RACSAM DOI 10.1007/s13398-016-0337-8 ORIGINAL PAPER

Weighted boundedness of multilinear operator associated to singular integral operator with variable Calderón–Zygmund Kernel Yanxiang Tan1 · Lanzhe Liu1

Received: 5 April 2016 / Accepted: 7 September 2016 © Springer-Verlag Italia 2016

Abstract In this paper, we establish the weighted sharp maximal function inequalities for the multilinear operator associated to the singular integral operator with variable Calderón– Zygmund kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces. Keywords Multilinear operator · Singular integral operator · Variable Calderón–Zygmund Kernel · Sharp maximal function · Weighted BMO · Weighted Lipschitz function Mathematics Subject Classification 42B20 · 42B25

1 Introduction and preliminaries As the development of singular integral operators (see [13,31,32]), their commutators and multilinear operators have been well studied. In [8,9,27,28], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on L p (R n ) for 1 < p < ∞. Chanillo (see [3]) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [17,24], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and L p (R n )(1 < p < ∞) spaces are obtained. In [1,16], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on L p (R n )(1 < p < ∞) spaces are obtained (also see [15]). In [2], Calderón and Zygmund introduce some singular integral operators with variable kernel and discuss their boundedness. In [19–21],[33], the authors obtain the boundedness for the commutators and multilinear operators generated by the singular integral operators with variable kernel and BMO functions. In [22], the authors prove the boundedness for the multilinear

B 1

Lanzhe Liu [email protected] Department of Mathematics, Changsha University of Science and Technology, Changsha 410114, People’s Republic of China

Y. Tan and L. Liu

oscillatory singular integral operators generated by the operators and BMO functions. Motivated by these, in this paper, we will study the multilinear operator generated by the singular integral operator with variable Calderón–Zygmund kernel and the weighted Lipschitz and BMO functions. First, let us introduce some notations. Throughout this paper, Q will denote a cube of R n with sides parallel to the axes. For any locally integrable function f , the sharp maximal function of f is defined by    1  f (y) − f Q  dy, M # ( f )(x) = sup Qx |Q| Q  where, and in what follows, f Q = |Q|−1 Q f (x)d x. It is well-known that (see [13,31])  1 | f (y) − c|dy. M # ( f )(x) ≈ sup inf Qx c∈C |Q| Q Let M( f )(x) = sup Qx

1 |Q|

 | f (y)|dy. Q

For η > 0, let Mη# ( f )(x) = M # (| f |η )1/η (x) and Mη ( f )(x) = M(| f |η )1/η (x). For 0 < η < n, 1 ≤ p < ∞ and the non-negative weight function w, set  1/ p  1 p Mη, p,w ( f )(x) = sup | f (y)| w(y)dy . 1− pη/n Q Qx w(Q) We write Mη, p,w ( f ) = M p,w ( f ) if η = 0. The A p weight is defined by (see [13]), for 1 < p < ∞, 



1 A p = w ∈ L loc (R n ), 0 < w < ∞ : sup Q

1 |Q|



 w(x)d x Q

1 |Q|



w(x)−1/( p−1) d x

 p−1

 <∞

Q

and 1 A1 = {w ∈ L loc (R n ), 0 < w < ∞ : M(w)(x) ≤ Cw(x), a.e.}.

Given 1 ≤ p < ∞ and w a non-negative weight function, the weighted Lebesgue space L p (R n , w) is the space of functions f such that  1/ p || f || L p (w) = | f (x)| p w(x)d x < ∞. Rn

For 0 < β < 1 and the non-negative weight function w, the weighted Lipschitz space Li pβ (w) is the space of functions b such that 1/ p   1 1 p 1− p ||b|| Li pβ (w) = sup |b(y) − b Q | w(x) dy < ∞, β/n w(Q) Q Q w(Q) and the weighted BMO space BMO(w) is the space of functions b such that  1/ p  1 p 1− p |b(y) − b Q | w(x) dy < ∞. ||b||BMO(w) = sup w(Q) Q Q

Weighted boundedness of multilinear operator…

Remark (1) It has been known that (see [1,12]), for b ∈ Li pβ (w), w ∈ A1 and x ∈ Q, |b Q − b2k Q | ≤ Ck||b|| Li pβ (w) w(x)w(2k Q)β/n . (2) It has been known that (see [12,17]), for b ∈ BMO(w), w ∈ A1 and x ∈ Q, |b Q − b2k Q | ≤ Ck||b||BMO(w) w(x). (3) Let b ∈ Li pβ (w) or b ∈ BMO(w) and w ∈ A1 . By [12], we know that spaces Li pβ (w) or BMO(w) coincide and the norms ||b|| Li pβ (w) or ||b|| B M O(w) are equivalent with respect to different values 1 ≤ p < ∞. Definition 1 Let ϕ be a positive, increasing function on R + and there exists a constant D > 0 such that ϕ(2t) ≤ Dϕ(t) for t ≥ 0. Let w be a non-negative weight function on R n and f be a locally integrable function on R n . Set, for 0 ≤ η < n and 1 ≤ p < n/η, 1/ p   1 p || f || L p,η,ϕ (w) = sup | f (y)| w(y)dy , 1− pη/n Q(x,d) x∈R n , d>0 ϕ(d) where Q(x, d) = {y ∈ R n : |x − y| < d}. The generalized fractional weighted Morrey space is defined by

