IJMMS 2003:2, 87–96
PII. S0161171203201150
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ
OPERATORS ON CERTAIN HARDY SPACES
LIU LANZHE
Received 14 January 2002
The boundedness for the multilinear Marcinkiewicz operators on certain Hardy
and Herz-Hardy spaces are obtained.
2000 Mathematics Subject Classification: 42B25, 42B20.
1. Introduction and definitions. Suppose that S n−1 is the unit sphere of Rn
(n ≥ 2) equipped with normalized Lebesgue measure dσ = dσ (x ). Let Ω be
homogeneous of degree zero and satisfy the following two conditions:
(i) Ω(x) is continuous on S n−1 and satisfies the Li pγ condition on S n−1
(0 ≤ γ ≤ 1), that is,
Ω x − Ω y ≤ M x − y γ ,
x , y ∈ S n−1 ;
(1.1)
(ii) S n−1 S(x )dx = 0.
Let m be a positive integer and A be a function on Rn . The multilinear
Marcinkiewicz integral operator is defined by
A
(f )(x) =
µΩ
∞
0
F A (f )(x)2 dt
t
t3
1/2
,
(1.2)
where
Ω(x − y) Rm+1 (A; x, y)
f (y)dy,
|x
− y|n−1
|x − y|m
|x−y|≤t
1
D α A(y)(x − y)β .
Rm+1 (A; x, y) = A(x) −
α!
|α|≤m
FtA (f )(x) =
(1.3)
We denote that Ft (f )(x) = f|x−y|≤t (Ω(x − y)/|x − y|n−1 )f (y)dy. We also
denote that
µΩ (f )(x) =
Ft (f )(x)2 dt
t3
∞
0
1/2
,
which is the Marcinkiewicz integral operator (see [5, 6, 12]).
(1.4)
88
LIU LANZHE
A
Note that when m = 0, µΩ
is just the commutator of Marcinkiewicz operator
(see [5, 12]). It is well known that multilinear operators are of great interest in
harmonic analysis and have been widely studied by many authors (see [1, 2,
3, 4, 5]). The main purpose of this paper is to consider the continuity of the
multilinear Marcinkiewicz operators on certain Hardy and Herz-Hardy spaces.
We first introduce some definitions (see [7, 8, 9, 10, 11]).
Definition 1.1. Let A be a function on Rn , m a positive integer, and 0 <
p ≤ 1. A bounded measurable function a on Rn is said to be a (p, D m A)-atom
if
(i) supp a ⊂ B = B(x0 , r ),
(ii) aL∞ ≤ |B|−1/p ,
(iii) a(y)dy = a(y)D α A(y)dy = 0, |α| = m.
p
A temperate distribution f is said to belong to HDm A (Rn ), if, in the Schwartz
distributional sense, it can be written as
f (x) =
∞
λj aj (x),
(1.5)
j=0
where aj ’s are (p, D m A)-atoms, λj ∈ C, and
∞
f H p m (Rn ) ∼ ( j=0 |λj |p )1/p .
D
∞
j=0 |λj |
p
< ∞. Moreover,
A
Let Bk = {x ∈ Rn : |x| ≤ 2k }, Ck = Bk \ Bk−1 , k ∈ Z, and mk (λ, f ) = |{x ∈
Ck : |f (x)| > λ}|; for k ∈ N, let m̃k (λ, f ) = mk (λ, f ) and m̃0 (λ, f ) = |{x ∈ B0 :
|f (x)| > λ}|.
Definition 1.2. Let 0 < p, q < ∞, and α ∈ R.
(1) The homogeneous Herz space is defined by
α,p
K̇q
q = f ∈ Lloc Rn \ {0} : f K̇qα,p (Rn ) < ∞ ,
(1.6)
where

f K̇qα,p (Rn )
=
∞
2
1/p
p
f χk Lq  .
kαp (1.7)
k=−∞
(2) The nonhomogeneous Herz space is defined by
q Rn = f ∈ Lloc Rn : f Kqα,p (Rn ) < ∞ ,
α,p Kq
(1.8)
where

f Kqα,p (Rn ) = 
∞
k=1
2
1/p
p p
f χk Lq + f χB0 Lq  ,
kαp (1.9)
89
BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ OPERATORS . . .
where

