boundedness of sublinear operators with rough kernels on weighted morrey spaces

Journal of Function Spaces and ApplicationsVolume 2013, Article ID 784983, 9 pages http://dx.doi.org/10.1155/2013/784983 Research Article Boundedness of Sublinear Operators with Rough Ke

Journal of Function Spaces and Applications

Volume 2013, Article ID 784983, 9 pages

http://dx.doi.org/10.1155/2013/784983

Research Article

Boundedness of Sublinear Operators with Rough Kernels on

Weighted Morrey Spaces

Shaoguang Shi and Zunwei Fu

Department of Mathematics, Linyi University, Linyi 276005, China

Correspondence should be addressed to Zunwei Fu; zwfu@mail.bnu.edu.cn

Received 18 January 2013; Accepted 11 March 2013

Academic Editor: Dashan Fan

Copyright © 2013 S Shi and Z Fu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited The aim of this paper is to get the boundedness of a class of sublinear operators with rough kernels on weighted Morrey spaces under generic size conditions, which are satisfied by most of the operators in classical harmonic analysis Applications to the corresponding commutators formed by certain operators and BMO functions are also obtained

1 Introduction and Main Results

Given a functionΩ over the unit sphere 𝑆𝑛−1ofR𝑛 (𝑛 ≥ 2)

equipped with the normalized Lebesgue measure𝑑𝜎 and 𝑥󸀠=

𝑥/|𝑥|, a Calder´on-Zygmund singular integral operator with

rough kernel was given by

𝑇Ω𝑓 (𝑥) = p.v ∫

R 𝑛

Ω (𝑥 − 𝑦)

󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨𝑛 𝑓 (𝑦) 𝑑𝑦 (1) and a related maximal operator

𝑀Ω𝑓 (𝑥) = sup

𝑟>0

1

𝑟𝑛∫𝐵(𝑥,𝑟)Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦, (2) whereΩ is homogeneous of degree zero and satisfies

Ω ∈ 𝐿𝑟(𝑆𝑛−1) , 1 < 𝑟 ≤ ∞ (3)

∫

𝑆 𝑛−1Ω (𝑥󸀠) 𝑑𝑥󸀠= 0 (4) WhenΩ is a smooth kernel and 𝑇Ω a standard

Calder´on-Zygmund singular integral operator which has been fully

studied by many papers, a classical survey work; see, for

example, [1]

For simplicity of notation, Ω is always homogeneous

of degree zero and satisfies (3) and (4) throughout this

paper if there are no special instructions Here and in what

follows, for 𝑥0 ∈ R𝑛, 𝑟 > 0, and 𝜆 > 0, 𝐵 = 𝐵(𝑥0, 𝑟)

denotes the ball centered at 𝑥0 with radius 𝑟 and 𝜆𝐵 = 𝐵(𝑥0, 𝜆𝑟) When Ω satisfies some size conditions, the kernel

of the operator 𝑇Ω has no regularity, and so the operator

𝑇Ωis called rough singular integral operator In recent years,

a variety of operators related to the singular integrals for Calder´on-Zygmund, but lacking the smoothness required in the classical theory, have been studied Duoandikoetxea [2] studied the norm inequalities for𝑇Ω in homogeneous case

on weighted𝐿𝑝 (1 < 𝑝 < ∞) spaces For more corresponding works, we refer the reader to [3–8] and the references therein

In [9], Hu et al considered some more general sublinear operators with rough kernels which satisfy

󵄨󵄨󵄨󵄨TΩ𝑓 (𝑥)󵄨󵄨󵄨󵄨

≤ 𝐶 ∫

R 𝑛

󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓(𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨𝑛 𝑑𝑦, 𝑥 ∉ supp 𝑓 (5) for𝑓 ∈ 𝐿1(R𝑛) with compact support Condition (5) was first introduced by Soria and Weiss [10] Inequality (5) is satisfied

by many operators with rough kernels in classical harmonic analysis, such as 𝑇Ω (see [11]) and the oscillatory singular integral operator

𝑇Ω𝑓 (𝑥)

= p.v ∫

R 𝑛𝑒𝑖𝑃(𝑥,𝑦)Ω (𝑥 − 𝑦)

󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨𝑛 𝑓 (𝑦) 𝑑𝑦, 𝑥 ∉ supp 𝑓,

(6)

where the phase is a polynomial The boundedness of𝑇Ωon

weighted𝐿𝑝(R𝑛) (1 ≤ 𝑝 < ∞) spaces was fully studied by

Ojanen in his doctoral dissertation [12]

Let𝐷𝑘 = {𝑥 ∈ R𝑛 : |𝑥| ≤ 2𝑘} and let 𝐴𝑘 = 𝐷𝑘 \ 𝐷𝑘−1

for𝑘 ∈ 𝑍 Throughout this paper, we will denote by 𝜒𝐸the

characteristic function of the set𝐸 Inspired by the works of

[6, 13], in this paper, we consider some sublinear operators

under some size conditions (the following (7) and (8)) which

are more general than (5):

󵄨󵄨󵄨󵄨TΩ𝑓 (𝑥)󵄨󵄨󵄨󵄨 ≤ 𝐶|𝑥|−𝑛∫

R 𝑛󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓(𝑦)󵄨󵄨󵄨󵄨𝑑𝑦, (7) when supp𝑓 ⊆ 𝐴𝑘and|𝑥| ≥ 2𝑘+1with𝑘 ∈ Z and

󵄨󵄨󵄨󵄨TΩ𝑓 (𝑥)󵄨󵄨󵄨󵄨 ≤ 𝐶2−𝑘𝑛∫

R 𝑛󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓(𝑦)󵄨󵄨󵄨󵄨𝑑𝑦, (8) when supp𝑓 ⊆ 𝐴𝑘and|𝑥| ≤ 2𝑘−1with𝑘 ∈ Z, respectively

It is worth pointing out that𝑀Ωsatisfies conditions (7) and

(8) Also, condition (5) implies the size conditions (7) and (8)

since|𝑥 − 𝑦| > |𝑥|/2 when |𝑥| ≥ 2𝑘+1and supp𝑓 ⊆ 𝐴𝑘while

supp𝑓 ⊆ 𝐴𝑘and|𝑥| ≤ 2𝑘−1imply|𝑥 − 𝑦| > |𝑦|/2

The topic of this paper is intended as an attempt to study

the boundedness of sublinear operators with rough kernels

which satisfy (7) and (8) on weighted Morrey spaces We first

recall some definitions and notations for weighted spaces The

Muckenhoupt classes𝐴𝑝and𝐴(𝑝,𝑞)[14] contain the functions

𝑤 which satisfy

𝐴𝑝: sup

𝐵 ( 1

|𝐵|∫𝐵𝑤 (𝑥) 𝑑𝑥)

