Báo cáo hóa học: "CONTINUITY OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES" potx

CONTINUITY OF MULTILINEAR OPERATORS ON
TRIEBEL-LIZORKIN SPACES
LANZHE LIU
Received 4 February 2006; Revised 20 September 2006; Accepted 28 September 2006
The continuity of some multilinear operators related to certain convolution operators on
the Triebel-Lizorkin space is obtained. The operators include Littlewood-Paley operator
and Marcinkiewi cz operator.
Copyright © 2006 Lanzhe Liu. This is an open access article distributed under the Cre-
ative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let T be the Calder
´
on-Zygmund singular integral operator, a well-known result of Coif-
man et al. (see [6]) states that the commutator [b,T]( f )
= T(bf) −bT(f )(whereb ∈
BMO) is bounded on L
p
(R
n
)(1<p<∞); Chanillo (see [1]) proves a similar result when
T is replaced by the fractional integr al operator; in [8, 9], these results on the Triebel-
Lizorkin spaces and the case b
∈ Lipβ (where Lipβ is the homogeneous Lipschitz space)
are obtained. The main purpose of this paper is to study the continuity of some multi-
linear operators related to certain convolution operators on the Triebel-Lizorkin spaces.
In fact, we will obtain the continuity on the Triebel-Lizorkin spaces for the multilinear
operators only under certain conditions on the size of the operators. As the applications,
the continuity of the multilinear operators related to the Littlewood-Paley oper ator and
Marcinkiewicz operator on the Triebel-Lizorkin spaces are obtained.
2. Notations and results

Throughout this paper, Q will denote a cube of R
n
with side parallel to the axes, and
for a cube Q,let f
Q
=|Q|
−1

Q
f (x)dx and f
#
(x) = sup
x∈Q
|Q|
−1

Q
|f (y) − f
Q
|dy.For
1
≤ r<∞ and 0 ≤δ<n,let
M
δ,r
( f )(x) =sup
x∈Q

1
|Q|
1−δr/n


Q


f (y)


r
dy

1/r
, (2.1)
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 58473, Pages 1–11
DOI 10.1155/JIA/2006/58473
2 Continuity of multilinear operators
we denote M
δ,r
( f ) = M
r
( f )ifδ = 0, which is the Hardy-Littlewood maximal function
when r
= 1 (see [10]). For β>0andp>1, let
˙
F
β,∞
p
be the homogeneous Triebel-Lizorkin
space, and let the Lipschitz space

˙
∧
β
be the space of functions f such that
f 
˙
∧
β
= sup
x, h∈R
n
, h=0


Δ
[β]+1
h
f (x)


|h|
β
< ∞
, (2.2)
where Δ
k
h
denotes the kth difference operator (see [9]).
We are going to study the multilinear operator as follows.
Let m be a positive integer and let A be a function on R

n
. We denote
R
m+1
(A;x, y) =A(x) −

|α|≤m
1
α!
D
α
A(y)(x − y)
α
. (2.3)
Definit ion 2.1. Let F(x,t)defineonR
n
×[0,+∞), denote
F
t
( f )(x) =

R
n
F(x − y,t) f (y)dy,
F
A
t
( f )(x) =

R

n
R
m+1
(A;x, y)
|x − y|
m
F(x − y,t) f (y)dy.
(2.4)
Let H be the Hilbert space H
={h : h< ∞}such that, for each fixed x ∈ R
n
, F
t
( f )(x)
and F
A
t
( f )(x) may be viewed as a mapping from [0,+∞)toH. Then, the multilinear
operators related to F
t
is defined by
T
A
( f )(x) =


F
A
t
( f )(x)



; (2.5)
and also define T(f )(x)
=F
t
( f )(x).
In particular, consider the following two sublinear operators.
Definit ion 2.2. Fix ε>0, n>δ
≥ 0. Let ψ be a fixed function which satisfies the following
properties:
(1)