f W Kqα,p (Rn )
= sup λ
λ>0
∞
1/p
2
kαp
p/q 
m̃k (λ, f )
.
(1.10)
k=0
Definition 1.3. Let m be a positive integer and A a function on Rn , α ∈ R,
and 1 < q ≤ ∞. A function a(x) on Rn is called a central (α, q, D m A)-atom (or
a central (α, q, D m A)-atom of restrict type), if
(1) supp a ⊂ B(0, r ) for some r > 0 (or for some r ≥ 1),
(2) aLq ≤ |B(0, r )|−α/n ,
(3) a(x)dx = a(x)D β A(x)dx = 0, |β| = m.
α,p
α,p
A temperate distribution f is said to belong to H K̇q,Dm A (Rn ) (or HKq,Dm A (Rn ))
∞
∞
if it can be written as f = j=−∞ λj aj (or f = j=0 λj aj ) in the S (Rn ) sense,
where aj is a central (α, q, D m A)-atom (or a central (α, q, D m A)-atom of restrict
∞
∞
type) supported on B(0, 2j ) and j=−∞ |λj |p < ∞ (or j=0 |λj |p < ∞). Moreover,
f H K̇ α,pm (or f HK α,pm ) ∼ ( j |λj |p )1/p .
q,D
A
q,D
A
2. Theorems and proofs. We begin with some preliminary lemmas.
Lemma 2.1 (see [2]). 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)
Q̃(x,y)
α
D A(z)q dz
1/q
,
(2.1)
√
where Q̃ is the cube centered at x and having side length 5 n|x − y|.
Lemma 2.2. Let 1 < p < ∞ and D α A ∈ Lr (Rn ), |α| = m, 1 < r ≤ ∞, and
A
is bound from Lp (Rn ) to Lq (Rn ), that is,
1/q = 1/p + 1/r . Then, µΩ
A
D α A r f Lp .
µ (f ) q ≤ C
Ω
L
L
(2.2)
|α|=m
Proof. By Minkowski inequality and the condition of Ω, we have
f (y)Rm+1 (A; x, y) ∞
A
(f )(x) ≤
µΩ
Rn
≤C
Rn
|x − y|m
|x−y|
Rm+1 (A; x, y) f (y)dy.
m+n
|x − y|
Thus, the lemma follows from [3, 4].
dt
t3
1/2
dy
(2.3)
90
LIU LANZHE
Theorem 2.3. Let 1 ≥ p > n/(n + γ), and let D β A ∈ BMO(Rn ) for |β| = m.
p
A
is bounded from HDm A (Rn ) to Lp (Rn ).
Then, µΩ
Proof. It suffices to show that there exists a constant c > 0 such that, for
every (p, D m A)-atom a,
A
µ (a)
Ω
Lp
≤ C.
(2.4)
Let a be a (p, D m A)-atom supported on a ball B = B(x0 , r ). We write
Rn
p
A
µΩ (a)(x) dx =
|x−x0 |≤2r
p
A
(a)(x) dx
µΩ
+
|x−x0 |>2r
A
p
µΩ (a)(x) dx
(2.5)
≡ I + II.
A
For I, taking q > 1 and by Hölder’s inequality and the Lq -boundedness of µΩ
(see Lemma 2.2), we see that
p A
1−p/q
(a)Lq · B x0 , 2r I ≤ C µΩ
p
≤ CaLq |B|1−p/q
(2.6)
≤ C.
A
To obtain the estimate of II, we need to estimate µΩ
(a)(x) for x ∈ (2B)c . Let
√
α
B̃ = 5 nB, and let Ã(x) = A(x)− |α|=m (1/α!)(D A)B̃ ·x α . Then, Rm (A; x, y) =
Rm (Ã; x, y). By the vanishing moment of a, we write
FtA (a)(x) =
|x−y|≤t
+
−
|x−y|≤t
1
α!
|α|=m
Ω x − x0
Ω(x − y)
Rm Ã; x, y a(y)dy
−
|x − y|m+n−1 x − x0 m+n−1
Ω x − x0
m+n−1 Rm Ã; x, y − Rm Ã; x, x0 a(y)dy
x − x0 |x−y|≤t
Ω(x − y)(x − y)α α
D A(y) − D α A B a(y)dy,
m+n−1
|x − y|
(2.7)
BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ OPERATORS . . .
91
thus,

A
(a)(x)
µΩ
Ω(x − y)
Ω x − x0
−
m+n−1 m+n−1
x − x0 0
|x−y|≤t |x − y|
1/2
2
dt 
a(y) dy
× Rm Ã; x, y
t3

∞ Ω x − x0 +
m+n−1
0
|x−y|≤t x − x0 ≤
∞ 
+
2 1/2
dt 
× Rm Ã; x, y − Rm Ã; x, x0
a(y) dy
t3
∞
1 Ω(x − y)(x − y)α
|x − y|m+n−1
0 |α|=m α! |x−y|≤t
α
α
× D A(y) − D A
B
1/2
2
dt
a(y)dy 3 
t
≡ II 1 + II 2 + II 3 .
(2.8)
By Lemma 2.1, for y ∈ B and x ∈ 2k+1 B \ 2k B, we know
D α A(x) − D α A k .
Rm Ã; x, y ≤ C|x − y|m
2 B
(2.9)
|α|=m
By the condition of Ω and Minkowski’s inequality, and noting that |x − y| ∼
|x − x0 | for y ∈ B and x ∈ Rn \ B, we obtain


Ω(x − y)
Ω x − x0
rγ
r
.