× ( 1

|𝐵|∫𝐵𝑤(𝑥)1−𝑝󸀠𝑑𝑥)𝑝−1≤ 𝐶, 1 < 𝑝 < ∞,

𝐴(𝑝,𝑞): sup

𝐵 ( 1

|𝐵|∫𝐵𝑤(𝑥)𝑞𝑑𝑥)1/𝑞

× ( 1

|𝐵|∫𝐵𝑤(𝑥)−𝑝󸀠𝑑𝑥)1/𝑝

󸀠

≤ 𝐶, 1 < 𝑝, 𝑞 < ∞,

(9) respectively, where1/𝑝 + 1/𝑝󸀠 = 1 For 𝑝 = 1, the 𝐴1 and

𝐴(1,𝑞) (1 < 𝑞 < ∞) weights are defined by

𝑀𝑤 (𝑥) ≤ 𝐶𝑤 (𝑥) ,

𝐴(1,𝑞) : sup

𝐵 (|𝐵|1 ∫

𝐵𝑤(𝑥)𝑞𝑑𝑥)1/𝑞

× ( ess sup

𝐵

1

𝑤 (𝑥)) ≤ 𝐶,

(10)

respectively Here ess sup and the following essinf are the

abbreviations of essential supremum and essential infimum,

respectively Clearly,𝑤 ∈ 𝐴1if and only if there is a constant

𝐶 > 0 such that

1

|𝐵|∫𝐵𝑤 (𝑥) 𝑑𝑥 ≤ Cess inf

𝐵 𝑤 (𝑥) (11)

In [15], Komori and Shirai introduced a weighted Morrey space, which is a natural generalization of weighted Lebesgue space, and investigated the boundedness of classical operators

in harmonic analysis Let1 ≤ 𝑝 < ∞, 0 < 𝜆 < 1 and let 𝑤 be

a weight function Then the weighted Morrey space𝑀𝑝,𝜆(𝑤)

is defined by

𝑀𝑝,𝜆(𝑤) =

{ { {

𝑓 : 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 𝑝,𝜆 (𝑤)

= sup

𝐵 ( 1 𝑤(𝐵)𝜆∫

𝐵󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥)

1/𝑝

< ∞

} } }

, (12) where𝑤(𝐵) = ∫𝐵𝑤(𝑥)𝑑𝑥 For 𝑤 ∈ 𝐴𝑝 (1 ≤ 𝑝 < ∞), if

𝜆 = 0, then 𝑀𝑝,0(𝑤) = 𝐿𝑝(𝑤) while 𝜆 = 1 implies 𝑀𝑝,1(𝑤) =

𝐿∞(𝑤)

Now, we formulate our major results of this paper as follows

Theorem 1 Let 0 < 𝜆 < 1, 1 < 𝑟 ≤ ∞, and 𝑟󸀠 ≤ 𝑝 < ∞

and let a sublinear operatorTΩsatisfy (7) and (8) IfTΩis bounded on𝐿𝑝(𝑤) with 𝑤 ∈ 𝐴𝑝/𝑟󸀠, thenTΩis bounded on

𝑀𝑝,𝜆(𝑤).

When𝑝 = 1, we have the following theorem

Theorem 2 Let 1 < 𝑟 < ∞ and 1/𝑟+𝜆 < 1 and let TΩsatisfy

(7) and (8) Then ifTΩis bounded from𝐿1(𝑤) to 𝐿1,∞(𝑤) with

𝑤 ∈ 𝐴1, there exists a constant 𝐶 > 0 such that for all 𝜇 > 0

and all balls 𝐵,

𝑤 ({𝑥 ∈ 𝐵 : TΩ𝑓 (𝑥) > 𝜇})

≤ 𝐶𝜇−1󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 1,𝜆 (𝑤)𝑤(𝐵)𝜆 (13)

In the fractional case, we need to consider a weighted Morrey space with two weights which is also introduced by Komori and Shirai in [15] Let1 ≤ 𝑝 < ∞, 0 < 𝜆 < 1 For two weights𝑤1and𝑤2,

𝑀𝑝,𝜆(𝑤1, 𝑤2) = {𝑓 : 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀𝑝,𝑘(𝑤1,𝑤2)

= sup

𝐵 ( 1

𝑤2(𝐵)𝜆∫𝐵󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤1(𝑥) 𝑑𝑥)

1/𝑝

< ∞}

(14)

If𝑤1 = 𝑤2 = 𝑤, we write 𝑀𝑝,𝜆(𝑤1, 𝑤1) = 𝑀𝑝,𝜆(𝑤2, 𝑤2) =

𝑀𝑝,𝜆(𝑤)

We can get similar results for fractional integrals follow-ing the line of Theorems1and2

Theorem 3 Let 0 < 𝛼 < 𝑛, 1 ≤ 𝑟 < ∞, and 0 < 𝜆 <

1 Suppose that a sublinear operator T𝛼,Ω satisfies the size

conditions

󵄨󵄨󵄨󵄨T𝛼,Ω𝑓 (𝑥)󵄨󵄨󵄨󵄨

≤ 𝐶|𝑥|−(𝑛−𝛼)∫

R 𝑛󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓(𝑦)󵄨󵄨󵄨󵄨𝑑𝑦 (15)

when supp𝑓 ⊆ 𝐴𝑘and|𝑥| ≥ 2𝑘+1with 𝑘 ∈ 𝑍 and

󵄨󵄨󵄨󵄨T𝛼,Ω𝑓 (𝑥)󵄨󵄨󵄨󵄨

≤ 𝐶2−𝑘(𝑛−𝛼)∫

R 𝑛󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓(𝑦)󵄨󵄨󵄨󵄨𝑑𝑦 (16)

when supp𝑓 ⊆ 𝐴𝑘and|𝑥| ≤ 2𝑘−1with 𝑘 ∈ 𝑍 Then one has

the following.

