Sharp Function Estimates for Vector-Valued Multilinear Operator of Multiplier Operator

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Lanzhe Liu

College of Mathematics

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

Abstract: In this paper, we establish a sharp function estimate for the vector-valued multilinear operator of the multiplier. As the application, we obtain the weighted Lp(1 < p < ∞) norm inequality for the multilinear operator.

Keywords: Vector-valued multilinear operator; Multiplier operator; Sharp estimate; BMO.

MR Subject Classification:42B20, 42B25.

1. Introduction and Results

Let b ∈ BM O(Rn) and T be the Calderón-Zygmund singular integral operator. The commutator [b, T] generated by band T is defined by [b, T](f)(x) =b(x)T(f)(x)−T(bf)(x). As the development of the Calderón-Zygmund singular integral operators, their commutators and multilinear operators have been well studied(see [1-7][15-18]). In [10], Hu and Yang proved a variant sharp function estimate for the multilinear singular integral operators. In [18], C.Pérez and R.Trujillo-Gonzalez obtained a sharp weighted estimates for the singular integral operators and their commutators. In [23], You proved that the commutator [b, T] is bounded in Lp(Rn) when T is a multiplier operator and b ∈ ∧˙β(Rn). In [24][25], Zhang studied the (Lp_,_F˙β,∞

p )− boundedness of the commutator of the multipliers. In this paper, we will introduce the vector-valued multilinear operator associated to the multiplier operator and study the sharp function inequality of the vector-valued multilinear operator. By using the sharp inequality, we obtain the weighted Lp₋ _{norm inequality for the vector-valued} multilinear operator.

First, let us introduce some notations. In this paper,Qwill denote a cube ofRn_{with sides}

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parallel to the axes. For a cubeQand a locally integrable functionb, letbQ =|Q|−1

R

Qb(x)dx. The sharp function of bis defined by

b#(x) = sup Q3x

1

|Q|

Z

Q|b(y)−bQ|dy.

It is well-known that (see [9])

b#(x)≈sup Q3x

inf c∈C

1

|Q|

Z

Q|b(y)−c|dy

and

||b−b₂k_Q||_{BM O} ≤Ck||b||_{BM O} fork≥1.

We say that b belongs to BM O(Rn), ifb# belongs to L∞(Rn) and define

kbk_{BM O} =kb#k_L∞.

Let M be the Hardy-Littlewood maximal operator defined by

M(f)(x) = sup x∈Q

1

|Q|

Z

Q|f(y)|dy,

we write that Mp(f) = (M(fp))1/p for 0< p <∞.

We denote the Muckenhoupt weights by A1 (see [9]), that is

A1 ={w:M(w)(x)≤Cw(x), a.e.}.

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 of f and S(Rn_{) is the Schwartz test function class.} Now, we recall the definition of the class M(s, l). Denote by|x| ∼tthe 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 ksatisfies the conditionM(s, l), if

ÃZ

|ξ|∼R|D

α_k₍_ξ₎_|s_dξ

!1

s

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for all R >0 and multi-indices α with |α| ≤l, when lis a positive integer, and, in addition, if

ÃZ

|ξ|∼R|D

α_k₍_ξ₎₋_Dα_k₍_ξ₋_z₎_|s_dξ

!1

s

≤C(|z|

R)

γ_Rn s−|α|

for all |z| < R/2 and all multi-indices α with |α| = [l], the integer part of l,i.e., [l] is the greatest integer less than or equal to l, and l= [l] +γ when l is not an integer.

Denote D(Rn_{) =} _{_φ_∈ _S₍_Rn_{) : supp(}_φ_{) is compact}_} _{and ˆ}_D

0(Rn) = {φ ∈ S(Rn) : ˆφ ∈

D(Rn_{) and ˆ}_φ_{vanishes in a neighbourhood of the origin}_}_{. The following boundedness property} of T on Lp₍_Rn_{) is proved by Str¨omberg and Torkinsky (see [11-14]).}

Lemma 1.([11]) Letk∈M(s, l),1≤s≤2, andl > n

s. Then the associated mappingT, 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_{) = ˇ}_k₍_x_).