≤C −
+
|x − y|m+n−1 x − x0 m+n x − x0 m+n−1 x − x0 m+n+γ−1
(2.10)
Thus,
II 1 ≤ C
B
Rm Ã; x, y a(y)

∞
|x−y|
1/2
dt
t3

r
rγ
dy
× 
+
x − x0 m+n x − x0 m+n+γ−1


γ
r
r
α
|B|1−1/p
D A(x) − D α A 2k B .
+
≤ C x − x n+1 x − x0 n+γ
0
|α|=m
(2.11)
92
LIU LANZHE
On the other hand, by the following formula (see [2]):
Rm Ã; x, y − Rm Ã; x, x0 =
β
1
Rm−|β| D β Ã; y, x0 x − x0
β!
|β|<m
(2.12)
and Lemma 2.1, we get
Rm Ã; x, y − Rm Ã; x, x0 x0 − y m−|β| x − x0 |β| D α A
≤C
BMO ,
(2.13)
|β|<m |α|=m
so that
II 2 ≤ C
B
≤C
B
x − x0 −(n+m)
Rm−|β| D β Ã; y, x0 x − x0 |β| a(y)dy
|β|<m
x − x0 −(n+m)
x0 − y m−|β| x − x0 |β|
|β|<m
D α A ×
BMO a(y) dy
|α|=m
x0 − y α a(y)dy
D A BMO
≤C
x − x0 n+1
B
|α|=m
−n−1 1/n−1/p+1
D α A |B|
.
≤C
BMO x − x0
|α|=m
(2.14)
For II 3 , and by the vanishing moment of a, we write,
Ω(x − y)(x − y)α α
D A(y) − D α A B a(y)dy
m
|x − y|
α Ω(x − y)(x − y)α Ω x − x0 x − x0
−
=
D α A(y) − D α A B a(y)dy.
m+n−1
m
x − x0 |x − y|
(2.15)
Similar to the estimate of II 1 , we obtain

r
rγ



+
x − x0 n+1 x − x0 n+γ
× x0 − y D α A(y) − D α A B a(y)dy
B


γ
r
r
α
1−1/p
D A 
.
+
≤C
BMO |B|
x − x0 n+1 x − x0 n+γ
II 3 ≤ C
|α|=m
|α|=m
(2.16)
93
BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ OPERATORS . . .
Recalling that p > n/(n + γ), therefore,
II ≤
∞ k=1
≤C
2k+1 B\2k B
∞ k=1
A
p
µΩ (a)(x) dx

p
rγ
r

 |B|p−1
+
x − x0 n+1 x − x0 n+γ
2k+1 B\2k B

p
α α
D A(x) − D A 2k+1 B  dx
×
|α|=m
p

D α A + C
BMO
∞ 
|α|=m

≤ C
D α A p
BMO

|α|=m

≤ C
D α A p
BMO
k=1
2k+1 B\2k B
x − x0 −p(n+1) |B|p(1+1/n−1/p) dx
∞
k(n−p−pn)
+ 2k(n−pn−pγ)
2
k=1
 ,
|α|=m
(2.17)
which, together with the estimate for I, yields the desired result. This finishes
the proof of Theorem 2.3.
Theorem 2.4. Let 0 < p < ∞, 1 < q < ∞, n(1 − 1/q) ≤ α < n(1 − 1/q) + γ,
α,p
A
is bounded from H K̇q,Dm A (Rn ) to
and D β A ∈ BMO(Rn ) for |β| = m. Then, µΩ
α,p
K̇q (Rn ).
∞
α,p
Proof. Let f ∈ H K̇q,Dm A (Rn ) and f (x) = j=−∞ λj aj (x) be the atomic
decomposition for f as in Definition 1.3. We write
A
µ (f )
Ω

α,p
K̇q
(Rn )
≤ C
∞

2
kαp 
k=−∞

∞
+ C
k−3
p 1/p
A λj µ aj χk q  
Ω
L
j=−∞

2
kαp 
k=−∞
p 1/p
∞
A λj µ aj χk q  
Ω
(2.18)
L
j=k−2
≡ I + II.
A
on Lq (Rn ) (see Lemma 2.2), we have
For II, and by the boundedness of µΩ

