Paper-25-20-2009. 150KB Jun 04 2011 12:03:13 AM

BOUNDEDNESS FOR MULTILINEAR OPERATORS OF MULTIPLIER OPERATORS ON TRIEBEL-LIZORKIN AND

LEBESGUE SPACES

Xiaosha Zhou and Lanzhe Liu

Abstract. _{The boundedness for the multilinear operators associated to some} multiplier operators and the Lipschitz functions on Triebel-Lizorkin and Lebesgue spaces are obtained.

2000Mathematics Subject Classification: 42B20, 42B25. 1. Introduction

As the development of singular integral operators, their commutators and mul-tilinear operators have been well studied (see [1-7]). From [2][13], we know that the commutators and multilinear operators generated by the singular integral oper-ators and the Lipschitz functions are bounded on the Triebel-Lizorkin and Lebesgue spaces. The purpose of this paper is to introduce some multilinear operators as-sociated to some multiplier operators and the Lipschitz functions, and prove the boundedness properties for the multilinear operators on the Triebel-Lizorkin and Lebesgue spaces.

2. Preliminaries and Theorems

First, let us introduce some notations. Throughout this paper,Q will denote a cube of Rn with sides parallel to the axes. For a locally integrable function f, let f(Q) =R

Qf(x)dx,fQ=|Q|−1RQf(x)dxand

f#(x) = sup

Q∋x

1

|Q_| Z

It is well-known that(see [14][15]) f#(x)_≈sup

Q∋x

inf

c∈C

1

|Q_| Z

Q|f(y)−c|dy.

For 1_≤p <_∞ and 0_≤δ < n, let

Mδ,p(f)(x) = sup Q∋x

₁

|Q_|1−pδ/n

Z

Q|

f(y)_|pdy 1/p

,

which is the Hardy-Littlewood maximal function when p= 1 and δ= 0. For β >0 and p >1, let ˙Fβ,∞

p (Rn) be the homogeneous Triebel-Lizorkin space

(see [13]). The Lipschitz space ˙_∧β(Rn) is the space of functionsf such that

||f_||∧˙β = sup

x, h∈Rn

h6= 0

∆

[β]+1

h f(x)

/|h|

β _<∞,

where ∆k_h denotes thek-th difference operator (see [13]).

In this paper, we will study a class of multilinear operators associated to some multiplier operators as follows.

A bounded measurable functionkdefined onRn_{\ {}0_}is called a multiplier. The multiplier operatorT associated withkis defined by

T(f)(x) =k(x) ˆf(x), forf _∈S(Rn),

where ˆf denotes the Fourier transform off andS(Rn) is the Schwartz test function class. Now, we recall the definition of the class M(s, l). Denote by _|x_{| ∼} t the fact that the value of x lies in the annulus _{x _∈ Rn _{: at < |x| < bt}, where}

0< a_≤1< b <_∞ are values specified in each instance.

Definition 1. _{([11]) Let l≥0 be a real number and 1≤s≤2. we say that the}

multiplier k satisfies the condition M(s, l), if

Z

|ξ|∼R|D α_k(ξ)

|sdξ !1

s

< CRn/s−|α|

for all R > 0 and multi-indices α with _|α_{| ≤}l, when l is a positive integer, and, in addition, if

Z |ξ|∼R|

Dαk(ξ)₋Dαk(ξ₋z)_|sdξ !1_s

≤C(|z| R)

for all _|z_|< R/2and all multi-indicesα with_|α_|= [l], the integer part ofl,i.e.,[l]is the greatest integer less than or equal to l, and l= [l] +γ when l is not an integer.

DenoteD(Rn) =_{φ_∈S(Rn) : supp(φ) is compact_}and ˆD0(Rn) ={φ∈ S(Rn) :

ˆ

φ_∈D(Rn) and ˆφvanishes in a neighbourhood of the origin_}. The following bound-edness property ofT onLp_(Rn_{) is proved by Str¨omberg and Torkinsky (see [11-14]).}

.