Definition 2.([11]) For a real number ˜l≥0 and 1≤˜s <∞, we say thatK verifies the condition ˜M(˜s,˜l), and write K∈M˜(˜s,˜l), if

ÃZ

|x|∼R|D

˜

α_K₍_x₎_|˜s_dx

!1 ˜

s

≤CRn/s˜−n−|α˜|, R >0

for all multi-indices |α˜| ≤˜land, in addition, if

ÃZ

|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,

ÃZ

|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 ˜αwith|α˜|=u, whereudenotes 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{1s,1− 1˜s}. ThenK ∈M˜(˜s,˜l), where ˜l=l−nr .

Lemma 3. Let 1 ≤ s ≤ 2, suppose that l is a positive real number with l > n/r, 1/r= max{1/s,1−1/s˜}, and k∈M(s, l). Then there is a positive constanta, such that

µZ

Bk

|K(x−z)−K(xQ−z)|˜sdz

¶₁_/_s_˜

≤C2−ka(2kh)−n/˜s0.

Proof.We split our proof into two cases:

Case 1. 1≤s≤2 and 0< l−n/s≤1. We choose a real number 1<s <˜ ∞ such that

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When ˜l = l− n_s < 1, noting that l is a positive real number and l > n_s. Applying the condition K ∈M˜(˜s,˜l) forv=l−n

s and u= 0, one has

µZ

Bk

|K(x−z)−K(x_Q−z)|˜sdz

¶1 ˜

s

≤C2−k(l−ns)(2kh)− n

˜

s0,

let a=l− n_s,

µZ

Bk

|K(x−z)−K(xQ−z)|˜sdz

¶1 ˜

s

≤C2−ka(2kh)−˜sn0.

When ˜l = l− n_s = 1, we choose 0 < ξ < 1, such that t1−ξ_log(1_/t₎ _≤ _C _{for 0} _{< t <} ₁_/_2. Noting that K ∈M˜(˜s,˜l), by Definition 2, for u= 0, v= 1,

µZ

Bk

|K(x−z)−K(xQ−z)|˜sdz

¶1 ˜

s

≤ C(|y−xQ| 2k_h )

ξ₍|y−xQ| 2k_h )

1−ξ_(log 2kh

|y−xQ|

)(2kh)n/˜s−n

≤ C2−kξ(2kh)−n/s˜0,

let a=ξ, then

µZ

Bk

|K(x−z)−K(xQ−z)|˜sdz

¶1 ˜

s

≤C2−ka(2kh)−n/s˜0.

Case 2. 1 ≤s≤2 and l−n/s >1. Set d= [l−n/s], if l−n/s >1 is not an integer, and d=l−n/s−1 if l−n/s >1 is an integer. Choose l1 = 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 replacingl by l1, we can obtain the

same result under the assumption l−n/s >1. We omit the details here. Certainly when 0<s˜0_{< s}_{, which is the same as the above.}

Now we can define the vector-valued multilinear operator associated to the multiplier operator T. Letmj be the positive integers(j = 1,· · ·, l),m1+· · ·+ml =m and bj be the functions on Rn₍_j_{= 1}_,_{· · ·}_{, l}_{). Set, for 1}_≤_j _≤_m_,

R_m_j₊₁(b_j;x, y) =b_j(x)− X

|α|≤mj

1

α!D α_b

j(y)(x−y)α.

