Paper-10-Acta24.2010. 141KB Jun 04 2011 12:03:09 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 24/2010
pp. 95-107

BOUNDEDNESS FOR MULTILINEAR COMMUTATOR OF
LITTLEWOOD-PALEY OPERATOR ON HARDY AND
HERZ-HARDY SPACES

Jiasheng Zeng
, K̇qα,p ) type boundedness
Abstract. In this paper, the (H~p , Lp ) and (H K̇ α,p
b
q,~b
for the multilinear commutator associated with the Littlewood-Paley operator are
obtained.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Introduction and definition
Let 0 < q < ∞ and Lqloc (Rn ) = {f q is locally integrable on Rn }. Suppose
f ∈ L1loc (Rn ), B = B(x0 , r) = {x ∈ Rn : |x − x0 | < Rr} denotes a ball of Rn

centered at xR0 and having radius r, write fB = |B|−1 B f (x)dx and f # (x) =
supx∈B |B|−1 B |f (x) − fB |dx < ∞. f is said to belong to BM O(Rn ), if f # ∈
L∞ (Rn ) and define ||f ||BM O = ||f # ||L∞ .
Let T be the Calderón-Zygmund singular integral operator and b ∈ BM O(Rn ).
The commutator [b, T ] generated by b and T is defined by
[b, T ](f )(x) = b(x)T (f )(x) − T (bf )(x).
A classical result of Coifman, Rochberg and Weiss (see[2]) proved that the commutator [b, T ] is bounded on Lr (Rn ) for any 1 < r < ∞. However, it was observed
that the [b, T ] is not bounded, in general, from H p (Rn ) to Lp (Rn ) when 0 < p ≤ 1.
But if H p (Rn ) is replaced by a suitable atomic space H~p (Rn )(see [1][7][12]), then
b
[b, T ] maps continuously H~p (Rn ) into Lp (Rn ), and a similar results holds for Herzb
type spaces. In addition we have easily known that H~p (Rn ) ⊂ H p (Rn ). The main
b
purpose of this paper is to consider the continuity of the multilinear commutators
related to the Littlewood-Paley operators and BM O(Rn ) functions on certain Hardy
and Herz-Hardy spaces. Let us first introduce some definitions(see [1][3-10][12][13]).
Definition 1. Let ε > 0, n > δ > 0 and ψ be a fixed function which satisfies the
following properties:
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J. Zeng - Boundedness for multilinear commutator of Littlewood-Paley operator...

R
(1)
Rn ψ(x)dx = 0,
(2) |ψ(x)| ≤ C(1 + |x|)−(n+1−δ) ,
(3) |ψ(x + y) − ψ(x)| ≤ C|y|ε (1 + |x|)−(n+1+ε−δ) when 2|y| < |x|;
The Littlewood-Paley multilinear commutator is defined by
"Z Z
#1/2
~b
~b
2 dydt
|Ft (f )(x, y)| n+1
Sδ (f )(x) =
,
t
Γ(x)
where
~

Ftb (f )(x, y) =

Z

Rn

n+1
R+




m
Y

j=1



(bj (x) − bj (z)) ψt (y − z)f (z)dz,

Γ(x) = {(y,R t) ∈
: |x − y| < t} and ψt (x) = t−n+δ ψ(x/t) for t > 0. Set
Ft (f )(y) = Rn ψt (y − x)f (x)dx. We also define that
Sδ (f )(x) =

dydt
|Ft (f )(x, y)|2 n+1
t
Γ(x)

Z Z

!1/2

,

which is the Littlewood-Paley operator with δ = 0(see [15]).
Given a positive integer m and 1 ≤ j ≤ m, we denote by Cjm the family of all

finite subsets σ = {σ(1), · · ·, σ(j)} of {1, · · ·, m} of j different elements. For σ ∈ Cjm ,
set σ c = {1, · · ·, m} \ σ. For ~b = (b1 , · · ·, bm ) and σ = {σ(1), · · ·, σ(j)} ∈ Cjm , set
~bσ = (bσ(1) , · · ·, bσ(j) ), bσ = bσ(1) · · · bσ(j) and ||~bσ ||BM O = ||bσ(1) ||BM O · · · ||bσ(j) ||BM O .
Definition 2. Let bi (i = 1, · · · , m) be a locally integrable functions and 0 < p ≤
1. A bounded measurable function a on Rn is called a (p, ~b) atom, if
(1) supp a ⊂ B = B(x0 , r)
−1/p
(2) R||a||L∞ ≤ |B(x
R 0 , r)| Q
m
(3)
l∈σ bl (y)dy = 0 for any σ ∈ Cj , 1 ≤ j ≤ m .
B a(y)dy = B a(y)
A temperate distribution(see [14][15]) f is said to belong to H~p (Rn ), if, in the
b
Schwartz distribution sense, it can be written as
f (x) =

