Paper-14-20-2009. 147KB Jun 04 2011 12:03:10 AM

ACTA UNIVERSITATIS APULENSIS

No 20/2009

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

Ren Sheng and Lanzhe Liu
, K˙ qα,p ) type boundedness for
Abstract. In this paper,the (H~p , Lp ) and (H K˙ α,p
q,~b
b
the multilinear commutator associated with some integral operator are obtained.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Introduction
Let b ∈ BM O(Rn ), and T be the Calder´on-Zygmund 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).
A classical result of Coifman, Rochberg and Weiss (see [3]) 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
(Rn ), then [b, T ] maps continuously

by a suitable atomic space H~bP (Rn ) and H K˙ α,p
q,~b
H P (Rn ) into Lp (Rn ) and H K˙ α,p (Rn ) into K˙ α,p . In addition we easily known that
~b
H~p (Rn )
b

q

q,~b

(Rn ) ⊂ H K˙ qα,p (Rn ). The main purpose of this paper is to
⊂
K˙ α,p
q,~b
consider the continuity of the multilinear commutators related to the LittlewoodPaley operators and BM O(Rn ) functions on certain Hardy and Herz-Hardy spaces.
Let us first introduce some definitions (see [1][4-16][18][19]).
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 ),

127

R. Sheng, L. Liu-Boundedness for multilinear commutator of integral operator...

Definition 1. Let bi (i = 1, · · · , m) be a locally integrable function and 0 < p ≤
1. A bounded measurable function a on Rn is said a (p, ~b) atom, if
(1) suppa ⊂ B = B(x0 , r)
−1/p
∞ ≤ |B|
(2) ||a||
R
R L
Q
m
(3)
B a(y)dy = B a(y) l∈σ bl (y)dy = 0 for any σ ∈ Cj , 1 ≤ j ≤ m .
p

A temperate distribution f is said to belong to H~ (Rn ), if, in the Schwartz disb
tribution sense, it can be written as
f (x) =

∞
X

λj aj (x).

j=1

where a′j s are (p, ~b) atoms, λ ∈ C and
P∞

P∞

p
j=1 |λ|

< ∞. Moreover, ||f ||H p ≈

~
b

p 1/p .
j=1 |λj | )
Definition 2.Let 0 < p, q < ∞, α ∈ R. For k ∈ Z, set Bk = {x ∈ Rn : |x| ≤ 2k }
and Ck = Bk \ Bk−1 . Denote by χk the characteristic function of Ck and χ0 the
characteristic function of B0 .
(1) The homogeneous Herz space is defined by

(

n

o

K˙ qα,p (Rn ) = f ∈ Lqloc (Rn \ {0}) : ||f ||K˙ qα,p < ∞ ,
where



||f ||K˙ qα,p = 

∞
X

k=−∞

1/p

2kαp ||f χBk ||pLq 

.

(2) The nonhomogeneous Herz space is defined by
n

o

Kqα,p (Rn ) = f ∈ Lqloc (Rn ) : ||f ||Kqα,p < ∞ ,

where
||f ||Kqα,p =

"

∞
X

2

kαp

||f χk ||pLq

+

||f χB0 ||pLq

k=1

#1/p

.

Definition 3.Let α ∈ Rn , 1 < q < ∞, α ≥ n(1 − 1q ), 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) suppa ∈ B = B(x0 , r)(or for some r ≥ 1),
−α/n
q ≤ |B|
(2) ||a||
R L
R
β
m
βQ
(3)
i∈σ bi (x)dx = 0 for any σ ∈ Cj , 1 ≤ j ≤ m.
B a(x)x dx = B a(x)x
n

A temperate distribution f is said to belong to H K˙ α,~ (Rn )(or HK α,p
~ (R )), if it
can be written as f =

P∞

j=−∞ λj aj

(or f =
128

P∞

q,b

j=0 λj aj ),

q,b

in the S ′ (Rn ) sense, where

R. Sheng, L. Liu-Boundedness for multilinear commutator of integral operator...
aj is a central (α, q, ~b)-atom(or a central (α, q, ~b)-atom of restrict type ) supported
P
P∞
p
on B(0, 2j ) and ∞
−∞ |λj | < ∞(or
j=0 |λj | < ∞). Moreover,
X

||f ||H K˙ α,p ( or ||f ||HK α,p ) = inf(
q,~
b

q,~
b

|λj |p )1/p ,

j

where the infimum are taken over all the decompositions of f as above.
Definition 4.Suppose bj (j = 1, · · · , m) are the fixed locally integrable functions
on Rn . Let Ft (x, y) be the function defined on Rn × Rn × [0, +∞). Set
St (f )(x) =
and
~