By Lemma 1, T(f)(x) = (K∗f)(x) for K(x) = ˇk(x). Given the functions fi defined on

Rn_,_i_{= 1}_,₂_{, ...}_{, for 1}_{< r <}_∞_{, the the vector-valued multilinear operator associated to} _T _is defined by

|Tb(f)(x)|r=

Ã_∞ X

i=1

(Tb(fi)(x))r

!₁_/r

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where

T_b(f_i)(x) =

Z

Rn

Q_l

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

|x−y|m K(x−y)fi(y)dy. Set

|T(f)(x)|r =

Ã_∞ X

i=1

|T(fi)(x)|r

!₁_/r

and |f(x)|r=

Ã_∞ X

i=1

|fi(x)|r

!₁_/r

.

Note that when m = 0, |T_b|_r is just the vector-valued multilinear commutator ofT and

bj(see [18]). While whenm >0, |Tb|r is non-trivial generalizations of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [2-5][7]). Hu and Yang (see [10]) proved a variant sharp estimate for the multilinear singular integral operators. In [17], P´erez and Trujillo-Gonzalez prove a sharp estimate for the multilinear commutator when bj ∈OscexpLrj(Rn). The main purpose of this paper is to prove a sharp function inequality for the vector-valued multilinear multiplier operator when Dα_b

j ∈BM O(Rn) for all α with|α|=mj. As the application, we obtain the Lp₍_{p >}_{1) norm inequality for the vector-valued multilinear operator.}

We shall prove the following theorems.

Theorem 1. Let 1 < r < ∞ and Dαbj ∈ BM O(Rn) for all α with |α| = mj and

j = 1,· · ·, l. Then there exists a constant C >0 such that for everyf ∈C∞

0 (Rn), 1< s <∞

and ˜x∈Rn,

(|Tb(f)|r)#s(˜x)≤C l

Y

j=1 

 X

|αj|=mj

||Dαj_b

j||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

Theorem 2. Let 1 < r < ∞ and Dα_b

j ∈ BM O(Rn) for all α with |α| = mj and

j = 1,· · ·, l. Then|Tb|r is bounded onLp(w) for any w∈A1 and 1< p <∞, that is

|||Tb(f)|r||Lp₍_w₎≤C

l

Y

j=1 

 X

|αj|=mj

||Dαj_b

j||BM O



_|||_f_|_r_||_Lp₍_w₎.

2. Proofs of Theorems

To prove the theorems, we need the following lemmas.

Lemma 4.([4]) Let bbe a function onRn _and_Dα_b_∈_Lq₍_Rn_{) for all}_α _with_|_α_|₌_m _and someq > n. Then

|Rm(b;x, y)| ≤C|x−y|m

X

|α|=m

Ã

1

|Q˜(x, y)|

Z

˜

Q(x,y)|D

α_b₍_z₎_|q_dz

!₁_/q

,

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Lemma 5.([8][11]) Let T be the multiplier operator. Then, for everyf ∈ Lp₍_Rn₎_,₁ _<

p < ∞,

|||T(f)|r||Lp ≤C|||f|_r||_Lp.

Proof of Theorem 1. It suffices to prove for f ∈C∞

0 (Rn) and some constant C0, the

following inequality holds:

1

|Q|

Z

Q||T

A₍_f₎₍_x₎_|

r−C0|dx≤C

l

Y

j=1 

 X

|αj|=mj

||Dαj_b

j||BM O



_M_s₍_|_f_|_r₎₍_x₎_.

Without loss of generality, we may assume l= 2. Fix a cube Q=Q(x₀, d) and ˜x∈ Q. Let ˜

