BULLETINof the MALAYSIANMATHEMATICAL
SCIENCESSOCIETY http://math.usm.my/bulletin
Bull. Malays. Math. Sci. Soc. (2)35(4) (2012), 1075–1086
Multilinear Singular Integral Operators on Triebel-Lizorkin and Lebesgue Spaces
LIULANZHE
Department of Mathematics, Hunan University, Changsha 410082, P. R. of China [email protected]
Abstract. The boundedness for the multilinear operators associated to some singular in- tegral operators with non-smooth kernels on Triebel-Lizorkin and Lebesgue spaces is ob- tained.
2010 Mathematics Subject Classification: 42B20, 42B25
Keywords and phrases: Multilinear operators, singular integral operator with non-smooth kernel, Triebel-Lizorkin space, Lipschitz space.
1. Introduction
As the development of singular integral operators, their commutators and multilinear oper- ators have been well studied (see [1–4]). From [8,10], we know that the commutators and multilinear operators generated by the singular integral operators and the Lipschitz func- tions are bounded on the Triebel-Lizorkin and Lebesgue spaces. The purpose of this paper is to introduce some multilinear operators associated to certain singular integral operators with non-smooth kernels and prove the boundedness properties for the multilinear operators on the Triebel-Lizorkin and Lebesgue spaces.
2. Preliminaries and theorems
In this paper, we will study a class of multilinear operators associated to some singular integral operators with non-smooth kernels as follows.
Definition 2.1. A family of operators Dt,t >0 is said to be an ”approximations to the identity” if, for every t>0, Dt can be represented by the kernel at(x,y)in the following sense:
Dt(f)(x) = Z
Rn
at(x,y)f(y)dy
for every f ∈Lp(Rn)with p≥1and every x∈Rn, and at(x,y)satisfies:
|at(x,y)| ≤ht(x,y) =Ct−n/2s(|x−y|2/t),
Communicated byLee See Keong.
Received:May 30, 2010;Revised:October 13, 2011.
where s is a positive, bounded and decreasing function satisfying
r→∞limrn+εs(r2) =0 for someε>0.
Letmj be positive integers(j=1,· · ·,l),m1+· · ·+ml=mandbjbe functions onRn withbj∈Cmj for 1≤j≤l. Set
Rmj+1(bj;x,y) =bj(x)−
∑
|α|≤m
1
α!Dαbj(y)(x−y)α. We have
Rmj+1(bj;x,y) =bj(x)−
∑
|α|≤m−1
1
α!Dαbj(y)(x−y)α−
∑
|α|=m
1
α!Dαbj(y)(x−y)α
=Rmj(bj;x,y)−
∑
|α|=m
1
α!Dαbj(y)(x−y)α.
Definition 2.2. A linear operator T is called the singular integral operators with non- smooth kernels if T is bounded on L2(Rn)and associated with a kernel K(x,y)such that
T(f)(x) = Z
Rn
K(x,y)f(y)dy
for every continuous function f with compact support, and for almost all x not in the support of f . Moreover, the following conditions hold:
(1) There exists an ”approximations to the identity” {Bt,t >0} such that T Bt has associated kernel kt(x,y)and there exist c1,c2>0so that
Z
|x−y|>c1t1/2
|K(x,y)−kt(x,y)|dx≤c2 for all y∈Rn.
(2) There exists an ”approximations to the identity” {At,t >0} such that AtT has associated kernel Kt(x,y)which satisfies
|Kt(x,y)| ≤c4t−n/2 if |x−y| ≤c3t1/2, and
|K(x,y)−Kt(x,y)| ≤c4tδ/2|x−y|−n−δ if |x−y| ≥c3t1/2, for some c3,c4>0,δ >0.
Letmj be positive integers(j=1,· · ·,l),m1+· · ·+ml=mandbjbe functions onRn withbj∈Cmj for 1≤j≤l. The multilinear operator associated toT is defined by
Tb(f)(x) = Z
Rn
∏lj=1Rmj+1(bj;x,y)
|x−y|m K(x,y)f(y)dy.
Note that whenm=0,Tbis just the multilinear commutator ofT andb(see [5,10–11]), while whenm>0, it is non-trivial generalizations of the commutators. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [2–4]). The purpose of this paper is to study the boundedness properties for the multilinear operator Tb on Triebel-Lizorkin and Lebesgue spaces. In Section 4, some applications of Theorems in this paper are given.
