Boundedness of fractional integral operators with rough kernels on

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Mar 7, 2012 - Later, Ding and Lu [4] considered the weighted norm inequalities ... define the commutators generated by fractional maximal and integral ...
arXiv:1203.1441v1 [math.CA] 7 Mar 2012

Boundedness of fractional integral operators with rough kernels on weighted Morrey spaces Hua Wang



Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Abstract Let MΩ,α and TΩ,α be the fractional maximal and integral operators with rough kernels, where 0 < α < n. In this paper, we shall study the continuity properties of MΩ,α and TΩ,α on the weighted Morrey spaces Lp,κ (w). The boundedness of their commutators with BMO functions is also obtained. MSC(2010): 42B20; 42B25 Keywords: Fractional integral operators; rough kernels; weighted Morrey spaces; commutator

1

Introduction

Let Ω ∈ Ls (S n−1 ) be homogeneous of degree zero on Rn , where S n−1 denotes the unit sphere of Rn (n ≥ 2) equipped with the normalized Lebesgue measure dσ and s > 1. For any 0 < α < n, then the fractional integral operator with rough kernel TΩ,α is defined by Z Ω(y ′ ) TΩ,α f (x) = f (x − y) dy n−α Rn |y| and a related fractional maximal operator MΩ,α is defined by Z 1 Ω(y ′ )f (x − y) dy, MΩ,α f (x) = sup n−α r>0 r |y|≤r

where y ′ = y/|y| for any y 6= 0. In 1971, Muckenhoupt and Wheeden [17] studied the weighted norm inequalities for TΩ,α with the weight w(x) = |x|β . The weak type estimates with power weights for MΩ,α and TΩ,α was obtained by Ding in [3]. Later, Ding and Lu [4] considered the weighted norm inequalities for MΩ,α and TΩ,α with more general weights. More precisely, they proved Theorem A ([4]). Let 0 < α < n, 1 ≤ s′ < p < n/α and 1/q = 1/p − α/n. If ′ Ω ∈ Ls (S n−1 ) and ws ∈ A(p/s′ , q/s′ ), then the operators MΩ,α and TΩ,α are all bounded from Lp (wp ) to Lq (wq ). ∗ E-mail

address: [email protected].

1

Let b be a locally integrable function on Rn , then for 0 < α < n, we shall define the commutators generated by fractional maximal and integral operators with rough kernels and b as follows. Z 1 [b, MΩ,α ](f )(x) = sup n−α |b(x) − b(y)||Ω(x − y)f (y)| dy, r>0 r |y−x|≤r [b, TΩ,α ](f )(x) = b(x)TΩ,α f (x) − TΩ,α (bf )(x) Z Ω(x − y) = [b(x) − b(y)]f (y) dy. n−α Rn |x − y|

In 1993, by using the Rubio de Francia extrapolation theorem, Segovia and Torrea [21] obtained the weighted boundedness of commutator [b, TΩ,α ], where b ∈ BM O(Rn ) and Ω satisfies some Dini smoothness condition (see also [20]). In 1999, Ding and Lu [5] improved this result by removing the smoothness condition imposed on Ω. More specifically, they showed (see also [14]). Theorem B ([5]). Let 0 < α < n, 1 ≤ s′ < p < n/α and 1/q = 1/p − α/n. ′ Assume that Ω ∈ Ls (S n−1 ), ws ∈ A(p/s′ , q/s′ ) and b ∈ BM O(Rn ), then the commutator [b, TΩ,α ] is bounded from Lp (wp ) to Lq (wq ). The classical Morrey spaces Lp,λ were first introduced by Morrey in [15] to study the local behavior of solutions to second order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator and the Calder´ on-Zygmund singular integral operator on these spaces, we refer the readers to [1, 2, 19]. For the properties and applications of classical Morrey spaces, see [7, 8, 9] and references therein. In 2009, Komori and Shirai [13] first defined the weighted Morrey spaces Lp,κ (w) which could be viewed as an extension of weighted Lebesgue spaces, and studied the boundedness of the above classical operators on these weighted spaces. Recently, in [22] and [23], we have established the continuity properties of some other operators on the weighted Morrey spaces Lp,κ (w). The purpose of this paper is to discuss the boundedness properties of MΩ,α and TΩ,α on the weighted Morrey spaces. Here, and in what follows we shall use the notation s′ = s/(s − 1) when 1 < s < ∞ and s′ = 1 when s = ∞. Our main results in the paper are formulated as follows. Theorem 1.1. Suppose that Ω ∈ Ls (S n−1 ) with 1 < s ≤ ∞. If 0 < α < n, 1 ≤ ′ s′ < p < n/α, 1/q = 1/p − α/n, 0 < κ < p/q and ws ∈ A(p/s′ , q/s′ ), then the fractional maximal operator MΩ,α is bounded from Lp,κ (wp , wq ) to Lq,κq/p (wq ). Theorem 1.2. Suppose that Ω ∈ Ls (S n−1 ) with 1 < s ≤ ∞. If 0 < α < n, 1 ≤ ′ s′ < p < n/α, 1/q = 1/p − α/n, 0 < κ < p/q and ws ∈ A(p/s′ , q/s′ ), then the fractional integral operator TΩ,α is bounded from Lp,κ (wp , wq ) to Lq,κq/p (wq ). Theorem 1.3. Suppose that Ω ∈ Ls (S n−1 ) with 1 < s ≤ ∞ and b ∈ BM O(Rn ). If 0 < α < n, 1 ≤ s′ < p < n/α, 1/q = 1/p − α/n, 0 < κ < p/q and ′ ws ∈ A(p/s′ , q/s′ ), then the commutator [b, TΩ,α ] is bounded from Lp,κ (wp , wq ) to Lq,κq/p (wq ). 2

