − x0 |1/2 |ψt (y − z)||f2 (z)|
dydt
|Rm (Ã; x0 , z)|
dz
tn+1
(t + |x − y|)(nµ+1)/2 |x0 − z|m
|f2 (z)||x − x0 |1/2 |Rm (Ã; x0 , z)|
|x0 − z|m
!1/2
nµ+1
Z Z
t
t−n dydt
·
dz
n+1
t + |x − y|
(t + |y − z|)2n+2
R+
Z ∞
1/2
dt
|f2 (z)||x − x0 |1/2 |Rm (Ã; x0 , z)|
dz
|x0 − z|m
(t + |x − z|)2n+2
0
|f2 (z)||x − x0 |1/2 |Rm (Ã; x0 , z)|
dz
|x0 − z|m+n+1/2
Rn
Z
∞
X
kl1/2 (2k l)m X
α
≤ C
||D
A||
(
|f (z)|dz)
BMO
(2k l)n+m+1/2
2k+1 Q̃
k=0
Z
≤ C
|α|=m
≤ C
X
|α|=m
≤ C
X
||Dα A||BMO ||f ||L∞ (w)
∞
X
k2−k/2
k=0
||Dα A||BMO ||f ||L∞ (w) ;
|α|=m
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For II4t (x), similar to the estimates of II1t (x) and II3t (x), we have
X
Z
|x − x0 |
|x − x0 |1/2
t
||II4 (x)|| ≤ C
+
|Dα Ã(z)||f (z)|dz
|x − z|n+1
|x − z|n+1/2
Rn \Q̃
|α|=m
≤ C
X
||Dα A||BMO ||f ||L∞ (w)
|α|=m
≤ C
X
∞
X
k(2−k + 2−k/2 )
k=0
α
||D A||BMO ||f ||L∞ (w) .
|α|=m
Combining these estimates, we complete the proof of Theorem 1.
Proof of Theorem 2. It suffices to show that there exists a constant C R> 0 such that for every H 1 (w)-atom a
(that is that a satisfies: supp a ⊂ Q = Q(x0 , r), ||a||L∞ (w) ≤ w(Q)−1 and a(y)dy = 0 (see [8])), we have
||g̃µA (a)||L1 (w) ≤ C.
We write
Z
Rn
g̃µA (a)(x)w(x)dx
"Z
=
#
Z
g̃µA (a)(x)w(x)dx := J + JJ.
+
|x−x0 |≤2r
|x−x0 |>2r
For J, by the following equality
Qm+1 (A; x, y) = Rm+1 (A; x, y) −
X 1
(x − y)α (Dα A(x) − Dα A(y)),
α!
|α|=m
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we have, similar to the proof of Lemma 2,
g̃µA (a)(x) ≤ gµA (a)(x) + C
X Z |Dα A(x) − Dα A(y)|
|a(y)|dy,
|x − y|n
|α|=m
thus, g̃µA is L∞ -bounded by Lemma 2 and [3]. We see that
J ≤ C||g̃µA (a)||L∞ (w) w(2Q) ≤ C||a||L∞ (w) w(Q) ≤ C.
P
1
To obtain the estimate of JJ, we denote that Ã(x) = A(x) − |α|=m α!
(Dα A)2B xα . Then Qm (A; x, y) =
P
1
Qm (Ã; x, y). We write, by the vanishing moment of a and Qm+1 (A; x, y) = Rm (A; x, y) − |α|=m α!
(x −
y)α Dα A(x), for x ∈ (2Q)c ,
F̃tA (a)(x, y)
Z
X 1 Z ψt (y − z)Dα Ã(z)(x − z)α
ψt (y − z)Rm (Ã; x, z)
=
a(z)dz −
a(z)dz
|x − z|m
α!
|x − z|m
|α|=m
#
Z "
ψt (y − z)Rm (Ã; x, z) ψt (y − x0 )Rm (Ã; x, x0 )
−
a(z)dz
=
|x − z|m
|x − x0 |m
X 1 Z ψt (y − z)(x − z)α
ψt (y − x0 )(x − x0 )α
−
−
Dα Ã(x)a(z)dz,
α!
|x − z|m
|x − x0 |m
|α|=m
thus, similar to the proof of II in Theorem 1, we obtain
|Q|1+1/n X
||F̃tA (a)(x, y)|| ≤ C
(
||Dα A||BMO |x − x0 |−n−1 + |x − x0 |−n−1 |Dα Ã(x)|),
w(Q)
|α|=m
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2 ) |Q1 |
note that if w ∈ A1 , then w(Q
|Q2 | w(Q1 ) ≤ C for all cubes Q1 , Q2 with Q1 ⊂ Q2 . Thus, by Holder’ inequality and
the reverse of Holder’ inequality for w ∈ A1 , taking p > 1 and 1/p + 1/p0 = 1, we obtain
JJ
≤ C
X
α
||D A||BMO
|α|=m
+C
∞
X
−k
2
k=1
∞
X X
−k
2
|Q|
w(Q)
|Q| w(2k+1 Q)
w(Q) |2k+1 Q|
1
1/p |D Ã(x)| dx
·
Z
α
p
|2k+1 Q| 2k+1 Q
∞
X
X
w(2k+1 Q) |Q|
α
−k
≤ C
||D A||BMO
k2
≤ C,
|2k+1 Q| w(Q)
|α|=m k=1
|α|=m
1
|2k+1 Q|
Z
1/p0
w(x) dx
p0
2k+1 Q
k=1
which together with the estimate for J yields the desired result. This finishes the proof of Theorem 2.
