IJMMS 2004:5, 259–271
PII. S016117120430147X
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
BOUNDEDNESS OF MULTILINEAR OPERATORS
ON TRIEBEL-LIZORKIN SPACES
LIU LANZHE
Received 20 January 2003
The purpose of this paper is to study the boundedness in the context of Triebel-Lizorkin
spaces for some multilinear operators related to certain convolution operators. The operators include Littlewood-Paley operator, Marcinkiewicz integral, and Bochner-Riesz operator.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Introduction. Let T be a Calderon-Zygmund operator. A well-known result of Coifman et al. [6] states that the commutator [b, T ] = T (bf ) − bT f (where b ∈ BMO) is
bounded on Lp (Rn ) for 1 < p < ∞; Chanillo [1] proves a similar result when T is replaced by the fractional integral operator. In [7, 9], Janson and Paluszyński extend
these results to the Triebel-Lizorkin spaces and the case b ∈ Lipβ (where Lip β is the
homogeneous Lipschitz space). The main purpose of this paper is to discuss the boundedness of some multilinear operators related to certain convolution operators in the
context of Triebel-Lizorkin spaces. In fact, we will establish the boundedness on the
Triebel-Lizorkin spaces for some multilinear operators related to certain convolution
operator only under certain conditions on the size of the operators. As applications, we
obtain the boundedness of the multilinear operators related to the Marcinkiewicz integral, Littlewood-Paley operator, and Bochner-Riesz operator in the context of TriebelLizorkin spaces.
2. Preliminaries. Throughout this paper, M(f ) will denote the Hardy-Littlewood
maximal function of f , Mp f = (M(f p ))1/p for p > 0, and Q will denote a cube of Rn
with sides parallel to the axes. For a cube Q, let fQ = |Q|−1 Q f (x)dx and f # (x) =
β,∞
supx∈Q |Q|−1 Q |f (y) − fQ |dy. For β > 0 and p > 1, let Ḟp be the homogeneous
˙ β is the space of functions f such that
Triebel-Lizorkin space. The Lipschitz space ∧
f ∧˙ β = sup
[β]+1
∆
f (x)
h
|h|β
x,h∈Rn
h≠0
< ∞,
(2.1)
where ∆kh denotes the kth difference operator (see [9]).
The operators considered in this paper are following several sublinear operators.
Let m be a positive integer and let A be a function on Rn . We denote
Rm+1 (A; x, y) = A(x) −
1 α
D A(y)(x − y)α .
α!
|α|≤m
(2.2)
260
LIU LANZHE
Definition 2.1. Let ε > 0 and let ψ be a fixed function which satisfies the following
properties:
(1) |ψ(x)| ≤ C(1 + |x|)−(n+1) ,
(2) |ψ(x + y) − ψ(x)| ≤ C|y|ε (1 + |x|)−(n+1+ε) when 2|y| < |x|.
The multilinear Littlewood-Paley operator is defined by
∞
A
(f )(x) =
gψ
0
A
F (f )(x)2 dt
t
t
1/2
,
(2.3)
where
FtA (f )(x) =
Rn
ψt (x − y)
Rm+1 (A; x, y)
f (y)dy
|x − y|m
(2.4)
and ψt (x) = t −n ψ(x/t) for t > 0. Denote Ft (f ) = ψt ∗ f . Also define
gψ (f )(x) =
∞
0
Ft (f )(x)2 dt
t
1/2
(2.5)
which is the Littlewood-Paley g function (see [10]).
∞
Let H be the space H = {h : h = ( 0 |h(t)|2 dt/t)1/2 < ∞}. Then, for each fixed
x ∈ Rn , FtA (f )(x) may be viewed as a mapping from [0, +∞) to H, and it is clear that
gψ (f )(x) = Ft (f )(x),
A
gψ
(f )(x) = FtA (f )(x).