(a) If T𝛼,Ω maps 𝐿𝑝(𝑤𝑝) into 𝐿𝑞(𝑤𝑞) with 𝑤 ∈

𝐴(𝑝/𝑟󸀠 ,𝑞), thenT𝛼,Ωis bounded from𝑀𝑝,𝜆(𝑤𝑝, 𝑤𝑞) to

𝑀𝑞,𝑞𝜆/𝑝(𝑤𝑞), where 𝑟󸀠 ≤ 𝑝 < 𝑛/𝛼, 1/𝑞 = 1/𝑝 − 𝛼/𝑛

and 𝑝 ≤ 𝑞 ≤ ∞.

(b) IfT𝛼,Ωis bounded from𝐿1(𝑤) to 𝐿𝑞,∞(𝑤𝑞) with 𝑤 ∈

𝐴(1,𝑞) and 1/𝑟 + 𝜆 < 1, then there exists a constant

𝐶 > 0 such that for all 𝜇 > 0 and all balls 𝐵,

𝑤({𝑥 ∈ 𝐵 : T𝛼,Ω𝑓 (𝑥) > 𝜇})1/𝑞

≤ 𝐶𝜇−1󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀1,𝜆(𝑤,𝑤 𝑞 )𝑤(𝐵)𝜆, (17)

where 1 < 𝑞 < ∞.

We emphasize that (15) and (16) are weaker conditions

than the following condition:

󵄨󵄨󵄨󵄨T𝛼,Ω𝑓 (𝑥)󵄨󵄨󵄨󵄨

≤ 𝐶 ∫

R 𝑛

󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓(𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨𝑛−𝛼 𝑑𝑦, 0 < 𝛼 < 𝑛 (18) for any integral function𝑓 with compact support Condition

(18) is satisfied by most fractional integral operators with

rough kernels, such as the fractional integral operators of

Muckenhoupt and Wheeden [16]:

T𝛼,Ω𝑓 (𝑥)

= ∫

R 𝑛

Ω (𝑥 − 𝑦) 𝑓 (𝑦)

󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨𝑛−𝛼 𝑑𝑦, 0 < 𝛼 < 𝑛 (19) For some mapping properties ofT𝛼,Ωon various kinds of

function spaces, see [17–19] and the references therein

We end this section with the outline of this paper

Section 2contains the proofs of Theorems1and3; this part

is partly motivated by the methods in [20] dealing with the

case of the Lebesgue measure InSection 3, we extend the

corresponding results to commutators of certain sublinear

operators

2 Boundedness of Sublinear Operators

Proofs of Theorems1and 3depend heavily on some prop-erties of𝐴𝑝 weights, which can be found in any papers or any books dealing with weighted boundedness for operators

in harmonic analysis, such as [1] For the convenience of the reader we collect some relevant properties of𝐴𝑝weights without proofs, thus making our exposition self-contained

Lemma 4 Let 1 ≤ 𝑝 < ∞ and 𝑤 ∈ 𝐴𝑝 Then the following statements are true.

(a) There exists a constant 𝐶 such that

𝑤 (2𝐵) ≤ 𝐶𝑤 (𝐵) , (20)

where 𝑤 satisfies this condition; one says 𝑤 satisfies the

doubling condition.

(b) There exists a constant 𝐶 > 1 such that

𝑤 (2𝐵) ≥ 𝐶𝑤 (𝐵) , (21)

where 𝑤 satisfies this condition; one says 𝑤 satisfies the

reverse doubling condition.

(c) There exist two constants 𝐶 and 𝑟 > 1 such that the

following reverse H¨older inequality holds for every ball

𝐵 ⊂ R𝑛:

(|𝐵|1 ∫

𝐵𝑤(𝑥)𝑟𝑑𝑥)1/𝑟≤ 𝐶 (|𝐵|1 ∫

𝐵𝑤 (𝑥) 𝑑𝑥) (22)

(d) For all 𝜆 > 1, one has

𝑤 (𝜆𝐵) ≤ 𝐶𝜆𝑛𝑝𝑤 (𝐵) (23)

(e) There exist two constants 𝐶 and 𝛿 > 0 such that for any

measurable set𝑄 ⊂ 𝐵

𝑤 (𝑄)

𝑤 (𝐵) ≤ 𝐶(||𝐵|𝑄|)

𝛿

if 𝑤 satisfies (24); one says𝑤 ∈ 𝐴∞.

(f) For all 𝑝 < 𝑞 < ∞, one has

𝐴∞= ∪𝑝𝐴𝑝, 𝐴𝑝 ⊂ 𝐴𝑞 (25)

The following lemma about the rough kernelΩ is essential

to our proofs One can find its proof in [21]

Lemma 5 Let Ω ∈ 𝐿𝑟(𝑆𝑛−1) with 1 ≤ 𝑟 < ∞ Then the

following statements are true.

(a) If𝑥 ∈ 𝐴𝑘and 𝑗 ≥ 𝑘 +1, then ∫𝐴

𝑗|Ω(𝑥−𝑦)|𝑟𝑑𝑦 ≤ 𝐶2𝑗𝑛.

(b) If𝑦 ∈ 𝐴𝑘and 𝑘 ≥ 𝑗 + 1, then ∫𝐴

𝑗|Ω(𝑥 − 𝑦)|𝑟𝑑𝑥 ≤ 𝐶2𝑘(𝑛−1)+𝑗.

Proof of Theorem 1 Let1 < 𝑟󸀠 ≤ 𝑝 < ∞, 𝑤 ∈ 𝐴𝑝/𝑟󸀠, and

0 < 𝜆 < 1 Our task is to show

1

𝑤(𝐵)𝜆∫𝐵󵄨󵄨󵄨󵄨TΩ𝑓 (𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥 ≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤) (26)

For a fixed ball𝐵 = 𝐵(𝑥0, 𝑟), there is no loss of generality in

assuming𝑟 = 1 We decompose 𝑓 = 𝑓𝜒2𝐵+ 𝑓𝜒(2𝐵)𝑐 := 𝑓1+ 𝑓2

SinceTΩis a sublinear operator, so we get

1

𝑤(𝐵)𝜆∫𝐵󵄨󵄨󵄨󵄨TΩ𝑓 (𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

𝑤(𝐵)𝜆∫

𝐵󵄨󵄨󵄨󵄨TΩ𝑓1(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

𝑤(𝐵)𝜆∫

𝐵󵄨󵄨󵄨󵄨TΩ𝑓2(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥 := 𝐼 + 𝐼𝐼

(27)

