Paper-13-20-2009. 155KB Jun 04 2011 12:03:10 AM

ACTA UNIVERSITATIS APULENSIS

No 20/2009

BOUNDEDNESS OF MULTILINEAR COMMUTATOR OF
SINGULAR INTEGRAL IN MORREY SPACES ON
HOMOGENEOUS SPACES

Qian Zhang and Lanzhe Liu
Abstract: In this paper, we prove the boundedness of the multilinear commutator related to the singular integral operator in Morrey and Morrey-Herz spaces on
homogeneous spaces.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Preliminaries
Sawano and Tanka(see [13]) introduced the Morrey spaces on the non-homogeneous
spaces and proved the boundedness of Hardy-Littlewood maximal operators, Calder´
onZygmund operators and fractional integral operators in Morrey spaces. On the base
of the above results, Yang and Meng(see [14]) considered the boundedness of the
commutators generated by Calder´on-Zygmund operators or fractional integral operators with RBMO(µ) in Morrey spaces. Motivated by these results, in this paper,
we will introduce the multilinear commutator related to the singular operator on homogeneous spaces, and prove the boundedness properties of the operator in Morrey
and Morrey-Herz spaces on homogeneous spaces.
Give a set X, a function d : X × X → R+ is called a quasi-distance on X if the

following conditions are satisfied:
(i) for every x and y in X, d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y,
(ii) for every x and y in X, d(x, y) = d(y, x),
(iii) there exists a constant l ≥ 1 such that
d(x, y) ≤ l(d(x, z) + d(z, y))

(1)

for every x, y and z in X.
Let µ be a positive measure on the σ-algebra of subsets of X which contains the
r-balls Br (x) = {y : d(x, y) < r}. We assume that µ satisfies a doubling condition,
that is, there exists a constant A such that
111

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

0 < µ(B2r (x)) ≤ Aµ(Br (x)) < ∞

(2)

holds for all x ∈ X and r > 0.
A structure (X, d, µ), with d and µ as above, is called a space of homogeneous
type. The constants k and A in (1) and (2) will be called the constants of the space.
A homogeneous space (X, d, µ) is said to be normal if there exist positive constants c1 , c2 and α > 0 such that c1 rα ≤ µ(Br (x)) ≤ c2 rα for every x ∈ X and
every r satisfying that µ(x) < r < µ(X).
By [11], we know that for any homogeneous space (X, d, µ), there exists a normal
homogeneous space (X, δ, µ) such that δ and d are topologically equivalent.
Then let us introduce some notations(see [1][4][11]). In this paper,
B will denote
R
a ball of X, and for a ball B and a function b, let bB = µ(B)−1 B b(x)dµ(x) and
the sharp function of b is defined by
1
B∋x µ(B)

b# (x) = sup

Z

B

|b(y) − bB |dµ(y).

It is well-known that (see [4])
1
b (x) ≈ sup inf
B∋x c∈C µ(B)
#

Z

|b(y) − C|dµ(y).

B

We say that b belongs to BM O(X) if b# belongs to L∞ (X) and define ||b||BM O =
||b# ||L∞ . It has been known that(see [4])
||b − b2k B ||BM O ≤ Ck||b||BM O .
Definition 1. Suppose bi (i = 1, · · · , m) are the fixed locally integrable functions
on X. Let T be the singular integral as

T (f )(x) =

Z

K(x, y)f (y)dµ(y),

X

where K is a locally integrable function on X × X \ {(x, y) : x = y} and satisfies the
following properties:
C
(1) |K(x, y)| ≤
,
µ(B(x, d(x, y)))
(d(y, y ′ )δ
(2) |K(x, y) − K(x, y ′ )| + |K(y, x) − K(y ′ , x)| ≤ C
,
µ(B(y, d(x, y)))(d(x, y))δ
when d(x, y) ≥ 2d(y, y ′ ), with some δ ∈ (0, 1].
The multilinear commutator of the singular integral is defined by