2

Notations and definitions

Let us first recall some standard definitions and notations. The classical Ap weight theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of Hardy-Littlewood maximal functions in [16]. A weight w is a nonnegative, locally integrable function on Rn , B = B(x0 , rB ) denotes the ball with the center x0 and radius rB . Given a ball B and λ > 0, λB denotes the ball with the same center as B whose radius is λ times that of B. For a given weight function w, we also denote the Lebesgue Rmeasure of B by |B| and the weighted measure of B by w(B), where w(B) = B w(x) dx. We say that w ∈ Ap , 1 < p < ∞, if 

1 |B|

 p−1 Z 1 −1/(p−1) w(x) dx w(x) dx ≤C |B| B B

Z

for every ball B ⊆ Rn ,

where C is a positive constant which is independent of B. For the case p = 1, w ∈ A1 , if Z 1 w(x) dx ≤ C · ess inf w(x) for every ball B ⊆ Rn . x∈B |B| B For the case p = ∞, w ∈ A∞ if it satisfies the Ap condition for some 1 < p < ∞. We also need another weight class A(p, q) introduced by Muckenhoupt and Wheeden in [18]. A weight function w belongs to A(p, q) for 1 < p < q < ∞ if there exists a constant C > 0 such that  1/q  1/p′ Z Z 1 1 q −p′ w(x) dx w(x) dx ≤ C for every ball B ⊆ Rn . |B| B |B| B A weight function w is said to belong to the reverse H¨ older class RHr if there exist two constants r > 1 and C > 0 such that the following reverse H¨ older inequality holds 

1 |B|

1/r   Z 1 w(x) dx w(x) dx ≤C |B| B B

Z

r

for every ball B ⊆ Rn .

We state the following results that we will use frequently in the sequel. Lemma 2.1 ([10]). Let w ∈ Ap with p ≥ 1. Then, for any ball B, there exists an absolute constant C > 0 such that w(2B) ≤ C w(B). In general, for any λ > 1, we have w(λB) ≤ C · λnp w(B), where C does not depend on B nor on λ. 3

Lemma 2.2 ([11]). Let w ∈ RHr with r > 1. Then there exists a constant C > 0 such that (r−1)/r  w(E) |E| ≤C w(B) |B| for any measurable subset E of a ball B. Next we shall introduce the Hardy-Littlewood maximal operator, its variant and BMO spaces. The Hardy-Littlewood maximal operator M is defined by Z 1 M (f )(x) = sup |f (y)| dy, x∈B |B| B where the supremum is taken over all balls B containing x. For 0 < α < n, s ≥ 1, we define the fractional maximal operator Mα,s by Mα,s (f )(x) = sup

x∈B



1 |B|1−

αs n

Z

s

|f (y)| dy

B

1/s

.

Moreover, we denote simply by Mα when s = 1. A locally integrable function b is said to be in BM O(Rn ) if Z 1 |b(x) − bB | dx < ∞, kbk∗ = sup B |B| B R 1 where bB stands for the average of b on B, i.e. bB = |B| B b(y) dy and the supremum is taken over all balls B in Rn . Theorem C ([6, 12]). Assume that b ∈ BM O(Rn ). Then for any 1 ≤ p < ∞, we have 1/p  Z 1 b(x) − bB p dx sup ≤ Ckbk∗ . |B| B B

We are going to conclude this section by defining the weighted Morrey space and giving the known result relevant to this paper. For further details, we refer the readers to [13]. Definition 2.3 ([13]). Let 1 ≤ p < ∞, 0 < κ < 1 and w be a weight function. Then the weighted Morrey space is defined by  Lp,κ (w) = f ∈ Lploc (w) : kf kLp,κ (w) < ∞ ,

where

kf kLp,κ (w) = sup B



1 w(B)κ

Z

B

 1/p |f (x)| w(x) dx p

and the supremum is taken over all balls B in Rn .

In order to deal with the fractional order case, we need to consider the weighted Morrey space with two weights. 4

Definition 2.4 ([13]). Let 1 ≤ p < ∞ and 0 < κ < 1. Then for two weights u and v, the weighted Morrey space is defined by  Lp,κ (u, v) = f ∈ Lploc (u) : kf kLp,κ(u,v) < ∞ , where

kf kLp,κ(u,v) = sup B



1 v(B)κ

Z

B

1/p |f (x)| u(x) dx . p

Theorem D. If 0 < α < n, 1 < p < n/α, 1/q = 1/p − α/n, 0 < κ < p/q and w ∈ A(p, q), then the fractional maximal operator Mα is bounded from Lp,κ (wp , wq ) to Lq,κq/p (wq ). Throughout this article, we will use C to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. By A ∼ B, we mean that there exists a constant C > 1 such that A 1 C ≤ B ≤ C.

3

Proof of Theorem 1.1

Proof of Theorem 1.1. For Ω ∈ Ls (S n−1 ), we set 1/s Z ′ s ′ . Ω(y ) dσ(y ) kΩkLs(S n−1 ) = S n−1

From H¨ older’s inequality, it follows that Z 1/s  Z 1/s′ s ′ 1 ′ s Ω(y ) dy MΩ,α f (x) ≤ sup n−α |f (x − y)| dy r>0 r |y|≤r |y|≤r  1/s′ Z ′ 1 |f (x − y)|s dy ≤ CkΩkLs (S n−1 ) · sup n−αs′ r r>0 |y|≤r !1/s′ Z 1 s′ ≤ CkΩkLs (S n−1 ) · sup |f (y)| dy αs′ r>0 |B(x, r)|1− n B(x,r) = CkΩkLs (S n−1 ) Mα,s′ (f )(x). ′

If we let p1 = p/s′ , q1 = q/s′ and ν = ws , then for 0 < α < n, 1 ≤ s′ < n/α, we have 1/q1 = 1/p1 − (αs′ )/n and 0 < κ < p1 /q1 . Also observe that ′



Mα,s′ (f ) = Mαs′ (|f |s )1/s . Hence, by Theorem D, we obtain





Mα,s′ (f ) q,κq/p q = Mαs′ (|f |s′ ) 1/s L (w ) Lq1 ,κq1 /p1 (ν q1 )

s′ 1/s′ ≤ C |f | p ,κ p q L

1



1 ,ν 1 )

≤ Ckf kLp,κ(wp ,wq ) .

This finishes the proof of Theorem 1.1. 5

4

Proof of Theorem 1.2

Proof of Theorem 1.2. Fix a ball B = B(x0 , rB ) ⊆ Rn and decompose f = f1 + f2 , where f1 = f χ2B and χ2B denotes the characteristic function of 2B. Since TΩ,α is a linear operator, then we can write 1 q w (B)κ/p

Z

1 ≤ q w (B)κ/p = I1 + I2 .

Z

1/q |TΩ,α f (x)| w(x) dx q

B

B

q

1/q |TΩ,α f1 (x)| w(x) dx + q

q

1 wq (B)κ/p

Z

1/q |TΩ,α f2 (x)| w(x) dx q

B

q



As in the proof of Theorem 1.1, we also set p1 = p/s′ , q1 = q/s′ and ν = ws . Since ν ∈ A(p1 , q1 ), then we get ν q1 = wq ∈ A1+q1 /p′1 (see [18]). Hence, by Theorem A and Lemma 2.1, we have Z 1/p 1 p p I1 ≤ C · q |f (x)| w(x) dx w (B)κ/p 2B ≤ Ckf kLp,κ(wp ,wq ) ·

wq (2B)κ/p wq (B)κ/p

≤ Ckf kLp,κ(wp ,wq ) . We now turn to deal with the term I2 . An application of H¨ older’s inequality gives us that Z |Ω(x − y)| TΩ,α (f2 )(x) ≤ |f (y)| dy (1) |x − y|n−α c (2B) 1/s  Z 1/s′ ′ ∞ Z X |f (y)|s s |Ω(x − y)| dy ≤ dy . (n−α)s′ 2k+1 B\2k B |x − y| 2k+1 B\2k B k=1 When x ∈ B and y ∈ 2k+1 B\2k B, then we can easily see that 2k−1 rB ≤ |y−x| < 2k+2 rB . Thus, by a simple computation, we deduce Z 1/s 1/s |Ω(x − y)|s dy ≤ CkΩkLs (S n−1 ) 2k+1 B . (2) 2k+1 B\2k B

We also note that if x ∈ B, y ∈ (2B)c , then |y − x| ∼ |y − x0 |. Consequently Z

2k+1 B\2k B



|f (y)|s dy |x − y|(n−α)s′

1/s′

1 ≤ C· k+1 1−α/n |2 B|

Z

s′

|f (y)| dy

2k+1 B

1/s′

(3)

Substituting the above two inequalities (2) and (3) into (1), we obtain ∞ X TΩ,α (f2 )(x) ≤ CkΩkLs (S n−1 ) k=1

1 |2k+1 B|1−α/n−1/s 6

Z

2k+1 B

s′

|f (y)| dy

1/s′

.

.

By using H¨ older’s inequality and the definition of ν ∈ A(p1 , q1 ), we can get Z 1/s′  Z 1/(p1 s′ )  Z 1/(p′1 s′ ) s′ p1 s′ p1 −p′1 |f (y)| dy ≤ |f (y)| ν(y) dy ν(y) dy 2k+1 B

2k+1 B

≤C

2k+1 B

Z

p

p

|f (y)| w(y) dy

2k+1 B

1/p 

|2

k+1

B|1−1/p1 +1/q1 ν q1 (2k+1 B)1/q1 ′

1/s′

κ/p |2k+1 B|1/s −1/p+1/q · wq (2k+1 B)1/q k+1 1−1/s−α/n κ/p−1/q = Ckf kLp,κ(wp ,wq ) 2 B . · wq 2k+1 B (4)

≤ Ckf kLp,κ(wp ,wq ) wq 2k+1 B

So we have

∞ X κ/p−1/q TΩ,α (f2 )(x) ≤ Ckf kLp,κ (wp ,wq ) wq 2k+1 B , k=1

which implies

I2 ≤ Ckf kLp,κ(wp ,wq )

∞ X

k=1

wq (B)1/q−κ/p wq (2k+1 B)1/q−κ/p

.

Observe that wq = ν q1 ∈ A1+q1 /p′1 , then we know that there exists r > 1 such that wq ∈ RHr . Thus, it follows directly from Lemma 2.2 that  (r−1)/r |B| wq (B) . (5) ≤ C wq (2k+1 B) |2k+1 B| Therefore (1−1/r)(1/q−κ/p) ∞  X 1 I2 ≤ Ckf kLp,κ (wp ,wq ) 2kn k=1

≤ Ckf kLp,κ (wp ,wq ) ,

where the last series is convergent since r > 1 and 0 < κ < p/q. Combining the above estimates for I1 and I2 and taking the supremum over all balls B ⊆ Rn , we complete the proof of Theorem 1.2.

5

Proof of Theorem 1.3

Proof of Theorem 1.3. Fix a ball B = B(x0 , rB ) ⊆ Rn . Let f = f1 + f2 , where f1 = f χ2B . Since [b, TΩ,α ] is a linear operator, then we have Z 1/q 1 [b, TΩ,α ]f (x) q w(x)q dx wq (B)κ/p B Z Z 1/q 1/q q q 1 1 q q [b, TΩ,α ]f1 (x) w(x) dx [b, TΩ,α ]f2 (x) w(x) dx + q ≤ q w (B)κ/p w (B)κ/p B B = J1 + J2 . 7



As before, we set p1 = p/s′ , q1 = q/s′ and ν = ws , then ν q1 = wq ∈ A1+q1 /p′1 . Theorem B and Lemma 2.1 imply Z 1/p 1 p p J1 ≤ Ckbk∗ · q |f (x)| w(x) dx w (B)κ/p 2B ≤ Ckbk∗ kf kLp,κ(wp ,wq ) ·

wq (2B)κ/p wq (B)κ/p

≤ Ckbk∗ kf kLp,κ(wp ,wq ) .

(6)

In order to estimate the term J2 , for any x ∈ B, we first write Z    Ω(x − y)  b, TΩ,α f2 (x) = b(x) − b(y) f (y) dy n−α |x − y| c (2B) Z |Ω(x − y)| |f (y)| dy ≤ b(x) − bB · n−α (2B)c |x − y| Z |Ω(x − y)| |b(y) − bB ||f (y)| dy + n−α (2B)c |x − y| = I+II.

For the term I, it follows from the previous estimates (2) and (4) that I ≤ Ckf kLp,κ(wp ,wq ) |b(x) − bB | ·

∞ X

k=1

Hence 1 wq (B)κ/p

Z

.

1/q I w(x) dx q

B

1 wq (2k+1 B)1/q−κ/p

q

Z 1/q ∞ X 1 1 q q · · |b(x) − b | w(x) dx B wq (B)κ/p k=1 wq (2k+1 B)1/q−κ/p B 1/q  Z ∞ X 1 wq (B)1/q−κ/p q q |b(x) − bB | w(x) dx . · = Ckf kLp,κ (wp ,wq ) wq (B) B wq (2k+1 B)1/q−κ/p ≤ Ckf kLp,κ (wp ,wq )

k=1

We now claim that for any 1 < q < ∞ and µ ∈ A∞ , the following inequality holds  1/q Z 1 q |b(x) − bB | µ(x) dx ≤ Ckbk∗ . (7) µ(B) B In fact, since µ ∈ A∞ , then there must exist r > 1 such that µ ∈ RHr . Thus, by H¨ older’s inequality and Theorem C, we obtain Z 1/(qr′ )  Z 1/(qr) 1/q  Z 1 1 qr ′ r q |b(x) − bB | dx µ(x) dx |b(x) − bB | µ(x) dx ≤ µ(B) B µ(B)1/q B B  1/(qr′ ) Z 1 qr ′ ≤C |b(x) − bB | dx |B| B ≤ Ckbk∗ , 8

which is our desired result. Note that wq ∈ A1+q1 /p′1 ⊂ A∞ . In addition, we have wq ∈ RHr with r > 1. Hence, by the inequalities (5) and (7), we get Z 1/q (1−1/r)(1/q−κ/p) ∞  X 1 1 q q p,κ p q I w(x) dx ≤ Ckbk kf k ∗ L (w ,w ) 2kn wq (B)κ/p B k=1 ≤ Ckbk∗ kf kLp,κ(wp ,wq ) .

(8)

On the other hand II ≤

∞ Z X

2k+1 B\2k B



k=1 ∞ Z X

2k+1 B\2k B

k=1 ∞ Z X

+

|Ω(x − y)| |b(y) − bB ||f (y)| dy |x − y|n−α |Ω(x − y)| b(y) − b2k+1 B |f (y)| dy |x − y|n−α

2k+1 B\2k B

k=1

= III+IV.

|Ω(x − y)| b2k+1 B − bB |f (y)| dy n−α |x − y|

To estimate III and IV, we observe that when x ∈ B, y ∈ (2B)c , then |y − x| ∼ |y − x0 |. Thus, it follows from H¨ older’s inequality and (2) that Z ∞ X 1 Ω(x − y) b(y) − b2k+1 B |f (y)| dy · III ≤ C k+1 1−α/n |2 B| 2k+1 B\2k B k=1 Z 1/s′ ∞ X s′ 1 s′ · b(y) − b2k+1 B |f (y)| dy . ≤C |2k+1 B|1−α/n−1/s 2k+1 B k=1

An application of H¨ older’s inequality yields Z 1/s′ ′ b(y) − b2k+1 B s |f (y)|s′ dy 2k+1 B



Z



Z



|f (y)|p1 s ν(y)p1 dy

2k+1 B

2k+1 B

1/(p1 s′ )  Z

2k+1 B

p

p

|f (y)| w(y) dy

1/p  Z

2k+1 B

′ ′ b(y) − b2k+1 B p1 s ν(y)−p′1 dy

′ ′ b(y) − b2k+1 B p1 s ν(y)−p′1 dy

1/(p′1 s′ )

1/(p′1 s′ )

.