By the assumption onTΩand (25), we can obtain

𝐼 ≤ 𝐶

𝑤(𝐵)𝜆∫R 𝑛󵄨󵄨󵄨󵄨TΩ𝑓1(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

𝑤(𝐵)𝜆∫

2𝐵󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀𝑝,𝜆(𝑤)

(28)

For the term𝐼𝐼, by (8) we have

𝐼𝐼 ≤ 𝐶

𝑤(𝐵)𝜆∫

𝐵

󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨

󵄨󵄨

∞

∑

𝑘=1

2−𝑘𝑛TΩ,𝑘𝑓 (𝑥)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨

󵄨󵄨

𝑝

𝑤 (𝑥) 𝑑𝑥, (29)

where

TΩ,𝑘𝑓 (𝑥) = ∫

𝐴 𝑘+1󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓(𝑦)󵄨󵄨󵄨󵄨𝑑𝑦 (30)

We distinguish two cases according to the size of𝑝 and 𝑟

to get the estimates forTΩ,𝑘

Case 1 (𝑝 > 𝑟󸀠) In this case,𝑤 ∈ 𝐴𝑝/𝑟󸀠implies that

∫

𝐵𝑤−𝑟󸀠/(𝑝−𝑟󸀠)𝑑𝑦 ≤ |𝐵|𝑝/(𝑝−𝑟 󸀠 )

𝑤(𝐵)𝑟󸀠/(𝑝−𝑟󸀠). (31)

By (31), H¨older’s inequality, andLemma 5, we have

TΩ,𝑘𝑓 (𝑥) ≤ 𝐶(∫

𝐴k+1󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)󵄨󵄨󵄨󵄨𝑟𝑑𝑦)1/𝑟

× (∫

2 𝑘+1 𝐵󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑟 󸀠

𝑑𝑦)1/𝑟

󸀠

≤ 𝐶2(𝑘+1)𝑛/𝑟

× (∫

2 𝑘+1 𝐵󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑟 󸀠

𝑤(𝑦)𝑟󸀠/𝑝𝑤(𝑦)−𝑟󸀠/𝑝𝑑𝑦)1/𝑟

󸀠

≤ 𝐶2(𝑘+1)𝑛/𝑟(∫

2 𝑘+1 𝐵󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑦) 𝑑𝑦)1/𝑝

× (∫

2 𝑘+1 𝐵𝑤(𝑦)−𝑟󸀠/(𝑝−𝑟󸀠)𝑑𝑦)(𝑝−𝑟

󸀠 )/(𝑝𝑟 󸀠 )

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 𝑝,𝜆 (𝑤)2(𝑘+1)𝑛/𝑟 󵄨󵄨󵄨󵄨󵄨2𝑘+1

𝐵󵄨󵄨󵄨󵄨󵄨1/𝑟󸀠 𝑤(2𝑘+1𝐵)(1−𝜆)/𝑝

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 𝑝,𝜆 (𝑤) 2(𝑘+1)𝑛

𝑤(2𝑘+1𝐵)(1−𝜆)/𝑝.

(32)

Case 2 (𝑝 = 𝑟󸀠) In this case,𝑤 ∈ 𝐴1implies that

(ess inf

𝑥∈2 𝑘+1 𝐵𝑤 (𝑥))−1≤ 󵄨󵄨󵄨󵄨󵄨2𝑘+1

𝐵󵄨󵄨󵄨󵄨󵄨

𝑤 (2𝑘+1𝐵), (33) which in combination with the H¨older inequality and

Lemma 5yields that

TΩ,𝑘𝑓 (𝑥) ≤ 𝐶2(𝑘+1)𝑛/𝑟

× (∫

2 𝑘+1 𝐵󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑦) 𝑤(𝑦)−1𝑑𝑦)1/𝑝

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 𝑝,𝜆 (𝑤) 2(𝑘+1)𝑛

𝑤(2𝑘+1𝐵)(1−𝜆)/𝑝.

(34)

Substituting (32) and (34) into (29), we can assert that

𝐼𝐼 ≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤)(∑∞

𝑘=1

𝑤(𝐵)(1−𝜆)/𝑝

𝑤(2𝑘+1𝐵)(1−𝜆)/𝑝)

𝑝

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀𝑝,𝜆(𝑤),

(35)

where we have used (21) in the last inequality Combining (28) and (29), we obtain the proof ofTheorem 1

Proof of Theorem 2 The task is now to show the following

inequality:

sup

𝜇>0

𝜇 𝑤(𝐵)𝜆𝑤 ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨󵄨TΩ𝑓 (𝑥)󵄨󵄨󵄨󵄨 > 𝜇})

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤)

(36)

In order to get this inequality, it will be necessary to

decom-pose𝑓 = 𝑓𝜒2𝐵+ 𝑓𝜒(2𝐵)𝑐 := 𝑓1+ 𝑓2with𝐵 as inTheorem 1

SinceTΩis a sublinear operator, we can rewrite

𝑤 ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨󵄨TΩ𝑓 (𝑥)󵄨󵄨󵄨󵄨 > 𝜇})

≤ 𝑤 ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨󵄨TΩ𝑓1(𝑥)󵄨󵄨󵄨󵄨 > 𝜇2})

+ 𝑤 ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨󵄨TΩ𝑓2(𝑥)󵄨󵄨󵄨󵄨 >𝜇2})

:= 𝐽 + 𝐽𝐽

(37)

An application of (20) and the weighted weak type

estimates forTΩyield that

𝐽 ≤ 𝐶𝜇−1󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 1,𝜆(𝑤) 𝑤(𝐵)𝜇 (38)

To estimate the term𝐽𝐽, we note that

𝐽𝐽 ≤ 𝐶𝜇∫

{𝑥∈𝐵:|T Ω 𝑓(𝑥)|>𝜇/2}

×󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨

󵄨󵄨

∞

∑

𝑘=1

2−𝑘𝑛TΩ,𝑘𝑓 (𝑥)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨

󵄨󵄨𝑤 (𝑥) 𝑑𝑥.