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 multilinear Littlewood-Paley operator is defined by
g
A
δ
( f )(x) =



∞
0


F
A
t
( f )(x)


2
dt
t

1/2
, (2.6)
where
F
A
t
( f )(x) =

R
n
R
m+1
(A;x, y)
|x − y|
m

ψ
t
(x − y) f (y)dy (2.7)
Lanzhe Liu 3
and ψ
t
(x) =t
−n+δ
ψ(x/t)fort>0. Denote that F
t
( f ) =ψ
t
∗ f , and also define that
g
δ
( f )(x) =


∞
0


F
t
( f )(x)


2
dt
t


1/2
, (2.8)
which is the Littlewood-Paley g function w hen δ
= 0 (see [11]).
Let H be the space H
={h : h=(

∞
0
|h(t)|
2
dt/t)
1/2
< ∞},then,foreachfixedx ∈ R
n
,
F
A
t
( f )(x) may be viewed as a mapping from [0,+∞)toH, and it is clear that
g
δ
( f )(x) =


F
t
( f )(x)



, g
A
δ
( f )(x) =


F
A
t
( f )(x)


. (2.9)
Definit ion 2.3. Let 0
≤ δ<n,0<γ≤ 1andΩ be homogeneous of degree zero on R
n
such that

S
n−1
Ω(x

)dσ(x

) = 0. Assume that Ω ∈ Lip
γ
(S
n−1
), that is, there exists a con-

stant M>0 such that for any x, y
∈ S
n−1
, |Ω(x) −Ω(y)|≤M|x − y|
γ
. The multilinear
Marcinkiewicz operator is defined by
μ
A
δ
( f )(x) =


∞
0


F
A
t
( f )(x)


2
dt
t
3

1/2
, (2.10)

where
F
A
t
( f )(x) =

|x−y|≤t
Ω(x − y)
|x − y|
n−1−δ
R
m+1
(A;x, y)
|x − y|
m
f (y)dy; (2.11)
denote
F
t
( f )(x) =

|x−y|≤t
Ω(x − y)
|x − y|
n−1−δ
f (y)dy, (2.12)
and also define that
μ
δ
( f )(x) =



∞
0


F
t
( f )(x)


2
dt
t
3

1/2
, (2.13)
which is the Marcinkiewicz operator when δ
= 0 (see [12]).
Let H be the space H
={h : h=(

∞
0
|h(t)|
2
dt/t
3
)

1/2
< ∞}. Then, it is clear that
μ
δ
( f )(x) =


F
t
( f )(x)


, μ
A
δ
( f )(x) =


F
A
t
( f )(x)


. (2.14)
It is clear that Definitions 2.2 and 2.3 are the particular examples of Definition 2.1.
Note that when m
= 0, T
A
is just the commutator of F

t
and A, while when m>0, it is
nontrivial generalizations of the commutators. It is well known that multilinear oper-
ators are of great interest in harmonic analysis and have been widely studied by many
authors (see [2–5, 7]). The main purpose of this paper is to study the continuity for the
multilinear opera tors on the Triebel-Lizorkin spaces. We will prove the following theo-
rems in Section 3.
Theorem 2.4. Let g
A
δ
be the mult ilinear Littlewood-Paley operator as in Definition 2.2.If
0 <β<min(1,ε) and D
α
A ∈
˙
∧
β
for |α|=m, then
4 Continuity of multilinear operators
(a) g
A
δ
maps L
p
(R
n
) continuously into
˙
F
β,∞

q
(R
n
),for1 <p<n/δand 1/q = 1/p−δ/n;
(b) g
A
δ
maps L
p
(R
n
) continuously into L
q
(R
n
) for 1 <p<n/(δ + β) and 1/p− 1/q =
(δ +β)/n.
Theorem 2.5. Let μ
A
δ
be the multilinear Marcinkiewiz operator as in Definition 2.3.If0 <
β<min(1/2, γ) and D
α
A ∈
˙
∧
β
for |α|=m, then
(a) μ
A