Definition 2.3. Let0<β <1and1≤p<∞. The Triebel-Lizorkin space associated with the ”approximations to the identity”{At,t>0}is defined by
F˙p,Aβ,∞(Rn) ={f∈L1loc(Rn):||f||˙
Fp,Aβ,∞ <∞}, where
||f||˙
Fp,Aβ,∞ =
sup
Q3·
1
|Q|1+β/n Z
Q
|f(x)−AtQ(f)(x)|dx Lp
,
and the supremum is taken over all cubes Q ofRnwith sides parallel to the axes, tQ=l(Q)2 and l(Q)denotes the side length of Q.
Now, let us introduce some notations. Throughout this paper,Qwill denote a cube ofRn with sides parallel to the axes. For a locally integrable function f, the sharp function off is defined by
f#(x) =sup
Q3x
1
|Q|
Z
Q
|f(y)−fQ|dy,
where, and in what follows, fQ=|Q|−1RQf(x)dx. It is well-known that (see [7,12]) f#(x)≈sup
Q3x c∈Qinf
1
|Q|
Z
Q
|f(y)−c|dy.
Let
M(f)(x) =sup
Q3x
1
|Q|
Z
Q
|f(y)|dy.
Forη>0, letMη(f)(x) =M(|f|η)1/η(x).
For 1≤p<∞and 0≤δ <n, let Mδ,p(f)(x) =sup
Q3x
1
|Q|1−pδ/n Z
Q
|f(y)|pdy 1/p
.
Forβ >0, the Lipschitz spaceLipβ(Rn)is the space of functions f such that (see [10])
||f||Lipβ = sup
x,h∈Rn,h6=0
|f(x+h)−f(x)|/|h|β<∞.
The sharp maximal functionMA(f)associated with the ”approximations to the identity”
{At,t>0}is defined by(see [6,9]) MA#(f)(x) =sup
Q3x
1
|Q|
Z
Q
|f(y)−AtQ(f)(y)|dy, wheretQ= (l(Q))2andl(Q)denotes the side length ofQ.
Now we can state our theorem as following.
Theorem 2.1. Suppose T is a singular integral operator with non-smooth kernel as Defi- nition 2.2. Let0<β <min(1,ε,δ/l),{At,t>0}be the ”approximations to the identity”
and Dαbj∈Lipβ(Rn)for allα with|α|=mjand j=1,· · ·,l. Then Tbis bounded from Lp(Rn)toF˙p,Alβ,∞(Rn)for any1<p<∞.
Theorem 2.2. Suppose T is a singular integral operator with non-smooth kernel as Defini- tion 2.2. Let0<β<1and Dαbj∈Lipβ(Rn)for allαwith|α|=mjand j=1,· · ·,l. Then Tbis bounded from Lp(Rn)to Lq(Rn)for any1<p<n/lβand1/p−1/q=lβ/n.
3. Proof of theorems
To prove the theorem, we need the following lemmas.
Lemma 3.1. (see [1])Suppose that0≤η<n,1≤r<p<n/η and1/q=1/p−η/n.
Then
||Mη,r(f)||Lq≤C||f||Lp.
Lemma 3.2. (see [4])Let A be a function onRnand DαA∈Lq(Rn)for|α|=m and some q>n. Then
|Rm(A;x,y)| ≤C|x−y|m
∑
|α|=m
1
|Q(x,˜ y)|
Z
Q(x,y)˜
|DαA(z)|qdz 1/q
, whereQ(x,˜ y)is the cube centered at x and having side length5√
n|x−y|.
Lemma 3.3. [6,9]For anyγ>0, there exists a constant C>0independent ofγsuch that
|{x∈Rn:M(f)(x)>Dλ,MA#(f)(x)≤γ λ}| ≤Cγ|{x∈Rn:M(f)(x)>λ}|
forλ >0, where D is a fixed constant which only depends on n. So that
||M(f)||Lp≤C||MA#(f)||Lp
for every f ∈Lp(Rn),1<p<∞.
Lemma 3.4. [9]Let1<p<∞and T a singular integral operator with non-smooth kernels as Definition 2.2. Then T is bounded on Lp(Rn).