Proof of Theorem3. By the equality
Rm+1 (A; x, y) = Qm+1 (A; x, y) +
X 1
(x − y)α (Dα A(x) − Dα A(y))
α!
|α|=m
and similar to the proof of Lemma 2, we get
gµA (f )(x)
≤
g̃µA (f )(x)
X Z |Dα A(x) − Dα A(y)|
+C
|f (y)|dy,
|x − y|n
|α|=m
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by Theorem 1 and 2 with [3], we obtain
w({x ∈ Rn : gµA (f )(x) > λ})
≤ w({x ∈ Rn : g̃µA (f )(x) > λ/2}) + w({x ∈ Rn :
X Z |Dα A(x) − Dα A(y)|
|f (y)|dy > Cλ})
|x − y|n
|α|=m
≤ C||f ||H 1 (w) /λ.
This completes the proof of Theorem 3.
1
Proof of Theorem 4. (i) It suffices to show that there exists a constant C > 0 such that for every H (w)-atom
a with suppa ⊂ Q = Q(x0 , d), there is
||gµA (a)||L1 (w) ≤ C.
P 1
α
α
Let Ã(x) = A(x) −
α! (D A)Q̃ x . We write, by the vanishing moment of a and for u ∈ 3Q \ 2Q,
|α|=m
FtA (a)(x, y)
= χ4Q (x)FtA (a)(x, y)
#
Z "
Rm (Ã; x, z)ψt (y − z) Rm (Ã; x, u)ψt (y − u)
+ χ(4Q)c (x)
−
a(z)dz
|x − z|m
|x − u|m
Rn
X 1 Z (x − z)α
(x − u)α
−χ(4Q)c (x)
−
ψt (y − z)Dα A(z)a(z)dz
α! Rn |x − z|m
|x − u|m
|α|=m
X 1 Z (x − u)α
−χ(4Q)c (x)
ψt (y − z)Dα A(z)a(z)dz,
α! Rn |x − u|m
|α|=m
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then
nµ/2
t
A
= Ft (a)(x, y)
t + |x − y|
nµ/2
t
FtA (a)(x, y)
≤ χ4Q (x) t + |x − y|
"
#
Z
Rm (Ã; x, u)ψt (y − u)
t
Rm (Ã; x, z)ψt (y − z)
nµ/2
)
−
a(z)dz
+ χ(4Q)c (x) (
t + |x − y|
|x − z|m
|x − u|m
Rn
nµ/2 X
Z α
α
1
(x − z)
(x − u)
t
α
−
·
ψ
(y
−
z)D
A(z)a(z)dz
+ χ(4Q)c (x) t
m
m
α! Rn |x − z|
|x − u|
t + |x − y|
|α|=m
nµ/2 X
Z
1
(x − u)α
t
+ χ(4Q)c (x) ψ
(y
−
z)a(z)dz
t
m
α! Rn |x − u|
t + |x − y|
gµA (a)(x)
|α|=m
= L1 (x) + L2 (x) + L3 (x, u) + L4 (x, u).
By the L∞ (w)-boundedness of gµA , we get
Z
Z
L1 (x)w(x)dx =
gµA (a)(x)w(x)dx ≤ ||gµA (a)||L∞ (w) w(4Q) ≤ C||a||L∞ (w) w(Q) ≤ C;
Rn
4Q
Similar to the proof of Theorem 1, we obtain
Z
L2 (x)w(x)dx ≤ C
Rn
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and
Z
L3 (x, u)w(x)dx ≤ C.
Rn
Thus, using the condition of L4 (x, u), we obtain
Z
gµA (a)(x)w(x)dx ≤ C.
Rn
(ii) For any cube Q = Q(x0 , d), let Ã(x) = A(x)−
P
|α|=m
1
α
α
α! (D A)Q̃ x .
We write, for f = f χ4Q +f χ(4Q)c = f1 +f2
and u ∈ 3Q \ 2Q,
F̃tA (f )(x, y)
Z
Rm (Ã; x, z)
ψt (y − z)f2 (z)dz
|x − z|m
Z X 1
(x − z)α
(u − z)α
−
ψt (y − z)f2 (z)dz
−
(Dα A(x) − (Dα A)Q )
m
α!
|u − z|m
Rn |x − z|
|α|=m
Z
X 1
(u − z)α
α
α
−
(D A(x) − (D A)Q )
ψ (y − z)f2 (z)dz,
m t
α!
Rn |u − z|
= F̃tA (f1 )(x, y) +
Rn
|α|=m
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then
!