(2.6)
Definition 2.2. Let 0 < γ ≤ 1 and let Ω be homogeneous of degree zero on Rn
such that S n−1 Ω(x )dσ (x ) = 0. Assume that Ω ∈ Lipγ (S n−1 ), that is, there exists a
constant M > 0 such that for any x, y ∈ S n−1 , |Ω(x) − Ω(y)| ≤ M|x − y|γ . The multilinear Marcinkiewicz integral operator is defined by
∞
A
µΩ
(f )(x) =
0
A
F (f )(x)2 dt
t
t3
1/2
,
(2.7)
where
FtA (f )(x) =
|x−y|≤t
Ω(x − y) Rm+1 (A; x, y)
f (y)dy.
|x − y|n−1
|x − y|m
(2.8)
Ω(x − y)
f (y)dy.
|x − y|n−1
(2.9)
Denote
Ft (f )(x) =
|x−y|≤t
Also define
µΩ (f )(x) =
∞
0
Ft (f )(x)2 dt
t3
1/2
(2.10)
which is the Marcinkiewicz integral (see [11]).
∞
Let H be the space H = {h : h = ( 0 |h(t)|2 dt/t 3 )1/2 < ∞}. Then, it is clear that
µΩ (f )(x) = Ft (f )(x),
A
µΩ
(f )(x) = FtA (f )(x).
(2.11)
261
BOUNDEDNESS OF MULTILINEAR OPERATORS . . .
Definition 2.3. Let Btδ (fˆ)(ξ) = (1 − t 2 |ξ|2 )δ+ fˆ(ξ). Denote
A
(f )(x) =
Bδ,t
Rn
Btδ (x − y)
Rm+1 (A; x, y)
f (y)dy,
|x − y|m
(2.12)
where Btδ (z) = t −n B δ (z/t) for t > 0. The maximal multilinear Bochner-Riesz operator
is defined by
A
A
(f )(x) = sup Bδ,t
(f )(x).
Bδ,∗
(2.13)
δ
(f )(x) = sup Btδ (f )(x)
B∗
(2.14)
t>0
Also define
t>0
which is the Bochner-Riesz operator (see [7, 8]).
Let H be the space H = {h : h = supt>0 |h(t)| < ∞}, then it is clear that
δ
B∗
(f )(x) = Btδ (f )(x),
A
A
(f )(x) = Bδ,t
(f )(x).
Bδ,∗
(2.15)
More generally, we consider the following multilinear operators related to certain
convolution operators.
Definition 2.4. Let K(x, t) be defined on Rn × [0, +∞). Denote that
Kt f (x) =
Rn
KtA f (x) =
Rn
K(x − y, t)f (y)dy,
Rm+1 (A; x, y)
K(x − y, t)f (y)dy.
|x − y|m
(2.16)
Let H be the normed space H = {h : h < ∞}. For each fixed x ∈ Rn , Kt f (x) and
KtA (f )(x) are viewed as a mapping from [0, +∞) to H. Then, the multilinear operators
related to Kt is defined by
TA f (x) = KtA (f )(x);
(2.17)
also define T f (x) = Kt f (x).
It is clear that Definitions 2.1, 2.2, and 2.3 are the particular examples of Definition
2.4. Note that when m = 0, TA is just the commutator of Kt and A. It is well known that
multilinear operators are of great interest in harmonic analysis and have been widely
studied by many authors (see [2, 3, 4, 5]). The main purpose of this paper is to consider
the continuity of the multilinear operators on Triebel-Lizorkin spaces. We will prove
the following theorems in Section 3.
A
be the multilinear Littlewood-Paley operator as in Definition 2.1
Theorem 2.5. Let gψ
˙ β for |α| = m. Then
and let 0 < β < min(1, ε), 1 < p < ∞, and D α A ∈ ∧
β,∞
A
(a) gψ
is bounded from Lp (Rn ) to Ḟp (Rn ),
A
(b) gψ is bounded from Lp (Rn ) to Lq (Rn ) for 1/p − 1/q = β/n and 1/p > β/n.
262
LIU LANZHE
A
Theorem 2.6. Let µΩ
be the multilinear Marcinkiewicz integral operator as in Defini˙ β for |α| = m.
tion 2.2 and let 0 < γ ≤ 1, 0 < β < min(1/2, γ), 1 < p < ∞, and D α A ∈ ∧
Then
β,∞
A
is bounded from Lp (Rn ) to Ḟp (Rn ),
(a) µΩ
A
p
n
q
(b) µΩ is bounded from L (R ) to L (Rn ) for 1/p − 1/q = β/n and 1/p > β/n.