Stb (f )(x) =

Z

m
Y

Rn j=1

Z

Rn

Ft (x, y)f (y)dy

(bj (x) − bj (y))Ft (x, y)f (y)dy

for every bounded and compactly supported function f . Let H be the Banach space
~
H = {h : ||h|| < ∞} such that, for each fixed x ∈ Rn , St (f )(x) and Stb (f )(x) may
be viewed as the mappings from [0, +∞) to H. The multilinear commutator related
to St is defined by
~
~
Tδb (f )(x) = ||Stb (f )(x)||,
where Ft satisfies: for fixed ε > 0 and 0 < δ < n,
||Ft (x, y)|| ≤ C|x − y|−n+δ
and
||Ft (x, y) − Ft (x, z)|| + ||Ft (y, x) − Ft (z, x)|| ≤ C|y − z|ǫ |x − z|−n−ε+δ
if 2|y − z| ≤ |x − z|. We also define Tδ (f )(x) = ||St (f )(x)||.
2. Theorems and Proofs
Lemma.(see [18]) Let 1 < r < ∞, bj ∈ BM O(Rn ) for j = 1, · · · , k and k ∈ N .
Then, we have
1
|Q|

Z Y
k

Q j=1

|bj (y) − (bj )Q |dy ≤ C

1

|Q|

Z Y
k

Q j=1

||bj ||BM O

j=1

and


k
Y

1/r

|bj (y) − (bj )Q |r dy 
129

≤C

k
Y

j=1

||bj ||BM O .

R. Sheng, L. Liu-Boundedness for multilinear commutator of integral operator...
Theorem 1.Let bi ∈ BM O(Rn ), 1 ≤ i ≤ m, ~b = (b1 , · · · , bm ), 0 < δ < n,
~
n/(n+ε−δ) < q ≤ 1, 1/q = 1/p−δ/n. Suppose that Tδb is the multilinear commutator
as in Definition 4 such that T is bounded from Ls (Rn ) to Lr (Rn ) for any 1 < s < n/δ
~
and 1/r = 1/s − δ/n. Then Tδb is bounded from H~p (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,
~
||Tδb (a)||Lq ≤ C.
Let a be a (p, ~b) atom supported on a ball B = B(x0 , l). We write
Z

Rn

~

|Tδb (a)(x)|q dx =

Z

~

|x−x0 |≤2l

|Tδb (a)(x)|q dx +

Z

~

|x−x0 |>2l

|Tδb (a)(x)|q dx = I + II.

For I, taking r, s > 1 with q < s < n/δ and 1/r = 1/s − δ/n, by H¨
older’s inequality
~
and the (Ls , Lr )- boundedness of Tδb , we see that
Z

I ≤

|x−x0 |≤2l

!q/r

~
|Tδb (a)(x)|r dx

· |B(x0 , 2l)|1−q/r

~

≤ C||Tδb (a)(x)||qLs · |B(x0 , 2l)|1−q/r
≤ C||a||qLs |B|1−q/r
≤ C|B|−q/p+q/s+1−q/r
≤ C.
For II, Rdenoting λ = (λ1 , · · · , λm ) with λi = (bi )B , 1 ≤ i ≤ m, where (bi )B =
1
older’s inequality and the vanishing moment of a, we
|B(x0 ,l)| B(x0 ,l) bi (x)dx, by H¨
get
II =

∞ Z
X

~

2k+1 B\2k B

k=1
∞
X

≤ C

≤ C

k=1
∞
X

|2

k+1

|Tδb (a)(x)|q dx
Z

1−q

B|

2k+1 B\2k B

!q

~
|Tδb (a)(x)|dx

|2k+1 B|1−q

k=1


Z
×

2k+1 B\2k B

≤ C

∞
X

||

Z

|2k+1 B|1−q

m
Y

B j=1

q

(bj (x) − bj (y))Ft (x, y)a(y)dy||dx

k=1

130

R. Sheng, L. Liu-Boundedness for multilinear commutator of integral operator...

Z
×

2k+1 B\2k B

Z

||Ft (x, y) − Ft (x, 0)||

B

m
Y

j=1

noting that y ∈ B, x ∈ 2k+1 B\2k B, then
Z

B

≤ C
≤ C

||Ft (x, y) − Ft (x, 0)||

m
Y

q

|(bj (x) − bj (y))||a(y)|dydx ;