T~b (f )(x) =

Z

m
Y

X i=1

(bi (x) − bi (y))K(x, y)f (y)dµ(y)
112

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...
Given a positive integer m and 1 ≤ i ≤ m, we denote by Cim the family of all finite
subsets σ = {σ(1), · · ·, σ(i)} of {1, · · ·, m} of i different elements. For σ ∈ Cim ,
set σ c = {1, · · ·, m} \ σ. For ~b = (b1 , · · ·, bm ) and σ = {σ(1), · · ·, σ(i)} ∈ Cim , set
~bσ = (bσ(1) , · · ·, bσ(i) ), bσ = bσ(1) · · · bσ(i) and ||~bσ ||BM O = ||bσ(1) ||BM O · · · ||bσ(i) ||BM O .
Fixed x0 ∈ X, Bk = {x ∈ X : d(x0 , x) < 2k }, Ak = Bk \ Bk−1 , for any k ∈ Z.
The notation χk (x) = χAk (x) is the characteristic function of the set Ak . In addition,
for a function f ∈ L1loc (X), denote fk = f χk .

In what follows, C > 0 always denotes a constant that is independent of main
parameters involved but whose value may differ from line to line. For any index
p ∈ [1, ∞], we denote by p′ its conjugate index, namely, 1/p + 1/p′ = 1.
2. Boundedness on Generalized Morrey space
In this section, the generalized Morrey spaces will be introduced and the boundedness in generalized Morrey spaces for the multilinear commutator of the singular
integral will be discussed.
Definition 2. For a positive growth function φ on X, which satisfies φ(2r) ≤
Dφ(r) for all r > 0 with D > 0 being a constant independent of r. The generalized
Morrey space Lp,φ with 1 ≤ p < ∞ is defined as follows:
Lp,φ (X) = {f ∈ Lploc (X) : ||f ||Lp,φ < ∞}
where
||f ||Lp,φ =

sup
x∈X,r>0

1
φ(r)

Z

!1/p

p

|f (y)| dµ(y)

Br (x)

.

Remark. If φ(r) = rδ , δ ≥ 0, then Lp,φ = Lp,δ , which is the classical Morrey
space(see[12]).
Lemma 1. Let 1 < r < ∞, bi ∈ BM O(X) for i = 1, · · · , k and k ∈ N . Then,
we have
Z Y
k
k
Y
1

||bi ||BM O
|bi (y) − (bi )B |dµ(y) ≤ C
µ(B) B i=1
i=1
and

1
µ(B)

Z Y
k

B i=1

r

!1/r

|bi (y) − (bi )B | dµ(y)

≤C

k
Y

||bi ||BM O .

i=1

Lemma 2. (see[3]) Let (X, d, µ) be a normal homogeneous space and p ∈ (1, ∞),
1 ≤ D < 2α . If a sublinear operator T is bounded on Lp (X) and for any f ∈ L1 (X)
with compact support and x ∈
/ c0 suppf ,
|T f (x)| ≤ C

Z

X

|f (y)|

dµ(y),
µ(B(x, y))

113

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

where c0 ≥ 1 and c > 0 are positive constants and B(x, y) is the ball {z ∈ X :
d(z, x) < d(x, y)}, then T is also bounded on Lp,φ (X).
Theorem 1. Let bj ∈ BM O(X) for j = 1, · · · , m, p ∈ (1, ∞), (X, d, µ) be a
normal homogeneous space and T be the singular integral as Definition 1. Suppose
φ is the function as in Definition 2 with 1 ≤ D < 2α . Then T~b is bounded on
Lp,φ (X).
Proof. Without loss of generality, we may assume c0 = 1. For any x0 ∈ X, r > 0,
and any complex-valued measurable function f on X, we write
f (y) = (f χB2N r (x0 ) )(y) +

X

k≥N

(f χB2k+1 r (x0 )\B2k r (x0 ) ) ≡ f0 (y) +

X

fk (y),

k≥N

where N is a positive integer to be chosen. When m = 1, set br = µ(Br (x0 ))−1
Br (x0 ) b1 (y)dµ(y), it is to see

R

Z

!1/p

p

≤

|Tb1 f (x)| dµ(x)

Br (x0 )

Z

Br (x0 )

+

!1/p

p

|Tb1 f0 (x)| dµ(x)

Z

X

Br (x0 )

k≥N

!1/p

p

|Tb1 fk (x)| dµ(x)

= I + II

For I, we have
Z

I =

Br (x0 )

