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ACTA UNIVERSITATIS APULENSIS

No 19/2009

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

Changhong Wu, Lanzhe Liu
, K˙ qα,p ) type boundedness
Abstract. In this paper, the (H~p , Lp ) and (H K˙ α,p
q,~b
b
for the multilinear commutator associated with the Littlewood-paley operator are
obtained.
Keywords: Littlewood-paley operator; Multilinear commutator; BMO; Hardy
space; Herz-Hardy space.
MR Subject Classification: 42B20, 42B25.
1. Introduction and definition
Let Lqloc (Rn ) = {f q is locally integrable on Rn }. Suppose f ∈ L1loc (Rn ), B =
n

B(x0 , r) = {x ∈ Rn : |x − x
R at x0 and having
R 0 | < r} denotes#a ball of R centered
−1
−1
radius r, write fB = |B|
B |f (x) − fB |dx <
B f (x)dx and f (x) = supx∈B |B|
∞. f is said to belong to BM O(Rn ), if f # ∈ L∞ (Rn ) and define ||f ||BM O =
||f # ||L∞ .
Let T be a linear operator and K be a function on Rn × Rn ,
Z
T (f )(x) =
K(x, y)f (y)dy for f ∈ C0∞ ,
Rn

where K satisfies:
(1) |K(x, y + h) − K(x, y)| ≤ C · |h|α · |x − y|−n−α for 2|h| < |x − y|, 0 < α ≤ 1;
(2) ||T (f )||Lp0 ≤ C||f ||Lp0 for some 1 < p0 ≤ ∞;
Then we call T is the Calder´on-Zygmund singular integral operator.

Let b ∈ BM O(Rn ) and T be the Calder´on-Zygmund singular integral operator.
The commutator [b, T ] generated by b and T is defined by
[b, T ](f )(x) = b(x)T (f )(x) − T (bf )(x).
67

C. Wu, L. Liu - Boundedness for multilinear commutator of Littlewood...

A classical result of Coifman, Rochberg and Weiss (see[2]) proved that the commutator [b, T ] is bounded on Lp (Rn ) (1 < p < ∞). However, it was observed that the
[b, T ] is not bounded, in general, from H p (Rn ) to Lp (Rn ). But if H p (Rn ) is replaced
by a suitable atomic space H~p (Rn ) and H K˙ α,p
(Rn )(see[1], [7], [12]), then [b, T ] maps
b
q,~b
continuously H p (Rn ) into Lp (Rn ) and H K˙ α,p (Rn ) into K˙ qα,p . In addition we have
~b

H~p (Rn )
b

q,~b

easily known that
⊂
K˙ α,p
(Rn ) ⊂ H K˙ qα,p (Rn ). The main purpose
q,~b
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]).
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 .
H p (Rn ),

Definition 1. 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
(3) B a(y)dy = B a(y) l∈σ bl (y)dy = 0 for any σ ∈ Cjm , 1 ≤ j ≤ m .
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

where a′j s are (p, ~b) atoms, λj ∈ C and
P
p 1/p .
( ∞
j=1 |λj | )

P∞

p
j=1 |λj |

< ∞. Moreover, ||f ||H p ≈
~
b

Definition 2. Let 0 < p, q < ∞, α ∈ R. For k ∈ Z, set Bk = {x ∈ Rn : |x| ≤
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 .
(1) The homogeneous Herz space is defined by
o
n
K˙ qα,p (Rn ) = f ∈ Lqloc (Rn \ {0}) : ||f ||K˙ qα,p < ∞ .
2k }

where

||f ||K˙ qα,p =

"

∞
X

k=−∞

68

2

kαp

||f χk ||pLq

#1/p

.

C. Wu, L. Liu - Boundedness for multilinear commutator of Littlewood...

(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 3. 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||
0 , r)|
R
R Lq ≤ |B(x
Q
m

β
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 K˙ 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 K˙ 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.
Definition 4. Let ε > 0 and ψ be a fixed function which satisfies the following
properties:
R

(1) Rn ψ(x)dx = 0,
(2) |ψ(x)| ≤ C(1 + |x|)−(n+1) ,
(3) |ψ(x + y) − ψ(x)| ≤ C|y|ε (1 + |x|)−(n+1+ε) when 2|y| < |x|;
The Littlewood-Paley multilinear commutator is defined by
~
Sψb (f )(x)