∞
X

λj aj (x).

j=1

P∞
p
p
where a′j s are (p, ~b) atoms, λj ∈ C and
j=1 |λj | < ∞. Moreover, ||f ||H~ ≈
b
P∞
( j=1 |λj |p )1/p .
Definition 3. Let 0 < p, q < ∞, α ∈ R. For k ∈ Z, set Bk = {x ∈ Rn : |x| ≤ 2k }
and Ck = Bk \ Bk−1 ,andχk = χCk for k ∈ Z,where χCk is the characteristic function
of set Ck . Denote the characteristic function of B0 by χ0 .
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J. Zeng - Boundedness for multilinear commutator of Littlewood-Paley operator...

(1) The homogeneous Herz space is defined by
n
o
K̇qα,p (Rn ) = f ∈ Lqloc (Rn \ {0}) : ||f ||K̇qα,p < ∞ .
where

||f ||K̇qα,p =

"

∞
X

2kαp ||f χk ||pLq

k=−∞

#1/p

.

(2) The nonhomogeneous Herz space is defined by
n
o
Kqα,p (Rn ) = f ∈ Lqloc (Rn ) : ||f ||Kqα,p < ∞ .

where

||f ||Kqα,p =

"

∞
X

2kαp ||f χk ||pLq + ||f χ0 ||pLq

k=1

#1/p

.

Definition 4. Let α ∈ Rn , 1 < q < ∞, α ≥ n(1 − 1/q), bi ∈ BM O(Rn ), 1 ≤
i ≤ m. A function a(x) is called a central (α, q, ~b) -atom (or a central (α, q, ~b)-atom
of restrict type ), if
(1) supp a ⊂ B = B(x0 , r)(or for some r ≥ 1),
−α/n ,
(2) ||a||
R
R Lq ≤ β|B(x0 , r)|
Q
m
β
(3)
i∈σ bi (x)dx = 0 for any σ ∈ Cj , 1 ≤ j ≤ m,
B a(x)x dx = B a(x)x
0 ≤ |β| ≤ P
α, where β = (β1 , ..., βn ) is the multi-indices with βi ∈ N for 1 ≤ i ≤ n
and |β| = ni=1 βi .

(Rn )), if it
(Rn )(or HK α,p
A temperate distribution f is said to belong to H K̇ α,p
q,~b
q,~b
P∞
P∞
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, ~b)-atom(or a central
(α, q, ~b)-atom of restrict type ) supported
P
P
∞
∞
on B(0, 2j ) and −∞ |λj |p < ∞(or j=0 |λj | < ∞). Moreover,
X
||f ||H K̇ α,p ( or ||f ||HK α,p ) = inf(
|λj |p )1/p ,
q,~

b

q,~
b

j

where the infimum are taken over all the decompositions of f as above.
2. Theorems and Proofs
To prove the theorems, we need the following lemmas.
Lemma 1.(see [12]) Let 1 < p < q < n/α, 1/q = 1/p − α/n. Then Sδ is bounded
from Lp (Rn ) to Lq (Rn ).
~
Lemma 2.(see [12]) Let 1 < p < q < n/α, 1/q = 1/p − α/n. Then Sδb is bounded
p
n
q
n
from L (R ) to L (R ).
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J. Zeng - Boundedness for multilinear commutator of Littlewood-Paley operator...