Lemma 1._([11]) Let k _∈ M(s, l),1 _≤s _≤ 2, and l > n_s. Then the associated mapping T, defined a priori for f _∈ Dˆ0(Rn), T(f)(x) = (f ∗K)(x), extends to a

bounded mapping from Lp(Rn) into itself for 1< p <_∞ and K(x) = ˇk(x).

Definition 2._{([11]) For a real number} ˜_{l ≥ 0 and 1 ≤ s <˜ ∞, we say that K}

verifies the condition M˜(˜s,˜l), and write K _∈M˜(˜s,˜l), if

Z |x|∼R|

Dα˜K(x)_|s˜dx !1_s˜

≤CRn/˜s−n−|α˜|, R >0

for all multi-indices _|α˜_{| ≤}˜l and, in addition, if

Z

|x|∼R|D

˜

α_K(x)

−Dα˜K(x₋z)_|s˜dx !1_˜

s

≤C(|z| R)

v_Rn

˜

s−n−u, if 0< v <1,

Z

|x|∼R|D

˜

α_K(x)

−Dα˜K(x₋z)_|s˜dx !1_s˜

≤C(|z| R)(log

R

|z_|)R

n

˜

s−n−u, if v= 1,

for all _|z_| < R₂, R > 0, and all multi-indices α˜ with_|α˜_| = u, where u denotes the largest integer strictly less than ˜l with ˜l=u+v.

Lemma 2. _{([11]) Suppose k∈M(s, l), 1≤s≤2. Given 1≤s <˜ ∞, let r≥1}

be such that 1_r = max_{1_s,1₋1_˜s}. Then K _∈M˜(˜s,˜l), where ˜l=l₋n_r .

Lemma 3. _{Let 1≤s≤2, suppose that lis a positive real number withl > n/r,} 1/r= max_{1/s,1₋1/˜s_}, and k_∈M(s, l). Then there is a positive constanta, such that

Z

Qk

|K(x₋z)₋K(xQ−z)|s˜dz

1/s˜

≤C2−ka(2kh)−n/s˜′.

Proof.We split our proof into two cases:

When ˜l=l₋n_s <1, noting thatl is a positive real number andl > n_s. Applying the conditionK _∈M(˜˜ s,˜l) for v=l₋n_s and u= 0, one has

Z

Qk

|K(x₋z)₋K(xQ−z)|s˜dz

1_˜s

≤C2−k(l−ns)(2kh)− n

˜

s′,

let a=l₋ n_s, Z

Qk

|K(x₋z)₋K(xQ−z)|˜sdz

1_s˜

≤C2−ka(2kh)−sn˜′.

When ˜l=l₋n_s = 1, we choose 0< ξ <1, such thatt1−ξlog(1/t)_≤Cfor 0< t <1/2. Noting that K _∈M˜(˜s,˜l), by Definition 2, for u= 0, v= 1,

Z

Qk

|K(x₋z)₋K(xQ−z)|s˜dz

1_˜s

≤ C(|y−xQ| 2k_h )

ξ₍|y−xQ|

2k_h )

1−ξ_(log 2kh |y₋x_Q|)(2

k_h)n/˜s−n ≤ C2−kξ(2kh)−n/s˜′,

let a=ξ, then Z

Qk

|K(x₋z)₋K(xQ−z)|˜sdz

1_s˜

≤C2−ka(2kh)−n/s˜′.

Case 2. _{1≤s≤2 and l−n/s >1. Set d= [l−n/s], ifl−n/s >1 is not an} integer, and d= l₋n/s₋1 if l₋n/s > 1 is an integer. Choosel1 = l−d; then

0 < l1−n/s≤ 1 and 0< l1 < l. So, from k ∈M(s, l) we know k ∈M(s, l1). Set

˜

l=l1−n/s; by Lemma 3, K∈M˜(˜s,˜l). Repeating the proof ofCase 1, except for

replacing l byl1, we can obtain the same result under the assumption l−n/s >1.

We omit the details here.

Certainly when 0<s˜′_{< s, which is the same as the above.}

Now we can define the multilinear operator associated to the multiplier operator T. Let mj be the positive integers(j= 1,· · ·, l),m1+· · ·+ml=m and Aj be the

functions on Rn_{(j= 1,· · ·, l). Set}

Rmj+1(Aj;x, y) =Aj(x)−

X |α|≤mj

1 α!D

α_A

j(y)(x−y)α.