Rm (Ã; x0 , ·)
A
f
(x
)
g̃µ (f )(x) − gµ
2
0
|x0 − ·|m
R
(
Ã;
x
,
·)
t
t
m
0
= (
)nµ/2 F̃tA (f )(x, y) − (
)nµ/2 Ft
f
)(x
)
2
0
t + |x − y|
t + |x0 − y|
|x0 − ·|m
!
nµ/2
nµ/2
t
Rm (Ã; x0 , ·)
t
A
F̃t (f )(x, y) −
Ft
f
(x
)
≤ 2
0
t + |x − y|
t + |x0 − y|
|x0 − ·|m
nµ/2
t
F̃tA (f1 )(x, y)
≤ t + |x − y|
Z "
nµ/2
t
Rm (Ã; x, z)
ψt (y − z)
+ R n
t + |x − y|
|x − z|m
#
nµ/2
t
Rm (Ã; x0 , z)
−
ψt (x0 − z) f2 (z)dz m
t + |x0 − y|
|x0 − z|
nµ/2 X
Z α
α
t
1
(y
−
z)
(u
−
z)
α
α
+ ·
(D A(x)−(D A)Q )
−
ψt (y − z)f2 (z)dz m
m
t
+
|x
−
y|
α!
|y
−
z|
|u
−
z|
n
R
|α|=m
nµ/2 X
Z
1
(u − z)α
t
α
α
+ ·
(D A(x) − (D A)Q )
ψ (y − z)f2 (z)dz m t
t
+
|x
−
y|
α!
|u
−
z|
n
R
|α|=m
= M1 (x) + M2 (x) + M3 (x, u) + M4 (x, u).
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By the L∞ (w)-boundedness of g̃µA , we get
Z
1
M1 (x)w(x)dx ≤ ||g̃µA (f1 )||L∞ (w) ≤ C||f ||L∞ (w) ;
w(Q) Q
Similar to the proof of Theorem 1, we obtain
Z
1
M2 (x)w(x)dx ≤ C||f ||L∞ (w)
w(Q) Q
and
Z
1
M3 (x, u)w(x)dx ≤ C||f ||L∞ (w) .
w(Q) Q
Thus, using the condition of M4 (x, u), we obtain
!
Z 1
Rm (Ã; x0 , ·)
A
f
(x
)
g̃µ (f )(x) − gµ
w(x)dx ≤ C||f ||L∞ (w) .
2
0
w(Q) Q |x0 − ·|m
This completes the proof of Theorem 4.
Acknowledgement. The author would like to express his deep gratitude to the referee for his valuable
comments and suggestions.
1. Alvarez J., Babgy R. J., Kurtz D. S. and PerezC., Weighted estimates for commutators of linear operators, Studia Math., 104
(1993), 195–209.
2. Bui Huy Qui, Weighted Hardy spaces, Math. Nachr., 103 (1981), 45–62.
3. Chanillo S., A note on commutators, Indiana Univ. Math. J., 31 (1982), 7–16.
4. Chen W. and Hu G., Weak type (H 1 , L1 ) estimate for multilinear singular integral operator, Adv.in Math.(China), 30 (2001),
63–69.
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5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Cohen J., A sharp estimate for a multilinear singular integral on Rn , Indiana Univ. Math. J., 30 (1981), 693–702.
Cohen J. and Gosselin J., On multilinear singular integral operators on Rn , Studia Math., 72 (1982), 199–223.
Cohen J. and Gosselin J., A BMO estimate for multilinear singular integral operators, Illinois J. Math., 30 (1986), 445–465.
Ding Y., A note on multilinear fractional integrals with rough kernel, Adv. in Math. (China), 30 (2001), 238–246.
Ding Y. and Lu S. Z., Weighted boundedness for a class rough multilinear operators, Acta Math. Sinica, 3 (2001), 517–526.
Garcia-Cuerva J. and Rubio de Francia J.L., Weighted norm inequalities and related topics, North-Holland Math. 16, Amsterdam,
1985.
Harboure E., Segovia C. and Torrea J. L., Boundedness of commutators of fractional and singular integrals for the extreme
values of p, Illinois J. Math., 41 (1997), 676–700.
Hu G. and Yang D.C., A variant sharp estimate for multilinear singular integral operators, Studia Math., 141 (2000), 25–42.
Hu G. and Yang D.C., Multilinear oscillatory singular integral operators on Hardy spaces, Chinese J. of Contemporary Math.,
18 (1997), 403–413.
Liu Lanzhe, Weighted weak type estimates for commutators of Littlewood-Paley operator, Japanese J. of Math., 29(1) (2003),
1–16.
Liu Lanzhe, Weighted weak type (H 1 , L1 ) estimates for commutators of Littlewood-Paley operator, (to appear in Indian J. of
Math.).
Perez C., Endpoint estimate for commutators of singular integral operators, J. Func. Anal., 128 (1995), 163–185.
Torchinsky A., The real variable methods in harmonic analysis, Pure and Applied Math. 123, Academic Press, New York, 1986.
Liu Lanzhe, College of Mathematics and Computer, Changsha University of Science and Technology, Changsha 410077, P. R. of
China, e-mail: lanzheliu@263.net
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