Q = 5√nQ and ˜bj(x) = bj(x)−

P

|α|=mj

1

α!(Dαbj)Q˜xα, then Rmj+1(bj;x, y) =Rmj+1(˜bj;x, y)

and Dα˜_b_j ₌_Dα_b

j −(Dαbj)_Q˜ for |α|=mj. We split f =g+h ={gi}+{hi} forgi =fiχ_Q˜

and hi=fiχ_Rn_\_Q˜. Write

Tb(fi)(x) =

Z

Rn

Q₂

j=1Rmj(˜bj;x, y)

|x−y|m K(x−y)gi(y)dy

− X

|α1|=m1

1

α1! Z

Rn

R_m₂(˜b₂;x, y)(x−y)α1

|x−y|m Dα1˜b1(y)K(x−y)gi(y)dy

− X

|α2|=m2

1

α2! Z

Rn

Rm1(˜b1;x, y)(x−y)α2

|x−y|m Dα2˜b2(y)K(x−y)gi(y)dy

+ X

|α1|=m1,|α2|=m2

1

α1!α2! Z

Rn

(x−y)α1+α2_Dα1˜_b

1(y)Dα2˜b2(y)

|x−y|m K(x−y)gi(y)dy

+

Z

Rn

Q₂

j=1Rmj+1(˜bj;x, y)

|x−y|m K(x−y)hi(y)dy

=T

Ã Q₂

j=1Rmj(˜bj;x,·)

|x− ·|m gi

!

(x)

−T



 X

|α1|=m1

1

α1!

Rm2(˜b2;x,·)(x− ·)α1Dα1˜b1

|x− ·|m gi

 ₍_x₎

−T



 X

|α2|=m2

1

α2!

Rm1(˜b1;x,·)(x− ·)α2Dα2˜b2

|x− ·|m gi

 ₍_x₎

+T



 X

|α1|=m1,|α2|=m2

1

α1!α2!

(x− ·)α1+α2_Dα1˜_b

1Dα2˜b2

|x− ·|m gi



₍_x_{) +}_T_˜

b(hi)(x),

then, by Minkowski’ inequality,

1

|Q|

Z

Q

¯

¯_|_T_b₍_f₎₍_x₎_|_r_{− |}_T_˜

b(h)(x0)|r

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≤ 1 |Q|

Z

Q

Ã_∞ X

i=1

|T_b(f_i)(x)−T˜_b(hi)(x0)|r !₁_/r

dx

≤ 1

|Q|

Z Q Ã_∞ X i=1 ¯ ¯ ¯ ¯ ¯T

Ã Q₂

j=1Rmj(˜bj;x,·)

|x− ·|m gi

!

(x)

¯ ¯ ¯ ¯ ¯

r!1/r

dx

+ 1

|Q|

Z

Q

 X∞

i=1 ¯ ¯ ¯ ¯ ¯ ¯T   X

|α1|=m1

1

α1!

Rm2(˜b2;x,·)(x− ·)α1Dα1˜b1

|x− ·|m gi

 ₍_x₎

¯ ¯ ¯ ¯ ¯ ¯ r  1/r dx + 1

|Q|

Z

Q

 X∞

i=1 ¯ ¯ ¯ ¯ ¯ ¯T   X

|α2|=m2

1

α2!

R_m₁(˜b₁;x,·)(x− ·)α2_Dα2˜_b

2

|x− ·|m gi

 ₍_x₎

¯ ¯ ¯ ¯ ¯ ¯ r  1/r dx + 1

|Q|

Z

Q

 X∞

i=1 ¯ ¯ ¯ ¯ ¯ ¯T   X

|α1|=m1,|α2|=m2

1

α1!α2!

(x− ·)α1+α2_Dα1˜_b

1Dα2˜b2

|x− ·|m gi

 ₍_x₎

¯ ¯ ¯ ¯ ¯ ¯ r  1/r dx + 1

|Q|

Z Q Ã_∞ X i=1 ¯ ¯_T_˜

b(hi)(x)−T˜b(hi)(x0)

¯ ¯r

!₁_/r

dx

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

Now, let us estimateI1,I2,I3,I4andI5, respectively. First, forx∈Qandy∈Q˜, by Lemma