By Lemma 1, T(f)(x) = (K _∗f)(x) for K(x) = ˇk(x). The multilinear operator associated to T is defined by

TA(f)(x) = Z

Rn

Ql

j=1Rmj+1(Aj;x, y)

Note that whenm= 0,TA os just the multilinear commutators ofT andA(see [20-21]). 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 [4-8][10]). The purpose of this paper is to study the boundedness properties for the multilinear operator TA_{. We}

shall prove the following theorems in Section 3. Now we can state our theorems as following.

Theorem 1. _{Let 0 < β < min(1/l, a/l) and D}α_{Aj ∈ ∧β}˙ _(Rn_{) for all α with}

|α_|=mj and j= 1,· · ·, l. Then TA is bounded from Lp(Rn) to F˙plβ,∞(Rn) for any

1< p <_∞.

Theorem 2. _{Let 0 < β ≤1 and D}α_{Aj ∈ ∧}˙_β(Rn_{) for all α with |α| =mj and} j = 1,_{· · ·}, l. Then TA is bounded from Lp(Rn) toLq(Rn) for any 1< p < n/lβ and

1/p₋1/q =lβ/n.

3. Proof of Theorem

To prove the theorem, we need the following lemmas. Lemma 4_{(see [13]). For0< β <1, 1< p <∞, we have}

||f_||F˙β,∞

p ≈

sup

Q

1

|Q_|1+β/n

Z

Q|

f(x)₋f_Q|dx _Lp

≈

sup ·∈Q

inf

c

1

|Q_|1+β/n

Z

Q|

f(x)₋c_|dx _Lp

.

Lemma 5_{(see [13]).} For0< β <1,1_≤p_{≤ ∞}, we have ||b_||∧˙β ≈sup

Q

1

|Q_|1+β/n

Z

Q|

b(x)₋b_Q|dx_≈sup

Q

1

|Q_|β/n

₁

|Q_| Z

Q|

b(x)₋b_Q|pdx 1/p

.

Lemma 6_{(see [1]).} Suppose that 0 _≤ δ < n, 1 _≤ r < p < n/δ and 1/q = 1/p₋δ/n. Then

||Mδ,r(f)||Lq ≤C||f||Lp.

Lemma 7_{(see [5]). Let A be a function on R}n _{and D}α_{A ∈L}q_(Rn_{) for |α|=m}

and some q > n. Then

|Rm(A;x, y)| ≤C|x−y|m

X |α|=m

1

|Q(x, y)˜ _| Z

˜

Q(x,y)|D

α_A(z) |qdz

!1/q ,

where ˜Q(x, y) is the cube centered atx and having side length 5√n_|x₋y_|.

Lemma 8_{(see [16]).Forb∈∧β}˙ _{,0< β <1, 0≤δ < nand 1< r <∞, we have}

||(b₋bQ)f χQ||Lr ≤C||b||_∧˙ β|Q|

Lemma 9._{([3])LetT be the multiplier operator. Then, for everyf ∈L}p_(Rn_),1< p <_∞,

||T(f)_||Lp ≤C||f||Lp.

Proof of Theorem 1._{We first prove the sharp estimate for T}A_{as following} 1

|Q_|1+lβ/n

Z

Q|

TA(f)(x)₋C0|dx≤C

l Y j=1   X |αj|=mj

||Dαj_A

j||BM O



M_s(f)(˜x). for 1 < s < p and some constant C0. Without loss of generality, we may assume

l = 2. Fix a cube Q = Q(x0, d) and ˜x ∈ Q. Let ˜Q = 5√nQ and ˜Aj(x) =

Aj(x)− P

|α|=mj 1

α!(DαAj)Q˜xα, then Rmj+1(Aj;x, y) =Rmj+1( ˜Aj;x, y) andDαA˜j =

Dα_A

j−(DαAj)_Q˜ for|α|=mj. We write, for f1=f χ_Q˜ and f2=f χ_Rn_\Q˜,

TA(f)(x) = Z

Rn

Q2

j=1Rmj+1( ˜Aj;x, y)

|x₋y_|m K(x−y)f2(y)dy

+ Z

Rn

Q2

j=1Rmj( ˜Aj;x, y)

|x₋y_|m K(x−y)f1(y)dy

− X

|α1|=m1 1 α1!