II ≤ C 
∞
k=−∞

2
kαp 
p 1/p
∞
λj aj q  
L
j=k−2
94
LIU LANZHE

≤ C
∞

2
p 1/p
∞
−jα
λ j 2
 
kαp 
k=−∞
j=k−2

1/p
∞
∞


p −jαp


kαp
 ,


2
0<p≤1
λj 2



k=−∞
j=k−2
≤C 


p/p 1/p

∞
∞
∞


p −jαp/2
/2
 kαp  −jαp

 , p>1




2
2
λj 2


k=−∞
j=k−2
j=k−2



1/p

∞

p j+2
p −jαp


(k−j)αp
 ,




2
λj
λj 2



j=−∞
k=−∞

1/p
≤C 

j+2
∞


p

(k−j)αp/2 


2
,
λj


j=−∞

≤ C
0<p≤1
p>1
k=−∞
1/p
∞
p
λ j 
≤ Cf H K̇ α,pm
q,D
j=−∞
A
(Rn ) .
(2.19)
For I, and similar to the proof of Theorem 2.3, we have, for x ∈ Ck , j ≤ k − 3,
1/n
γ/n A
µΩ
aj (x) ≤ C |x|−n−m−1 Bj + |x|−n−m−γ Bj ×
Bj
aj (y)Rm Ã; x, y dy
D β A +C
1/n
−n−1 Bj BMO |x|
|β|=m
Bj
a(y)dy
1+1/n
1+γ/n + |x|−n−γ Bj + C |x|−n−1 Bj β
D A(y) − D β A Bj a(y)dy
×
|β|=m
Bj
≤ C 2−k(n+1) 2j(1+n(1−1/q)−α) + 2−k(n+γ) 2j(γ+n(1−1/q)−α)

D β A(x) − D β A Bk 
×

|β|=m
+C
D α A BMO (k − j)
|α|=m
× 2−k(n+1) 2j(1+n(1−1/q)−α) + 2−k(n+γ) 2j(γ+n(1−1/q)−α)
(2.20)
BOUNDEDNESS FOR MULTILINEAR MARCINKIEWICZ OPERATORS . . .
95
thus,

∞
I ≤ C

2
k−3
kαp 
k=−∞
−k(n+1)+j(1+n(1−1/q)−α)
λ j 2
+ 2−k(n+γ)+j(γ+n(1−1/q)−α)
j=−∞
×
|α|=m


∞
+ C
2
kαp 
k=−∞
k−3
Bk
α
q
D A(x) − D α A Bk dx
1/q p 1/p
 
λj (k − j) 2−k(n+1)+j(1+n(1−1/q)−α)
j=−∞
p 1/p
D α A  
+ 2−k(n+γ)+j(γ+n(1−1/q)−α) 2kn/q
BMO
|α|=m
≡ I1 + I2 .
(2.21)
To estimate I1 and I2 , we consider two cases.
Case 1 (0 < p ≤ 1). We have

I1 ≤ C 
∞
2kαp
k=−∞
k−3
|λj |p 2[−k(n+1)+j(1+n(1−1/q)−α)]p
j=−∞
+2
=C
D β A BMO
|β|=m
[−k(n+γ)+j(γ+n(1−1/q)−α)]p



p 1/p
knp/q β D A 
 
2
BMO
|β|=m
∞
j=−∞
p
λ j ∞
k=j+3
(j−k)(1+n(1−1/q)−α)p
2
1/p
+ 2(j−k)(γ+n(1−1/q)−α)p 

1/p
∞
p
β
D A λ j 

≤C
BMO
|β|=m
j=−∞
≤ Cf H K̇ α,pm
q,D
A
(Rn ) .
(2.22)
Similarly,
I2 ≤ Cf H K̇ α,pm
q,D
A
(Rn ) .
(2.23)
96
LIU LANZHE
Case 2 (p > 1). By Hölder’s inequality, we deduce that



∞
k−3
p
β (j−k)p(γ+n(1−1/q)−α)/2


I1 ≤ C
D A BMO 
λj 2
|β|=m
j=−∞
j=−∞

×

≤ C
1/p
∞
p
λ j 
j=−∞
I2 ≤ Cf H K̇ α,pm
q,D
A
k−3
p/p 1/p
2
(j−k)p (γ+n(1−1/q)−α)/2 
≤ Cf H K̇ α,pm
q,D

(2.24)
j=−∞
A
(Rn ) ,
(Rn ) .
This finishes the proof of Theorem 2.
Remark 2.5. Theorem 2.4 also holds for nonhomogeneous Herz-type space.
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Liu Lanzhe: Department of Applied Mathematics, Hunan University, Changsha
410082, China
E-mail address: lanzheliu@263.net