δ
maps L
p
(R
n
) continuously into
˙
F
β,∞
q
(R
n
) for 1 <p<n/δand 1/q =1/p−δ/n,
(b) μ
A
δ
maps L
p
(R
n
) continuously into L
q
(R
n
) for 1 <p<n/(δ + β) and 1/p−1/q =
(δ +β)/n.
3. Main theorem and proof
We first prove a general theorem.
Theorem 3.1 (main theorem). Let 0
≤ δ<n, 0 <β<1,andD

α
A ∈
˙
∧
β
for |α|=m.Sup-
pose F
t
, T,andT
A
are the same as in Definition 2.1,ifT is bounded from L
p
(R
n
) to L
q
(R
n
)
for 1 <p<n/δand 1/q
= 1/p−δ/n,andT satisfies the following size condition:


F
A
t
( f )(x) −F
A
t
( f )


x
0



≤
C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
M
δ,1
f (x) (3.1)
for any cube Q with supp f
⊂ (2Q)
c
and x ∈Q, then
(a) T

A
is bounded from L
p
(R
n
) to
˙
F
β,∞
q
(R
n
) for 1 <p<n/δand 1/q =1/p−δ/n,
(b) T
A
is bounded from L
p
(R
n
) to L
q
(R
n
) for 1 <p<n/(δ + β) and 1/q = 1/p−(δ +
β)/n.
To prove the theorem, we need the following lemmas.
Lemma 3.2 (see [9]). For 0 <β<1, 1 <p<
∞,
f 
˙

F
β,∞
p
≈




sup
Q
1
|Q|
1+β/n

Q


f (x) − f
Q


dx




L
p
≈





sup
·∈Q
inf
c
1
|Q|
1+β/n

Q


f (x) −c


dx




L
p
.
(3.2)
Lemma 3.3 (see [9]). For 0 <β<1, 1
≤ p ≤∞,
f 
˙

∧
β
≈ sup
Q
1
|Q|
1+β/n

Q


f (x) − f
Q


dx
≈ sup
Q
1
|Q|
β/n

1
|Q|

Q


f (x) − f
Q



p
dx

1/p
.
(3.3)
Lemma 3.4 (see [1, 2]). Suppose that 1
≤ r<p<n/δand 1/q = 1/p−δ/n. Then


M
δ,r
( f )


L
q
≤ Cf 
L
p
. (3.4)
Lanzhe Liu 5
Lemma 3.5 (see [5]). Let A be a function on R
n
and D
α
A ∈L
q

(R
n
) for |α|=m and some
q>n. Then


R
m
(A;x, y)


≤
C|x − y|
m

|α|=m

1



Q(x, y)




Q(x,y)


D

α
A(z)


q
dz

1/q
, (3.5)
where

Q(x, y) isthecubecenteredatx and has side length 5
√
n|x − y|.
Proof of Theorem 3.1 (main theorem). Fix a cube Q
= Q(x
0
,l)andx ∈ Q.Let

Q = 5
√
nQ
and

A(x) = A(x) −

|α|=m
(1/α!)(D
α
A)


Q
x
α
,thenR
m
(A;x, y) = R
m
(

A;x, y)andD
α

A =
D
α
A −(D
α
A)

Q
for |α|=m.Wewrite,for f
1
= fχ

Q
and f
2
= fχ
R

n
\

Q
,
F
A
t
( f )(x) =

R
n
R
m+1


A;x, y

|x − y|
m
F(x − y,t) f (y)dy
=

R
n
R
m+1


A;x, y


|x − y|
m
F(x − y,t) f
2
(y)dy
+

R
n
R
m


A;x, y

|x − y|
m
F(x − y,t) f
1
(y)dy
−

|α|=m
1
α!