|(bj (x) − bj (y))||a(y)|dy

j=1
m
Y

Z

B j=1
Z Y
m

B j=1

|(bj (x) − bj (y))|||Ft (x, y) − Ft (x, 0)|||a(y)|dy
|(bj (x) − bj (y))|

|y|ε
|a(y)|dy,
|x|n+ε−δ

thus
II ≤ C
≤ C

∞
X

k=1
∞
X


Z
k+1
1−q 
|2 B|

2k+1 B\2k B

|2k+1 B|1−q


Z
−(n+ε−δ) 
|x|

m
Y

B j=1

q



|bj (x) − bj (y)||y|ε |a(y)|dy  dx

k=1



×


m
X

X Z

2k+1 B\2k B

j=0 σ∈Cjm

≤ C

m X Z
X

B

j=0 σ∈Cjm

×
≤ C

∞
X

k=1
m
X

k+1

|2

X

"Z

2k+1 B\2k B

−(n+ε−δ)

|x|

||~bσc ||qBM O · ||~bσ ||qBM O

j=0 σ∈Cjm

≤ C||~b||qBM O

|(~b(y) − λ)σc ||y|ε |a(y)|dy

1−q

B|

|x|−(n+ε−δ) |(~b(x) − λ)σ |dx

∞
X

∞
X

q

Z

B

q


|(~b(y) − λ)σc ||y|ε |a(y)|dy 

#q

|(~b(x) − λ)σ |dx

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

k=1

k q · 2−knq(1+ε/n−δ/n−1/q)

k=1

≤ C||~b||qBM O .

This finish the proof of Theorem 1.
Theorem 2.Let 0 < p < ∞, 0 < δ < n, 1 < q1 , q2 < ∞, 1/q1 − 1/q2 = δ/n,
n(1 − 1/q1 ) + δ ≤ α < n(1 − 1/q1 ) + ε + δ and bi ∈ BM O(Rn ), 1 ≤ i ≤ m, ~b =
131

R. Sheng, L. Liu-Boundedness for multilinear commutator of integral operator...
~

(b1 , · · · , bm ). Suppose that Tδb is the multilinear commutator as in Definition 4 such
that T is bounded from Ls (Rn ) to Lr (Rn ) for any 1 < s < n/δ and 1/r = 1/s − δ/n.
~
(Rn ).
Then Tδb is bounded from H K˙ α,p~ (Rn ) to K˙ qα,p
2
q 1 ,b
P
λj aj (x) be the atomic decomProof. Let f ∈ H K˙ α,p (Rn ) and f (x) = ∞
j=−∞

q1 ,~b

position for f as in Definition 3, we write
~

||Tδb (f )(x)||K˙ qα,p
2



∞
X



∞
X

≤ C

2kαp (

j=−∞

k=−∞

≤ C

2kαp (

k=−∞



+C 

∞
X

∞
X

k−3
X

j=−∞

2kαp (

k=−∞

1/p
~b
|λj |||Tδ (aj )χk ||Lq2 )p 
~

|λj |||Tδb (aj )χk ||Lq2 )p 

∞
X

j=k−2

= I + II.

1/p

1/p
~b
|λj |||Tδ (aj )χk ||Lq2 )p 

For II, noting that suppaj ⊆ B(0, 2j ), ||aj ||Lq1 ≤ |B(0, 2j )|−α/n , by the boundedness
~
of Tδb on (Lq1 (Rn ), Lq2 (Rn )) and the H¨
older’s inequality, we get


II = C 

∞
X

2kαp (

k=−∞



∞
X



∞
X

≤ C

k=−∞

≤ C

k=−∞

∞
X

j=k−2



∞
X



∞
X

2kαp 

~

|λj |||Tδb (aj )χk ||Lq2 )p 

j=k−2

2kαp 

1/p

j=k−2

p 1/p

|λj |||aj ||Lq1  

p 1/p

|λj | · 2−jα  

 h
i1/p
P∞

p Pj+2
(k−j)αp

, 0