(39)

By (22), (33), and the H¨older inequality, we can estimate𝐽𝐽 as

𝐽𝐽 ≤ 𝐶

𝜇

∞

∑

𝑘=1

2−𝑘𝑛

× ∫

2 𝑘+1 𝐵∫

𝐵󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)󵄨󵄨󵄨󵄨𝑤(𝑥)𝑑𝑥󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑑𝑦

≤ 𝐶𝑤 (B)

𝜇

∞

∑

𝑘=1

2−𝑘𝑛+𝑘(𝑛−1)/𝑟

× ∫

2 𝑘+1 𝐵󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑤(𝑦)𝑤(𝑦)−1𝑑𝑦

≤ 𝐶𝑤 (𝐵)

𝜇 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 1,𝜆(𝑤)

×∑∞

𝑘=1

𝑤(2𝑘+1𝐵)𝜆(ess inf

𝑥∈2 𝑘+1 𝐵𝑤)−1

≤ 𝐶

𝜇󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 1,𝜆(𝑤)

×∑∞

𝑘=1

2𝑘𝑛/𝑟−𝑘𝑛(1−𝜆)𝑤(𝐵)𝜆

≤ 𝐶𝜇󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 1,𝜆(𝑤) 𝑤(𝐵)𝜆

(40)

Combining these inequalities for𝐽 and 𝐽𝐽, we have completed

the proof ofTheorem 2

Proof of Theorem 3 We can use the similar arguments as in

the proof ofTheorem 1andTheorem 2 For the proof of(𝑎),

it suffices to show that

1

𝑤𝑞(𝐵)𝑞𝜆/𝑝∫𝐵󵄨󵄨󵄨󵄨T𝛼,Ω𝑓(𝑥)󵄨󵄨󵄨󵄨𝑞𝑤(𝑥)𝑞𝑑𝑥 ≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑞

𝑀 𝑝,𝜆 (𝑤 𝑝 ,𝑤 𝑞 ) (41)

For a fixed ball𝐵 = 𝐵(𝑥0, 1), we decompose 𝑓 = 𝑓𝜒2𝐵+

𝑓𝜒(2𝐵)𝑐 := 𝑓1+ 𝑓2 SinceT𝛼,Ωis a sublinear operator, we get

1

𝑤𝑞(𝐵)𝑞𝜆/𝑝∫

𝐵󵄨󵄨󵄨󵄨T𝛼,Ω𝑓(𝑥)󵄨󵄨󵄨󵄨𝑞𝑤(𝑥)𝑞𝑑𝑥

𝑤𝑞(𝐵)𝑞𝜆/𝑝∫

𝐵(󵄨󵄨󵄨󵄨T𝛼,Ω𝑓1(𝑥)󵄨󵄨󵄨󵄨𝑞 +󵄨󵄨󵄨󵄨T𝛼,Ω𝑓2(𝑥)󵄨󵄨󵄨󵄨𝑞) 𝑤𝑞(𝑥) 𝑑𝑥 := 𝐾 + 𝐾𝐾

(42)

To estimate the term 𝐾, using the fact that T𝛼,Ω is bounded from𝐿𝑝(𝑤𝑝) to 𝐿𝑞(𝑤𝑞) with 𝑤 ∈ 𝐴(𝑝/𝑟󸀠 ,𝑞), we can get

∫

𝐵󵄨󵄨󵄨󵄨T𝛼,Ω𝑓1(𝑥)󵄨󵄨󵄨󵄨𝑞𝑤𝑞(𝑥) 𝑑𝑥

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑞

𝑀𝑝,𝜆(𝑤 𝑝 ,𝑤 𝑞 )𝑤𝑞(𝐵)𝑞𝜆/𝑝,

(43)

which implies that

𝐾 ≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 𝑝,𝜆 (𝑤 𝑝 ,𝑤 𝑞 ) (44) For the term 𝐾𝐾, by the similar arguments as that of

Theorem 1, we obtain

𝐾𝐾 ≤ 𝐶∑

𝑘

(2−𝑘(𝑛−𝛼)∫

𝐴 𝑘󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑑𝑦)𝑞

× 𝑤𝑞(𝐵)1−𝑞𝜆/𝑝

≤ 𝐶∑

𝑘

[2−𝑘(𝑛−𝛼)+𝑛𝑘/𝑟(󵄨󵄨󵄨󵄨𝑓 (𝑦)󵄨󵄨󵄨󵄨𝑟 󸀠

𝑑𝑦)1/𝑟

󸀠 ]

𝑞

× 𝑤𝑞(𝐵)1−𝑞𝜆/𝑝

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 𝑝,𝜆 (𝑤 𝑝 ,𝑤 𝑞 )

× (∑∞

𝑘=1

𝑤𝑞(𝐵)(1/𝑞−𝜆/𝑝)

𝑤𝑞(2𝑘+1𝐵)(1/𝑞−𝜆/𝑝))

𝑞

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑞

𝑀 𝑝,𝜆 (𝑤 𝑝 ,𝑤 𝑞 )

(45)

We have completed the proof of(𝑎)

We will omit the proof of(𝑏) since we can prove it by using

𝐴(1,𝑞)condition and the weak type estimates ofT𝛼,Ωsimilar

to the proof ofTheorem 2

3 Boundedness of Commutators

We say that𝑏 is a BMO(R𝑛) function if the following sharp maximal function is finite:

𝑏#(𝑥) = sup

𝐵

1

|𝐵|∫𝐵󵄨󵄨󵄨󵄨𝑏(𝑦) − 𝑏𝐵󵄨󵄨󵄨󵄨𝑑𝑦, (46) where the supreme is taken over all balls𝐵 ⊂ R𝑛and𝑓𝐵 = (1/|𝐵|) ∫𝐵𝑓(𝑦)𝑑𝑦 This means ‖𝑏‖BMO (R 𝑛 ) = ‖𝑏#‖𝐿∞ < +∞

An early work about BMO(R𝑛) space can be attributed to

John and Nirenberg [22] For1 < 𝑝 < ∞, there is a close

relation between BMO(R𝑛) and 𝐴𝑝weights:

BMO(R𝑛) = {𝛼 log 𝑤 : 𝑤 ∈ 𝐴𝑝, 𝛼 ≥ 0} (47)

Given an operator𝑇 acting on a generic function 𝑓 and a

function𝑏, the commutator 𝑇𝑏is formally defined as

𝑇𝑏𝑓 = [𝑏, 𝑇] 𝑓 = 𝑏𝑇 (𝑓) − 𝑇 (𝑏𝑓) (48)

Since 𝐿∞(R𝑛) ⊊ BMO(R𝑛), the boundedness of 𝑇𝑏 is

worse than𝑇 (e.g., the singularity; see also [23]) Therefore,

many authors want to know whether𝑇𝑏 shares the similar

boundedness with 𝑇 There are a lot of articles that deal

with the topic of commutators of different operators with

BMO functions on Lebesgue spaces The first results for this

commutator were obtained by Coifman et al [24] in their

study of certain factorization theorems for generalized Hardy

spaces In the present section, we will extend the boundedness

ofTΩandT𝛼,ΩtoTΩ,𝑏andT𝛼,Ω,𝑏, respectively

Theorem 6 Let 𝑟 , 𝑝, 𝜆, and 𝑤 be as in Theorem 1 Suppose

that the sublinear operatorTΩsatisfies condition (5) for any

integral function 𝑓 with compact support If TΩ,𝑏is bounded

on 𝐿𝑝(𝑤) with 𝑏 ∈ BMO(R𝑛), then TΩ,𝑏 is bounded on

𝑀𝑝,𝜆(𝑤).