Lemma 3.5. Let{At,t>0}be an ”approximations to the identity”, 0<β <1 and b∈ Lipβ(Rn). Then for every f ∈Lp(Rn)andx˜∈Rn, we have
(a) supQ3x˜1/|Q|RQ|AtQ((b−bQ)f)(x)|dx≤C||b||Lip
βMβ,1(f)(x);˜ (b) supQ3x˜1/(|Q|1+β/n)RQ|AtQ((b−bQ)f)(x)|dx≤C||b||Lip
βM(f)(x)˜ if β <ε;
(c) supQ3x˜1/|Q|RQ|AtQ(f)(x)|dx≤CM(f)(x).˜ Proof. (a).Write
1
|Q|
Z
Q
|AtQ((b−bQ)f)(x)|dx≤ 1
|Q|
Z
Q Z
Rn
htQ(x,y)|(b(y)−bQ)f(y)|dydx
= 1
|Q|
Z
Q Z
2Q
htQ(x,y)|(b(y)−bQ)f(y)|dydx +
∞ k=1
∑
1
|Q|
Z
Q Z
2k+1Q\2kQ
htQ(x,y)|(b(y)−bQ)f(y)|dydx
=I1+I2. We have, by the H¨older’s inequality,
I1≤ C
|Q||2Q|
Z
Q Z
2Q
|(b(y)−bQ)f(y)|dydx≤C 1
|2Q|
Z
2Q
|(b(y)−bQ)f(y)|dy
≤C||b||Lip
β|Q|βn 1
|2Q|
Z
2Q
|f(y)|dy≤C||b||Lip
β
1
|2Q|1−βn Z
2Q
|f(y)|dy
≤C||b||Lip
βMβ,1(f)(x).˜
ForI2, notice forx∈Qandy∈2k+1Q\2kQ, then|x−y| ≥2k−1tQ1/2andhtQ(x,y)≤CtQ−n/2 s(22(k−1)). Thus
I2≤C
∞
∑
k=1
2kns(22(k−1)) 1
|2k+1Q|
Z
2k+1Q
|(b(y)−bQ)f(y)|dy
≤C
∞ k=1
∑
2kns(22(k−1))||b||Lip
β|2k+1Q|βn 1
|2k+1Q|
Z
2k+1Q
|f(y)|dy
≤C
∞ k=1
∑
2(k−1)ns(22(k−1))||b||Lip
β
1
|2k+1Q|1−βn Z
2k+1Q
|f(y)|dy
≤C
∞
∑
k=1
2kns(22(k−1))||b||Lip
βMβ,1(f)(x)≤C||b||Lip
βMβ,1(f)(˜x), where the last inequality follows from
∞
∑
k=1
2kns(22(k−1))≤C
∞
∑
k=1
2(k−1)ns(22(k−1))≤C
∞
∑
k=1
2−kε<∞
forε>0.
(b).
1
|Q|1+βn Z
Q
|AtQ((b−bQ)f)(x)|dx
≤ 1
|Q|1+βn Z
Q Z
2Q
htQ(x,y)|(b(y)−bQ)f(y)|dydx
+
∞ k=1
∑
1
|Q|1+βn Z
Q Z
2k+1Q\2kQ
htQ(x,y)|(b(y)−bQ)f(y)|dydx
≤C||b||Lipβ 1
|2Q|
Z
2Q
|f(y)|dy+C
∞
∑
k=1
2(n+β)ks(22(k−1))||b||Lipβ 1
|2k+1Q|
Z
2k+1Q
|f(y)|dy
≤C||b||Lip
βM(f)(x),˜
where the last inequality follows from
∞
∑
k=1
2(n+β)ks(22(k−1))≤C
∞
∑
k=1
2βk2(k−1)ns(22(k−1))≤C
∞
∑
k=1
2k(β−ε)<∞
forβ <ε. This completes the proof.
Proof of Theorem 2.1. Following [9], we first prove the sharp estimate forTbas following:
1
|Q|1+lβ/n Z
Q
|Tb(f)(x)−AtQTb(f)(x)|dx≤CMr(f)(˜x)
for any cubeQand 1<r<p<∞. Without loss of generality, we may assumel=2. Fix a cubeQ=Q(x0,d)and ˜x∈Q. Let ˜Q=5√
nQand ˜bj(x) =bj(x)−∑|α|=m1/(α!)(Dαbj)Q˜xα, then, by [4],Rm(bj;x,y) =Rm(b˜j;x,y)andDαb˜j=Dαbj−(Dαbj)Q˜for|α|=mj. We write, for f1=fχQ˜and f2=fχRn\Q˜,
Tb(f)(x) = Z
Rn
∏2j=1Rmj+1(b˜j;x,y)
|x−y|m K(x,y)f(y)dy
= Z
Rn
∏2j=1Rmj(b˜j;x,y)
|x−y|m K(x,y)f1(y)dy
−
∑
|α1|=m1
1 α1!