1 L p,η,ϕ (R n , w) = f ∈ L loc (R n ) : || f || L p,η,ϕ (w) < ∞ . We write L p,η,ϕ (R n , w) = L p,ϕ (R n , w) if η = 0, which is the generalized weighted Morrey space. If ϕ(d) = d δ , δ > 0, then L p,ϕ (R n , w) = L p,δ (R n , w), which is the classical Morrey spaces (see [25,26]). If ϕ(d) = 1, then L p,ϕ (R n , w) = L p (R n , w), which is the weighted Lebesgue spaces (see [13]). As the Morrey space may be considered an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [10,14, 18,23,29]). In this paper, we will study some singular integral operators as following (see [2,30]). Definition 2 Let K (x) = (x)/|x|n : R n \{0} → R. K is said to be a Calderón–Zygmund kernel if (a)  ∈ C ∞ (R n \{0}); (b)  is homogeneous of degree zero; (c)  (x)x α dσ (x) = 0 for all multi-indices α ∈ (N ∪ {0})n with |α| = N , where  = {x ∈ R n : |x| = 1} is the unit sphere of R n . Definition 3 Let K (x, y) = (x, y)/|y|n : R n × (R n \{0}) → R. K is said to be a variable Calderón–Zygmund kernel if (d) K (x, ·) is a Calderón–Zygmund kernel for a.e. x ∈ R n ;  |    (e) max|γ |≤2n  ∂∂ γγy| (x, y) ∞ n = L < ∞. L (R ×)

Moreover, let m be the positive integer and b be the function on R n . Set  1 Rm+1 (b; x, y) = b(x) − D α b(y)(x − y)α . α! |α|≤m

Y. Tan and L. Liu

Let T be the singular integral operator with variable Calderón–Zygmund kernel as  K (x, x − y) f (y)dy, T ( f )(x) = Rn

n where K (x, x − y) = (x,x−y) |x−y|n and that (x, y)/|y| is a variable Calderón–Zygmund kernel. The multilinear operator related to the operator T is defined by  Rm+1 (b; x, y) T b ( f )(x) = K (x, x − y) f (y)dy. |x − y|m Rn

Note that the commutator [b, T ]( f ) = bT ( f ) − T (b f ) is a particular operator of the multilinear operator T b if m = 0. The multilinear operator T b are the non-trivial generalizations of the commutator. It is well known that commutators and multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [6,7,11]). The main purpose of this paper is to prove the sharp maximal inequalities for the multilinear operator T b . As the application, we obtain the weighted L p -norm inequality and Morrey space boundedness for the multilinear operator T b .

2 Theorems and Lemmas We shall prove the following theorems. Theorem 1 Let T be the singular integral operator as Definition 3, w ∈ A1 , 0 < η < 1, 1 < r < ∞ and D α b ∈ BMO(w) for all α with |α| = m. Then there exists a constant C > 0 such that, for any f ∈ C0∞ (R n ) and x˜ ∈ R n ,  ˜ ≤C ||D α b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜ Mη# (T b ( f ))(x) |α|=m

Theorem 2 Let T be the singular integral operator as Definition 3, w ∈ A1 , 0 < η < 1, 1 < r < ∞, 0 < β < 1 and D α b ∈ Li pβ (w) for all α with |α| = m. Then there exists a constant C > 0 such that, for any f ∈ C0∞ (R n ) and x˜ ∈ R n ,  ˜ ≤C ||D α b|| Li pβ (w) w(x)M ˜ β,r,w ( f )(x). ˜ Mη# (T b ( f ))(x) |α|=m

Theorem 3 Let T be the singular integral operator as Definition 3, w ∈ A1 , 1 < p < ∞ and D α b ∈ BMO(w) for all α with |α| = m. Then T b is bounded from L p (R n , w) to L p (R n , w 1− p ). Theorem 4 Let T be the singular integral operator as Definition 3, w ∈ A1 , 1 < p < ∞, 0 < D < 2n and D α b ∈ BMO(w) for all α with |α| = m. Then T b is bounded from L p,ϕ (R n , w) to L p,ϕ (R n , w 1− p ). Theorem 5 Let T be the singular integral operator as Definition 3, w ∈ A1 , 0 < β < 1, 1 < p < n/β, 1/q = 1/ p − β/n and D α b ∈ Li pβ (w) for all α with |α| = m. Then T b is bounded from L p (R n , w) to L q (R n , w 1−q ). Theorem 6 Let T be the singular integral operator as Definition 3, w ∈ A1 , 0 < β < 1, 0 < D < 2n , 1 < p < n/β, 1/q = 1/ p − β/n and D α b ∈ Li pβ (w) for all α with |α| = m. Then T b is bounded from L p,β,ϕ (R n , w) to L q,ϕ (R n , w 1−q ).