Theorem 7 Let 𝑝, 𝑟, 𝑞, 𝛼, 𝑤, and 𝜆 be as in Theorem 3(𝑎)

and let the sublinear operator T𝛼,Ω satisfy condition (18)

for any integral function 𝑓 with compact support If T𝛼,Ω,𝑏

maps𝐿𝑝(𝑤𝑝) into 𝐿𝑞(𝑤𝑞)with 𝑏 ∈ BMO(R𝑛), then T𝛼,Ω,𝑏is

bounded from𝑀𝑝,𝜆(𝑤𝑝, 𝑤𝑞) to 𝑀𝑞,𝑞𝜆/𝑝(𝑤𝑞).

The following lemmas about BMO(R𝑛) functions will

help us to prove Theorems6and7

Lemma 8 (see [25, Theorem3.8]) Let 1 ≤ 𝑝 < ∞ and 𝑏 ∈

BMO(R𝑛) Then for any ball 𝐵 ⊂ R𝑛, the following statements

are true.

(a) There exist constants𝐶1and𝐶2such that for all𝛼 > 0

󵄨󵄨󵄨󵄨{𝑥 ∈ 𝐵 : 󵄨󵄨󵄨󵄨𝑏(𝑥) − b𝐵󵄨󵄨󵄨󵄨 > 𝛼}󵄨󵄨󵄨󵄨 ≤ 𝐶1|𝐵| 𝑒−𝐶2 𝛼/‖𝑏‖BMO(R𝑛) (49)

(b) Inequality (49) is called John-Nirenberg inequality:

󵄨󵄨󵄨󵄨𝑏2 𝜆 𝐵− 𝑏𝐵󵄨󵄨󵄨󵄨 ≤ 2𝑛𝜆‖𝑏‖BMO(R 𝑛 ) (50)

Lemma 9 ([1, Proposition7.1.2] (see also [14, Theorem5]))

Let𝑤 ∈ 𝐴∞and 1 < 𝑝 < ∞ Then the following statements

are equivalent:

(a)‖𝑏‖𝐵𝑀𝑂(R𝑛 )∼ sup𝐵((1/|𝐵|) ∫𝐵|𝑏(𝑥) − 𝑏𝐵|𝑝𝑑𝑥)1/𝑝,

(b)‖𝑏‖𝐵𝑀𝑂(R𝑛 )∼ sup𝐵inf𝑎∈R(1/|𝐵|) ∫𝐵|𝑏(𝑥) − 𝑎|𝑑𝑥,

(c)‖𝑏‖𝐵𝑀𝑂(𝑤) = sup𝐵(1/𝑤(𝐵)) ∫𝐵|𝑏(𝑥) − 𝑏𝐵,𝑤|𝑤(𝑥)𝑑𝑥,

where𝐵𝑀𝑂(𝑤) = {𝑏 : ‖𝑏‖𝐵𝑀𝑂(𝑤) < ∞} and 𝑏𝐵,𝑤 =

(1/𝑤(𝐵)) ∫𝐵𝑏(𝑦)𝑤(𝑦)𝑑𝑦.

Lemma 10 Let 𝑝, 𝑟, 𝑏, and 𝑤 be as in Theorem 6 and let𝐵 = 𝐵(𝑥0, 1) be a generic fixed ball Then the inequality

(∫

|𝑥 0 −𝑦|>2

󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥0− 𝑦󵄨󵄨󵄨󵄨𝑛 󵄨󵄨󵄨󵄨𝑏𝐵,𝑤− 𝑏 (𝑦)󵄨󵄨󵄨󵄨 𝑑𝑦)

𝑝

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀𝑝,𝜆(𝑤)𝑤(𝐵)𝜆−1

(51)

holds for every𝑦 ∈ (2𝐵)𝑐, where(2𝐵)𝑐= R𝑛\ (2𝐵).

Proof We will consider two cases.

Case1 (𝑃 > 𝑟󸀠) In this case, 𝑤 ∈ 𝐴𝑝/𝑟󸀠 Using H¨older’s inequality andLemma 5to the left-hand side of (51), we have

∫

|𝑥0−𝑦|>2

󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥0− 𝑦󵄨󵄨󵄨󵄨𝑛 󵄨󵄨󵄨󵄨𝑏𝐵,𝑤− 𝑏 (𝑦)󵄨󵄨󵄨󵄨 𝑑𝑦

≤∑∞

𝑗=1

∫

2 𝑗 <|𝑥0−𝑦|<2 𝑗+1

󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥0− 𝑦󵄨󵄨󵄨󵄨𝑛

× 󵄨󵄨󵄨󵄨𝑏𝐵,𝑤− 𝑏 (𝑦)󵄨󵄨󵄨󵄨 𝑑𝑦

≤∑∞

𝑗=1

1

󵄨󵄨󵄨󵄨2𝑗𝐵󵄨󵄨󵄨󵄨∫𝐴 𝑗󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑏𝐵,𝑤− 𝑏 (𝑦)󵄨󵄨󵄨󵄨 𝑑𝑦

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑀 𝑝,𝜆 (𝑤)

∞

∑

𝑗=1

2𝑛𝑗/𝑟𝑤(2𝑗+1𝐵)𝜆/𝑝

󵄨󵄨󵄨󵄨2𝑗𝐵󵄨󵄨󵄨󵄨

× (∫

2 𝑗+1 𝐵󵄨󵄨󵄨󵄨𝑏𝐵,𝑤− 𝑏 (𝑦)󵄨󵄨󵄨󵄨𝑝𝑟󸀠/(𝑝−𝑟󸀠)