A
be the maximal multilinear Bochner-Riesz operator as in
Theorem 2.7. Let Bδ,∗
Definition 2.3 and let δ > (n − 1)/2, 0 < β < min(1, δ − (n − 1)/2), 1 < p < ∞, and
˙ β for |α| = m. Then
Dα A ∈ ∧
β,∞
A
(a) Bδ,∗
is bounded from Lp (Rn ) to Ḟp (Rn );
A
(b) Bδ,∗ is bounded from Lp (Rn ) to Lq (Rn ) for 1/p − 1/q = β/n and 1/p > β/n.
3. Main theorem and proof. First, we will establish the following theorem.
˙ β for |α| = m. Let Kt , T , and
Theorem 3.1. Let 0 < β < 1, 1 < p < ∞, and D α A ∈ ∧
TA be the same as in Definition 2.4. If T is bounded on Lq (Rn ) for q ∈ (1, +∞) and TA
satisfies the size condition
A
D α A |Q|β/n M(f )(x)
K (f )(x) − K A (f ) x0 ≤ C
˙β
t
t
∧
(3.1)
|α|=m
for any cube Q with supp f ⊂ (2Q)c and x ∈ Q, then
β,∞
(a) TA is bounded from Lp (Rn ) to Ḟp (Rn ),
(b) TA is bounded from Lp (Rn ) to Lq (Rn ) for 1/p − 1/q = β/n and 1/p > β/n.
To prove the theorem, we need the following lemmas.
Lemma 3.2 (see [9]). For 0 < β < 1 and 1 < p < ∞,
1
f (x) − fQ dx f Ḟ β,∞ ≈ sup
Q |Q|1+β/n Q
p
p
L
1
≈ sup inf
f (x) − c dx .
·∈Q c |Q|1+β/n Q
p
L
(3.2)
Lemma 3.3 (see [9]). For 0 < β < 1 and 1 ≤ p ≤ ∞,
f ∧˙ β ≈ sup
Q
≈ sup
Q
1
|Q|1+β/n
Q
f (x) − fQ dx
1
1
|Q|β/n |Q|
Q
f (x) − fQ p dx
1/p
(3.3)
.
Lemma 3.4 (see [1]). For 1 ≤ r < ∞ and δ > 0, let
Mδ,r (f )(x) = sup
x∈Q
1
|Q|1−δr /n
f (y)p dy
1/p
.
Q
Suppose that r < p < δ/n and 1/q = 1/p − δ/n. Then Mδ,r (f )Lq ≤ Cf Lp .
(3.4)
263
BOUNDEDNESS OF MULTILINEAR OPERATORS . . .
Lemma 3.5 (see [9]). Let Q1 ⊂ Q2 . Then
fQ − fQ ≤ Cf ∧˙ Q2 β/n .
1
2
β
(3.5)
Lemma 3.6 (see [4]). Let A be a function on Rn and D α A ∈ Lq (Rn ) for |α| = m and
some q > n. Then
Rm (A; x, y) ≤ C|x − y|m
|α|=m
1
Q̃(x, y)
α
D A(z)q dz
1/q
,
(3.6)
Q̃(x,y)
√
where Q̃(x, y) is the cube centered at x and having side length 5 n|x − y|.
√
Proof of Theorem 3.1. (a) Fix a cube Q = Q(x0 , l) and x̃ ∈ Q. Let Q̃ = 5 nQ and
Ã(x) = A(x) − |α|=m (1/α!)(D α A)Q̃ x α , then Rm (A; x, y) = Rm (Ã; x, y) and D α Ã =
D α A − (D α A)Q̃ for |α| = m. For f1 = f χQ̃ and f2 = f χRn \Q̃ ,
Rm+1 Ã; x, y
K(x − y, t)f (y)dy
|x − y|m
Rn
Rm+1 Ã; x, y
K(x − y, t)f (y)dy
=
|x − y|m
Rn
Rm Ã; x, y
K(x − y, t)f1 (y)dy
+
|x − y|m
Rn
1
K(x − y, t)(x − y)α α
D Ã(y)f1 (y)dy,
−
n
α!
|x − y|m
R
|α|=m
KtA (f )(x) =
(3.7)
then
TA (f )(x) − T f2 x0 = K A (f )(x) − K Ã f2 x0 Ã
t
t
Rm Ã; x, ·
(x)
f
≤
K
1
t
|x − ·|m
1 (x − ·)α
α
+
Kt |x − ·|m D Ãf1 (x)
α!
|α|=m
+ Ktà f2 (x) − Ktà f2 x0 (3.8)
= A(x) + B(x) + C(x).