Weighted boundedness of multilinear operator…

To prove the theorems, we need the following Lemmas. Lemma 1 (See [[13], p. 485]) Let 0 < p < q < ∞ and for any function f ≥ 0. We define that, for 1/r = 1/ p − 1/q,  1/q N p,q ( f ) = sup || f χ Q || L p /||χ Q || L r , || f ||W L q = sup λ {x ∈ R n : f (x) > λ} , λ>0

Q

where the sup is taken for all measurable sets Q with 0 < |Q| < ∞. Then || f ||W L q ≤ N p,q ( f ) ≤ (q/(q − p))1/ p || f ||W L q . Lemma 2 (See [2]) Let T be the singular integral operator as Definition 3. Then T is bounded on L p (R n , w) for w ∈ A p with 1 < p < ∞, and weak (L 1 , L 1 ) bounded. Lemma 3 (See [12,13]) Let 0 ≤ η < n, 1 ≤ s < p < n/η, 1/q = 1/ p − η/n and w ∈ A1 . Then ||Mη,s,w ( f )|| L q (w) ≤ C|| f || L p (w) . Lemma 4 (See [13]) Let 0 < p, η < ∞ and w ∈ ∪1≤r <∞ Ar . Then, for any smooth function f for which the left-hand side is finite,   Mη ( f )(x) p w(x)d x ≤ C Mη# ( f )(x) p w(x)d x. Rn

Rn

Lemma 5 (See [13]) Let 0 < p < ∞, 0 < η < ∞, 0 < D < 2n and w ∈ A1 . Then, for any smooth function f for which the left-hand side is finite, ||Mη ( f )|| L p,ϕ (w) ≤ C||Mη# ( f )|| L p,ϕ (w) . Lemma 6 (See [13]) Let 0 ≤ η < n, 0 < D < 2n , 1 ≤ s < p < n/η, 1/q = 1/ p − η/n and w ∈ A1 . Then ||Mη,s,w ( f )|| L q,ϕ (w) ≤ C|| f || L p,η,ϕ (w) . Lemma 7 (See [7]) Let b be a function on R n and D α A ∈ L q (R n ) for all α with |α| = m and any q > n. Then 1/q    1 m α q |D b(z)| dz , |Rm (b; x, y)| ≤ C|x − y| ˜ ˜ | Q(x, y)| Q(x,y) |α|=m √ where Q˜ is the cube centered at x and having side length 5 n|x − y|.

3 Proofs of Theorems Proof of Theorem 1 We apply a method used by Chiarenza, Frasca and Longo in [4,5]. According to it, it suffices to prove for f ∈ C0∞ (R n ) and some constant C0 , the following inequality holds:    η 1/η  1  b  ≤C ||D α b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜ T ( f )(x) − C0  d x |Q| Q |α|=m

Y. Tan and L. Liu

√ ˜ Fix Q = Q(x0 , d) and x˜ ∈ Q. Let Q˜ = 5 n Q and b(x) = b(x) − a cube 1 α α α α α ˜ ˜ |α|=m α! (D b) Q˜ x , then Rm (b; x, y) = Rm (b; x, y) and D b = D b − (D b) Q˜ for |α| = m. We write, for f 1 = f χ Q˜ and f 2 = f χ R n \ Q˜ , 

˜ x, y) Rm (b; K (x, x − y) f 1 (y)dy m R n |x − y|  1  (x − y)α D α b(y) ˜ K (x, x − y) f 1 (y)dy − m α! R n |x − y| |α|=m  ˜ x, y) Rm+1 (b; + K (x, x − y) f 2 (y)dy |x − y|m Rn ⎞ ⎛

  1 (x − ·)α D α b˜ ˜ x, ·) Rm (b; ˜ f1 − T ⎝ f 1 ⎠ + T b ( f 2 )(x), =T |x − ·|m α! |x − ·|m

T b ( f )(x) =

|α|=m

then 

1 |Q|

  η 1/η ˜   b T ( f )(x) − T b ( f 2 )(x0 ) d x Q

η 1/η    ˜ x, ·) Rm (b; 1   ≤C f1  d x T m  |Q| Q  |x − ·|  ⎛ ⎞η ⎞1/η ⎛     (x − ·)α D α b˜   1  ⎝ ⎠ ⎝ T +C f 1  d x ⎠ |Q| Q  |x − ·|m 

|α|=m



  η 1/η 1  b˜  b˜ +C ( f )(x) − T ( f )(x ) T 2 2 0  dx |Q| Q = I1 + I2 + I3 .