Z
Rn
Rm2(b˜2;x,y)(x−y)α1Dα1b˜1(y)
|x−y|m K(x,y)f1(y)dy
−
∑
|α2|=m2
1 α2!
Z
Rn
Rm1(b˜1;x,y)(x−y)α2Dα2b˜2(y)
|x−y|m K(x,y)f1(y)dy
+
∑
|α1|=m1,|α2|=m2
1 α1!α2!
Z
Rn
(x−y)α1+α2Dα1b˜1(y)Dα2b˜2(y)
|x−y|m K(x,y)f1(y)dy +
Z
Rn
∏2j=1Rmj+1(b˜j;x,y)
|x−y|m K(x,y)f2(y)dy
=T ∏2j=1Rmj(b˜j;x,·)
|x− ·|m f1
!
−T
∑
|α1|=m1
1 α1!
Rm2(b˜2;x,·)(x− ·)α1Dα1b˜1
|x− ·|m f1
!
−T
∑
|α2|=m2
1 α2!
Rm1(b˜1;x,·)(x− ·)α2Dα2b˜2
|x− ·|m f1
!
+T
∑
|α1|=m1,|α2|=m2
1 α1!α2!
(x− ·)α1+α2Dα1b˜1Dα2b˜2
|x− ·|m f1
!
+T ∏2j=1Rmj+1(b˜j;x,·)
|x− ·|m f2
!
and
AtQTb(f)(x)
= Z
Rn
∏2j=1Rmj(b˜j;x,y)
|x−y|m KtQ(x,y)f1(y)dy
−
∑
|α1|=m1
1 α1!
Z
Rn
Rm2(b˜2;x,y)(x−y)α1Dα1b˜1(y)
|x−y|m KtQ(x,y)f1(y)dy
−
∑
|α2|=m2
1 α2!
Z
Rn
Rm1(b˜1;x,y)(x−y)α2Dα2b˜2(y)
|x−y|m KtQ(x,y)f1(y)dy
+
∑
|α1|=m1,|α2|=m2
1 α1!α2!
Z
Rn
(x−y)α1+α2Dα1b˜1(y)Dα2b˜2(y)
|x−y|m KtQ(x,y)f1(y)dy +
Z
Rn
∏2j=1Rmj+1(b˜j;x,y)
|x−y|m KtQ(x,y)f2(y)dy
=AtQT ∏2j=1Rmj(b˜j;x,·)
|x− ·|m f1
!
−AtQT
∑
|α1|=m1
1 α1!
Rm2(b˜2;x,·)(x− ·)α1Dα1b˜1
|x− ·|m f1
!
−AtQT
∑
|α2|=m2
1 α2!
Rm1(b˜1;x,·)(x− ·)α2Dα2b˜2
|x− ·|m f1
!
+AtQT
∑
|α1|=m1,|α2|=m2
1 α1!α2!
(x− ·)α1+α2Dα1b˜1Dα2b˜2
|x− ·|m f1
!
+AtQT ∏2j=1Rmj+1(b˜j;x,·)
|x− ·|m f2
! ,
thus
1
|Q|1+2β/n Z
Q
Tb(f)(x)−AtQTb(f)(x) dx
≤ 1
|Q|1+2β/n Z
Q
T ∏2j=1Rmj(b˜j;x,·)
|x− ·|m f1
!
dx
+ C
|Q|1+2β/n Z
Q
T
∑
|α1|=m1
Rm2(b˜2;x,·)(x− ·)α1Dα1b˜1
|x− ·|m f1
!
dx
+ C
|Q|1+2β/n Z
Q
T
∑
|α2|=m2
Rm1(b˜1;x,·)(x− ·)α2Dα2b˜2
|x− ·|m f1
!
