55
Acta Math. Univ. Comenianae
Vol. LXXIII, 1(2004), pp. 55–67
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 LittlewoodPaley operator is defined by
"Z Z
#1/2
nµ
t
dydt
gµA (f )(x) =
|FtA (f )(x, y)|2 n+1
, µ > 1,
n+1
t + |x − y|
t
R+
where
Z
Rm+1 (A; x, z)
ψt (y − z)f (z)dz,
|x − z|m
Rn
X 1
Rm+1 (A; x, y) = A(x) −
Dα A(y)(x − y)α ,
α!
FtA (f )(x, y)
=
|α|≤m
and ψt (x) = t
define
−n
ψ(x/t) for t > 0. We denote by Ft (f )(y) = f ∗ ψt (y). We also
Z Z
gµ (f )(x) =
n+1
R+
t
t + |x − y|
nµ
dydt
|Ft (f )(y)|2 n+1
t
!1/2
,
which is the Littlewood-Paley operator (see [17]).
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).
56
LIU LANZHE
Let H be the Hilbert space H =
R R
1/2
2
n+1
h : ||h|| =
<∞
.
n+1 |h(t)| dydt/t
R
+
n
Then for each fixed x ∈ R ,
to H, and it is clear that
gµA (f )(x)
gµ (f )(x)
FtA (f )(x, y)
may be viewed as a mapping from (0, +∞)
nµ/2
t
FtA (f )(x, y) ,
= t + |x − y|
nµ/2
t
= Ft (f )(y) .
t + |x − y|
We also consider the variant of gµA , which is defined by
"Z Z
#1/2
nµ
t
A
A
2 dydt
g̃µ (f )(x) =
|F̃t (f )(x, y)| n+1
,
n+1
t + |x − y|
t
R+
µ > 1,
where
F̃tA (f )(x, y) =
Z
Rn
Qm+1 (A; x, z)
ψt (y − z)f (z)dz
|x − z|m
and
Qm+1 (A; x, z) = Rm (A; x, z) −
X
Dα A(x)(x − z)α .
|α|=m
gµA
Note that when m = 0,
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 CalderonZygmund operator and BMO functions are obtained. The main purpose of this
paper is to study the weighted endpoint boundedness of the multilinear LittlewoodPaley operators. Throughout this paper, M (f ) will denote the Hardy-Littlewood
maximal function of f , Q will denote a cube of Rn with side parallel to the
axes. For
locally
integral function f on Rn ,R we denote that
R a cube Q and any
R
−1
f (Q) = Q f (x)dx, fQ = |Q|
f (x)dx and f # (x) = sup |Q|−1 Q |f (y)−fQ |dy.
Q
x∈Q
Moreover, for a weight 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. A function a is called a H 1 (w) atom if there
exists a cube Q such that
R
a is supported on Q, ||a||L∞ (w) ≤ w(Q)−1 and 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).
57
ENDPOINT ESTIMATES FOR LITTLEWOOD-PALEY OPERATORS
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).
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
α! |x − u|m Q
(4Q)c t + |x − y|
|α|=m
then gµA is bounded from H 1 (w) to L1 (w);
(ii) If for any cube Q and u ∈ 3Q \ 2Q, there is
nµ/2 X
Z t
1
1
(Dα A(x) − (Dα A)Q )
w(Q) Q t + |x − y|
α!
|α|=m
Z
α
(u − z)
ψ
(y
−
z)f
(z)dz
·
w(x)dx
t
m
(4Q)c |u − z|
≤ 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).
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
m
|Rm (A : x, y)| ≤ C|x − y|
X
|α|=m
1
|Q̃(x, y)|
Z
!1/q
α
q
|D A(z)| dz
,
Q̃(x,y)
√
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
||gµA (f )||Lq (w) ≤ C
X
|α|=m
||Dα A||BMO ||f ||Lp (w) .
58
LIU LANZHE
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
Z
|f (z)||Rm+1 (A; x, z)|
t−2n
≤ C
2n+2
|x − z|m
Rn
0
Rn (1 + |y − z|/t)
nµ
1/2
dydt
t
·
dz
t + |x − y|
t1+n
Z ∞ nµ
Z
Z |f (z)||Rm+1 (A; x, z)|
t
−n
≤ C
t
|x − z|m
t + |x − y|
Rn
0
Rn
1/2
dy
·
tdt
dz,
(t + |y − z|)2n+2
noting that
nµ
dy
t
t
t
+
|x
−
y|
(t
+
|y
−
z|)2n+2
n
R
1
1
(x) ≤ C
≤ CM
(t + | · −z|)2n+2
(t + |x − z|)2n+2
−n
Z
and
Z
0
∞
tdt
= C|x − z|−2n ,
(t + |x − z|)2n+2
we obtain
gµA (f )(x)
Z
≤ C
Rn
Z
= C
Rn
|f (z)|
|Rm+1 (A; x, z)|
|x − z|m
|f (z)||Rm+1 (A; x, z)|
dz,
|x − z|m+n
Z
∞
0
thus, the lemma follows from [8] [9].