For I1 , noting that w ∈ A1 , w satisfies the reverse of Hölder’s inequality:  1/ p0   1 C w(x) p0 d x ≤ w(x)d x |Q| Q |Q| Q for all cube Q and some 1 < p0 < ∞ (See [13]). We take q = r p0 /(r + p0 − 1) in Lemma 7 and have 1 < q < r and p0 = q(r − 1)/(r − q), then by the Lemma 7 and Hölder’s inequality, we gain 1/q    1 q ˜ x, y)| ≤ C|x − y|m ˜ |D α b(z)| dz |Rm (b; ˜ ˜ | Q(x, y)| Q(x,y) |α|=m 1/r   m −1/q α˜ r 1−r ˜ ≤ C|x − y| | Q| |D b(z)| w(z) dz |α|=m

 ×

˜ Q(x,y)

≤ C|x − y|m

˜ Q(x,y)

w(z)q(r −1)/(r −q) dz  |α|=m

(r −q)/rq

˜ −1/q ||D α b||BMO(w) w( Q) ˜ 1/r | Q| ˜ (r −q)/rq | Q|

Weighted boundedness of multilinear operator…





1

×

w(z) p0 dz

(r −q)/rq

˜ ˜ | Q(x, y)| Q(x,y)  m ˜ −1/q w( Q) ˜ 1/r | Q| ˜ 1/q−1/r ≤ C|x − y| ||D α b||BMO(w) | Q| |α|=m





1

×

(r −1)/r

w(z)dz ˜ ˜ | Q(x, y)| Q(x,y)  ˜ −1/q w( Q) ˜ 1/r | Q| ˜ 1/q−1/r w( Q) ˜ 1−1/r | Q| ˜ 1/r −1 ≤ C|x − y|m ||D α b|| B M O(w) | Q| |α|=m

≤ C|x − y|m



||D α b||BMO(w)

|α|=m

≤ C|x − y|

m



˜ w( Q) ˜ | Q|

||D α b||BMO(w) w(x), ˜

|α|=m

thus, by the L s -boundedness of T (See Lemma 2) for 1 < s < r and w ∈ A1 ⊆ Ar/s , we obtain     ˜ x, ·) Rm (b;   f T 1  dx m  |x − ·| Q  1/s   1 ≤C ||D α b||BMO(w) w(x) ˜ |T ( f 1 )(x)|s d x |Q| R n |α|=m  1/s  α −1/s s ≤C ||D b||BMO(w) w(x)|Q| ˜ | f 1 (x)| d x

C I1 ≤ |Q|

Rn

|α|=m



≤C

−1/s ||D α b||BMO(w) w(x)|Q| ˜

|α|=m



≤C

−1/s ||D α b||BMO(w) w(x)|Q| ˜

|α|=m

≤C

 |α|=m



 Q˜

| f (x)|s w(x)s/r w(x)−s/r d x



1/r Q˜

| f (x)|r w(x)d x

−1/s ˜ 1/r ||D α b||BMO(w) w(x)|Q| ˜ w( Q)



(r −s)/r s 

1 1 w(x)−s/(r −s) d x ˜ Q˜ ˜ | Q| | Q|  α ≤C ||D b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜ ×



1 ˜ w( Q)

Q˜

w(x)−s/(r −s) d x

(r −s)/r s

1/r

 Q˜

| f (x)|r w(x)d x

1/r

 Q˜

1/s

w(x)d x

˜ 1/s w( Q) ˜ −1/r | Q|

|α|=m

For I2 , by the weak (L 1 , L 1 ) boundedness of T (See Lemma 2) and Kolmogorov’s inequality (See Lemma 1), we obtain 1/η   1  α˜ η I2 ≤ C |T (D b f 1 )(x)| d x |Q| Q |α|=m

≤C

 |Q|1/η−1 ||T (D α b˜ f 1 )χ Q || L η |Q|1/η ||χ Q || L η/(1−η)

|α|=m

Y. Tan and L. Liu



1 ||T (D α b˜ f 1 )||W L 1 |Q| |α|=m  1  ˜ ≤C |D α b(x) f 1 (x)|d x |Q| R n ≤C

|α|=m

≤C

 |α|=m

≤C

 |α|=m

≤C



1 |Q|





Q˜

1 ˜ 1/r w( Q) ˜ 1/r ||D α b||BMO(w) w( Q) |Q| ||D α b||BMO(w)

|α|=m

≤C





|(D α b(x) − (D α b) Q˜ )|r w(x)1−r d x 

1 ˜ w( Q)