Z

=

|Tb1 f0 (x)|p dµ(x)
|

Br (x0 )

≤ C

!1/p

Z

Z

Br (x0 )

X

(b1 (x) − b1 (y))K(x, y)f0 (y)dµ(y)| dµ(x)
−p

µ(B2N r (x0 ))
−1

≤ Cµ(B2N r (x0 ))

≤ Cµ(B2N r (x0 ))

1/p

+µ(Br (x0 ))

|

Z

" Z

B2N r (x0 )

!1/p

P

(b1 (x) − b1 (y))f (y)dµ(y)| dµ(x)

Z

p

B2N r (x0 )

!1/p

|b1 (x) − br | dµ(x)

#

|b1 (y) − br ||f (y)|dµ(y)

114

p

!1/p

!

|b1 (x) − b1 (y)| |f (y)| dµ(y) dµ(x)
p

Br (x0 )

Z

Z

B2N r (x0 )

Br (x0 )

−1

= ν1 + ν2 .

!1/p

P

Z

B2N r (x0 )

!

|f (y)|dµ(y)

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

For ν1 , we get
ν1 ≤ C



×

µ(Br (x0 ))
µ(B2N r (x0 ))
Z

1/p

1
µ(Br (x0 ))

Z

!1/p

p

|b1 (x) − br | dµ(x)

Br (x0 )

!1/p

p

|f (y)| dµ(y)

B2N r (x0 )

≤ C||b1 ||BM O ||f ||Lp,φ φ1/p (2N r)
≤ CDN ||b1 ||BM O ||f ||Lp,φ φ1/p (r)
≤ C||b1 ||BM O ||f ||Lp,φ .
For ν2 , by H¨
older’s inequality, we get
ν2

µ(Br (x0 ))1/p
≤ C
µ(B2N r (x0 ))

Z

q

!1/q

|b1 (y) − br | dµ(y)

B2N r (x0 )

µ(Br (x0 ))1/p µ(B2N r (x0 ))1/q
≤ C
µ(B2N r (x0 ))

1
µ(B2N r (x0 ))

Z

Z

p

!1/p

q

!1/q

|f (y)| dµ(y)

B2N r (x0 )

B2N r (x0 )

|b1 (y) − br | dµ(y)

×||f ||Lp,φ φ1/p (2N r)
≤ CN ||b1 ||BM O DN ||f ||Lp,φ φ1/p (r)
≤ C||b1 ||BM O ||f ||Lp,φ .
For II, k ≥ N , we have
Z

p

Br (x0 )

≤ C

Z

!1/p

|Tb1 fk (x)| dµ(x)

Br (x0 )

Z

B2k+1 r (x0 )\B2k r (x0 )

!p

|b1 (x) − b1 (y)||f (y)|
dµ(y)
µ(B(x, y))

dµ(x)

Similar to [3], we choose N be a fixed large positive integer such that
l+1
1
− N >0
2
l
2 l
and there is an integer k0 ∈ [1, N ] satisfying
2−k0 <

1
l+1
− N .
2
l
2 l

115

!1/p

.

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...
For these k0 and N chosen, we claim that B(x0 , 2k−k0 r) ⊆ B(x, y) for any x ∈ Br (x0 )
and y ∈ B2k+1 (x0 ) \ B2k r (x0 ) if k = N, N + 1, N + 2, · · · . Therefore, we obtain that
Z

Br (x0 )

≤ C

|Tb1 fk (x)| dµ(x)
1

Z

µ(B2k−k0 r (x0 ))

Br (x0 )

C

≤

!1/p

p

µ(B2k−k0 r (x0 ))
1/p

+µ(Br (x0 ))
= S1 + S2 .