tdt
(t + |x − z|)2n+2
1/2
dz
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
√
holds for any cube Q. Fix a cube Q = Q(x0 , l). Let Q̃ = 5 nQ and
P 1
α
α
α
Ã(x) = A(x) −
α! (D A)Q̃ x , then Rm (A; x, y) = Rm (Ã; x, y) and D Ã =
|α|=m
Dα A − (Dα A)Q̃ for |α| = m. We write FtA (f ) = FtA (f1 ) + FtA (f2 ) for f1 = f χQ̃
ENDPOINT ESTIMATES FOR LITTLEWOOD-PALEY OPERATORS
59
and f2 = f χRn \Q̃ , then
1
w(Q)
=
Z
|gµA (f )(x) − gµA (f2 )(x0 )|w(x)dx
Z 1
t
nµ/2 A
||(
)
Ft (f )(x, y)||
w(Q) Q
t + |x − y|
Q
− ||(
≤
1
w(Q)
t
)nµ/2 FtA (f2 )(x0 , y)|| w(x)dx
t + |x0 − y|
Z
gµA (f1 )(x)w(x)dx
nµ/2
Z t
1
FtA (f2 )(x, y)
+
w(Q) Q t + |x − y|
nµ/2
t
−
FtA (f2 )(x0 , y) w(x)dx
t + |x0 − y|
Q
:= 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) .
To estimate II, we write
nµ/2
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
t
(x0 − z)α 
−
ψt (y − z)Dα Ã(z)f2 (z)dz
t + |x0 − y|
|x0 − z|m
=
t
t + |x − y|
:= II1t (x) + II2t (x) + II3t (x) + II4t (x),
60
LIU LANZHE
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||
(
|f (z)|dz)
BMO
(2k l)n+m+1
2k+1 Q̃
|α|=m
k=0
X
≤ C
∞
X
||Dα A||BMO ||f ||L∞ (w)
|α|=m
X
≤ C
k2−k
k=0
α
||D A||BMO ||f ||L∞ (w) ;
|α|=m
For
II2t (x),
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)|
X
≤ C
||Dα A||BMO (|x − x0 |m +
|α|=m
X
|x0 − z|m−|β| |x − x0 ||β| ),
0<|β|<m
thus, for x ∈ Q,
||II2t (x)||
Z
≤ C
|f2 (z)|
|Rm (Ã; x, z) − Rm (Ã; x0 , z)|dz
|x
− z|m+n
Rn
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
|α|=m
kDα A||BMO
∞
X
k=0
klm
k
(2 l)m+n
||Dα A||BMO ||f ||L∞ (w)
∞
X
k=1
||Dα A||BMO ||f ||L∞ (w) ;
Z
|f (z)|dz
2k+1 Q̃
k2−km
ENDPOINT ESTIMATES FOR LITTLEWOOD-PALEY OPERATORS
61
For II3t (x), 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 ,
||II3t (x)||
Z
≤ C
n+1
R+
Rn
Z
≤ C
Rn
Z
≤ C
Rn
Z
!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
||Dα A||BMO ||f ||L∞ (w)
|α|=m
≤ C
X
∞
X
k2−k/2
k=0
α
||D A||BMO ||f ||L∞ (w) ;
|α|=m
For II4t (x), similar to the estimates of II1t (x) and II3t (x), we have
||II4t (x)||
Z
≤ C
Rn \Q̃
≤ C
X
|x − x0 |
|x − x0 |1/2
+
|x − z|n+1
|x − z|n+1/2
||Dα A||BMO ||f ||L∞ (w)
|α|=m
≤ C
X
∞
X
X
|Dα Ã(z)||f (z)|dz
|α|=m
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 > 0 such
that for every H 1 (w)-atom
R 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.
62
LIU LANZHE
We write
Z
Rn
g̃µA (a)(x)w(x)dx
"Z
Z
=
+
|x−x0 |≤2r
#
g̃µA (a)(x)w(x)dx := J + JJ.
|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
we have, similar to the proof of Lemma 2,
g̃µA (a)(x)
≤
gµA (a)(x)
X Z |Dα A(x) − Dα A(y)|
+C
|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) = Qm (Ã; x, y).
by the vanishing moment of a and
P We write,
1
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
||F̃tA (a)(x, y)||
≤ C
|Q|1+1/n X
(
||Dα A||BMO |x − x0 |−n−1 + |x − x0 |−n−1 |Dα Ã(x)|),
w(Q)
|α|=m
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 ,
ENDPOINT ESTIMATES FOR LITTLEWOOD-PALEY OPERATORS
63
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
|α|=m k=1
2−k
|Q|
w(Q)
|Q| w(2k+1 Q)
w(Q) |2k+1 Q|
Z
1
|2k+1 Q|
·
1
1/p
|Dα Ã(x)|p dx
2k+1 Q
Z
1/p0
w(x) dx
p0
|2k+1 Q| 2k+1 Q
∞
X
X
w(2k+1 Q) |Q|
≤ C,
≤ C
||Dα A||BMO
k2−k
|2k+1 Q| w(Q)
|α|=m
k=1
which together with the estimate for J yields the desired result. This finishes the
proof of Theorem 2.