=

"Z Z

Γ(x)

where
~

Ftb (f )(x, y) =

Z

Rn




m
Y

j=1

dydt
~
|Ftb (f )(x, y)|2 n+1
t



#1/2

,

(bj (x) − bj (z)) ψt (y − z)f (z)dz,

69

C. Wu, L. Liu - Boundedness for multilinear commutator of Littlewood...
n+1
−n
Γ(x) = {(y,
R t) ∈ R+ : |x − y| < t} and ψt (x) = t ψ(x/t) for t > 0. Set
Ft (f )(y) = Rn ψt (y − x)f (x)dx. We also define that
!1/2
Z Z
dydt
,
|Ft (f )(x, y)|2 n+1
Sψ (f )(x) =
t
Γ(x)

which is the Littlewood-Paley operator(see [15]).
2.Theorems and Proofs
Theorem 1. Let ε > 0,bi ∈ BM O, 1 ≤ i ≤ m, ~b = (b1 , · · · , bm ), n/(n + ε) <
~
p ≤ 1. Then the multilinear commutator Sψb is bounded from H~p (Rn ) to Lp (Rn ).
b

Proof. It suffices to show that there exist a constant C > 0, such that for every
(p, ~b) atom a,
~
||Sψb (a)||Lp ≤ C.
Let a be a (p, ~b) atom supported on a ball B = B(x0 , r). When m = 1 see [7], and
now we prove m > 1. Write
Z
Z
Z
~b
~b
~
p
p
|Sψ (a)(x)| dx =
|Sψ (a)(x)| dx +
|Sψb (a)(x)|p dx = I + II.
Rn

|x−x0 |>2r

|x−x0 |≤2r

~

For I, taking q > 1, by H¨
older’s inequality and the Lq − boundedness of Sψb , we
have
!p/q
Z
~b
q
· |B(x0 , 2r)|1−p/q
|Sψ (a)(x)| dx
I ≤
|x−x0 |≤2r
~

≤ C||Sψb (a)(x)||pLq · |B(x0 , 2r)|1−p/q
≤ C||~b||p
||a||p q |B|1−p/q
BM O

L

≤ C||~b||pBM O .
For II, denoting
λ = (λ1 , · · · , λm ) with λi = (bi )B , 1 ≤ i ≤ m, where (bi )B =
R
older’s inequality and the vanishing moment of a,
|B(x0 , r)|−1 B(x0 ,r) bi (x)dx, by H¨
we get
∞ Z
X
~
II =
|Sψb (a)(x)|p dx
k=1 2
∞
X

≤ C

k+1 r≥|x−x

0 |>2

k+1

|B(x0 , 2

kr

1−p

r)|

Z

2k+1 r≥|x−x0 |>2k r

k=1

70

~
|Sψb (a)(x)|dx

!p

C. Wu, L. Liu - Boundedness for multilinear commutator of Littlewood...

≤ C

∞
X

|B(x0 , 2k+1 r)|1−p

k=1


Z

·

2k+1 r≥|x−x0 |>2k r

dydt
−z)a(z)dz|2 n+1
t

≤ C

∞
X


Z Z


|

Γ(x)

1/2

#p

Z Y
m

(bj (x) − bj (z))ψt (y

B j=1

dx

|B(x0 , 2k+1 r)|1−p ×

k=1

"Z

2k+1 r≥|x−x0 |>2k r

−ψt (y − x0 )|

m
Y

j=1

Z Z

Γ(x)

Z

|ψt (y − z)

B

2

|(bj (x) − bj (z))||a(z)|dz 

1/2

dydt 
tn+1

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

Sψb (a)(x) =


dx ;

|ψt (y − z) − ψt (y − x0 )|

B

Γ(x)

1/2
dydt
·
|bj (x) − bj (z)||a(z)|dz  n+1 
t
j=1


Z
Z Z
m
Y


t−n |a(z)|
|bj (x) − bj (z)|
≤ C
2

m
Y

Γ(x)

B

(|x0 − z|/t)ε
·
dz
(1 + |x0 − y|/t)n+1+ε

≤ C

Z Z

Γ(x)

71

j=1

2

dydt
tn+1

#1/2

t1−n
dydt
(t + |x0 − y|)2(n+1+ε)

p

!1/2

C. Wu, L. Liu - Boundedness for multilinear commutator of Littlewood...

·

Z Y
m

|bj (x) − bj (z)||x0 − z|ε |a(z)|dz

B j=1

Z Z

≤ C

Γ(x)