" Z

Z

B2k+1 r (x0 )

!p

|b1 (x) − b1 (y)||f (y)|dµ(y)
!1/p

p

|b1 (x) − br | dµ(x)

Br (x0 )

Z

Z

B2k+1 r (x0 )

!1/p

dµ(x)

!

|f (y)|dµ(y)

B2k+1 r (x0 )

#

|b1 (y) − br ||f (y)|dµ(y)

For S1 , we have
S1

µ(Br (x0 ))1/p µ(B2k+1 r (x0 ))1/q
≤ C
µ(B2k−k0 r (x0 ))
Z

×

1
µ(Br (x0 ))

Z

p

Br (x0 )

!1/p

|b1 (x) − br | dµ(x)

!1/p

p

|f (y)| dµ(y)

B2k+1 r (x0 )

≤ Crα/p (2k+1 r)−α/p ||b1 ||BM O φ1/p (2k+1 r)||f ||Lp,φ (X)
≤ C



D
2α

(k+1)/p

||b1 ||BM O φ1/p (r)||f ||Lp,φ

For S2 , by H¨
older’s inequality, we have
S2

µ(Br (x0 ))1/p µ(B2k+1 r (x0 ))1/q
≤ C
µ(B2k−k0 r (x0 ))
Z

×


D
2α

µ(B2k+1 r (x0 ))

!1/p

|f (y)|p dµ(y)

B2k+1 r (x0 )

≤ C

1

(k+1)/p

k||b1 ||BM O φ1/p (r)||f ||Lp,φ .

Thus
X

k≥N

Z

Br (x0 )

p

!1/p

|Tb1 fk (x)| dµ(x)
116

Z

B2k+1 r(x0 )

q

!1/q

|b1 (y) − br | dµ(y)

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

≤ C

X  D (k+1)/p

2α

k≥N

(k + 1)||b1 ||BM O φ1/p (r)||f ||Lp,φ (X)

≤ C||b1 ||BM O ||f ||Lp,φ (X) .
This completes the proof of the case m = 1.
When m > 1, set ~br = ((b1 )Br (x0 ) , · · · , (bm )Br (x0 ) ), where (bi )Br (x0 ) = µ(Br (x0 ))−1
R
Br (x0 ) bj (y)dµ(y), 1 ≤ i ≤ m, it is easy to see
Z

Br (x0 )

!1/p

|T~b f (x)|p dµ(x)

!1/p

Z

≤

|T~b f0 (x)|p dµ(x)

Br (x0 )

+

X

k≥N

Z

!1/p

p

Br (x0 )

|T~b fk (x)| dµ(x)

= H1 + H 2 .

Since
T~b (f )(x) =
Z

=

m h
Y

X i=1

m
X

=

X

m
Y

Z

X i=1

(bi (x) − bi (y))K(x, y)f (y)dµ(y)
i

(bj (x) − (bj )Br (x0 ) ) − (bj (y) − (bj )Br (x0 ) ) K(x − y)f (y)dµ(y)
(−1)m−i (~b(x) − ~br )σ

i=0 σ∈Cim

Z

X

(~b(y) − ~br )σc K(x, y)f (y)dµ(y),

so
H1 (x)
=

Z

p

Br (x0 )

=

Z

|T~b f0 (x)| dµ(x)
|

Br (x0 )


Z
≤ 

Br (x0 )

!1/p

Z

m
Y

X i=1

(bi (x) − bi (y))K(x, y)f0 (y)dµ(y)| dµ(x)


Z
m X
X

|(~b(x) − ~br )σ |
i=0

!1/p

p

B2N r (x0 )

σ∈Cim

≤ Cµ(B2N r (x0 ))−1

m X
X
i=0

σ∈Cim

Z

Br (x0 )

p

|(~b(y) − ~br )σc ||K(x, y)||f (y)|dµ(y) dµ(x)
!1/p Z

|(~b(x) − ~br )σ |p dµ(x)

m X
µ(Br (x0 ))1/p µ(B2N r (x0 ))1/q X
≤ C
µ(B2N r (x0 ))
i=0 σ∈C m
i

117

1/p

1
µ(Br (x0 ))

Z

Br (x0 )

B2N r (x0 )

|(~b(y) − ~br )σc ||f (y)|dµ(y)
!1/p

|(~b(x) − ~br )σ |p dµ(x)

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

1

×

µ(B2N r (x0 ))

Z

B2N r (x0 )

!1/q

|(~b(y) − ~br )σc |q dµ(y)

Z

!1/p

p

|f (y)| dµ(y)

B2N r (x0 )

≤ C||~bσ ||BM O N m ||~bσc ||BM O φ1/p (2N r)||f ||Lp,φ (X)
≤ CN m DN ||~b||BM O ||f ||Lp,φ (X)
≤ C||~b||BM O ||f ||Lp,φ
and
H2 (x) =