R
n
F(x − y,t)(x − y)

α
|x − y|
m
D
α

A(y) f
1
(y)dy,
(3.6)
then


T
A
( f )(x) −T

A

f
2

x
0



=





F
A
t
( f )(x)


−


F

A
t

f
2

x
0





≤





F
t

R
m


A;x,·

|x −·|
m
f
1

(x)




+

|α|=m
1
α!




F

t

(x −·)
α
|x −·|
m
D
α

Af
1

(x)




+


F

A
t

f
2

(x) −F


A
t

f
2

x
0



=
A(x)+B(x)+C(x),
(3.7)
thus,
1
|Q|
1+β/n

Q


T
A
( f )(x) −T

A
( f )

x

0



dx
≤
1
|Q|
1+β/n

Q
A(x)dx +
1
|Q|
1+β/n

Q
B(x)dx
+
1
|Q|
1+β/n

Q
C(x)dx :=I + II +III.
(3.8)
6 Continuity of multilinear operators
Now, let us estimate I, II,andIII, respectively. First, for x
∈ Q and y ∈


Q, using Lemmas
3.3 and 3.5,weget


R
m


A;x, y



≤
C|x − y|
m

|α|=m
sup
x∈

Q


D
α
A(x) −

D
α
A



Q


≤
C|x − y|
m
|Q|
β/n

|α|=m


D
α
A


˙
∧
β
,
(3.9)
thus, taking r, s such that 1
≤ r<pand 1/s =1/r −δ/n,bythe(L
r
,L
s
) boundedness of T

and Holder’ inequality, we obtain
I
≤ C

|α|=m


D
α
A


˙
∧
β
1
|Q|

Q


T

f
1

(x)


dx ≤ C


|α|=m


D
α
A


˙
∧
β


T

f
1



L
s
|Q|
−1/s
≤ C

|α|=m



D
α
A


˙
∧
β


f
1


L
r
|Q|
−1/s
≤ C

|α|=m


D
α
A


˙
∧

β
M
δ,r
( f )(x).
(3.10)
Secondly, using the following inequality (see [9]):



D
α
A −

D
α
A


Q

fχ

Q


L
r
≤ C|Q|
1/s+β/n



D
α
A


˙
∧
β
M
δ,r
( f )(x), (3.11)
and similar to the proof of I,wegain
II
≤ C

|α|=m


D
α
A


˙
∧
β
M
δ,r
( f )(x). (3.12)

For III, using the size condition of T,wehave
III
≤ C

|α|=m


D
α
A


˙
∧
β
M
δ,1
( f )(x). (3.13)
We now put these estimates together; and taking the supremum over all Q such that
x ∈Q, and using Lemmas 3.2 and 3.4,weobtain


T
A
( f )


˙
F
β,∞

q
≤ C

|α|=m


D
α
A


˙
∧
β
f 
L
p
. (3.14)
This completes the proof of (a).
(b) By same argument as in proof of (a), we have
1
|Q|

Q


T
A
( f )(x) −T


A

f
2

x
0



dx
≤ C

|α|=m


D
α
A


˙
∧
β

M
δ+β,r
( f )+M
δ+β,1
( f )


,
(3.15)
thus,

T
A
( f )

#
≤ C

|α|=m


D
α
A


˙
∧
β

M
δ+β,r
( f )+M
δ+β,1
( f )


. (3.16)
Lanzhe Liu 7
Now, using Lemma 3.4,wegain


T
A
( f )


L
q
≤ C



T
A
( f )

#


L
q
≤ C

|α|=m



D
α
A


˙
∧
β



M
δ+β,r
( f )


L
q
+


M
δ+β,1
( f )


L
q

≤

Cf 
L
p
.
(3.17)
This completes the proof of (b) and the theorem.