Z

Rn

Rm2( ˜A2;x, y)(x−y)

α1

|x₋y_|m D α1_A˜

1(y)K(x−y)f1(y)dy

− X

|α2|=m2 1 α2!

Z

Rn

Rm1( ˜A1;x, y)(x−y)

α2

|x₋y_|m D α2_A˜

2(y)K(x−y)f1(y)dy

+ X

|α1|=m1,|α2|=m2 1 α1!α2!

Z

Rn

(x₋y)α1+α2_Dα1_A˜

1(y)Dα2A˜2(y)

|x₋y_|m ·K(x₋y)f1(y)dy,

then 1

|Q_|1+2β/n

Z

Q

T

A_(f)(x)−TA˜_(f

2)(x0)

dx 1

|Q_|1+2β/n

Z Q Z Rn Q2

j=1Rmj( ˜Aj;x, y)

|x₋y_|m K(x−y)f1(y)dy

dx + C

|Q_|1+2β/n

Z Q X |α1|=m1

Z

Rn

Rm2( ˜A2;x, y)(x−y)

α1

|x₋y_|m D α1_A˜

1(y)K(x−y)f1(y)dy

dx + C

|Q_|1+2β/n

Z Q X |α2|=m2

Z

Rn

Rm1( ˜A1;x, y)(x−y)

α2

|x₋y_|m D α2_A˜

2(y)K(x−y)f1(y)dy

+ C

|Q_|1+2β/n

Z Q X |α1|=m1,|α2|=m2

Z

Rn

(x₋y)α1+α2_Dα1_A˜

1(y)Dα2A˜2(y)

|x₋y_|m K(x−y)f1(y)dy

dx + 1

|Q_|1+2β/n

Z Q T ˜ A_(f

2)(x)−TA˜(f2)(x0)

dx :=I1+I2+I3+I4+I5.

Now, let us estimate I1, I2, I3, I4 and I5, respectively. First, by Lemma 7 and

Lemma 5, we get, for x_∈Q and y_∈Q,˜

|Rm( ˜Aj;x, y)| ≤ C|x−y|m

X |α|=m

sup

x∈Q˜

|DαAj(x)−(DαAj)_Q˜|

≤ C_|x₋y_|m_|Q_|β/n X |α|=m

||DαA_j||∧˙β.

Now, by the H¨older’s inequality and Ls-boundedness ofT, we obtain

I1 ≤ C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β

 

1

|Q_| Z

Q|T(f1)(x)|dx

≤ C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β

 

₁

|Q_| Z

Rn|T(f1)(x)|

s_dx1/s

≤ C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β

 

1

|Q˜_| Z

˜

Q|f(x)| s_dx

!1/s

≤ C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β



M_s(f)(˜x). For I2, by Lemma 8, we get

I2 ≤ C

X |α2|=m2

||Dα2_A

2||∧˙β|Q|

−β/n X |α1|=m1

1

|Q_| Z

Q|T(D α1_Af˜

1)(x)|dx

≤ C X

|α2|=m2

||Dα2_A

2||∧˙β

X |α1|=m1

|Q_|−β/n−1/s_||T((Dα1_A

−(Dα1_A)

˜

Q)f1)||Ls

≤ C X

|α2|=m2

||Dα2_A

2||∧˙β|Q|

−β/n−1/s X |α1|=m

||(Dα1_A

−(Dα1_A)

˜

Q)f1||Ls

≤ C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β



For I3, similar to the proof ofI2, we get

I3 ≤C 2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



M_s(f)(˜x).