Thus,
TA f (x) − T (f ) x0 dx
Ã
Q
1
1
1
A(x)dx
+
B(x)dx
+
C(x)dx
≤
|Q|1+β/n Q
|Q|1+β/n Q
|Q|1+β/n Q
1
|Q|1+β/n
:= I+II+III.
(3.9)
264
LIU LANZHE
Now, we estimate I, II, and III, respectively. First, for x ∈ Q and y ∈ Q̃, using Lemmas
3.3 and 3.6, we get
Rm Ã; x, y ≤ C|x − y|m
sup D α A(x) − D α A Q̃ |α|=m x∈Q̃
m
≤ C|x − y| |Q|β/n
D α A |α|=m
(3.10)
˙β .
∧
Thus, by Holder’s inequality and the Lr boundedness of T for 1 < r < p, we obtain
1
T f1 (x)dx
|Q|
Q
|α|=m
α
D A T f1 r |Q|−1/r
≤C
˙β
L
∧
D α A I≤C
˙β
∧
|α|=m
D α A ≤C
|α|=m
D α A ≤C
|α|=m
˙β
∧
f1 r |Q|−1/r
L
(3.11)
˙ β Mr (f )(x̃).
∧
Secondly, for 1 < r < q, using the inequality (see [9])
α
D A − D α A f χ r ≤ C|Q|1/r +β/n D α A Mr (f )(x),
˙β
Q̃ L
∧
Q̃
(3.12)
and similar to the proof of I, we obtain
II ≤
C
|Q|1+β/n
≤ C|Q|
r |Q|1−1/r
T D α A − D α A
L
Q̃ f χQ̃
|α|=m
−β/n−1/r
Dα A − Dα A
Q̃ f χQ̃ Lr
(3.13)
|α|=m
≤C
D α A Mr (f )(x̃).
˙β
∧
|α|=m
For III, using the size condition of TA , we have
III ≤ C
D α A M(f )(x̃).
˙β
∧
(3.14)
|α|=m
Putting these estimates together, taking the supremum over all Q such that x̃ ∈ Q, and
using the Lp boundedness of Mr for r < p, we obtain
TA (f ) β,∞ ≤ C
D α A f Lp .
˙β
∧
Ḟ
p
(3.15)
|α|=m
This completes the proof of (a).
(b) By the same argument as in the proof of (a), we have
1
|Q|
Q
TA (f )(x) − T f2 x0 dx ≤ C
D α A ˙ β Mβ,r (f ) + Mβ,1 (f ) ,
Ã
∧
|α|=m
(3.16)
BOUNDEDNESS OF MULTILINEAR OPERATORS . . .
265
thus,
#
D α A TA (f ) ≤ C
˙ β Mβ,r (f ) + Mβ,1 (f ) .
∧
(3.17)
|α|=m
Now, using Lemma 3.4, we obtain
TA (f )
Lq
# ≤ C TA (f ) Lq
Mβ,r (f ) q + Mβ,1 (f ) q ≤ Cf Lp .
D α A ≤C
˙β
∧
L
L
(3.18)
|α|=m
This completes the proof of (b) and the theorem.
A
A
A
To prove Theorems 2.5, 2.6, and 2.7, it suffices to verify that gψ
, µΩ
, and Bδ,∗
satisfy
the size condition in the Theorem 3.1.
Suppose supp f ⊂ Q̃c and x ∈ Q = Q(x0 , l). Note that |x0 − y| ≈ |x − y| for y ∈ Q̃c .