Theorem 1. Let ε > 0, n/(n + ε − δ) < p ≤ 1, 1/q = 1/p − δ/n, bi ∈ BM O, 1 ≤
~
i ≤ m, ~b = (b1 , · · · , bm ). Then the multilinear commutator Sδb is bounded from
p
H~ (Rn ) to Lq (Rn ).
b
Proof. It suffices to show that there exist a constant C > 0, such that for every
~
(p, b) atom a,
~
||Sδb (a)||Lq ≤ C.
Let a be a (p, ~b) atom supported on a ball B = B(x0 , d). When m = 1 see [7], and
now we prove m > 1. Write
Z
Z
Z
~
~
~
|Sδb (a)(x)|q dx =
|Sδb (a)(x)|q dx +
|Sδb (a)(x)|q dx = I + II.
Rn

|x−x0 |≤2d

|x−x0 |>2d

For I, taking r, s > 1 with q < s < n/δ and 1/r = 1/s − δ/n, by Hölder’s
~
inequality and the (Ls , Lr )-boundedness of Sδb , we have
I ≤

Z

|x−x0 |≤2d

!q/r

~
|Sδb (a)(x)|r dx

· |B(x0 , 2d)|1−q/r

~

≤ C||Sδb (a)(x)||qLr · |B(x0 , d)|1−q/r
≤ C||a||qLs |B|1−q/r
≤ C.
For II, denoting λ = (λ1 , · · · , λm ) with λi = (bi )B , 1 ≤ i ≤ m, where (bi )B =

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J. Zeng - Boundedness for multilinear commutator of Littlewood-Paley operator...
|B(x0 , r)|−1

R

B(x0 ,r) bi (x)dx,

∞ Z
X

II =

~

2k+1 r≥|x−x

k=1
∞
X

0

|>2k d

|Sδb (a)(x)|q dx
Z

|B(x0 , 2k+1 d)|1−q

≤ C

≤ C

by the vanishing moment of a, we get

k=1
∞
X

2k+1 d≥|x−x0 |>2k d

!q

~
|Sδb (a)(x)|dx

|B(x0 , 2k+1 d)|1−q

k=1


Z

× 

2k+1 d≥|x−x

≤ C

∞
X

0

|>2k d

k+1

|B(x0 , 2

k=1



× 




Z Z

1−q

d)|

|

Γ(x)

Z Y
m

(bj (x) − bj (z))ψt (y − z)a(z)dz|2

B j=1

Z
[

1/2

dydt 
tn+1

q


dx

2k+1 d≥|x−x0 |>2k d

2

1/2
Z
m
Y
dydt
 |ψt (y − z) − ψt (y − x0 )|
|(bj (x) − bj (z))||a(z)|dz  n+1  dx]q ;
t
Γ(x)
B

Z Z

j=1

noting that z ∈ B, x ∈ B(x0 , 2k+1 d) \ B(x0 , 2k d), then
~

Sδb (a)(x)


2
1/2
Z Z
Z
m
Y
dydt
 |ψt (y − z) − ψt (y − x0 )|
= 
|bj (x) − bj (z)||a(z)|dz  n+1 
t
Γ(x)
B
j=1


Z Z

≤ C

Γ(x)

≤ C

Z Z

Γ(x)

≤ C

Z Z

Γ(x)




Z

B

t−n+δ |a(z)|

m
Y

j=1

2
1/2
dydt
(|x0 − z|/t)ε
dz  n+1 
|bj (x) − bj (z)|
t
(1 + |x0 − y|/t)n+1+ε−δ

t1−n
dydt
(t + |x0 − y|)2(n+1+ε−δ)

!1/2 Z

t1−n 22(n+1+ε−δ)
dydt
(2t + 2|x0 − y|)2(n+1+ε−δ)

m
Y

|bj (x) − bj (z)||x0 − z|ε |a(z)|dz

B j=1

!1/2 Z

m
Y

|bj (x) − bj (z)||x0 − z|ε |a(z)|dz;

B j=1

Notice that 2t + |x0 − y| > 2t + |x0 − x| − |x − y| > t + |x0 − x| when |x − y| < t,

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J. Zeng - Boundedness for multilinear commutator of Littlewood-Paley operator...