Similarly, for I4, set s = pq3 with p, q1, q2, q3 > 1 and 1/q1+ 1/q2+ 1/q3 = 1, we

obtain I4 ≤ C

X |α1|=m1,|α2|=m2

1

|Q_|1+2β/n

Z

Q|T(D α1_A˜

1Dα2A˜2f1)(x)|dx

≤ C X

|α1|=m1,|α2|=m2

|Q_|−2β/n ₁

|Q_| Z

Rn|T(D

α1_A˜

1Dα2A˜2f1)(x)|pdx

1/p

≤ C X

|α1|=m1,|α2|=m2

|Q_|−2β/n−1/p Z

Rn|D

α1_A˜

1(x)Dα2A˜2(x)f1(x)|pdx

1/p

≤ C X

|α1|=m1,|α2|=m2

|Q˜_|−2β/n 1

|Q˜_| Z

˜

Q|

f(x)_|pq3_dx !1/pq3

× 1

|Q˜_| Z

˜

Q|D α1_A˜

1(x)|pq1dx

!1/pq1 1

|Q˜_| Z

˜

Q|D α2_A˜

2(x)|pq2dx

!1/pq2

≤ C

2

Y

j=1

 

X |αj|=mj

||Dαj_Aj|| ˙

∧β



M_s(f)(˜x). For I5, we write

TA˜(f2)(x)−TA˜(f2)(x0)

= Z

Rn

_K(x−y)

|x₋y_|m −

K(x0−y)

|x0−y|m

2 Y

j=1

Rmj( ˜Aj;x, y)f2(y)dy

+ Z

Rn

Rm1( ˜A1;x, y)−Rm1( ˜A1;x0, y) R_m

2( ˜A2;x, y)

|x0−y|m

K(x0−y)f2(y)dy

+ Z

Rn

Rm2( ˜A2;x, y)−Rm2( ˜A2;x0, y)

R_m1( ˜A₁;x₀, y)

|x0−y|m

K(x0−y)f2(y)dy

− X

|α1|=m1 1 α1!

Z

RnD

α1_A˜

1(y)f2(y)

×

"

Rm2( ˜A2;x, y)(x−y)

α1

|x₋y_|m K(x−y)−

Rm2( ˜A2;x0, y)(x0−y)

α1

|x0−y|m

K(x0−y)

− X

|α2|=m2 1 α2!

Z

RnD

α2_A˜

2(y)f2(y)

×

"

Rm1( ˜A1;x, y)(x−y)

α2

|x₋y_|m K(x−y)−

Rm1( ˜A1;x0, y)(x0−y)

α2

|x0−y|m

K(x0−y)

# dy

+ X

|α1|=m1,|α2|=m2 1 α1!α2!

Z

Rn

"

(x₋y)α1+α2

|x₋y_|m K(x−y)−

(x0−y)α1+α2

|x0−y|m

K(x0−y)

#

×Dα1_A˜

1(y)Dα2A˜2(y)f2(y)dy

= I₅(1)+I₅(2)+I₅(3)+I₅(4)+I₅(5)+I₅(6).

By Lemma 8 and the following inequality, forb_∈∧β˙ (Rn_), |b(x)₋bQ| ≤ 1

|Q_| Z

Q||b||∧˙β|x−y| β_dy

≤ ||b_||∧˙β(|x−x0|+d)

β_,

we get

|Rmj( ˜Aj;x, y)| ≤

X |α|=mj

||DαAj||∧˙β(|x−y|+d)

mj+β_.

Note that _|x₋y_{| ∼ |}x0−y|forx∈Qand y∈Rn\Q, we obtain, by Lemma 2 and˜

3,

|I₅(1)_{| ≤} ∞ X

k=0

Z

2k+1_Q˜_\2k_Q˜|K(x−y)−K(x0−y)|

1

|x₋y_|m

2

Y

j=1

|Rmj( ˜Aj;x, y)||f(y)|dy

+ ∞ X

k=0

Z

2k+1_Q˜\2k_Q˜

1

|x₋y_|m −

1

|x0−y|m

|K(x0−y)|

2

Y

j=1

|Rmj( ˜Aj;x, y)||f(y)|dy

≤ C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β

 

∞ X

k=0

|2k+1Q˜_|2β/n Z

2k+1_Q˜|f(y)|

s_dy1/s

×

Z

2k+1_Q˜\2k_Q˜|K(x−y)−K(x0−y)|

s′

dy !1/s′

+C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β

 

∞ X

k=0

|2k+1Q˜_|2β/n2−k Z

2k+1_Q˜|f(y)|

s_dy1/s

×

Z

2k+1_Q˜\2k_Q˜|K(x0−y)|

s′

dy !1/s′

≤ C 2 Y j=1   X |αj|=mj

||Dαj_A

j||∧˙β



|Q|2β/n ∞ X

k=1

· 1 |2k_Q˜_|

Z

2k_Q˜|f(y)|

s_dy

!1/s

≤ C

2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



|Q|2β/nMs(f)(˜x).