A
For gψ
, we write
Ftà (f )(x) − Ftà (f ) x0
ψt (x − y) ψt x0 − y
m Rm Ã; x, y f (y)dy
−
=
m
n
|x
−
y|
x0 − y
R \Q̃
ψt x0 − y f (y) m
+
Rm Ã; x, y − Rm Ã; x0 , y dy
n
x0 − y
R \Q̃
α 1 ψt (x − y)(x − y)α ψt x0 − y x0 − y
−
−
x0 − y m
α! Rn \Q̃
|x − y|m
|α|=m
× D α Ã(y)f (y)dy
= I1 +I2 +I3 .
By the condition of ψ, we obtain
x − x0 I1 ≤ C
m+1 Rm Ã; x, y f (y)
Rn \Q̃ x0 − y 1/2
∞
t dt
dy
×
2(n+1)
0
t + x0 − y x − x0 ε m Rm Ã; x, y f (y)
+C
Rn \Q̃ x0 − y 1/2
∞
t dt
dy
×
2(n+1+ε)
0
t + x0 − y x − x0 ε
x − x0 Rm Ã; x, y f (y)dy
≤C
m+n+1 + m+n+ε
Rn \Q̃ x0 − y x0 − y D α A |Q|β/n
≤C
˙β
∧
|α|=m
×
∞ k=0
x − x0 x − x0 ε
n+ε f (y)dy
n+1 + 2k+1 Q̃\2k+1 Q̃ x0 − y x0 − y
(3.19)
266
LIU LANZHE
D α A ≤C
β/n
˙ β |Q|
∧
|α|=m
∞
k=1
1
2−k + 2−kε 2k Q̃
2k Q̃
f (y)dy
∞
−k
D α A |Q|β/n
≤C
2 + 2−kε M(f )(x)
˙β
∧
|α|=m
k=1
D α A |Q|β/n M(f )(x).
≤C
˙β
∧
|α|=m
(3.20)
For I2 , by the formula (see [4])
Rm Ã; x, y − Rm Ã; x0 , y =
1
Rm−|η| D η Ã; x, x0 (x − y)η
η!
|η|<m
(3.21)
and Lemma 3.6, we get
D α A |Q|β/n x − x0 x0 − y m−1 .
Rm Ã; x, y − Rm Ã; x0 , y ≤ C
˙β
∧
|α|=m
(3.22)
Thus, similar to the proof of I1 ,
Rm Ã; x, y − Rm Ã; x0 , y f (y)dy
m+n
x0 − y Rn \Q̃
∞ x − x0 α β/n
D A ∧˙ β |Q|
≤C
n+1 f (y)dy
k+1 Q̃\2k Q̃ 2
x0 − y
|α|=m
k=0
α β/n
≤C
D A ∧˙ β |Q| M(f )(x).
I2 ≤ C
(3.23)
|α|=m
For I3 , by Lemma 3.5, we get
α
D A(y) − D α A ≤ D α A x0 − y β .
˙β
∧
Q̃
(3.24)
Thus, similar to the proof of I1 , we obtain
x − x0 ε
x − x0 n+ε f (y)D α Ã(y)dy
n+1 + n
R \Q̃
x0 − y
x0 − y
I3 ≤ C
|α|=m
≤C
D α A |α|=m
≤C
˙ β |Q|
∧
D α A |α|=m
˙ β |Q|
∧
β/n
∞
k(β−1)
+ 2k(β−ε) M(f )(x)
2
(3.25)
k=1
β/n
M(f )(x).
So,
Ã
D α A |Q|β/n M(f )(x).
F (f )(x) − F Ã (f ) x0 ≤ C
˙β
t
t
∧
|α|=m
(3.26)
267
BOUNDEDNESS OF MULTILINEAR OPERATORS . . .