Z

X

Br (x0 )

k≥N

= C

X

Z

X

Z

k≥N

≤ C

k≥N

!1/p

p

|T~b fk (x)| dµ(x)
(bi (x)
X i=1
X
m X

|

Br (x0 )

Br (x0 )

m
Y

Z

σ∈Cim

p

×|K(x, y)||f (y)|dµ(y)
≤ C

X

−1

µ(B2k−k0 r (x0 ))

Z

B2k+1 r (x0 )

≤ C

≤ C

B2k+1 r (x0 )\B2k r (x0 )

m X
X

Z

!1/p

|(~b(x) − ~br )σ | dµ(x)
p

Br (x0 )

|(~b(y) − ~br )σc ||f (y)|dµ(y)

µ(B2k−k0 r (x0 ))

1
µ(B2k+1 r (x0 ))

Z

X  D (k+1)/p

k≥N

|(~b(y) − ~br )σc |

dµ(x)

m X
X µ(Br (x0 ))1/p µ(B k+1 (x0 ))1/q X
2
r

k≥N

×

Z

!1/p

i=0 σ∈Cim

k≥N

×

− bi (y))K(x, y)fk (y)dµ(y)| dµ(x)

|(~b(x) − ~br )σ |

i=0

!1/p

p

2α

B2k+1 r (x0 )

i=0 σ∈Cim

1
µ(Br (x0 ))
!1/q

|(~b(y) − ~br )σc |q dµ(y)

Z

Br (x0 )

Z

!1/p

|(~b(x) − ~br )σ |p dµ(x)
!1/p

|f (y)| dµ(y)

B2k+1 r (x0 )

k m ||~b||BM O φ1/p ||f ||Lp,φ (X)

≤ C||~b||BM O ||f ||Lp,φ .

This finishes the proof of Theorem 1.
3. Boundedness on homogeneous Morrey-Herz space
In this section, the homogeneous Morrey-Herz spaces will be introduced and the
118

p

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

boundedness in homogeneous Morrey-Herz spaces for the multilinear commutator
of the singular integral will be discussed.
Definition 3. Let α ∈ R, 0 ≤ λ < ∞, 0 < p < ∞ and 1 ≤ q < ∞. The
α,λ (X) is defined by
homogeneous Morrey-Herz space M K˙ p,q
α,λ
M K˙ p,q
(X) = {f ∈ Lq (X \ {x0 }) : ||f ||M K˙ α,λ < ∞},
p,q

where


k0
X

||f ||M K˙ α,λ = sup µ(Bk0 )−λ 
p,q

k0 ∈Z

k=−∞

1/p

µ(Bk )αp ||f χk ||pLq (X) 

with the usual modifications made when p = ∞.

α,λ (X) with the homogeneous
Compare the homogeneous Morrey-Herz space M K˙ p,q
Herz space K˙ qα,p (X)(see [9]) and the Morrey space Mqλ (X)(see [10]), where K˙ qα,p (X)
is defined by

K˙ qα,p (X) =







f ∈ Lqloc (X \ {x0 }) : 




∞
X

k=−∞

1/p

µ(Bk )αp ||f χk ||pLq 





0 t

Z

q

)

|f (y)| dµ(y) < ∞ .

d(x,y) 0.

Theorem 2. Let T be the singular integral as Definition 1, bi ∈ BM O(X) for
i = 1, · · · , m, 0 < λ < ∞, 0 < p < ∞, 1 < q < ∞ and −1/q + λ < α < 1/q ′ + λ. If
α,λ (X).
T~b is bounded on Lq (X), then T~b is also bounded on M K˙ p,q
α,λ (X) and write it as
Proof. Let f ∈ M K˙ p,q
f (x) =

∞
X

f (x)χj (x) ≡

∞
X

fj (x).

j=−∞

j=−∞

When m = 1
||Tb1 (f )||M K˙ α,λ (X) =
p,q



sup µ(Bk0 )−λ 

k0 ∈Z

k0
X

k=−∞

119

1/p

µ(Bk )αp ||Tb1 (f )χk ||pLq (X) 