For I₅(2), by the formula (see [5]): Rmj( ˜Aj;x, y)−Rmj( ˜Aj;x0, y) =

X |η|<mj

1

η!Rmj−|η|(D

η_A˜

j;x, x0)(x−y)η

and Lemma 4, we get

|I₅(2)_{| ≤} C

2

Y

j=1

 

X |α|=mj

||DαAj||∧˙β

 

∞ X

k=0

|2k+1Q˜_|2β/n

·

Z

2k+1_Q˜\2k_Q˜

|x₋x0|

|x0−y||

K(x0−y)||f(y)|dy

≤ C

2

Y

j=1

 

X |α|=mj

||DαAj||∧˙β

 

∞ X

k=0

|2k+1Q˜_|2β/n2−k Z

2k+1_Q˜\2k_Q˜|K(x0−y)|

s′

dy !1/s′

×

Z

2k+1_Q˜|f(y)|

s_dy1/s

≤ C

2

Y

j=1

 

X |α|=mj

||Dαbj||∧˙β

 

∞ X

k=0

2k(2β−1) 1

|2k_Q˜_|

Z

2k_Q˜|f(y)|

s_dy

!1/s

≤ C

2

Y

j=1

 

X |α|=mj

||DαAj||∧˙β



|Q|2β/nM_s(f)(˜x). Similarly,

|I₅(3)_{| ≤}C

2

Y

j=1

 

X |α|=mj

||DαAj||∧˙β



|Q|2β/nM_s(f)(˜x). For I₅(4), similar to the estimates of I₅(1) and I₅(2), we obtain,

|I₅(4)_{| ≤} C X |α1|=m1

Z

Rn_\Q˜

(x₋y)α1_K(x−y)

|x₋y_|m −

(x0−y)α1K(x0−y)

|x0−y|m

×|Rm2( ˜A2;x, y)||D

α1_A˜

+C X |α1|=m1

Z

Rn_\Q˜|Rm2( ˜A2;x, y)−Rm2( ˜A2;x0, y)|

×|(x0−y) α1_K(x

0−y)|

|x0−y|m |

Dα1_A˜

1(y)||f(y)|dy

≤ C

2

Y

j=1

 

X |α|=mj

||DαAj||∧˙β



|Q|2β/n ∞ X

k=0

(2k(2β−a)+ 2k(2β−1))

× 1

|2k_Q˜_| Z

2k_Q˜|f(y)|

s_dy

!1/s

≤ C

2

Y

j=1

 

X |α|=mj

||DαAj||∧˙β



|Q|2β/nM_s(f)(˜x). Similarly,

|I₅(5)_{| ≤}C

2

Y

j=1

 

X |α|=mj

||DαAj||∧˙β



|Q|2β/nM_s(f)(˜x). For I₅(6), we get

|I₅(6)_{| ≤} C X |α1|=m1,|α2|=m2

Z

Rn_\Q˜

(x₋y)α1+α2_K(x−y)

|x₋y_|m −

(x0−y)α1+α2K(x0−y)

|x0−y|m

×|Dα1_A˜

1(y)||Dα2A˜2(y)||f(y)|dy

≤ C

2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



|Q|2β/n ∞ X

k=0

(2k(2β−a)+ 2k(2β−1))

· 1

|2k_Q˜_|

Z

2k_Q˜|f(y)|

s_dy

!1/s

≤ C

2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



|Q|2β/nM_s(f)(x). Thus

|TA˜(f2)(x)−T ˜

A_(f

2)(x0)| ≤C

X |α|=m

||DαA_||∧˙β|Q| 2β/n_M

s(f)(˜x)

and

I5 ≤C

X |α|=m

We now put these estimates together, and taking the supremum over allQsuch that ˜

x_∈Q, and using Lemma 4 and 6, we obtain

||TA(f)_||F˙2β,∞

p ≤ C

2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



||Ms(f)||Lp

≤ C

2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



||f||Lp. This completes the proof of the theorem.