4, we get

Rm(˜bj;x, y)≤C|x−y|m

X

|αj|=m

||Dαj_b

j||BM O,

thus, by the H¨older’s inequality and Ls_{-boundedness of}_|_T_|

r(Lemma 5), we obtain

I1 ≤ C 2 Y

j=1 

 X

|αj|=mj

||Dαj_b

j||BM O

  1

|Q|

Z

Q|T(g)(x)|rdx

≤ C 2 Y j=1   X

|αj|=mj

||Dαj_b

j||BM O

 

µ

1

|Q|

Z

Rn|T(g)(x)|

s rdx

¶₁_/s

≤ C 2 Y j=1   X

|αj|=mj

||Dαj_b

j||BM O

 

µ

1

|Q|

Z

Rn|g(x)|

s rdx

¶₁_/s

≤ C 2 Y j=1   X

|αj|=mj

||Dαj_b

j||BM O

 

Ã

1

|Q˜|

Z

˜

Q|f(x)| s rdx

!₁_/s

≤ C 2 Y j=1   X

|αj|=mj

||Dαj_b

j||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

For I2, denotings=pqfor 1< p <∞,q >1 and 1/q+ 1/q0= 1, we have, by Lemma 5,

I2 ≤ C X

|α2|=m2 ||Dα2_b

2||BM O

X

|α1|=m1

1

|Q|

Z

Q|T(D α1˜_b

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≤ C X |α2|=m2

||Dα2_b

2||BM O

X

|α1|=m1

µ

1

|Q|

Z

Rn|T(D

α1˜_b

1g)(x)|prdx

¶₁_/p

≤ C X

|α2|=m2 ||Dα2_b

2||BM O

X

|α1|=m1

µ

1

|Q|

Z

Rn|D

α1˜_b

1(x)||g(x)|prdx

¶₁_/p

≤ C X

|α2|=m2 ||Dα2_b

2||BM O

X

|α1|=m1

Ã

1

|Q˜|

Z

˜

Q|D α1_b

1(x)−(Dαbj)_Q˜|pq

0 dx

!₁_/pq0Ã 1

|Q˜|

Z

˜

Q|f(x)| pq r dx

!₁_/pq

≤ C 2 Y j=1   X

|α|=mj

||Dαb_j||_{BM O}



_M_s₍_|_f_|_r_)(˜_x₎_.

For I3, similar to the proof of I2, we get

I3≤C 2 Y

j=1 

 X

|α|=mj

||Dαbj||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

Similarly, for I4, denoting s=pq3 for 1< p <∞,q1, q2, q3 >1 and 1/q1+ 1/q2+ 1/q3 = 1,

we obtain

I4 ≤ C

X

|α1|=m1,|α2|=m2

1

|Q|

Z

Q|T(D α1˜_b

1Dα2˜b2g)(x)|rdx

≤ C X

|α1|=m1,|α2|=m2

µ

1

|Q|

Z

Rn|T(D

α1˜_b

1Dα2˜b2g)(x)|prdx

¶₁_/p

≤ C X

|α1|=m1,|α2|=m2

µ

1

|Q|

Z

Rn|D

α1˜_b

1(x)Dα2˜b2(x)||g(x)|prdx

¶₁_/p

≤ C X

|α1|=m1,|α2|=m2

2 Y

j=1 Ã

1

|Q˜|

Z

˜

Q|D αj˜_b

j(x)|pqjdx

!₁_/pq_jÃ

1

|Q˜|

Z

˜

Q|f(x)| pq3

r dx

!₁_/pq₃

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

For I5, we write

T˜_b(hi)(x)−T˜_b(hi)(x0) = Z

Rn(K(x−y)−K(x0−y))

1

|x−y|m

2 Y

j=1

Rmj(˜bj;x, y)hi(y)dy

+

Z

Rn

µ

1

|x−y|m − 1

|x0−y|m ¶

K(x₀−y)

2 Y

j=1

R_m_j(˜b_j;x, y)h_i(y)dy

+

Z

Rn

³

Rm1(˜b1;x, y)−Rm1(˜b1;x0, y)

´_R

m2(˜b2;x, y)

|x0−y|m K(x0−y)hi(y)dy

+

Z

Rn

³

R_m₂(˜b₂;x, y)−R_m₂(˜b₂;x₀, y)´Rm1(˜b1;x0, y) |x0−y|m

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− X

|α1|=m1

1

α₁!