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

≤ C sup µ(Bk0 )−λ
k0 ∈Z


k0
 X

k0 ∈Z

k0 ∈Z

k+2
X



∞
X

µ(Bk )αp 



j=k−2


k0
 X

µ(Bk )αp 



k=−∞

= E1 + E2 + E3 .



j=−∞


k0
 X

k=−∞

≤ C sup µ(Bk0 )−λ

k−3
X

µ(Bk )αp 



k=−∞

≤ C sup µ(Bk0 )−λ



j=k+3

p 1/p

+
||Tb1 (f )χk ||Lq (X) 


p 1/p

+
||Tb1 (f )χk ||Lq (X) 


p 1/p

||Tb1 (f )χk ||Lq (X) 


We first estimate E1 . Note that j−k ≤ −3 and x ∈ Ak , denote bj = µ(B)−1
where B means the smallest doubling ball which is like 2k B(k ∈ N ∪ {0}).
||Tb1 (fj )χk ||Lq

=

(Z

Ak

≤ C

|b1 (x) − b1 (y)f (y)K(x, y)|dµ(y)

Aj

(Z

Z

Ak

≤ C

!q

Z

Aj

(Z

µ(B(x, d(x, y)))−q

Aj

−1

≤ Cµ(Bk )

dµ(x)

|b1 (x) − b1 (y)||f (y)||K(x, y)|dµ(y)

Ak

(Z

Ak

(

Z

≤ C µ(Bk )−1 ||fj ||L1

Aj

)1/q

dµ(x)

1/q

Z

Aj

!1/q′ )

q′

|b1 (y) − bj | dµ(y)

′

≤ C µ(Bk )−1 ||fj ||Lq µ(Bj )1/q µ(Bk )1/q ||b1 ||BM O
′

+µ(Bk )−1 µ(Bk )1/q µ(Bj )1/q ||fj ||Lq ||b1 ||BM O
≤ C



µ(Bj )
µ(Bk )

1/q′

o

||fj ||Lq ||b1 ||BM O ,

where we used the condition |K(x, y)| ≤

C
and H¨
older’s inequality.
µ(B(x, d(x, y)))

120

)1/q

dµ(x)

|b1 (x) − bj |q dµ(x)

+µ(Bk )−1 µ(Bk )1/q ||fj ||Lq
n

dµ(x)

|b1 (x) − b1 (y)||f (y)|dµ(y)

|b1 (x) − b1 (y)||f (y)|dµ(y)

Ak

)1/q
!q

!q

Z

b(y)dµ(y),

B

)1/q

!q

Z

R

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...
Note that α < 1/q ′ + λ, we get
E1 ≤ C sup µ(Bk0 )−λ
k0 ∈Z

−λ

≤ C sup µ(Bk0 )
k0 ∈Z

−λ

≤ C sup µ(Bk0 )
k0 ∈Z

(
(
(

k0
X

 k−3
X  µ(Bj ) 1/q′
αp

µ(Bk )

k0
X

λp

µ(Bk )

k0
X

λp

µ(Bk )

||fj ||Lq

 k−3
X  µ(Bj ) 1/q′ −α+λ

µ(Bk )

j=−∞

k=−∞

 k−3
X

(j−k)(λ−α+1/q ′ )

2

−λ

µ(Bj )

j
 X

αp

µ(Bj )

l=−∞

||f ||M K˙ α,λ
p,q

j=−∞

k=−∞

≤ C||f ||M K˙ α,λ .

µ(Bk )

j=−∞

k=−∞

p )1/p

1/p p
p
||fj ||Lq )

p )1/p

p,q

Next we estimate E2 , by the Lq (X) boundedness of Tb1 , we have
−λ

E2 ≤ C sup µ(Bk0 )
k0 ∈Z

−λ

≤ C sup µ(Bk0 )
k0 ∈Z

(
(

k0
X

αp

µ(Bk )

k=−∞
k0
X

 k+2
X

αp

µ(Bk )

||fj ||pLq

k=−∞

≤ C||f ||M K˙ α,λ .

||fj ||Lq

j=k−2

p )1/p

)1/p

p,q

Finally, we estimate E3 . Similar to the proof of E1 , we easily obtain
||Tb1 (fj )χk ||Lq

=

(Z

Ak

≤ C

|b1 (x) − b1 (y)f (y)K(x, y)|dµ(y)