Proof of Theorem 2._{By using the same argument as in proof of Theorem 1,} we obtain, for 1< s < p,

1

|Q_| Z

Q|T

A_(f)(x)

−TA˜(f2)(x0)|dx≤C 2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



M_2β,s(f),

thus, we get the sharp estimate of TA as following

(TA(f))#_≤C

2

Y

j=1

 

X |αj|=mj

||Dαj_Aj|| ˙

∧β



M_2β,s(f). Now, using Lemma 6, we get

||TA(f)_||Lq ≤ C||(TA(f))#||Lq ≤C 2

Y

j=1

 

X |αj|=mj

||Dαj_Aj|| ˙

∧β



||M_2β,r(f)||Lq

≤ C

2

Y

j=1

 

X |αj|=mj

||Dαj_A

j||∧˙β



||f||Lp. This completes the proof of the theorem.

References

[1] S. Chanillo, A note on commutators, Indiana Univ. Math. J., 31, (1982), 7-16.

[2] W. G. Chen, Besov estimates for a class of multilinear singular integrals, Acta Math. Sinica, 16, (2000), 613-626.

[3] J. Cohen,A sharp estimate for a multilinear singular integral onRn_{, Indiana}

[4] J. Cohen and J. Gosselin, On multilinear singular integral operators on Rn, Studia Math., 72, (1982), 199-223.

[5] J. Cohen and J. Gosselin, A BMO estimate for multilinear singular integral operators, Illinois J. Math., 30, (1986), 445-465.

[6] R. Coifman and Y. Meyer, Wavelets,Calder´on-Zygmund and multilinear op-erators, Cambridge Studies in Advanced Math., 48, Cambridge University Press, Cambridge, 1997.

[7] J. Garcia-Cuerva and J. L. Rubio de Francia,Weighted norm inequalities and related topics, North-Holland Math.16, Amsterdam, 1985.

[8] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Math., 16, (1978), 263-270.

[9] D. S. Kurtz and R. L. Wheeden, Results on weighted norm inequalities for multiplies, Trans. Amer. Math. Soc., 255, (1979), 343-362.

[10] D. S. Kurtz, Sharp function estimates for fractional integrals and related operators, J. Austral. Math. Soc., 49(A), (1990), 129-137.

[11] S. Z. Lu and D. C. Yang, Multiplier theorem for Herz type Hardy spaces, Proc. Amer. Math. Soc., 126, (1998), 3337-3346.

[12] B. Muckenhoupt, R. L. Wheeden and W. S. Young, Sufficient conditions for Lp multipliers with general weights, Trans. Amer. Math. Soc., 300(2), (1987), 463-502.

[13] M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., 44, (1995), 1-17.

[14] C. P´erez, Endpoint estimate for commutators of singular integral operators, J. Func. Anal., 128, (1995), 163-185.

[15] C. P´erez and R. Trujillo-Gonzalez,Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65, (2002), 672-692.

[16] E. M. Stein, Harmonic Analysis: real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 1993.

[17] J. O. Str¨omberg and A. Torkinsky, Weighted Hardy spaces, Lecture Notes in Math., Vol.1381, Springer Verlag, Berlin, 1989.

[18] Z. You,Results of commutators obtained norm inequalities, Adv. in Math.(in Chinese), 17(1), (1988), 79-84.

[19] P. Zhang and J. C. Chen, The (Lp,F˙_pβ,∞)-boundedness of commutators of multiplies, Acta Math. Sinica, 21(4), (2005), 765-772.

[20] P. Zhang and J. C. Chen,Boundedness properties for commutators of mul-tiplies, Acta Math. Sinica (Chinese Series), 49(6), (2006), 1387-1396.

Changsha University of Science and Technology Changsha, 410077, P.R.of China