To prove Th e or e m s 2.4 and 2.5, since g
δ
and μ
δ
are all bounded from L
p
(R
n
)toL
q
(R
n
)
for 1 <p<n/δand 1/q
= 1/p−δ/n (see [11, 12]), it suffices to verify that g
A
δ
and μ
A
δ
satisfy the size condition in Theorem 3.1 (main theorem).
Suppose supp f
⊂ (2Q)

c
and x ∈ Q = Q(x
0
,l). Note that |x
0
− y|≈|x − y| for y ∈
(2Q)
c
.
For g
A
δ
,wewrite
F

A
t
( f )(x) −F

A
t
( f )

x
0

=

R
n

\

Q

ψ
t
(x − y)
|x − y|
m
−
ψ
t

x
0
− y



x
0
− y


m

R
m



A;x, y

f (y)dy
+

R
n
\

Q
ψ
t

x
0
− y

f (y)


x
0
− y


m

R
m



A;x, y

−
R
m


A;x
0
, y

dy
−

|α|=m
1
α!

R
n
\

Q

ψ
t
(x − y)(x − y)
α
|x − y|

m
−
ψ
t

x
0
− y

x
0
− y

α


x
0
− y


m

D
α

A(y) f (y)dy
= I
1
+ I

2
+ I
3
.
(3.18)
By the condition on ψ,weobtain


I
1


≤
C

R
n
\

Q


x −x
0




x
0

− y


m+1


R
m


A;x, y





f (y)




∞
0
tdt

t +


x
0

− y



2(n+1−δ)

1/2
dy
+ C

R
n
\

Q


x −x
0


ε


x
0
− y


m



R
m


A;x, y





f (y)




∞
0
tdt

t +


x
0
− y




2(n+1+ε−δ)

1/2
dy
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
∞

k=0

2
k+1

Q\2
k+1

Q




x −x
0




x
0
− y


n+1−δ
+


x −x
0


ε


x
0
− y



n+ε−δ



f (y)


dy
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
∞

k=1

2
−k
+2

−kε


1


2
k

Q


1−δ/n

2
k

Q


f (y)


dy

≤
C

|α|=m



D
α
A


˙
∧
β
|Q|
β/n
M
δ,1
( f )(x).
(3.19)
8 Continuity of multilinear operators
For I
2
, by the formula (see [5]):
R
m


A;x, y

−
R
m



A;x
0
, y

=

|η|<m
1
η!
R
m−|η|

D
η

A;x,x
0

(x − y)
η
(3.20)
and Lemma 3.5,weget


R
m


A;x, y


−
R
m


A;x
0
, y



≤
C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n


x −x

0




x
0
− y


m−1
, (3.21)
thus, similar to the proof of I
1
,


I
2


≤
C

R
n
\

Q



R
m


A;x, y

−
R
m


A;x
0
, y





x
0
− y


m+n−δ


f (y)



dy
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
∞

k=0

2
k+1

Q\2
k

Q



x −x
0




x
0
− y


n+1−δ


f (y)


dy
≤ C

|α|=m


D
α
A


˙
∧

β
|Q|
β/n
M
δ,1
( f )(x).
(3.22)
For I
3
, similar to the proof of I
1
,weobtain


I
3


≤
C

|α|=m

R
n
\

Q




x −x
0




x
0
− y


n+1−δ
+


x −x
0


ε


x
0
− y


n+ε−δ




f (y)




D
α

A(y)


dy
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
∞


k=1

2
k(β−1)
+2
k(β−ε)

M
δ,1
( f )(x)
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
M
δ,1
( f )(x)
(3.23)

so that


F

A
t
( f )(x) −F

A
t
( f )

x
0



≤
C

|α|=m


D
α
A


˙

∧
β
|Q|
β/n
M
δ,1
( f )(x). (3.24)
Lanzhe Liu 9
For μ
A
δ
,wewrite


F

A
t
( f )(x) −F

A
t
( f )

x
0



≤



∞
0


|x−y|≤t, |x
0
−y|>t


Ω(x − y)