1/r 

1/r Q˜

| f (x)|r w(x)d x 1/r

 Q˜

| f (x)|r w(x)d x

˜ w( Q) ˜ Mr,w ( f )(x) ˜ | Q|

||D α b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜

|α|=m

For I3 , note that |x − y| ≈ |x0 − y| for x ∈ Q and y ∈ R n \Q, we write ˜

˜

|T b ( f 2 )(x) − T b ( f 2 )(x0 )|     ˜ x, y) − Rm (b; ˜ x0 , y) |(x, x − y)| | f 2 (y)|dy ≤ Rm (b; |x − y|m+n Rn     (x, x − y) (x0 , x0 − y)    |Rm (b; ˜ x0 , y)|| f 2 (y)|dy + −  n+m |x0 − y|n+m  R n |x − y|    1   (x, x − y) (x0 , x0 − y)    |(x − y)α ||D α b(y)|| ˜ − + f 2 (y)|dy α! R n  |x − y|n+m |x0 − y|n+m  |α|=m    1   (x − y)α (x0 − y)α   ˜ |(x0 , x0 − y)||D α b(y)|| + − f 2 (y)|dy α! R n  |x − y|m+n |x0 − y|m+n  |α|=m

=

(1) (2) (3) (4) I3 (x) + I3 (x) + I3 (x) + I3 (x).

(1)

For I3 (x), by the formula (See [7]): ˜ x, y) − Rm (b; ˜ x0 , y) = Rm (b;

 1 ˜ x, x0 )(x − y)γ Rm−|γ | (D γ b; γ!

|γ |
and Lemma 7, we have, similar to the proof of I1 ,   ˜ x, y) − Rm (b; ˜ x0 , y)| ≤ C |x − x0 |m−|γ | |x − y||γ | ||D α b|| B M O(w) w(x), ˜ |Rm (b; |γ |
thus, by w ∈ A1 ⊆ Ar , (1)

∞  

   ˜ x, y) − Rm (b; ˜ x0 , y) |(x, x − y)| | f (y)|dy Rm (b; k+1 Q\2 |x − y|m+n ˜ k Q˜ k=0 2 ∞    |x − x0 | ||D α b||BMO(w) w(x) ˜ | f (y)|dy ≤C ˜ k Q˜ |x 0 − y|n+1 2k+1 Q\2

I3 (x) ≤

|α|=m

k=0

Weighted boundedness of multilinear operator…

≤C



||D α b||BMO(w) w(x) ˜

|α|=m

≤C



k=1

||D α b||BMO(w) w(x) ˜

|α|=m

≤C

2k Q˜



w(y)−1/(r −1) dy



1

×

d (2k d)n+1 d k (2 d)n+1

 2k Q˜

| f (y)|w(y)1/r w(y)−1/r dy 1/r

 2k Q˜

| f (y)|r w(y)dy

(r −1)/r

||D α b||BMO(w) w(x) ˜

|α|=m



∞  k=1



×

∞ 

∞ 

d

k=1

(2k d)n+1

˜ 1/r w(2k Q)

1/r

| f (y)| w(y)d x ˜ 2k Q˜ w(2k Q)  (r −1)/r  1 × w(y)−1/(r −1) dy ˜ 2k Q˜ |2k Q|  1/r  1 k ˜ −1/r ˜ × w(y)dy |2k Q|w(2 Q) ˜ 2k Q˜ |2k Q| ≤C



r

||D α b||BMO(w) w(x)M ˜ r,w ( f )(x) ˜

|α|=m

≤C



∞ 

2−k

k=1 α

||D b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜

|α|=m (2)

For I3 (x), by [2], we know that ∞

 |(x, x − y)| Yuv (x − y) ≤C auv (x) , m+n |x − y| |x − y|n gu

u=1 v=1

where gu ≤

Cu −2n ,

||auv || L ∞ ≤ |Yuv (x − y)| ≤ Cu n/2−1 and    Yuv (x − y) Yuv (x0 − y)  n/2 n+1    |x − y|n − |x − y|n  ≤ Cu |x − x0 |/|x0 − y| 0

Cu n−2 ,

for |x − y| > 2|x0 − x| > 0. Thus, we get ∞   (2) ˜ x0 , y)| |Rm (b; I3 (x) ≤ C k+1 Q\2 ˜ k Q˜ k=0 2 gu ∞  

   Yuv (x − y) Yuv (x0 − y)  | f (y)|dy |auv (x)|  − |x − y|n+m |x0 − y|n+m  u=1 v=1 ∞  ∞    ≤C ||D α b||BMO(w) w(x) ˜ u −2n · u n/2 ×

≤C

|α|=m

u=1



∞ 

d

k=1

(2k d)n+1

|α|=m

≤C



|α|=m

||D α b||BMO(w) w(x) ˜

||D α b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜



k=0

2k Q˜

˜ k Q˜ 2k+1 Q\2

| f (y)|dy

|x − x0 | | f (y)|dy |x0 − y|n+1

Y. Tan and L. Liu

Similarly, we have (3)