Z

Rn

"

Rm2(˜b2;x, y)(x−y)α1

|x−y|m K(x−y)−

Rm2(˜b2;x0, y)(x0−y)α1

|x₀−y|m K(x0−y)

#

×Dα1˜_b

1(y)hi(y)dy

− X

|α2|=m2

1

α2! Z

Rn

"

Rm1(˜b1;x, y)(x−y)α2

|x−y|m K(x−y)−

Rm1(˜b1;x0, y)(x0−y)α2

|x0−y|m K(x0−y) #

×Dα2˜_b

2(y)hi(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˜_b

1(y)Dα2˜b2(y)hi(y)dy

= I₅(1)+I₅(2)+I₅(3)+I₅(4)+I₅(5)+I₅(6)+I₅(7).

By Lemma 4 and the following inequality(see [20])

|bQ1 −bQ2| ≤Clog(|Q2|/|Q1|)||b||BM O forQ1 ⊂Q2,

we know that, for x∈Qand y∈2k+1_Q˜_\₂k_Q˜_,

|Rm(˜b;x, y)| ≤ C|x−y|m

X

|α|=m

(||Dαb||BM O+|(Dαb)_Q˜₍_x,y₎−(Dαb)_Q˜|)

≤ Ck|x−y|m X

|α|=m

||Dαb||BM O.

Note that|x−y| ∼ |x₀−y|forx∈Q and y∈Rn_\_Q˜_{, we have}

|I₅(1)| ≤C

2 Y

j=1 

 X

|α|=mj

||Dαb_j||_{BM O}

 X∞

k=0 Z

2k+1_Q˜_\₂k_Q˜k

2_|_K₍_x₋_y₎₋_K₍_x

0−y)||fi(y)|dy,

thus, by the Minkowski’ inequality and Lemma 3, we obtain

Ã_∞ X

i=1

||I₅(1)||r

!₁_/r

≤ C

2 Y

j=1 

 X

|α|=mj

||Dαbj||BM O

 X∞

k=0 Z

2k+1_Q˜_\₂k_Q˜k

2_|_K₍_x₋_y₎₋_K₍_x

0−y)||f(y)|rdy

≤ C

2 Y

j=1 

 X

|α|=mj

||Dαb_j||_{BM O}

 X∞

k=0

k2

µZ

2k+1_Q˜|f(y)|

s rdy

¶₁_/s

×

ÃZ

2k+1_Q˜_\₂k_Q˜|K(x−y)−K(x0−y)|

s0 dy

!₁_/s0

≤ C

2 Y

j=1 

 X

|α|=mj

||Dαbj||BM O

 X∞

k=1

k22−ka

Ã

1

|2k_Q˜_|

Z

2k_Q˜|f(y)|

s rdy

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≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O

 X∞

k=1

k22−kaMs(|f|r)(˜x)

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

For I₅(2), by the Minkowski’ inequality and Lemma 2, we obtain

Ã_∞ X

i=1

||I₅(2)||r

!₁_/r

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O

 X∞

k=0 Z

2k+1_Q˜_\₂k_Q˜k

2|x−x0|

|x0−y||K(x0−y)||f(y)|rdy

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O

 X∞

k=0

k22−k

ÃZ

2k+1_Q˜_\₂k_Q˜|K(x0−y)|

s0 dy

!₁_/s0

×

µZ

2k+1_Q˜|f(y)|

s rdy

¶₁_/s

≤ C 2 Y j=1   X

|α|=mj

||Dαb_j||_{BM O}

 X∞

k=0

k22−k

Ã

1

|2k_Q˜_|

Z

2k_Q˜|f(y)|

s rdy

!₁_/s

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

For I₅(3), by the formula (see [5]):