A
, we write
For µΩ
Ã
F (f )(x) − F Ã (f ) x0 t
t

≤
∞
Ω(x − y)Rm Ã; x, y
f (y)dy
|x − y|m+n−1
0 |x−y|≤t
−

+C
|x0 −y|≤t

|α|=m
1/2
2
dt
Ω x0 − y Rm Ã; x0 , y
f (y)dy 3 
t
x0 − y m+n−1
∞
Ω(x − y)(x − y)α
|x − y|m+n−1
0 |x−y|≤t
−
|x0 −y|≤t
α Ω x0 − y x0 − y
x0 − y m+n−1
1/2
2
dt
× D Ã(y)f (y)dy 3 
t
α

≤
∞ |x−y|≤t, |x0 −y|>t
0

+

+
1/2
2
Ω(x − y)Rm Ã; x, y f (y)dy dt 
|x − y|m+n−1
t3
∞ |x−y|>t, |x0 −y|≤t
0
∞ 0
1/2
2
Ω x0 − y Rm Ã; x0 , y dt
f (y)dy

x0 − y m+n−1
t3
Ω(x − y)R Ã; x, y Ωx − y R Ã; x , y m
0
m
0
−
x0 − y m+n−1
|x − y|m+n−1
|x−y|≤t, |x0 −y|≤t × f (y)dy
+C
|α|=m


2
1/2
dt 
t3
∞
Ω(x − y)(x − y)α
|x − y|m+n−1
0 |x−y|≤t
−
|x0 −y|≤t
α Ω x0 − y x0 − y
m+n−1
x0 − y 1/2
2
dt
× D Ã(y)f (y)dy 3 
t
α
:= J1 + J2 + J3 + J4 .
(3.27)
268
LIU LANZHE
Thus
f (y)Rm Ã; x, y J1 ≤ C
dt
t3
1/2
dy
|x − y|m+n−1
|x−y|≤t<|x0 −y|
f (y)Rm Ã; x, y x0 − x 1/2
dy
≤C
|x − y|m+n−1
|x − y|3/2
Rn \Q̃
∞
−1
D α A |Q|β/n
f (y)dy
≤C
2−k/2 2k Q̃
Rn \Q̃
˙β
∧
|α|=m
(3.28)
2k Q̃
k=1
D α A |Q|β/n M(f )(x).
≤C
˙β
∧
|α|=m
Similarly, we have J2 ≤ C |α|=m D α A∧˙ β |Q|β/n M(f )(x).
For J3 , by the inequality (see [11])
Ω(x − y)
x − x0 γ
x − x0 Ω x0 − y
≤C −
+
,
|x − y|m+n−1 x0 − y m+n x0 − y m+n−1 x0 − y m+n−1+γ
(3.29)
we obtain
x − x0 x − x0 γ
n + x0 − y n−1+γ
Rn \Q̃ x0 − y D α A |Q|β/n
˙β
∧
J3 ≤ C
|α|=m
×
D α A |Q|β/n
≤C
˙β
∧
|α|=m
∞
|x0 −y|≤t, |x−y|≤t
dt
t3
1/2
f (y)dy
(3.30)
−k
2 + 2−γk M(f )(x)
k=1
D α A |Q|β/n M(f )(x).
≤C
˙β
∧
|α|=m
For J4 , similar to the proof of J1 , J2 , and J3 , we obtain
J4 ≤ C
|α|=m
≤C
x − x0 x − x0 1/2
x − x0 γ
+
n+1 + x0 − y n+1/2 x0 − y n+γ
Rn \Q̃ x0 − y × D α Ã(y)f (y)dy
D α A ˙ β |Q|
∧
|α|=m
×
∞
k=1
β/n
1
2k(β−1) + 2k(β−1/2) + 2k(β−γ) 2k Q̃
D α A |Q|β/n M(f )(x).
≤C
˙β
∧
|α|=m
(3.31)
2k Q̃
f (y)dy
BOUNDEDNESS OF MULTILINEAR OPERATORS . . .
269
A
, we write
For Bδ,∗
Ã
Ã
Bδ,t
(f )(x) − Bδ,t
(f ) x0
Btδ (x − y) Btδ x0 − y
m Rm Ã; x, y f (y)dy
=
−
m
n
|x − y|
x0 − y
R \Q̃
δ
Bt x0 − y m Rm Ã; x, y − Rm Ã; x0 , y f (y)dy
+
n
R \Q̃ x0 − y
α 1 Btδ (x − y)(x − y)α Btδ x0 − y x0 − y
−
−
x0 − y m
α! Rn \Q̃
|x − y|m
(3.32)
|α|=m
× D α Ã(y)f (y)dy
= L1 + L2 + L3 .