and it is easy to calculate that
Z ∞
0

tdt
= C|x − x0 |−2(n+ε−δ) ;
(t + |x − x0 |)2(n+1+ε−δ)

then, we deduce
~

Sδb (a)(x)
Z Z
≤ C

!1/2 Z m
Y
t1−n
dydt
|bj (x) − bj (z)||x0 − z|ε |a(z)|dz
2(n+1+ε−δ)
(2t
+
|x
−
y|)
0
B j=1
Γ(x)
!
1/2 Z
Z Z
m
Y
t1−n
dydt
|bj (x) − bj (z)||x0 − z|ε |a(z)|dz
≤ C
2(n+1+ε−δ)
B j=1
Γ(x) (t + |x − x0 |)
Z ∞
1/2 Z Y
m
tdt
≤ C
|bj (x) − bj (z)||x0 − z|ε |a(z)|dz
2(n+1+ε−δ)
(t
+
|x
−
x
|)
0
0
B j=1
Z Y
m
≤ C|B|ε/n−1/p |x − x0 |−(n+ε−δ)
|bj (x) − bj (z)|dz.
B j=1

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J. Zeng - Boundedness for multilinear commutator of Littlewood-Paley operator...

So
∞
X

II ≤ C|B|(ε/n−1/p)q

|B(x0 , 2k+1 r)|1−q

k=1


Z
×

2k+1 d≥|x−x0 |>2k d

≤ C|B|(ε/n−1/p)q

∞
X


 q
Z Y
m
|x − x0 |−(n+ε−δ) 
|bj (x) − bj (z)|dz  dx
B j=1

|B(x0 , 2k+1 r)|1−q

k=1


m X Z
X
×
j=0

σ∈Cjm

≤ C|B|(ε/n−1/p)q

2k+1 d≥|x−x

0

|>2k d

m X Z
X

×

|B(x0 , 2k+1 d)|1−q

k=1

≤ C

m X
X

|(~b(z) − λ)σc |dz

"Z

2k+1 d≥|x−x0 |>2k d

||~bσc ||qBM O · ||~bσ ||qBM O

j=0 σ∈Cjm

≤ C||~b||qBM O

B

B

j=0 σ∈Cjm

∞
X

|x − x0 |−(n+ε−δ) |(~b(x) − λ)σ |dx

∞
X

Z

∞
X

q

q

|(~b(z) − λ)σc |dz 

|x − x0 |−(n+ε−δ) |(~b(x) − λ)σ |dx

#q

|B(x0 , 2k+1 d)|1−(n+ε−δ)q/n k q |B|(1+ε/n−1/p)q

k=1

k q · 2k(n−q−qn+qδ)

k=1

≤ C.

This finishes the proof of Theorem 1.
Theorem 2. Let ε > 0, 0 < p < ∞, 1 < q1 , q2 < ∞, 1/q1 − 1/q2 = δ/n,
n(1 − 1/q1 ) ≤ α < n(1 − 1/q1 ) + ε and bi ∈ BM O(Rn ), 1 ≤ i ≤ m, ~b = (b1 , · · · , bm ).
~
α,p
n
n
Then Sδb is bounded from H K̇qα,p
m (R ) to K̇q2 (R ).
1 ,D A
P
Proof. Let f ∈ H K̇ α,p~ (Rn ) and f (x) = ∞
j=−∞ λj aj (x) be the atomic decomq 1 ,b

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J. Zeng - Boundedness for multilinear commutator of Littlewood-Paley operator...

position for f as in Definition 4, we write
~
||Sδb (f )(x)||K̇qα,p
2



∞
X



∞
X

≤ C

2kαp (

2kαp (

+C 

∞
X

∞
X

k−3
X

2kαp (

k=−∞

kαp

2

~
||Sδb (f )χk ||pLq2

k=−∞

j=−∞

k=−∞



=

j=−∞

k=−∞

≤ C

∞
X

~

1/p

~

1/p

|λj |||Sδb (aj )χk ||Lq2 )p 

|λj |||Sδb (aj )χk ||Lq2 )p 

∞
X

j=k−2

= I + II.

!1/p

~

1/p

|λj |||Sδb (aj )χk ||Lq2 )p 
~

For II, by the (Lq1 , Lq2 )-boundedness of Sδb and the Hölder’s inequality, we have


II ≤ C 

∞
X

k=−∞



2kαp 

∞
X

j=k−2

p 1/p

~
|λj |||Sδb (aj )χj ||Lq2 



p 1/p
∞
∞
X
X
2kαp 
|λj |||aj ||Lq1  
≤ C
−∞



≤ C



j=k−2

∞
X

k=−∞



2kαp 

∞
X

j=k−2

p 1/p

|λj | · 2−jα  

 h
i1/p
Pj+2

p
(k−j)αp
 P∞
|λ
|
2
, 0