Rmj(˜b;x, y)−Rmj(˜b;x0, y) =

X

|β|<mj

1

β!Rmj−|β|(D

β_˜_b_;_{x, x}

0)(x−y)β

and Lemma 4, we have

|R_m_j(˜b;x, y)−R_m_j(˜b;x₀, y)| ≤C X |β|<mj

X

|α|=mj

|x−x₀|mj−|β|_|_x₋_y_||β|_||_Dα_b_||

BM O,

thus, by the Minkowski’ inequality and Lemma 2, we obtain

Ã_∞ X

i=1

||I₅(3)||r

!₁_/r

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O

 X∞

k=0 Z

2k+1_Q˜_\₂k_Q˜k

2|x−x0|

|x0−y||K(x0−y)||f(y)|rdy

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O

 X∞

k=0

k22−k

ÃZ

2k+1_Q˜_\₂k_Q˜|K(x0−y)|

s0 dy

!₁_/s0

×

µZ

2k+1_Q˜|f(y)|

s rdy

¶₁_/s

≤ C 2 Y j=1   X

|α|=mj

||Dαb_j||_{BM O}

 X∞

k=0

(11)

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

Similarly,

Ã_∞ X

i=1

||I₅(4)||r

!₁_/r

≤C 2 Y j=1   X

|αj|=mj

||Dαj_b

j||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

For I₅(5), similar to the proofs of I₅(1), I₅(2), I₅(3) and I4, we get, for 1 < s1, s2 < ∞ with

1/s1+ 1/s2+ 1/s= 1, Ã_∞

X

i=1

||I₅(5)||r

!₁_/r

≤ C X

|α1|=m1

Z

( ˜Q)c

|(x−y)α1_|

|x−y|m |Rm2(˜b2;x, y)−Rm2(˜b2;x0, y)||K(x−y)D

α1˜_b

1(y)||f(y)|rdy

+C X

|α1|=m1

Z

( ˜Q)c

¯ ¯ ¯ ¯

(x−y)α1 |x−y|m −

(x₀−y)α1 |x0−y|m ¯ ¯ ¯

¯|Rm2(˜b2;x0, y)K(x−y)D

α1˜_b

1(y)||f(y)|rdy

+C X

|α1|=m1

Z

( ˜Q)c

|Rm2(˜b2;x0, y)||(x0−y)α1|

|x0−y|m |K(x−y)−K(x0−y)||D

α1˜_b

1(y)||f(y)|rdy

≤ C X

|α|=m2

||Dαb2||BM O

X

|α1|=m1

∞ X

k=0 Z

2k+1_Q˜_\₂k_Q˜k

|x−x0|

|x0−y||K(x−y)||D

α1˜_b

1(y)||f(y)|rdy

+C X

|α|=m2

||Dαb2||BM O

X

|α1|=m1

∞ X

k=0 Z

2k+1_Q˜_\₂k_Q˜k|K(x−y)−K(x0−y)||D

α1˜_b

1(y)||f(y)|rdy

≤ C X

|α|=m2

||Dαb2||BM O

X

|α1|=m1

∞ X

k=0

k2−k

ÃZ

2k+1_Q˜_\₂k_Q˜|K(x−y)|

s1_dy

!₁_/s₁

×

µZ

2k+1_Q˜|D

α1˜_b

1(y)|s2dy

¶₁_/s₂µZ

2k+1_Q˜|f(y)|

s rdy

¶₁_/s

+C X

|α|=m2

||Dαb₂||_{BM O} X

|α1|=m1

∞ X

k=0

k

ÃZ

2k+1_Q˜_\₂k_Q˜|K(x−y)−K(x0−y)|

s1_dy

!₁_/s₁

×

µZ

2k+1_Q˜|D

α1˜_b

1(y)|s2dy

¶₁_/s₂µZ

2k+1_Q˜|f(y)|

s rdy

¶₁_/s

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O

 X∞

k=1

k2(2−k+ 2−ka)