(4)

I3 (x) + I3 (x) ∞  

≤C

u

−2n

·u

n/2

∞  

|α|=m u=1

+C

d k (2 d)n+1

∞   |α|=m k=1 ∞  

≤C

|α|=m k=1



2k Q˜

d (2k d)n+1

2k Q˜

2k Q˜

|(D α b)2k Q˜ − (D α b) Q˜ || f (y)|w(y)1/r w(y)−1/r dy

 2k Q˜

 ×

|D α b(y) − (D α b)2k Q˜ |w(y)−1/r | f (y)|w(y)1/r dy



d (2k d)n+1

|x − x0 | ˜ | f (y)||D α b(y)|dy |x0 − y|n+1

|x − x0 | ˜ | f (y)||D α b(y)|dy |x0 − y|n+1

k+1 Q\2 ˜ k Q˜ |α|=m k=0 2

|α|=m k=1

+C

k=0

∞   

∞  

≤C

˜ k Q˜ 2k+1 Q\2





|(D α b(y) − (D α b)2k Q˜ )|r w(y)1−r dy

1/r

1/r

| f (y)|r w(y)dy

 1/r d r | f (y)| w(y)d x (2k d)n+1 2k Q˜ |α|=m k=1  (r −1)/r  1/r   1 1 −1/(r −1) k ˜ −1/r ˜ × w(y) dy w(y)dy |2k Q|w(2 Q) ˜ 2k Q˜ ˜ 2k Q˜ |2k Q| |2k Q| 1/r  ∞   ˜  1 w(2k Q) ≤C ||D α b||BMO(w) 2−k | f (y)|r w(y)d x ˜ ˜ 2k Q˜ |2k Q| w(2k Q) +C



||D α b||BMO(w) w(x) ˜

|α|=m

+C

 

k

k=1

α

||D b||BMO(w) w(x) ˜

|α|=m

≤C

∞ 

∞ 

k2

≤C



||D α b||BMO(w) w(x)M ˜ r,w ( f )(x) ˜

1 ˜ w(2k Q)

k=1

|α|=m



−k

∞ 

1/r

 | f (y)| w(y)d x r

2k Q˜

k2−k

k=1 α

||D b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜

|α|=m

Thus I3 ≤ C



||D α b||BMO(w) w(x)M ˜ r,w ( f )(x). ˜

|α|=m



These complete the proof of Theorem 1.

Proof of Theorem 2 It suffices to prove for f ∈ C0∞ (R n ) and some constant C0 , the following inequality holds:    η 1/η  1  b  ≤C ||D α b|| Li pβ (w) w(x)M ˜ β,r,w ( f )(x). ˜ T ( f )(x) − C0  d x |Q| Q |α|=m

Weighted boundedness of multilinear operator…

Fix a cube Q = Q(x0 , d) and x˜ ∈ Q. Similar to the proof of Theorem 1, we have, for f 1 = f χ Q˜ and f 2 = f χ R n \ Q˜ ,    η 1/η 1   b b˜ ( f )(x) − T ( f )(x ) T 2 0  dx |Q| Q

η 1/η    ˜ x, ·) Rm (b; 1   ≤C f T  dx 1  |Q| Q  |x − ·|m  ⎛ ⎛ ⎞η ⎞1/η     (x − ·)α D α b˜   1 T ⎝ ⎠ d x ⎠ +C ⎝ f 1   m |Q| Q  |x − ·|  |α|=m    η 1/η 1   b˜ b˜ +C T ( f 2 )(x) − T ( f 2 )(x0 ) d x |Q| Q = J1 + J2 + J3 . For J1 and J2 , by using the same argument as in the proof of Theorem 1, we get  ˜ x, y)| ≤ C|x − y|m ˜ −1/q | Q| |Rm (b; |α|=m

 ×

˜ Q(x,y)

≤ C|x − y|m



˜ Q(x,y)

≤ C|x − y|  ×

m

r ˜ |D α b(z)| w(z)1−r dz



˜ | Q|

−1/q

1/r  ˜ Q(x,y)

w(z)q(r −1)/(r −q) dz

(r −q)/rq

˜ β/n+1/r | Q| ˜ (r −q)/rq ||D b|| Li pβ (w) w( Q) α

|α|=m



1

1/q

˜ −1/q | Q|

|α|=m

 ×

q ˜ |D α b(z)| w(z)q(1−r )/r w(z)q(r −1)/r dz

w(z) p0 dz

(r −q)/rq

˜ ˜ | Q(x, y)| Q(x,y)  m ˜ −1/q w( Q) ˜ β/n+1/r | Q| ˜ 1/q−1/r ≤ C|x − y| ||D α b|| Li pβ (w) | Q|  ×

|α|=m



1

w(z)dz

(r −1)/r

˜ ˜ | Q(x, y)| Q(x,y)  m ˜ −1/q w( Q) ˜ β/n+1/r | Q| ˜ 1/q−1/r w( Q) ˜ 1−1/r | Q| ˜ 1/r −1 ≤ C|x − y| ||D α b|| Li pβ (w) | Q| |α|=m