× 𝑤(𝑦)−𝑟󸀠/(𝑝−𝑟󸀠)𝑑𝑦)(𝑝−𝑟

󸀠 )/𝑝𝑟 󸀠

(52)

Set

𝐴 = (∫

2 𝑗+1 𝐵󵄨󵄨󵄨󵄨𝑏𝐵,𝑤− 𝑏 (𝑦)󵄨󵄨󵄨󵄨𝑝󸀠𝑟󸀠𝑤1−𝑝󸀠𝑑𝑦)1/𝑝

󸀠 𝑟 󸀠 , (53) where𝑝 = 𝑝/𝑟󸀠> 1 Thus

𝐴 ≤ (∫

2 𝑗+1 𝐵(󵄨󵄨󵄨󵄨󵄨󵄨𝑏2𝑗+1 𝐵,𝑤 1−𝑝󸀠 − 𝑏 (𝑦)󵄨󵄨󵄨󵄨󵄨󵄨

+󵄨󵄨󵄨󵄨󵄨󵄨𝑏2𝑗+1 𝐵,𝑤 1−𝑝󸀠− 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨󵄨󵄨 )𝑝󸀠𝑟 󸀠

× 𝑤(𝑦)1−𝑝󸀠𝑑𝑦)1/𝑝

󸀠 𝑟 󸀠

≤ (∫

2 𝑗+1 𝐵󵄨󵄨󵄨󵄨󵄨󵄨𝑏2𝑗+1 𝐵,𝑤 1−𝑝󸀠− 𝑏 (𝑦)󵄨󵄨󵄨󵄨󵄨󵄨𝑝󸀠𝑟 󸀠

× 𝑤(𝑦)1−𝑝󸀠𝑑𝑦)1/𝑝

󸀠 𝑟 󸀠

+󵄨󵄨󵄨󵄨󵄨󵄨𝑏2𝑗+1 𝐵,𝑤 1−𝑝󸀠− 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨󵄨󵄨

× 𝑤1−𝑝󸀠(2𝑗+1𝐵)1/𝑝󸀠𝑟󸀠

=: 𝐴1+ 𝐴2

(54)

Lemma 9implies that

𝐴1≤ 𝐶w1−𝑝󸀠(2𝑗+1𝐵)1/𝑝󸀠𝑟󸀠 (55)

We are now in a position to deal with𝐴2; by (50), we have

󵄨󵄨󵄨󵄨

󵄨󵄨𝑏2 𝑗+1 𝐵,𝑤 1−𝑝󸀠− 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨󵄨󵄨

≤ 󵄨󵄨󵄨󵄨󵄨𝑏2 𝑗+1 𝐵,𝑤 1−𝑝󸀠− 𝑏2𝑗+1 𝐵󵄨󵄨󵄨󵄨󵄨

+ 󵄨󵄨󵄨󵄨𝑏2 𝑗+1 𝐵− 𝑏𝐵󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑏𝐵− 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨

𝑤1−𝑝󸀠(2𝑗+1𝐵)

× ∫

2 𝑗+1 𝐵󵄨󵄨󵄨󵄨𝑏(𝑦) − 𝑏2 𝑗+1 𝐵󵄨󵄨󵄨󵄨𝑤(𝑦)1−𝑝󸀠𝑑𝑦 + 2𝑛(𝑗 + 1) ‖𝑏‖BMO (R 𝑛 )

+ 1

𝑤 (𝐵)∫𝐵󵄨󵄨󵄨󵄨𝑏(𝑦) − 𝑏𝐵󵄨󵄨󵄨󵄨𝑤(𝑦)𝑑𝑦 := 𝐴21+ 𝐴22+ 𝐴23

(56)

Combining (23) with (49), we have

𝐴23= 1

𝑤 (𝐵)

× ∫∞

0 𝑤 ({𝑥 ∈ 𝐵 : 󵄨󵄨󵄨󵄨𝑏 (𝑦) − 𝑏𝐵󵄨󵄨󵄨󵄨 > 𝛼})𝑑𝛼

≤ 𝐶 ∫∞

0 𝑒−𝐶2 𝛼𝛿/‖𝑏‖ BMO(R𝑛)𝑑𝛼

≤ 𝐶

(57)

In the same manner we can see that

It follows immediately that

𝐴2≤ 𝐶 (2𝑛(𝑗 + 1) + 2) 𝑤1−𝑝󸀠(2𝑗+1𝐵)1/𝑝󸀠𝑟󸀠 (59)

Therefore

𝐴 ≤ 𝐶 (𝑗 + 1) 𝑤1−𝑝󸀠(2𝑗+1𝐵)1/𝑝󸀠𝑟󸀠 (60)

A further use of (21) and𝑤 ∈ 𝑝/𝑟󸀠allow us to obtain

∞

∑

𝑗=1

2𝑛𝑗/𝑟𝑤(2𝑗+1𝐵)𝜆/𝑝

󵄨󵄨󵄨󵄨2𝑗𝐵󵄨󵄨󵄨󵄨

× (∫

2 𝑗+1 𝐵󵄨󵄨󵄨󵄨𝑏(𝑦) − 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨𝑝 󸀠 𝑟 󸀠

𝑤(𝑦)1−𝑝󸀠𝑑𝑦)1/𝑝

󸀠 𝑟 󸀠

≤∑∞

𝑗=1

2𝑛𝑗/𝑟𝑤(2𝑗+1𝐵)𝜆/𝑝

󵄨󵄨󵄨󵄨2𝑗𝐵󵄨󵄨󵄨󵄨

× (𝑗 + 1) 𝑤(𝑦)1−𝑝󸀠(2𝑗+1𝐵)1/𝑝󸀠𝑟󸀠

≤ 𝐶∑∞

𝑗=1

2𝑛𝑗/𝑟󵄨󵄨󵄨󵄨󵄨2𝑗+1

𝐵󵄨󵄨󵄨󵄨󵄨1/𝑟󸀠(𝑗 + 1)

󵄨󵄨󵄨󵄨2𝑗𝐵󵄨󵄨󵄨󵄨

× 𝑤(𝐵)(1−𝜆)/𝑝 𝑤(2𝑗+1𝐵)(1−𝜆)/𝑝𝑤(𝐵)(𝜆−1)/𝑝

≤ 𝐶∑∞

𝑗=1

𝑗 + 1

𝐷(𝑗+1)(1−𝜆)/𝑝𝑤(𝐵)(𝜆−1)/𝑝

≤ 𝐶𝑤(𝐵)(𝜆−1)/𝑝,

(61) where𝐷 > 1 is a constant that appeared in (21)