dx
+ C
|Q|1+2β/n Z
Q
T
∑
|α1|=m1,|α2|=m2
(x− ·)α1+α2Dα1b˜1Dα2b˜2
|x− ·|m f1
!
dx
+ 1
|Q|1+2β/n Z
Q
AtQT ∏2j=1Rmj(b˜j;x,·)
|x− ·|m f1
!
dx
+ C
|Q|1+2β/n Z
Q
AtQT
∑
|α1|=m1
Rm2(b˜2;x,·)(x− ·)α1Dα1b˜1
|x− ·|m f1
!
dx
+ C
|Q|1+2β/n Z
Q
AtQT
∑
|α2|=m2
Rm1(b˜1;x,·)(x− ·)α2Dα2b˜2
|x− ·|m f1
!
dx
+ C
|Q|1+2β/n Z
Q
AtQT
∑
|α1|=m1,|α2|=m2
(x− ·)α1+α2Dα1b˜1Dα2b˜2
|x− ·|m f1
!
dx
+ 1
|Q|1+2β/n Z
Q
(T−AtQT) ∏2j=1Rmj+1(b˜j;x,·)
|x− ·|m f2
!
dx :=I0+I1+I2+I3+I4+I5+I6+I7+I8.
Now, let us estimateI0,I1,I2,I3andI4, respectively. First, by Lemma 3.2, we get, forx∈Q andy∈Q,˜
|Rm(b˜j;x,y)| ≤C|x−y|m
∑
|α|=m
sup
x∈Q˜
|Dαbj(x)−(Dαbj)Q˜| ≤C|x−y|m|Q|β/n
∑
|α|=m
||Dαbj||Lip
β, thus, by the H¨older’s inequality andLr-boundedness ofT, we obtain
I0≤ C
|Q|1+2β/n Z
Q
T
|Q|2β/n|x−y|m1+m2∏2j=1
∑|αj|=mj||Dαjbj||Lip
β
|x−y|m f1
(x)
dx
≤C
2
∏
j=1
∑
|αj|=mj
||Dαjbj||Lip
β
! 1
|Q|
Z
Q
|T(f1)(x)|dx
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lipβ
! 1
|Q|
Z
Rn
|T(f1)(x)|rdx 1/r
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
! 1
|Q|˜ Z
Q˜
|f(x)|rdx 1/r
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
Mr(f)(x);˜
I1≤C
∑
|α2|=m2
||Dα2b2||Lipβ
∑
|α1|=m1
1
|Q|1+β/n Z
Q
|T(Dα1b˜1f1)(x)|dx
≤C
∑
|α2|=m2
||Dα2b2||Lip
β
∑
|α1|=m1
1
|Q|β/n 1
|Q|
Z
Rn
|T(Dα1b˜1f1)(x)|rdx 1/r
≤C
∑
|α2|=m2
||Dα2b2||Lip
β
∑
|α1|=m1
1
|Q|β/n 1
|Q|
Z
Rn
|(Dα1b1(x)−(Dα1b1)Q˜)f1(x)|rdx 1/r
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
! 1
|Q|˜ Z
Q˜
|f(x)|rdx 1/r
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
Mr(f)(x).˜
ForI2, similar to the proof ofI2, we get
I2≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
Mr(f)(˜x).
Similarly, forI3, we obtain
I3≤C
∑
|α1|=m1,|α2|=m2
|Q|−2β/n 1
|Q|
Z
Rn
|T(Dα1b˜1Dα2b˜2f1)(x)|rdx 1/r
≤C
∑
|α1|=m1,|α2|=m2
|Q|−2β/n 1
|Q|
Z
Rn
|Dα1b˜1(x)Dα2b˜2(x)f1(x)|rdx 1/r
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lipβ
! 1
|Q|˜ Z
Q˜
|f(x)|rdx 1/r
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
Mr(f)(x).˜
ForI4,I5,I6andI7, by Lemma 3.5 and similar to the proof ofI0,I1,I2andI3, we get
I4+I5+I6+I7≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
! 1
|Q|
Z
Q
|AtQT(f1)(x)|dx
+C
∑
|α2|=m2
||Dα2b2||Lip
β
∑
|α1|=m1
1
|Q|1+β/n Z
Q
|AtQT(Dα1b˜1f1)(x)|dx
+C
∑
|α1|=m1
||Dα1b1||Lip
β
∑
|α2|=m2
1
|Q|1+β/n Z
Q
|AtQT(Dα2b˜2f1)(x)|dx
+C
∑
|α1|=m1,|α2|=m2
1
|Q|1+2β/n Z
Q
|AtQT(Dα1b˜1Dα2b˜2f1)(x)|dx
≤C
2
∏
j=1∑
|α|=mj
||Dαbj||Lip
β
!