Ã

1

|2k_Q˜_|

Z

2k_Q˜|f(y)|

s rdy

!₁_/s

≤ C 2 Y j=1   X

|α|=mj

||Dαb_j||_{BM O}



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Similarly,

Ã_∞ X

i=1

||I₅(6)||r

!₁_/r

≤C 2 Y j=1   X

|αj|=mj

||Dαj_b

j||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

ForI₅(7), similar to the proofs ofI₅(5)and I4, we get, for 1< s1, s2, s3<∞with 1/s1+ 1/s2+

1/s3+ 1/s= 1, Ã_∞

X

i=1

||I₅(7)||r

!₁_/r

≤ C X

|α1|=m1,|α2|=m2

Z

( ˜Q)c

¯ ¯ ¯ ¯ ¯

(x−y)α1+α2 |x−y|m −

(x₀−y)α1+α2 |x0−y|m

¯ ¯ ¯ ¯

¯|K(x−y)D

α1˜_b

1(y)Dα2˜b2(y)||f(y)|rdy

+C X

|α1|=m1,|α2|=m2

Z

( ˜Q)c

|(x₀−y)α1+α2_| |x0−y|m

|K(x−y)−K(x₀−y)||Dα1˜_b

1(y)Dα2˜b2(y)||f(y)|rdy

≤ C X

|α1|=m1,|α|=m2

∞ X

k=0 Z

2k+1_Q˜_\₂k_Q˜

|x−x0|

|x0−y||K(x−y)||D

α1˜_b

1(y)Dα2˜b2(y)||f(y)|rdy

+C X

|α1|=m1,|α|=m2

∞ X

k=0 Z

2k+1_Q˜_\₂k_Q˜|K(x−y)−K(x0−y)||D

α1˜_b

1(y)Dα2˜b2(y)||f(y)|rdy

≤ C X

|α1|=m1,|α|=m2

∞ X

k=0

2−k

ÃZ

2k+1_Q˜_\₂k_Q˜|K(x−y)|

s1_dy

!₁_/s₁_µZ

2k+1_Q˜|D

α1˜_b

1(y)|s2dy ¶₁_/s₂

×

µZ

2k+1_Q˜|D

α2˜_b

2(y)|s3dy

¶₁_/s₃µZ

2k+1_Q˜|f(y)|

s rdy

¶₁_/s

+C X

|α1|=m1,|α|=m2

∞ X

k=0 ÃZ

2k+1_Q˜_\₂k_Q˜|K(x−y)−K(x0−y)|

s1_dy

!₁_/s₁

×

µZ

2k+1_Q˜|D

α1˜_b

1(y)|s2dy

¶₁_/s₂µZ

2k+1_Q˜|D

α2˜_b

2(y)|s3dy

¶₁_/s₃µZ

2k+1_Q˜|f(y)|

s rdy

¶₁_/s

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O

 X∞

k=1

k2(2−k+ 2−ka)

Ã

1

|2k_Q˜_|

Z

2k_Q˜|f(y)|

s rdy

!₁_/s

≤ C 2 Y j=1   X

|α|=mj

||Dαbj||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

Thus

I5≤C 2 Y

j=1 

 X

|α|=mj

||Dαbj||BM O



_M_s₍_|_f_|_r_)(˜_x₎_.

This completes the proof of Theorem 1.

Proof of Theorem 2. We choose 1< s < pin Theorem 1 and by [8], we get

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≤ C

l

Y

j=1 

 X

|α|=mj

||Dαbj||BM O



_||_M_s₍_|_f_|_r₎_||_Lp₍_w₎

≤ C

l

Y

j=1 

 X

|α|=mj

||Dαbj||BM O



_|||_f_|_r_||_Lp₍_w₎.

This finishes the proof.

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