·

Z Y
m

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

!1/2

|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, and it is easy to calculate that
Z ∞
tdt
= C|x − x0 |−2(n+ε) ;
2(n+1+ε)
(t
+
|x
−
x
|)
0
0
then, we deduce
~
Sψb (a)(x)

≤ C

Z Z

Γ(x)

·

Z Y
m

t1−n
dydt
(2t + |x0 − y|)2(n+1+ε)

!1/2

|bj (x) − bj (z)||x0 − z|ε |a(z)|dz

B j=1

≤ C

Z Z

Γ(x)

·

Z Y
m

t1−n
dydt
(t + |x − x0 |)2(n+1+ε)

!1/2

|bj (x) − bj (z)||x0 − z|ε |a(z)|dz

B j=1

≤ C

Z

∞

0

·

Z Y
m

tdt
(t + |x − x0 |)2(n+1+ε)

1/2

|bj (x) − bj (z)||x0 − z|ε |a(z)|dz

B j=1

≤ C|B|ε/n−1/p |x − x0 |−(n+ε)

Z Y
m

B j=1

72

|bj (x) − bj (z)|dz.

C. Wu, L. Liu - Boundedness for multilinear commutator of Littlewood...

So
II ≤ C|B|ε/n−1/p

∞
X

|B(x0 , 2k+1 r)|1−p

k=1


Z
·


 p
Z Y
m
|x − x0 |−(n+ε) 
|bj (x) − bj (z)|dz  dx

2k+1 r≥|x−x0 |>2k r

≤ C|B|ε/n−1/p

∞
X

B j=1

|B(x0 , 2k+1 r)|1−p

k=1


m X Z
X
·
j=0 σ∈Cjm

·

Z

2k+1 r≥|x−x

|(~b(z) − λ)σc |dz

B

ε/n−1/p

≤ C|B|

m X Z
X
j=0 σ∈Cjm

×
≤ C

∞
X

k=1
m
X

|B(x0 , 2k+1 r)|1−p
X

0

|>2k r

p

|(~b(z) − λ)σc |dz

B

"Z

2k+1 r≥|x−x

||~bσc ||pBM O · ||~bσ ||pBM O

j=0 σ∈Cjm

≤ C||~b||pBM O

∞
X

|x − x0 |−(n+ε) |(~b(x) − λ)σ |dx

0

∞
X

|>2k r

p
#p

|x − x0 |−(n+ε) |(~b(x) − λ)σ |dx

|B(x0 , 2k+1 r)|1−(n+ε)p/n k p |B|(1+ε/n−1/p)p

k=1

k p · 2kn(1−p(n+ε)/n)

k=1

≤ C||~b||BM O .

This finishes the proof of Theorem 1.
Theorem 2. Let 0 < p < ∞, 1 < q < ∞, n(1 − 1/q) ≤ α < n(1 − 1/q) + ε,
~
ε > 0,and bi ∈ BM O(Rn ), 1 ≤ i ≤ m, ~b = (b1 , · · · , bm ). Then Sψb is bounded from
H K˙ α,p
(Rn ) to K˙ qα,p (Rn ).
q,~b
P∞
n
Proof. Let f ∈ H K˙ α,p
j=−∞ λj aj (x) be the atomic decom~ (R ) and f (x) =
q,b

73

C. Wu, L. Liu - Boundedness for multilinear commutator of Littlewood...

position for f as in Definition 3, we write
~
||Sψb (f )(x)||K˙ qα,p

∞
X

=

2

kαp

~
||Sψb (f )χk ||pLq

k=−∞



∞
X



∞
X

≤ C

2kαp (

j=−∞

k=−∞

≤ C

2kαp (

+C 

∞
X

k−3
X

j=−∞

k=−∞



∞
X

2kαp (

k=−∞

~

1/p

~

1/p

|λj |||Sψb (aj )χk ||Lq )p 
|λj |||Sψb (aj )χk ||Lq )p 

∞
X

j=k−2

= I + II.

!1/p

~

1/p

|λj |||Sψb (aj )χk ||Lq )p 

~

For II, by the boundedness of Sψb on Lq and the H¨
older’s inequality, we have


II ≤ C 

∞
X

k=−∞



2kαp 

∞
X

j=k−2

p 1/p

~
|λj |||Sψb (aj )χj ||Lq  

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



≤ C

j=k−2

∞
X

k=−∞



2kαp 

∞
X

j=k−2

p 1/p

|λj | · 2−jα  

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