Proof of Theorem 3. 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)|
|f (y)|dy,
+C
|x − y|n
|α|=m
by Theorem 1 and 2 with [3], we obtain
w({x ∈ Rn : gµA (f )(x) > λ})
≤ w({x ∈ Rn : g̃µA (f )(x) > λ/2})
X Z |Dα A(x) − Dα A(y)|
+ w({x ∈ Rn :
|f (y)|dy > Cλ})
|x − y|n
|α|=m
≤ C||f ||H 1 (w) /λ.
This completes the proof of Theorem 3.
Proof of Theorem 4. (i) It suffices to show that there exists a constant C > 0
such that for every H 1 (w)-atom a with suppa ⊂ Q = Q(x0 , d), there is
||gµA (a)||L1 (w) ≤ C.
64
LIU LANZHE
P
Let Ã(x) = A(x) −
|α|=m
1
α
α
α! (D A)Q̃ x .
We write, by the vanishing moment of a
and for u ∈ 3Q \ 2Q,
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)
−
a(z)dz
+ χ(4Q)c (x)
|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
then
gµA (a)(x)
nµ/2
t
A
= Ft (a)(x, y)
t + |x − y|
nµ/2
t
A
≤ χ4Q (x) Ft (a)(x, y)
t + |x − y|
"
Z
nµ/2
t
Rm (Ã; x, z)ψt (y − z)
+ χ(4Q)c (x) t + |x − y|
|x − z|m
Rn
#
Rm (Ã; x, u)ψt (y − u)
a(z)dz
−
|x − u|m
nµ/2 X
Z t
1
(x − z)α
(x − u)α
c
+ χ(4Q) (x) −
α! Rn |x − z|m
|x − u|m
t + |x − y|
|α|=m
· ψt (y − z)Dα A(z)a(z)dz||
nµ/2 X
Z
α
t
1
(x
−
u)
+ χ(4Q)c (x) ψt (y − z)a(z)dz m
α! Rn |x − u|
t + |x − y|
|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)
Rn
4Q
≤ C||a||L∞ (w) w(Q) ≤ C;
ENDPOINT ESTIMATES FOR LITTLEWOOD-PALEY OPERATORS
65
Similar to the proof of Theorem 1, we obtain
Z
L2 (x)w(x)dx ≤ C
Rn
and
Z
L3 (x, u)w(x)dx ≤ C.
Rn
Thus, using the condition of L4 (x, u), we obtain
Z
Rn
gµA (a)(x)w(x)dx ≤ C.
P
(ii) For any cube Q = Q(x0 , d), let Ã(x) = A(x) −
|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
−
(Dα A(x) − (Dα A)Q )
−
ψt (y − z)f2 (z)dz
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
then
!
Rm (Ã; x0 , ·)
A
f
(x
)
g̃µ (f )(x) − gµ
2
0
|x0 − ·|m
nµ/2
t
A
= F̃t (f )(x, y)
t + |x − y|
!
nµ/2
t
Rm (Ã; x0 , ·)
− Ft
f2 (x0 )
m
t + |x0 − y|
|x0 − ·|
nµ/2
t
≤ F̃tA (f )(x, y)
t + |x − y|
!
nµ/2
t
Rm (Ã; x0 , ·)
−
Ft
f2 (x0 )
m
t + |x0 − y|
|x0 − ·|
66
LIU LANZHE
nµ/2
t
A
≤ F̃t (f1 )(x, y)
t + |x − y|
Z "
nµ/2
Rm (Ã; x, z)
t
ψt (y − z)
+ R n
t + |x − y|
|x − z|m
#
nµ/2
Rm (Ã; x0 , z)
t
ψt (x0 − z) f2 (z)dz −
m
t + |x0 − y|
|x0 − z|
nµ/2
t
+ t + |x − y|
Z α
α
X 1
(y
−
z)
(u
−
z)
·
(Dα A(x)−(Dα A)Q )
−
ψ
(y
−
z)f
(z)dz
t
2
m
m
α!
|u − z|
Rn |y − z|
|α|=m
nµ/2
t
+ t + |x − y|
Z
α
X 1
(u − z)
α
α
·
(D A(x) − (D A)Q )
ψ
(y
−
z)f
(z)dz
t
2
m
α!
Rn |u − z|
|α|=m
= M1 (x) + M2 (x) + M3 (x, u) + M4 (x, u).
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
1
w(Q)
Z
M3 (x, u)w(x)dx ≤ C||f ||L∞ (w) .
Q
Thus, using the condition of M4 (x, u), we obtain
!
Z 1
Rm (Ã; x0 , ·)
A
f
(x
)
g̃µ (f )(x) − gµ
2
0 w(x)dx ≤ C||f ||L∞ (w) .
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.
ENDPOINT ESTIMATES FOR LITTLEWOOD-PALEY OPERATORS
67
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Liu Lanzhe, College of Mathematics and Computer, Changsha University of Science and Technology, Changsha 410077, P. R. of China, e-mail: lanzheliu@263.net