Case 2 (𝑃 = 𝑟󸀠) In this case,𝑤 ∈ 𝐴1 We can prove (51) by

a similar analysis as in the proof ofTheorem 1(in the case

𝑃 = 𝑟󸀠) and Case1

Having disposed of the previous preliminary step, we can now return to the proofs of Theorems6and7

Proof of Theorem 6 The task is now to find a constant𝐶 such that for fixed ball𝐵 = 𝐵(𝑥0, 1), we can obtain

1 𝑤(𝐵)𝜆∫𝐵󵄨󵄨󵄨󵄨TΩ,𝑏𝑓 (𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤)

(62)

We decompose𝑓 = 𝑓𝜒2𝐵+𝑓𝜒(2𝐵)𝑐 := 𝑓1+𝑓2and consider the corresponding splitting

∫

𝐵󵄨󵄨󵄨󵄨TΩ,𝑏𝑓 (𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

≤ 𝐶 (∫

𝐵󵄨󵄨󵄨󵄨TΩ,𝑏𝑓1(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥 + ∫

𝐵󵄨󵄨󵄨󵄨TΩ,𝑏𝑓2(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥)

=: 𝐿 + 𝐿𝐿

(63)

It follows from the𝐿𝑝(𝑤) boundedness of TΩ,𝑏and𝑤 ∈

𝐴𝑝/𝑟󸀠⊂ 𝐴𝑝that

𝐿 ≤ 𝐶 ∫

2𝐵󵄨󵄨󵄨󵄨𝑓(𝑥)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤)𝑤(𝐵)𝜆

(64)

Then a further use of (5) derives that

󵄨󵄨󵄨󵄨TΩ,𝑏𝑓2(𝑥)󵄨󵄨󵄨󵄨𝑝

≤ 𝐶(∫

R 𝑛

󵄨󵄨󵄨󵄨Ω(𝑥 − 𝑦)𝑓2(𝑦)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑏(𝑥) − 𝑏(𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨󵄨𝑛 𝑑𝑦)

𝑝

≤ 𝐶 (∫

|𝑥 0 −𝑦|>2

󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥0− 𝑦󵄨󵄨󵄨󵄨𝑛

× {󵄨󵄨󵄨󵄨𝑏 (𝑥) − 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑏𝐵,𝑤− 𝑏 (𝑦)󵄨󵄨󵄨󵄨} 𝑑𝑦)

𝑝

(65)

Therefore

𝐿𝐿 ≤ 𝐶(∫

|𝑥 0 −𝑦|>2

󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥0− 𝑦󵄨󵄨󵄨󵄨𝑛 𝑑𝑦)

𝑝

× ∫

𝐵󵄨󵄨󵄨󵄨𝑏(𝑥) − 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨𝑝

𝑤 (𝑥) 𝑑𝑥

+ 𝐶 (∫

|𝑥 0 −𝑦|>2

󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥0− 𝑦󵄨󵄨󵄨󵄨𝑛

× 󵄨󵄨󵄨󵄨𝑏 (𝑦) − 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨𝑑𝑦)𝑝𝑤 (𝐵) :

= 𝐿𝐿1+ 𝐿𝐿2

(66)

ByLemma 10, we have

𝐿𝐿2≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤)𝑤(𝐵)𝜆 (67)

We proceed to estimate𝐿𝐿1 Without loss of generality,

we assume that𝑝 > 𝑟󸀠 Taking into account (20), (22), and

Lemma 9, we have

𝐿𝐿1= (∑∞

𝑗=1

∫

2 𝑗 <|𝑥0−𝑦|<2 𝑗+1

󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦) 𝑓 (𝑦)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝑥0− 𝑦󵄨󵄨󵄨󵄨𝑛 𝑑𝑦)

𝑝

× ∫

𝐵󵄨󵄨󵄨󵄨𝑏(𝑥) − 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

≤ (∑∞

𝑗=1

1

󵄨󵄨󵄨󵄨2𝑗𝐵󵄨󵄨󵄨󵄨∫𝐴 𝑗󵄨󵄨󵄨󵄨Ω(𝑥0− 𝑦)󵄨󵄨󵄨󵄨𝑟𝑑𝑦)

1/𝑟

× (∫

2 𝑗+1 𝐵󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑟 󸀠

𝑑𝑦)1/𝑟

󸀠

× ∫

𝐵󵄨󵄨󵄨󵄨𝑏(𝑥) − 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

≤ 𝐶 [ [

∞

∑

𝑗=1

2𝑛𝑗/𝑟

󵄨󵄨󵄨󵄨2𝑗𝐵󵄨󵄨󵄨󵄨

× (∫

2 𝑗+1 𝐵󵄨󵄨󵄨󵄨𝑓(𝑦)󵄨󵄨󵄨󵄨𝑝𝑤 (𝑦) 𝑑𝑦)1/𝑝

×(∫

2 𝑗+1 𝐵𝑤(𝑦)−𝑟󸀠/(𝑝−𝑟󸀠)𝑑𝑦)(𝑝−𝑟

󸀠 )/𝑝𝑟 󸀠 ] ]

𝑝

× ∫

𝐵󵄨󵄨󵄨󵄨𝑏(𝑥) − 𝑏𝐵,𝑤󵄨󵄨󵄨󵄨𝑝𝑤 (𝑥) 𝑑𝑥

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤)

× (∑∞

𝑗=1

𝑤(𝐵)(1−𝜆)/𝑝

𝑤(2𝑗+1𝐵)(1−𝜆)/𝑝)

𝑝

𝑤(𝐵)𝜆

≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀𝑝,𝜆(𝑤)𝑤(𝐵)𝜆

(68) Hence

𝐿𝐿 ≤ 𝐶󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝

𝑀 𝑝,𝜆 (𝑤)𝑤(𝐵)𝜆 (69) According to (64) and (69), we have completed the proof

ofTheorem 6

Proof of Theorem 7 The proof ofTheorem 7is similar to that

ofTheorem 6, except using𝑤 ∈ 𝐴(𝑝,𝑞) We omit its proof here

Acknowledgments

The authors thank Professor Shanzhen Lu and the referee for their valuable suggestions This work was partially sup-ported by NSF of China (Grants nos 11271175, 10901076, and 11171345) and NSF of Shandong Province (Grant no ZR2012AQ026)

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