R
m


A;x, y



|x − y|
m+n−1−δ


f (y)



dy

2
dt
t
3

1/2
+


∞
0


|x−y|>t, |x
0
−y|≤t


Ω

x
0
− y






R
m


A;x
0
, y





x
0
− y


m+n−1−δ


f (y)


dy

2
dt
t
3


1/2
+


∞
0


|x−y|≤t,|x
0
−y|≤t





Ω(x − y)R
m
(

A;x, y)
|x − y|
m+n−1−δ
−
Ω

x
0
− y


R
m


A;x
0
, y



x
0
− y


m+n−1−δ







f (y)


dy

2

dt
t
3

1/2
+ C

|α|=m


∞
0





|x−y|≤t

Ω(x − y)(x − y)
α
|x − y|
m+n−1−δ
−

|x
0
−y|≤t
Ω


x
0
− y

x
0
− y

α


x
0
− y


m+n−1−δ

×
D
α

A(y) f (y)dy





2
dt

t
3

1/2
:= J
1
+ J
2
+ J
3
+ J
4
.
(3.25)
Then
J
1
≤ C

R
n
\

Q


f (y)





R
m


A;x, y



|x − y|
m+n−1−δ


|x−y|≤t<|x
0
−y|
dt
t
3

1/2
dy
≤ C

R
n
\

Q



f (y)




R
m


A;x, y



|x − y|
m+n−1−δ


x
0
−x


1/2
|x − y|
3/2
dy
≤ C

|α|=m



D
α
A


˙
∧
β
|Q|
β/n
∞

k=1
2
−k/2
1


2
k

Q


1−δ/n

2
k


Q


f (y)


dy
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
M
δ,1
( f )(x),
(3.26)
similarly, we have J
2
≤ C


|α|=m
D
α
A
˙
∧
β
|Q|
β/n
M
δ,1
( f )(x).
For J
3
, by the following inequality (see [12]):





Ω(x − y)
|x − y|
m+n−1−δ
−
Ω

x
0
− y




x
0
− y


m+n−1−δ





≤
C



x −x
0




x
0
− y



m+n−δ
+


x −x
0


γ


x
0
− y


m+n−1−δ+γ

,
(3.27)
10 Continuity of multilinear operators
we gain
J
3
≤ C

|α|=m


D

α
A


˙
∧
β
|Q|
β/n

R
n
\

Q



x −x
0




x
0
− y


n−δ

+


x −x
0


γ


x
0
− y


n−1−δ+γ

×


|x
0
−y|≤t, |x−y|≤t
dt
t
3

1/2



f (y)


dy
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
∞

k=1

2
−k
+2
−γk

M
δ,1

( f )(x)
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
M
δ,1
( f )(x).
(3.28)
For J
4
, similar to the proof of J
1
, J
2
,andJ
3
,weobtain
J

4
≤ C

|α|=m

R
n
\

Q



x −x
0




x
0
− y


n+1−δ
+


x −x
0



1/2


x
0
− y


n+1/2−δ
+


x −x
0


γ


x
0
− y


n+γ−δ

×



D
α

A(y)


f (y)


dy
≤ C

|α|=m


D
α
A


˙
∧
β
|Q|
β/n
∞

k=1


2
k(β−1)
+2
k(β−1/2)
+2
k(β−γ)

1


2
k

Q



2
k

Q


f (y)


dy
≤ C

|α|=m



D
α
A


˙
∧
β
|Q|
β/n
M
δ,1
( f )(x).
(3.29)
These yield the desired results.
Acknowledgment
The author would like to express his gratitude to the referee for his comments and sug-
gestions.
References
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7–16.
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[3] J. Cohen, A sharp est i mate for a multilinear singular integral in R
n
, Indiana University Mathe-
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n
, Studia Mathematica 72
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, A BMO estimate for multilinear singular integrals, Illinois Journal of Mathematics 30
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Lanzhe Liu 11
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or Matematik
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nski, Characterization of the Besov spaces via the commutator operator of Coifman,
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Lanzhe Liu: Department of Mathematics, Changsha University of Science and Technology,
Changsha 410077, China
E-mail address: