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Nov 26, 2010 - CA] 26 Nov 2010. The boundedness of some operators with rough kernel on the weighted Morrey spaces. Hua Wang∗. School of Mathematical ...
arXiv:1011.5763v1 [math.CA] 26 Nov 2010

The boundedness of some operators with rough kernel on the weighted Morrey spaces Hua Wang ∗ School of Mathematical Sciences, Peking University, Beijing 100871, China

Abstract Let Ω ∈ L (S ) with 1 < q ≤ ∞ be homogeneous of degree zero and has mean value zero on S n−1 . In this paper, we will study the boundedness of homogeneous singular integrals and Marcinkiewicz integrals with rough kernel on the weighted Morrey spaces Lp,κ (w) for q ′ ≤ p < ∞(or q ′ < p < ∞) and 0 < κ < 1. We will also prove that the commutator operators formed by a BM O(Rn ) function b(x) and these rough operators are bounded on the weighted Morrey spaces Lp,κ (w) for q ′ < p < ∞ and 0 < κ < 1. MSC(2000) 42B20; 42B25; 42B35 Keywords: Homogeneous singular integrals; Marcinkiewicz integrals; rough kernel; weighted Morrey spaces; commutator; Ap weights q

n−1

1. Introduction Suppose that S n−1 is the unit sphere in Rn (n ≥ 2) equipped with the normalized Lebesgue measure dσ. Let Ω ∈ Lq (S n−1 ) with 1 < q ≤ ∞ be homogeneous of degree zero and satisfy the cancellation condition Z Ω(x′ )dσ(x′ ) = 0, S n−1

where x′ = x/|x| for any x 6= 0. The homogeneous singular integral operator TΩ is defined by Z Ω(y ′ ) f (x − y) dy TΩ f (x) = lim ε→0 |y|>ε |y|n and a related maximal operator MΩ is defined by Z 1 |Ω(y ′ )f (x − y)| dy. MΩ f (x) = sup n r>0 r |y| 0 independent of f such that kMΩ (f )kLp,κ (w) ≤ Ckf kLp,κ (w) . 2

Theorem 2. Assume that Ω ∈ Lq (S n−1 ) with 1 < q < ∞. Then for every q ′ ≤ p < ∞, w ∈ Ap/q′ and 0 < κ < 1, there exists a constant C > 0 independent of f such that kTΩ (f )kLp,κ (w) ≤ Ckf kLp,κ (w) . Theorem 3. Assume that Ω ∈ Lq (S n−1 ) with 1 < q < ∞ and b ∈ BM O(Rn ). Then for every q ′ < p < ∞, w ∈ Ap/q′ and 0 < κ < 1, there exists a constant C > 0 independent of f such that

[b, TΩ ](f ) p,κ ≤ Ckf kLp,κ (w) . L (w)

Theorem 4. Assume that Ω ∈ Lq (S n−1 ) with 1 < q ≤ ∞. Then for every q ′ < p < ∞, w ∈ Ap/q′ and 0 < κ < 1, there exists a constant C > 0 independent of f such that kµΩ (f )kLp,κ (w) ≤ Ckf kLp,κ (w) .

Theorem 5. Assume that Ω ∈ Lq (S n−1 ) with 1 < q ≤ ∞ and b ∈ BM O(Rn ). Then for every q ′ < p < ∞, w ∈ Ap/q′ and 0 < κ < 1, there exists a constant C > 0 independent of f such that

[b, µΩ ](f ) p,κ ≤ Ckf kLp,κ (w) . L (w)

2. Notations and definitions

First let us 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 [11]. A weight w is a locally integrable function on Rn which takes values in (0, ∞) almost everywhere, B = B(x0 , r) denotes the ball with the center x0 and radius r. We say that w ∈ Ap , 1 < p < ∞, if 

1 |B|

Z

w(x) dx B



1 |B|

Z

1 − p−1

w(x)

dx

B

p−1

≤C

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

3

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|

Z

B

r

w(x) dx

1/r

≤C



1 |B|

Z

w(x) dx B



for every ball B ⊆ Rn .

It is well known that if w ∈ Ap with 1 < p < ∞, then w ∈ Ar for all r > p, and w ∈ Aq for some 1 < q < p. If w ∈ Ap with 1 ≤ p < ∞, then there exists r > 1 such that w ∈ RHr . We give the following results that we will use frequently in the sequel. Lemma A ([6]). Let w ∈ Ap , p ≥ 1. Then, for any ball B, there exists an absolute constant C such that w(2B) ≤ Cw(B). In general, for any λ > 1, we have w(λB) ≤ Cλnp w(B), where C does not depend on B nor on λ. Lemma B ([7]). Let w ∈ RHr with r > 1. Then there exists a constant C such that   |E| (r−1)/r w(E) ≤C w(B) |B| for any measurable subset E of a ball B. A locally integrable function b is said to be in BM O(Rn ) if Z 1 kbk∗ = sup |b(x) − bB | dx < ∞, B |B| B R 1 n where bB = |B| B b(y) dy and the supremum is taken over all balls B in R . Theorem C ([5,8]). 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

Next we shall define the weighted Morrey space and give one of the results relevant to this paper. For further details, we refer the readers to [9]. 4

Definition 1. 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

p

|f (x)| w(x) dx

B

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

1/p

In [9], the authors obtained the following result. Theorem D. If 1 < p < ∞, 0 < κ < 1 and w ∈ Ap , then the HardyLittlewood maximal operator M is bounded on Lp,κ (w). We are going to conclude this section by giving several results concerning the weighted boundedness of rough operators MΩ , TΩ and µΩ on the weighted Lp spaces. Given a Muckenhoupt’s weight function w on Rn , for 1 ≤ p < ∞, we denote by Lpw (Rn ) the space of all functions satisfying 1/p Z p p < ∞. |f (x)| w(x) dx kf kLw = Rn

Theorem E ([4]). Suppose that Ω ∈ Lq (S n−1 ), 1 < q < ∞. Then for every q ′ ≤ p < ∞ and w ∈ Ap/q′ , there is a constant C independent of f such that kMΩ (f )kLpw ≤ Ckf kLpw kTΩ (f )kLpw ≤ Ckf kLpw . Theorem F ([2]). Suppose that Ω ∈ Lq (S n−1 ), 1 < q ≤ ∞. Then for every q ′ < p < ∞ and w ∈ Ap/q′ , there is a constant C independent of f such that kµΩ (f )kLpw ≤ Ckf kLpw . Theorem G ([3]). Suppose that Ω ∈ Lq (S n−1 ) with 1 < q ≤ ∞ and b ∈ BM O(Rn ). Then for q ′ < p < ∞ and w ∈ Ap/q′ , there is a constant C > 0 independent of f such that

[b, µΩ ](f ) p ≤ Ckf kLp . w L w

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 A ≤ C. such that C1 ≤ B 5

3. Proof of Theorem 1 First, by using H¨older’s inequality, we can easily see that MΩ f (x) ≤ C · kΩkLq (S n−1 ) Mq′ (f )(x), ′

′

where Mq′ (f )(x) = M (|f |q )(x)1/q . Then for q ′ < p < ∞ and w ∈ Ap/q′ , it follows immediately from Theorem D that

′ 1/q ′

Mq′ (f ) p,κ = M (|f |q ) Lp/q′ ,κ (w) L (w)

′ 1/q ′ ≤ C |f |q Lp/q′ ,κ (w) ≤ Ckf kLp,κ (w) .

Now we consider the case p = q ′ . Fix a ball B = B(x0 , rB ) ⊆ Rn and decompose f = f1 + f2 , where f1 = f χ2B , χ2B denotes the characteristic function of 2B. Since MΩ is a sublinear operator, then we have 1/p Z 1 p |M f (x)| w(x) dx Ω w(B)κ/p B Z  1/p 1 p |M f (x)| w(x) dx ≤ Ω 1 w(B)κ/p B 1/p Z 1 p |M f (x)| w(x) dx + Ω 2 w(B)κ/p B = I1 + I2 . Theorem E and Lemma A imply  1 I1 ≤ C · w(B)κ/p ≤ Ckf kLp,κ (w) ·

Z

|f (x)|p w(x) dx 2B

w(2B)κ/p w(B)κ/p

1/p

(1)

≤ Ckf kLp,κ (w) . We turn to estimate the term I2 . For any given r > 0 and x ∈ B, by H¨older’s inequality and the A1 condition, we thus obtain Z 1 |Ω(y ′ )f2 (x − y)| dy r n |y| 1 such that w ∈ RHr . By using Lemma B, we thus get  (r−1)/r |B| w(B) ≤C . (6) w(2j+1 B) |2j+1 B| Therefore ∞  X 1 (1−κ)(r−1)/pr J2 ≤ Ckf kLp,κ (w) 2jn j=1

≤ Ckf kLp,κ (w) ,

where the last series is convergent since (1 − κ)(r − 1)/pr > 0. Using the estimates for J1 and J2 and taking the supremum over all balls B ⊆ Rn , we complete the proof of Theorem 2. Proof of Theorem 3. As in the proof of Theorem 2, we can write 1/p Z 1 [b, TΩ ]f (x) p w(x) dx w(B)κ/p B 1/p Z 1 [b, TΩ ]f1 (x) p w(x) dx ≤ w(B)κ/p B 1/p Z 1 [b, TΩ ]f2 (x) p w(x) dx + w(B)κ/p B = J1′ + J2′ . By Theorem E and the well-known boundedness criterion for the commutators of linear operators, which was obtained by Alvarez, Bagby, Kurtz and P´erez(see [1]), we see that [b, TΩ ] is bounded on Lpw for all q ′ < p < ∞ and w ∈ Ap/q′ . This together with Lemma A yield 1/p Z 1 p ′ |f (x)| w(x) dx J1 ≤ Ckbk∗ · w(B)κ/p 2B ≤ Ckbk∗ kf kLp,κ (w) ·

w(2B)κ/p w(B)κ/p

≤ Ckbk∗ kf kLp,κ (w) . 9

(7)

We now turn to deal with the term J2′ . For any given x ∈ B, we have Z |Ω((x − y)′ )| [b, TΩ ]f2 (x) ≤ b(x) − bB · |f (y)| dy |x − y|n (2B)c Z |Ω((x − y)′ )| |b(y) − bB ||f (y)| dy + |x − y|n (2B)c = I+II.

In the proof of Theorem 2, for any q ′ < p < ∞, we have already showed I ≤ C|b(x) − bB | · kf kLp,κ (w)

∞ X j=1

1 w(2j+1 B)(1−κ)/p

.

Consequently 1/p Z 1 p I w(x) dx w(B)κ/p B ∞ 1/p Z X 1 1 b(x) − bB p w(x) dx · · ≤ Ckf kLp,κ (w) w(B)κ/p j=1 w(2j+1 B)(1−κ)/p B Z ∞  1 1/p X w(B)(1−κ)/p b(x) − bB p w(x) dx = Ckf kLp,κ (w) · . w(B) B w(2j+1 B)(1−κ)/p j=1

Using the same arguments as that of Theorem 2, we can see that the above summation is bounded by a constant. Hence  1 Z Z 1/p 1/p p 1 p p,κ I w(x) dx ≤Ckf k b(x) − b w(x) dx . B L (w) w(B) B w(B)κ/p B Since w ∈ Ap/q′ , then w ∈ Ap . As before, there exists a number r > 1 such that w ∈ RHr . By the reverse H¨older’s inequality and Theorem C, we get 1/p  1 Z b(x) − bB p w(x) dx w(B) B Z 1/pr Z pr′ 1/pr′  1 r w(x) dx b(x) − b dx ≤C · B w(B)1/p B (8) B  1 Z pr′ 1/pr′ b(x) − bB dx ≤C · |B| B ≤ Ckbk∗ .

10

So we have  1 w(B)κ/p

Z

Ip w(x) dx

B

1/p

≤ Ckbk∗ kf kLp,κ (w) .

(9)

On the other hand, it follows from H¨older’s inequality and (3) that Z ∞  X ′ 1/q′ ′ 1 b(y) − bB q f (y) q dy II ≤ C |2j+1 B| 2j+1 B j=1 Z ∞  X ′ ′ 1/q′ 1 b(y) − b2j+1 B q f (y) q dy ≤C |2j+1 B| 2j+1 B j=1 Z ∞ X  1 ′ 1/q′ b2j+1 B − bB · f (y) q dy +C |2j+1 B| 2j+1 B j=1

= III+IV.

Set s = p/q ′ > 1. Then by using H¨older’s inequality, we thus obtain Z ′ ′ 1/q′ b(y) − b2j+1 B q f (y) q dy (10) 2j+1 B 1/p 1/q′ s′  Z Z q′ s′ −s′ /s f (y) p w(y) dy w (y) dy b(y) − b2j+1 B ≤ 2j+1 B 2j+1 B 1/q′ s′ Z ′ ′ b(y) − b2j+1 B q s w−s′ /s (y) dy . ≤ Ckf kLp,κ (w) · w(2j+1 B)κ/p 2j+1 B

−s′ /s

1−s′

(y). Then we have v ∈ As′ because w ∈ As (see Let v(y) = w (y) = w [6]), which implies v ∈ Aq′ s′ . Following along the same lines as that of (8), we can get Z  1/q′ s′ ′ ′ 1 b(y) − b2j+1 B q s v(y) dy ≤ Ckbk∗ . (11) v(2j+1 B) 2j+1 B Substituting the above inequality (11) into (10), we thus have Z ′ 1/q′ ′ b(y) − b2j+1 B q f (y) q dy 2j+1 B

′ ′

≤Ckbk∗ kf kLp,κ (w) · w(2j+1 B)κ/p · v(2j+1 B)1/q s

(12)

′

≤Ckbk∗ kf kLp,κ (w) · |2j+1 B|1/q · w(2j+1 B)(κ−1)/p . Hence  1 w(B)κ/p

Z

IIIp w(x) dx B

1/p

≤ Ckbk∗ kf kLp,κ (w)

∞ X j=1

≤ Ckbk∗ kf kLp,κ (w) . 11

w(B)(1−κ)/p w(2j+1 B)(1−κ)/p (13)

Now let’s deal with the last term IV. Since b ∈ BM O(Rn ), then a simple computation shows that |b2j+1 B − bB | ≤ C · jkbk∗ .

(14)

It follows immediately from the inequalities (5) and (14) that Z ∞  1 X q′ 1/q′ j· IV ≤ Ckbk∗ f (y) dy |2j+1 B| 2j+1 B j=1

≤ Ckbk∗ kf kLp,κ (w)

∞ X

j · w(2j+1 B)(κ−1)/p .

j=1

Therefore, by the estimate (6), we obtain ∞ Z 1/p X 1 w(B)(1−κ)/p p p,κ (w) IV w(x) dx j · ≤ Ckbk kf k ∗ L w(B)κ/p B w(2j+1 B)(1−κ)/p j=1 ≤ Ckbk∗ kf kLp,κ (w)

∞ X j 2jnθ j=1

≤ Ckbk∗ kf kLp,κ (w) ,

(15)

where w ∈ RHr and θ = (1 − κ)(r − 1)/pr. Summarizing the estimates (13) and (15) derived above, we can get 1/p Z 1 p II w(x) dx ≤ Ckbk∗ kf kLp,κ (w) . (16) w(B)κ/p B

Combining the inequalities (7), (9) with the inequality (16) and taking the supremum over all balls B ⊆ Rn , we conclude the proof of Theorem 3.

5. Proofs of Theorems 4 and 5 Proof of Theorem 4. Fix a ball B = B(x0 , rB ) ⊆ Rn . Let f = f1 + f2 , where f1 = f χ2B . Then we have Z 1/p 1 p |µ f (x)| w(x) dx Ω w(B)κ/p B Z  1/p 1 p |µ f (x)| w(x) dx ≤ Ω 1 w(B)κ/p B Z 1/p 1 p |µ f (x)| w(x) dx + Ω 2 w(B)κ/p B = K1 + K2 . 12

Theorem F and Lemma A imply  1 K1 ≤ C · w(B)κ/p ≤ Ckf kLp,κ (w) ·

Z

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

2B

w(2B)κ/p w(B)κ/p

1/p

≤ Ckf kLp,κ (w) . To estimate K2 , observe that when x ∈ B and y ∈ 2j+1 B\2j B(j ≥ 1), then t ≥ |x − y| ≥ |y − x0 | − |x − x0 | ≥ 2j−1 rB . Therefore ! 2 dt 1/2 Ω(x − y) f (y) dy 3 n−1 t (2B)c ∩{y:|x−y|≤t} |x − y| 0 Z Z ∞   ∞ X |Ω(x − y)| dt 1/2 ≤ |f (y)| dy · n−1 3 2j+1 B\2j B |x − y| 2j−1 rB t j=1 Z ∞ X |Ω(x − y)| 1 |f (y)| dy. · ≤C n−1 j+1 1/n |2 B| 2j+1 B\2j B |x − y|

µΩ (f2 )(x) =

Z

∞ Z



j=1

L∞ (S n−1 ),

When Ω ∈ then by assumption, we have w ∈ Ap , 1 < p < ∞. It follows from the H¨older’s inequality and the Ap condition that Z ∞ X 1 1 µΩ (f2 )(x) ≤ CkΩkL∞ (S n−1 ) |f (y)| dy · |2j+1 B|1/n |2j+1 B|(n−1)/n 2j+1 B j=1 Z ∞ 1/p X 1  p ≤ CkΩkL∞ (S n−1 ) |f (y)| w(y) dy |2j+1 B| 2j+1 B j=1 Z 1/p′ ′ (17) × w(y)−p /p dy 2j+1 B

≤ CkΩkL∞ (S n−1 ) kf kLp,κ (w)

∞ X

w(2j+1 B)(κ−1)/p .

j=1

When Ω ∈ Lq (S n−1 ), 1 < q < ∞, by the inequalities (3) and (5), we get Z ∞  1/q′ X 1 q′ µΩ (f2 )(x) ≤ CkΩkLq (S n−1 ) |f (y)| dy |2j+1 B| 2j+1 B j=1 (18) ∞ X j+1 (κ−1)/p ≤ CkΩkLq (S n−1 ) kf kLp,κ (w) w(2 B) . j=1

13

Hence, for 1 < q ≤ ∞, q ′ < p < ∞, by the estimates (17) and (18), we have K2 ≤ Ckf kLp,κ (w)

∞  X j=1

w(B) w(2j+1 B)

(1−κ)/p

≤ Ckf kLp,κ (w) .

Using the above estimates for K1 and K2 and taking the supremum over all balls B ⊆ Rn , we get our desired result. Proof of Theorem 5. As before, we can write 1/p Z 1 [b, µΩ ]f (x) p w(x) dx w(B)κ/p B 1/p Z 1 [b, µΩ ]f1 (x) p w(x) dx ≤ w(B)κ/p B 1/p Z 1 [b, µΩ ]f2 (x) p w(x) dx + w(B)κ/p B = K1′ + K2′ . Theorem G and Lemma A yield w(2B)κ/p w(B)κ/p ≤ Ckbk∗ kf kLp,κ (w) .

K1′ ≤ Ckbk∗ kf kLp,κ (w)

Finally, let us deal with the term K2′ . For any fixed x ∈ B, we have [b, µΩ ]f2 (x) ! Z ∞ Z 2 dt 1/2 Ω(x − y) ≤ |b(x) − bB | f (y) dy 3 n−1 t 0 (2B)c ∩{y:|x−y|≤t} |x − y| ! Z ∞ Z 2 dt 1/2 Ω(x − y) + [b(y) − bB ]f (y) dy 3 n−1 |x − y| t c (2B) ∩{y:|x−y|≤t} 0 = I+II.

In the proof of Theorem 4, for any q ′ < p < ∞, we have already proved I ≤ C|b(x) − bB | · kf kLp,κ (w)

∞ X j=1

14

w(2j+1 B)(κ−1)/p .

Following the same lines as in the proof of Theorem 3, we obtain 1/p Z 1 p I w(x) dx ≤ Ckbk∗ kf kLp,κ (w) . w(B)κ/p B

On the other hand, we note that when x ∈ B and y ∈ 2j+1 B\2j B(j ≥ 1), then we have t ≥ 2j−1 rB . Consequently Z ∞ X |Ω(x − y)| 1 · |b(y) − bB ||f (y)| dy II ≤ C n−1 j+1 1/n |2 B| 2j+1 B\2j B |x − y| j=1 Z ∞ X |Ω(x − y)| 1 ≤C · |b(y) − b2j+1 B ||f (y)| dy j+1 B|1/n |x − y|n−1 j+1 j |2 2 B\2 B j=1 Z ∞ X |Ω(x − y)| |b2j+1 B − bB | · +C |f (y)| dy n−1 j+1 1/n |2 B| 2j+1 B\2j B |x − y| j=1

= III+IV.