Aj

(Z

Z

(Z

Z
−1

≤ Cµ(Bj )

|b1 (x) − b1 (y)||f (y)|µ(B(x, d(x, y)))

(Z

Z

Ak

Aj

≤ C µ(Bj )−1 ||fj ||L1
+µ(Bj )

)1/q

dµ(x)

−1

(

−1

dµ(x)

|b1 (x) − b1 (y)||f (y)||K(x, y)|dµ(y)

Aj

Ak

)1/q

!q

Aj

Ak

≤ C

!q

Z

1/q

µ(Bk )

!q

|b1 (x) − b1 (y)||f (y)|dµ(y)

Z

Ak

dµ(x)

′

Z

Aj

q′

!1/q′ )

|b1 (y) − bj | dµ(y)

)1/q

dµ(x)

)1/q

1/q

≤ C µ(Bj )−1 ||fj ||Lq µ(Bj )1/q µ(Bk )1/q ||b1 ||BM O
121

dµ(y)

|b1 (x) − bj |q dµ(x)

||fj ||Lq (X)

n

!q

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

′

+µ(Bj )−1 µ(Bk )1/q µ(Bj )1/q ||fj ||Lq ||b1 ||BM O
µ(Bk )
µ(Bj )

≤ C

!1/q

o

||fj ||Lq ||b1 ||BM O .

Note that α > −1/q + λ and k − j ≤ −3, we get
−λ

E3 ≤ C sup µ(Bk0 )
k0 ∈Z

−λ

≤ C sup µ(Bk0 )
k0 ∈Z

−λ

≤ C sup µ(Bk0 )
k0 ∈Z

(
(
(

≤ C||f ||M K˙ α,λ (X) .

k0
X

αp

µ(Bk )

k=−∞
k0
X

µ(Bj )

j=k+3

λp

µ(Bk )

k=−∞
k0
X

 X
∞ 
µ(Bk ) 1/q

 X
∞ 
µ(Bk ) 1/q+α−λ

µ(Bj )

j=k+3

λp

µ(Bk )

k=−∞

||fj ||Lq

 X
∞

(k−j)(1/q+α−λ)

2

p )1/p
−λ

µ(Bj )

j
 X

αp

µ(Bj )

l=−∞

||f ||M K˙ α,λ
p,q

j=k+3

p )1/p

p,q

This completes the proof of the case m = 1.
When m > 1,
||T~b (f )||M K˙ α,λ
p,q

=



sup µ(Bk0 )−λ 

k0 ∈Z

k0
X

k=−∞

≤ C sup µ(Bk0 )−λ
k0 ∈Z

µ(Bk )αp ||T~b (f )χk ||pLq 


k0
 X


k=−∞

≤ C sup µ(Bk0 )−λ
k0 ∈Z


k0
 X


k=−∞

≤ C sup µ(Bk0 )−λ
k0 ∈Z


k0
 X


k=−∞

= G1 + G2 + G3 .

1/p



k−3
X



k+2
X



∞
X

µ(Bk )αp 

j=−∞

µ(Bk )αp 

j=k−2

µ(Bk )αp 

j=k+3

p 1/p

+
||T~b (f )χk ||Lq 


p 1/p

+
||T~b (f )χk ||Lq 


p 1/p

||T~b (f )χk ||Lq 


R
For G1 , similar to the proof of E1 , set ~b = (b1 , · · · , bm ), where bi = µ(Bi )−1 Bi bi (y)dµ(y),
1 ≤ i ≤ m, we have

||T~b (fj )χk ||Lq

=

(Z

Ak

Z

m
Y

Aj i=1

q

|bi (x) − bi (y)f (y)K(x, y)|dµ(y)

122

)1/q

dµ(x)

1/p p
||fj ||pLq

)

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

−1

≤ Cµ(Bk )

m X
X

(Z

Ak i=0 σ∈C m
i

)1/q Z

|(~b(x) − ~b)σ |q dµ(x)

Aj

|(~b(y) − ~b)σc ||f (y)|dµ(y)

′
≤ Cµ(Bk )−1 µ(Bk )1/q k m ||~bσ ||BM O µ(Bj )1/q k m ||~bσc ||BM O ||fj ||Lq

≤ C



µ(Bj )
µ(Bk )

1/q′

k 2m ||fj ||Lq ||~b||BM O ,

thus, notice α < 1/q ′ + λ,
−λ

G1 ≤ C sup µ(Bk0 )
k0 ∈Z

−λ

≤ C sup µ(Bk0 )
k0 ∈Z

−λ

×µ(Bj )

j
 X

(
(

k0
X

αp

µ(Bk )

λp

µ(Bk )
αp

µ(Bj )

≤ C sup µ(Bk0 )
k0 ∈Z

≤ C||f ||M K˙ α,λ .