Since ν ∈ A(p1 , q1 ), then we know that ν −p1 ∈ A1+p′1 /q1 ⊂ A∞ (see [18]). Hence, by using the inequality (7) and the fact that ν ∈ A(p1 , q1 ), we obtain Z 1/(p′1 s′ ) ′ ′ 1/(p′1 s′ ) ′ b(y) − b2k+1 B p1 s ν(y)−p′1 dy ≤ Ckbk∗ · ν −p1 2k+1 B 2k+1 B



≤ Ckbk∗ ·

9

|2k+1 B|1/q1 +1/p1 ν q1 (2k+1 B)1/q1

!1/s′



|2k+1 B|1/s −1/p+1/q . = Ckbk∗ · wq (2k+1 B)1/q

(9)

Consequently, by the above inequality (9), we deduce III ≤ Ckbk∗ kf kLp,κ(wp ,wq )

∞ X

k=1

1 wq (2k+1 B)1/q−κ/p

,

which implies 1 wq (B)κ/p

Z

B

1/q ∞ X IIIq w(x)q dx ≤ Ckbk∗ kf kLp,κ(wp ,wq )

wq (B)1/q−κ/p wq (2k+1 B)1/q−κ/p

k=1

≤ Ckbk∗ kf kLp,κ(wp ,wq ) .

(10)

Since b ∈ BM O(Rn ), then a direct calculation shows that b2k+1 B − bB ≤ C · kkbk∗ .

Moreover, by H¨ older’s inequality, the estimates (2) and (4), we can get IV ≤ Ckbk∗

∞ X

k=1

1 k · k+1 1−α/n |2 B|

≤ Ckbk∗ kf kLp,κ(wp ,wq )

∞ X



k=1

Z

|Ω(x − y)||f (y)| dy

2k+1 B\2k B

1 wq (2k+1 B)1/q−κ/p

.

Therefore 1 q w (B)κ/p

Z

B

1/q ∞ X k· IV w(x) dx ≤ Ckbk∗ kf kLp,κ (wp ,wq ) q

q

≤ Ckbk∗ kf kLp,κ (wp ,wq )

k=1 ∞ X k=1

wq (B)1/q−κ/p wq (2k+1 B)1/q−κ/p

k

2knδ

≤ Ckbk∗ kf kLp,κ (wp ,wq ) ,

(11)

where wq ∈ RHr and δ = (1 − 1/r)(1/q − κ/p). Summarizing the estimates (10) and (11) derived above, we thus obtain 1 wq (B)κ/p

Z

B

1/q II w(x) dx ≤ Ckbk∗ kf kLp,κ(wp ,wq ) . q

q

(12)

Combining the inequalities (6) and (8) with the above inequality (12) and taking the supremum over all balls B ⊆ Rn , we conclude the proof of Theorem 1.3. It should be pointed out that [b, MΩ,α ](f ) can be controlled pointwise by [b, T|Ω|,α ](|f |) for any f (x). In fact, for any 0 < α < n, x ∈ Rn and r > 0, we 10

have [b, T|Ω|,α ](|f |)(x) ≥

Z

|y−x|≤r



1

rn−α

|Ω(x − y)| |b(x) − b(y)||f (y)| dy |x − y|n−α

Z

|Ω(x − y)||b(x) − b(y)||f (y)| dy.

|y−x|≤r

Taking the supremum for all r > 0 on both sides of the above inequality, we get [b, MΩ,α ](f )(x) ≤ [b, T|Ω|,α ](|f |)(x),

for all x ∈ Rn .

Hence, as a direct consequence of Theorem 1.3, we finally obtain the following Corollary 5.1. Suppose that Ω ∈ Ls (S n−1 ) with 1 < s ≤ ∞ and b ∈ BM O(Rn ). If 0 < α < n, 1 ≤ s′ < p < n/α, 1/q = 1/p − α/n, 0 < κ < p/q and ′ ws ∈ A(p/s′ , q/s′ ), then the commutator [b, MΩ,α ] is bounded from Lp,κ (wp , wq ) to Lq,κq/p (wq ).

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