≤ C|x − y|m

 |α|=m

≤ C|x − y|

m



||D α b|| Li pβ (w)

˜ β/n+1 w( Q) ˜ | Q|

˜ β/n w(x), ||D α b|| Li pβ (w) w( Q) ˜

|α|=m

thus J1 ≤ C

 |α|=m

−1/s ˜ β/n w(x)|Q| ||D α b|| Li pβ (w) w( Q) ˜



1/s Rn

| f 1 (x)|s d x

Y. Tan and L. Liu

≤C



−1/s ˜ β/n w(x)|Q| ||D α b|| Li pβ (w) w( Q) ˜

|α|=m



× ≤C

Q˜



w(x)−s/(r −s) d x

1/r

 Q˜

| f (x)|r w(x)d x

(r −s)/r s

˜ −1/s w( Q) ˜ 1/r ||D α b|| Li pβ (w) w(x)| ˜ Q|



1/r



1

| f (x)|r w(x)d x

˜ 1−rβ/n Q˜ w( Q)  (r −s)/r s  1/r   1 1 −s/(r −s) ˜ 1/s w( Q) ˜ −1/r × w(x) dx w(x)d x | Q| ˜ Q˜ ˜ Q˜ | Q| | Q|  ≤C ||D α b|| Li pβ (w) w(x)M ˜ β,r,w ( f )(x), ˜ |α|=m

|α|=m

J2 ≤ C



|α|=m

≤C

 |α|=m

1 |Q| 1 |Q|

 Q˜

|D α b(x) − (D α b) Q˜ ||w(x)−1/r | f (x)|w(x)1/r d x





Q˜



|(D α b(x) − (D α b) Q˜ )|r w(x)1−r d x

1/r 

1/r Q˜

| f (x)|r w(x)d x



1 ˜ β/n+1/r w( Q) ˜ 1/r −β/n ||D α b|| Li pβ (w) w( Q) |Q| |α|=m  1/r  1 × | f (x)|r w(x)d x ˜ 1−rβ/n Q˜ w( Q)

≤C

≤C



||D α b|| Li pβ (w)

|α|=m

≤C



˜ w( Q) Mβ,r,w ( f )(x) ˜ ˜ | Q|

||D α b|| Li pβ (w) w(x)M ˜ β,r,w ( f )(x). ˜

|α|=m

For J3 , we have ˜ x, y) − Rm (b; ˜ x0 , y)| |Rm (b;   k ˜ β/n |x − x0 |m−|γ | |x − y||γ | ||D α b|| Li pβ (w) w(x)w(2 ˜ ≤C Q) , |γ |
thus ˜

˜

|T b ( f 2 )(x) − T b ( f 2 )(x0 )| ∞      ˜ x, y) − Rm (b; ˜ x0 , y) |(x, x − y)| | f (y)|dy ≤ Rm (b; k+1 Q\2 |x − y|m+n ˜ k Q˜ k=0 2   ∞    (x, x − y) (x0 , x0 − y)    |Rm (b; ˜ x0 , y)|| f (y)|dy − +  k+1 Q\2 |x0 − y|n+m  ˜ k Q˜ |x − y|n+m 2 k=0   ∞     (x, x − y) (x0 , x0 − y)    |(x − y)α ||D α b(y)|| ˜ − f (y)|dy +C  |x0 − y|n+m  ˜ k Q˜ |x − y|n+m 2k+1 Q\2 |α|=m k=0

Weighted boundedness of multilinear operator…

k+1 Q\2 ˜ k Q˜ |α|=m k=0 2



≤C

   (x − y)α (x0 − y)α  α˜  −  |x − y|m+n |x − y|m+n  |(x0 , x0 − y)||D b(y)|| f (y)|dy

∞   

+C

||D α b|| Li pβ (w) w(x) ˜

|α|=m ∞  

+C

|α|=m k=1 ∞  

+C

0

|α|=m k=1

d (2k d)n+1 d (2k d)n+1

∞ 

˜ β/n w(2k+1 Q)

k=0

 2k Q˜

 2k Q˜

 ˜ k Q˜ 2k+1 Q\2

|x − x0 | | f (y)|dy |x0 − y|n+1

|D α b(y) − (D α b)2k Q˜ |w(y)−1/r | f (y)|w(y)1/r dy |(D α b)2k Q˜ − (D α b) Q˜ || f (y)|w(y)1/r w(y)−1/r dy