Mr(f)(x).˜ ForI8, we write
(T−AtQT) ∏2j=1Rmj+1(b˜j;x,·)
|x− ·|m f2
!
= Z
Rn
∏2j=1Rmj+1(b˜j;x,y)
|x−y|m (K(x,y)−KtQ(x,y))f2(y)dy
= Z
Rn
∏2j=1Rmj(b˜j;x,y)
|x−y|m (K(x,y)−KtQ(x,y))f2(y)dy
−
∑
|α1|=m1
1 α1!
Z
Rn
Dα1b˜1(y)(x−y)α1Rm2(b˜2;x,y)
|x−y|m (K(x,y)−KtQ(x,y))f2(y)dy
−
∑
|α2|=m2
1 α2!
Z
Rn
Dα2b˜2(y)(x−y)α2Rm1(b˜1;x,y)
|x−y|m (K(x,y)−KtQ(x,y))f2(y)dy
+
∑
|α1|=m1,|α2|=m2
1 α1!α2!
Z
Rn
Dα1b˜1(y)Dα2b˜2(y)(x−y)α1+α2
|x−y|m (K(x,y)−KtQ(x,y))f2(y)dy.
By Lemma 3.2 and the following inequality, forb∈Lipβ(Rn),
|b(x)−bQ| ≤ 1
|Q|
Z
Q
||b||Lip
β|x−y|βdy≤C||b||Lip
β(|x−x0|+√ nd/2)β, we get
|Rmj(b˜j;x,y)| ≤C
∑
|α|=mj
||Dαbj||Lip
β(|x−x0|+d)mj+β.
Note that|x−y| ∼ |x0−y|forx∈Qandy∈Rn\Q, so that, we obtain, by the conditions on˜ KandKtQ,
(T−AtQT) ∏2j=1Rmj+1(b˜j;x,·)
|x− ·|m f2
!
≤C Z
Rn\Q˜
|x−x0|δ
|x0−y|m+n+δ
2
∏
j=1|Rmj(b˜j;x,y)||f(y)|dy
+C
∑
|α1|=m1 Z
Rn\Q˜
|Dα1b˜1(y)||x−x0|δ
|x0−y|m2+n+δ |Rm2(b˜2;x,y)||f(y)|dy
+C
∑
|α2|=m2 Z
Rn\Q˜
|Dα2b˜2(y)||x−x0|δ
|x0−y|m1+n+δ |Rm1(b˜1;x,y)||f(y)|dy
+C
∑
|α1|=m1,|α2|=m2 Z
Rn\Q˜
|Dα1b˜1(y)Dα2b˜2(y)||x−x0|δ
|x0−y|n+δ |f(y)|dy
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
! ∞
∑
k=0 Z
2k+1Q\2˜ kQ˜
|x−x0|δ
|x0−y|n+δ−2β|f(y)|dy
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
|Q|2β/n
∞ k=0
∑
2k(2β−δ) 1
|2k+1Q|˜ Z
2k+1Q˜
|f(y)|dy
≤C
2
∏
j=1
∑
|αj|=mj
||Dαjbj||Lip
β
!
|Q|2β/nMr(f)(x),˜
thus
I8≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
Mr(f)(˜x).
We now put these estimates together and take the supremum over allQsuch that ˜x∈Q, then, by Lemma 3.1,
||Tb(f)||˙
Fp,A2β,∞≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
||Mr(f)||Lp
≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
||f||Lp.
This completes the proof of Theorem 2.1.
Proof of Theorem 2.2. By using the same argument as in proof of Theorem 2.1, we obtain 1
|Q|
Z
Q
|Tb(f)(x)−AtQTb(f)(x)|dx≤C
2
∏
j=1
∑
|αj|=mj
||Dαjbj||Lip
β
!
M2β,r(f), for any cubeQand 1<r<p<∞, thus, we get the sharp estimate ofTbas following
MA#(Tb(f))≤C
2
∏
j=1∑
|αj|=mj
||Dαjbj||Lip
β
!