We consider the following two cases.
Case 1 (0 < t ≤ l). In this case, notice that (see [8])
δ
B (z) ≤ c 1 + |z| −(δ+(n+1)/2) .
(3.33)
We obtain
L1 ≤ Ct −n
Rn \Q̃
f (y)Rm Ã; x, y −(δ+(n+1)/2)
m
1 + |x − y|/t
dy
x0 − y D α A |Q|β/n (t/l)δ−(n−1)/2
≤C
˙β
∧
|α|=m
×
∞
k=1
2k((n−1)/2−δ) 1
2k Q̃
2k Q̃
f (y)dy
D α A |Q|β/n M(f )(x),
≤C
˙β
∧
|α|=m
L2 ≤ Ct −n
Rn \Q̃
f (y)Rm Ã; x, y − Rm Ã; x0 , y m
x0 − y −(δ+(n+1)/2)
× 1 + |x − y|/t
dy
≤C
D α A ˙ β |Q|
∧
|α|=m
×
∞
k=1
β/n
(t/l)δ−(n−1)/2
2k((n−1)/2−δ) 1
2k Q̃
D α A |Q|β/n M(f )(x).
≤C
˙β
∧
|α|=m
2k Q̃
f (y)dy
(3.34)
270
LIU LANZHE
For L3 , similar to the proof of L1 , we get
L 3 ≤ C
D α A |Q|β/n (t/l)δ−(n−1)/2
˙β
∧
|α|=m
×
∞
k=1
2k(β−δ+(n−1)/2) D α A ≤C
˙ β |Q|
∧
|α|=m
β/n
1
2k Q̃
2k Q̃
f (y)dy
(3.35)
M(f )(x).
Case 2 (t > l). In this case, we choose δ0 such that β + (n − 1)/2 < δ0 < min(δ, (n +
1)/2). Notice that (see [8])
(∂/∂z)B δ (z) ≤ C 1 + |z| −(δ+(n+1)/2) .
Similar to the proof of Case 1, we obtain
L1 ≤ Ct −n
Rn \Q̃
f (y)Rm Ã; x, y x0 − y m+1
−(δ0 +(n+1)/2)
× x0 − x 1 + x0 − y /t
dy
f (y)Rm Ã; x, y + Ct −n−1
x0 − y m
Rn \Q̃
−(δ0 +(n+1)/2)
× x0 − x 1 + x0 − y /t
dy
≤C
D α A |Q|β/n (l/t)(n+1)/2−δ0
˙β
∧
|α|=m
×
∞
k=1
2k((n−1)/2−δ0 ) 1
2k Q̃
2k Q̃
f (y)dy
D α A |Q|β/n M(f )(x),
≤C
˙β
∧
|α|=m
L2 ≤ Ct −n
≤C
f (y)Rm Ã; x, y − Rm Ã; x0 , y m
x0 − y Rn \Q̃
−(δ0 +(n+1)/2)
× 1 + x0 − y /t
dy
D α A |Q|β/n (l/t)(n+1)/2−δ0
˙β
∧
|α|=m
×
∞
k=1
2k((n−1)/2−δ0 ) 1
2k Q̃
D α A |Q|β/n M(f )(x),
≤C
˙β
∧
|α|=m
2k Q̃
f (y)dy
(3.36)
BOUNDEDNESS OF MULTILINEAR OPERATORS . . .
271
D α A |Q|β/n (l/t)(n+1)/2−δ0
L 3 ≤ C
˙β
∧
|α|=m
×
∞
k=1
2k(β+(n−1)/2−δ0 ) 1
2k Q̃
2k Q̃
f (y)dy
D α A |Q|β/n M(f )(x).
≤C
˙β
∧
|α|=m
(3.37)
These yield the desired results.
Acknowledgment. This work was supported by the National Natural Science Foundation (NNSF) Grant 10271071.
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Liu Lanzhe: Department of Applied Mathematics, Hunan University, Changsha 410082, China
E-mail address: lanzheliu@263.net