 1/r d k ˜ β/n r w(2 | f (y)| w(y)d x Q) (2k d)n+1 2k Q˜ |α|=m k=1  (r −1)/r  1/r   1 1 k ˜ −1/r ˜ × w(y)−1/(r −1) dy w(y)dy |2k Q|w(2 Q) ˜ 2k Q˜ ˜ 2k Q˜ |2k Q| |2k Q|  1/r ∞   d α α r 1−r +C |(D b(y) − (D b) )| w(y) dy k ˜ 2 Q (2k d)n+1 2k Q˜ |α|=m k=1  1/r × | f (y)|r w(y)dy 

≤C

||D α b|| Li pβ (w) w(x) ˜

∞ 

2k Q˜

+C  ×

 |α|=m

1

||D α b|| Li pβ (w) w(x) ˜

∞ 

˜ β/n kw(2k Q)

k=1



w(y)−1/(r −1) dy

(r −1)/r 

˜ 2k Q˜ |2k Q|  ∞   ≤C ||D α b|| Li pβ (w) w(x) ˜ k2−k |α|=m

k=1



d k (2 d)n+1

1 ˜ |2k Q|

1/r

2k Q˜

1/r

 2k Q˜

1 k ˜ w(2 Q)1−rβ/n

| f (y)|r w(y)d x

w(y)dy

k ˜ −1/r ˜ |2k Q|w(2 Q)

1/r

 2k Q˜

| f (y)|r w(y)d x

1/r  ∞ k ˜    1 α −k w(2 Q) r +C ||D b|| Li pβ (w) 2 | f (y)| w(y)d x ˜ ˜ 1−rβ/n 2k Q˜ |2k Q| w(2k Q) |α|=m k=1  ≤C ||D α b|| Li pβ (w) w(x)M ˜ β,r,w ( f )(x). ˜ |α|=m

This completes the proof of Theorem 2.



Proof of Theorem 3 Choose 1 < r < p in Theorem 1 and notice w 1− p ∈ A1 , then we have, by Lemmas 3 and 4, ||T b ( f )|| L p (w1− p ) ≤ Mη (T b ( f )) L p (w1− p ) ≤ CMη# (T b ( f )) L p (w1− p )  ≤C ||D α b||BMO(w) wMr,w ( f ) L p (w1− p ) |α|=m

=C



|α|=m

||D α b||BMO(w) Mr,w ( f ) L p (w)

Y. Tan and L. Liu

≤C



||D α b||BMO(w)  f  L p (w) .

|α|=m

This completes the proof of Theorem 3.



Proof of Theorem 4 Choose 1 < r < p in Theorem 1 and notice w 1− p ∈ A1 , then we have, by Lemmas 5 and 6, ||T b ( f )|| L p,ϕ (w1− p ) ≤ Mη (T b ( f )) L p,ϕ (w1− p ) ≤ CMη# (T b ( f )) L p,ϕ (w1− p )  ≤C ||D α b||BMO(w) wMr,w ( f ) L p,ϕ (w1− p ) |α|=m

=C



||D α b||BMO(w) Mr,w ( f ) L p,ϕ (w)

|α|=m

≤C



||D α b||BMO(w)  f  L p,ϕ (w) .

|α|=m

This completes the proof of Theorem 4.



Proof of Theorem 5 Choose 1 < r < p in Theorem 2 and notice w 1−q ∈ A1 , then we have, by Lemmas 3 and 4, ||T b ( f )|| L q (w1−q ) ≤ Mη (T b ( f )) L q (w1−q ) ≤ CMη# (T b ( f )) L q (w1−q )  ≤C ||D α b|| Li pβ (w) wMβ,r,w ( f ) L q (w1−q ) |α|=m

=C



||D α b|| Li pβ (w) Mβ,r,w ( f ) L q (w)

|α|=m

≤C



||D α b|| Li pβ (w)  f  L p (w) .

|α|=m

This completes the proof of Theorem 5.



Proof of Theorem 6 Choose 1 < r < p in Theorem 2 and notice w 1−q ∈ A1 , then we have, by Lemmas 5 and 6, ||T b ( f )|| L q,ϕ (w1−q ) ≤ Mη (T b ( f )) L q,ϕ (w1−q ) ≤ CMη# (T b ( f )) L q,ϕ (w1−q )  ≤C ||D α b|| Li pβ (w) wMβ,r,w ( f ) L q,ϕ (w1−q ) |α|=m

=C



||D α b|| Li pβ (w) Mβ,r,w ( f ) L q,ϕ (w)

|α|=m

≤C



|α|=m

||D α b|| Li pβ (w)  f  L p,β,ϕ (w) .

This completes the proof of Theorem 6.



Acknowledgments The authors would like to express their gratitude to the referee for his/her valuable comments and suggestions.

Weighted boundedness of multilinear operator…

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