(

k0
X

 k−3
X

k 2m

j=−∞

k=−∞

l=−∞
−λ

µ(Bk )

j=−∞

k=−∞
k0
X

 k−3
X  µ(Bj ) 1/q′

1/p p
||fj ||pLq
λp

µ(Bk )

k

2m

||fj ||Lq

 µ(B ) 1/q′ −α+λ
j

µ(Bk )

)1/p

 k−3
X

(j−k)(λ−α+1/q ′ ) 2m

2

k

||f ||M K˙ α,λ
p,q

j=−∞

k=−∞

p )1/p

p )1/p

p,q

For G2 , similar to the proof of E2 , by the Lq (X) boundedness of T~b , we get
−λ

G2 ≤ C sup µ(Bk0 )
k0 ∈Z

−λ

≤ C sup µ(Bk0 )
k0 ∈Z

(
(

≤ C||f ||M K˙ α,λ .

k0
X

αp

µ(Bk )

k=−∞
k0
X

 k+2
X

||fj ||Lq (X)

j=k−2

αp

µ(Bk )

k=−∞

||fj ||pLq

p )1/p

)1/p

p,q

For G3 , similar to the proof of G1 , we have
||T~b (fj )χk ||Lq

=

(Z

Ak

Z

m
Y

Aj i=1

≤ Cµ(Bj )−1

(Z

q

|bi (x) − bi (y)f (y)K(x, y)|dµ(y)
m X
X

Ak i=0 σ∈C m
i

)1/q

dµ(x)

)1/q Z

|(~b(x) − ~b)σ |q dµ(x)

Aj

|(~b(y) − ~b)σc ||f (y)|dµ(y)

≤ Cµ(Bj )−1 µ(Bk )1/q k m ||~bσ ||BM O µ(Bj )1/q k m ||~bσc ||BM O ||fj ||Lq
′

123

Q. Zhang, L. Liu-Boundedness of multilinear commutator of singular integral in ...

µ(Bk )
µ(Bj )

≤ C

!1/q

k 2m ||fj ||Lq ||~b||BM O .

thus, notice α > −1/q + λ,
−λ

G3 ≤ C sup µ(Bk0 )
k0 ∈Z

≤ C sup µ(Bk0 )−λ
k0 ∈Z

−λ

×µ(Bj )

j
 X

(
(

k0
X

αp

µ(Bk )

k=−∞
k0
X

−λ

αp

≤ C sup µ(Bk0 )
k0 ∈Z

≤ C||f ||M K˙ α,λ .

(

k0
X

k=−∞

 X
∞

k 2m

j=k+3

k=−∞

µ(Bj )

µ(Bj )

j=k+3

µ(Bk )λp

l=−∞

 X
∞ 
µ(Bk ) 1/q

1/p p
p
||fj ||Lq
λp

µ(Bk )

 X
∞

k

2m

||fj ||Lq

p )1/p

 µ(B ) 1/q+α−λ
k

µ(Bj )

)1/p

2

j=k+3

(k−j)(1/q+α−λ) 2m

k

||f ||M K˙ α,λ
p,q

p )1/p

p,q

The estimates for G1 , G2 and G3 indicate that ||T~b ||M K˙ α,λ ≤ C||f ||M K˙ α,λ . And
p,q
p,q
we complete the proof of the Theorem 2.
References
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Appl., 322, (2006), 825-846.
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[7] Y. Lin, Strongly singular Calder´
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Qian Zhang and Lanzhe Liu
College of Mathematics
Changsha University of Science and Technology
Changsha, 410077, P.R.of China
E-mail: lanzheliu@163.com

125