M2β,r(f).
By Lemma 3.3,||M(g)||Lp≤C||MA#(g)||Lpfor everyg∈Lp(Rn)and 1<p<∞. Now, using Lemma 3.1 and 3.4, we get
||Tb(f)||Lq≤ ||M(Tb(f))||Lq≤C||MA#(Tb(f))||Lq
≤C
2
∏
j=1
∑
|αj|=mj
||Dαjbj||Lip
β
!
||M2β,r(f)||Lq ≤C
2
∏
j=1
∑
|αj|=mj
||Dαjbj||Lip
β
!
||f||Lp.
This completes the proof of the theorem.
4. Applications
In this section we shall apply Theorems of the paper to the holomorphic functional calculus of linear elliptic operators. First, we review some definitions regarding the holomorphic functional calculus (see [6,9]). Given 0≤θ<π, define
Sθ={z∈C:|arg(z)| ≤θ}[{0}
and its interior bySθ0. Set ˜Sθ =Sθ\ {0}. An closed operatorLon some Banach spaceE is said to be of typeθ if its spectrumσ(L)⊂Sθ and for everyν∈(θ,π], there exists a constantCν such that
|η|||(ηI−L)−1|| ≤Cν, η∈/S˜θ. Forν∈(0,π], let
H∞(S0µ) ={f :S0θ→C: f is holomorphic and||f||L∞<∞}, where||f||L∞=sup{|f(z)|:z∈S0µ}. Set
Ψ(S0µ) =
g∈H∞(S0µ):∃s>0,∃c>0 such that|g(z)| ≤c |z|s 1+|z|2s
.
IfLis of typeθandg∈H∞(S0µ), we defineg(L)∈L(E)by g(L) =−(2πi)−1
Z
Γ
(ηI−L)−1g(η)dη,
whereΓis the contour{ξ =re±iφ :r≥0}parameterized clockwise aroundSθ withθ <
φ<µ. If, in addition,Lis one-one and has dense range, then, forf ∈H∞(S0µ), f(L) = [h(L)]−1(f h)(L),
whereh(z) =z(1+z)−2. Lis said to have a bounded holomorphic functional calculus on the sectorSµ, if
||g(L)|| ≤N||g||L∞
for someN>0 and for allg∈H∞(S0µ).
Now, letLbe a linear operator onL2(Rn)withθ<π/2 so that(−L)generates a holo- morphic semigroupe−zL, 0≤ |arg(z)|<π/2−θ. Applying Theorem 6 of [9], we get Theorem 4.1. Let0<β <1and1≤p<∞. The Triebel-Lizorkin space associated with the ”approximations to the identity”{At,t>0}is defined by
F˙p,Aβ,∞(Rn) ={f∈L1loc(Rn):||f||˙
Fp,Aβ,∞ <∞}, where
||f||˙
Fp,Aβ,∞ =
sup
Q3·
1
|Q|1+β/n Z
Q
|f(x)−AtQ(f)(x)|dx Lp
, Assume the following conditions are satisfied:
(i). The holomorphic semigroup e−zL, 0≤ |arg(z)|<π/2−θ is represented by the kernels az(x,y)which satisfy, for allν>θ, an upper bound
|az(x,y)| ≤cνh|z|(x,y)
for x,y∈Rn, and0≤ |arg(z)|<π/2−θ, where ht(x,y) =Ct−n/2s(|x−y|2/t)and s is a positive, bounded and decreasing function satisfying
r→∞limrn+εs(r2) =0.
(ii). The operator L has a bounded holomorphic functional calculus in L2(Rn), that is, for allν>θ and g∈H∞(S0µ), the operator g(L)satisfies
||g(L)(f)||L2 ≤cν||g||L∞||f||L2. Let Dαbj∈Lipβ(Rn)for allα with|α|=mjand j=1,· · ·,l.
(a). If0<β <min(1,ε,δ/l), then the multilinear operator g(L)bassociated to g(L)is bounded from Lp(Rn)toF˙p,Alβ,∞(Rn)for any1<p<∞;
(b). If0<β<1, then g(L)bis bounded from Lp(Rn)to Lq(Rn)for any1<p<n/lβ and1/p−1/q=lβ/n.
Acknowledgement. The author would like to express his gratitude to the referee for his comments and suggestions.
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