When Ω ∈ L∞ (S n−1 ), then it follows from H¨older’s inequality that Z ∞ X 1 |b(y) − b2j+1 B ||f (y)| dy III ≤ CkΩkL∞ (S n−1 ) |2j+1 B| 2j+1 B j=1 Z ∞ 1/p′ X ′ 1  b(y) − b2j+1 B p w−p′ /p (y) dy ≤ CkΩkL∞ (S n−1 ) |2j+1 B| 2j+1 B j=1 1/p Z f (y) p w(y) dy × 2j+1 B

≤ CkΩkL∞ (S n−1 ) kf kLp,κ (w)

∞ X j=1

×

Z

2j+1 B ′

1

|2j+1 B|

· w(2j+1 B)κ/p

1/p′ ′ b(y) − b2j+1 B p w−p′ /p (y) dy . ′

Set u(y) = w−p /p (y) = w1−p (y). In this case, since w ∈ Ap , then we have u ∈ Ap′ , it follows from the inequality (8) and the Ap condition that III ≤ CkΩkL∞ (S n−1 ) kbk∗ kf kLp,κ (w) ·

∞ X j=1

≤ CkΩkL∞ (S n−1 ) kbk∗ kf kLp,κ (w) ·

∞ X j=1

15

1 ′ w(2j+1 B)κ/p · u(2j+1 B)1/p |2j+1 B| w(2j+1 B)(κ−1)/p .

(19)

When Ω ∈ Lq (S n−1 ), then by using H¨older’s inequality, the inequalities (3) and (12), we can deduce III ≤ CkΩkLq (S n−1 )

∞ X j=1

 1 ′ |2j+1 B|1/q

≤ CkΩkLq (S n−1 ) kbk∗ kf kLp,κ (w) ·

Z

2j+1 B

∞ X

′ 1/q′ ′ b(y) − b2j+1 B q f (y) q dy

w(2j+1 B)(κ−1)/p .

(20)

j=1

Hence, for 1 < q ≤ ∞, q ′ < p < ∞, by the estimates (19) and (20), we get  1 w(B)κ/p

Z

IIIp w(x) dx B

1/p

≤ Ckbk∗ kf kLp,κ (w)

∞ X j=1

w(B)(1−κ)/p w(2j+1 B)(1−κ)/p

≤ Ckbk∗ kf kLp,κ (w) .

Again, in Theorem 4, we have already obtained the following inequality Z |Ω(x − y)| 1 · |f (y)| dy ≤ Ckf kLp,κ (w) · w(2j+1 B)(κ−1)/p . n−1 j+1 1/n |2 B| 2j+1 B\2j B |x − y| From (14), it follows immediately that IV ≤ Ckbk∗ kf kLp,κ (w)

∞ X

j · w(2j+1 B)(κ−1)/p .

j=1

The rest of the proof is exactly the same as that of (15), we finally obtain 1/p Z 1 p IV w(x) dx ≤ Ckbk∗ kf kLp,κ (w) . w(B)κ/p B Therefore, by combining the above estimates and taking the supremum over all balls B ⊆ Rn , we conclude the proof of Theorem 5.

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[3] Y. Ding, S. Lu, K. Yabuta, On commutators of Marcinkiewicz integrals with rough kernel, J. Math. Anal. Appl, 275(2002), 60–68. [4] J. Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc, 336(1993), 869–880. [5] J. Duoandikoetxea, Fourier Analysis, American Mathematical Society, Providence, Rhode Island, 2000. [6] J. Garcia-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. [7] R. F. Gundy and R. L. Wheeden, Weighted integral inequalities for nontangential maximal function, Lusin area integral, and Walsh-Paley series, Studia Math, 49(1974), 107–124. [8] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math, 14(1961), 415-426. [9] Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr, 282(2009), 219–231. [10] C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc, 43(1938), 126–166. [11] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc, 165(1972), 207–226.

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