Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 784983, 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 Introduction and Main Results Given a function Ω over the unit sphere 𝑆𝑛−1 of R𝑛 (𝑛 ≥ 2) equipped with the normalized Lebesgue measure 𝑑𝜎 and 𝑥 = 𝑥/|𝑥|, a Calder´on-Zygmund singular integral operator with rough kernel was given by 𝑇Ω 𝑓 (𝑥) = p.v ∫ R𝑛 Ω (𝑥 − 𝑦) 𝑛 𝑓 (𝑦) 𝑑𝑦 𝑥 − 𝑦 (1) and a related maximal operator 𝑀Ω 𝑓 (𝑥) = sup 𝑟>0 𝑟𝑛 ∫ 𝐵(𝑥,𝑟) Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦, (2) where Ω is homogeneous of degree zero and satisfies Ω ∈ 𝐿𝑟 (𝑆𝑛−1 ) , ∫ 𝑆𝑛−1 1 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 𝑇Ω is 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 for 𝑇Ω in homogeneous case on weighted 𝐿𝑝 (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 TΩ 𝑓 (𝑥) (5) Ω (𝑥 − 𝑦) 𝑓 (𝑦) ≤ 𝐶∫ 𝑑𝑦, 𝑥 ∉ supp 𝑓 𝑛 𝑥 − 𝑦 R𝑛 for 𝑓 ∈ 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 𝑇Ω 𝑓 (𝑥) Ω (𝑥 − 𝑦) = p.v ∫ 𝑒𝑖𝑃(𝑥,𝑦) 𝑓 (𝑦) 𝑑𝑦, 𝑥 − 𝑦𝑛 R𝑛 𝑥 ∉ supp 𝑓, (6) Journal of Function Spaces and Applications where the phase is a polynomial The boundedness of 𝑇Ω on weighted 𝐿𝑝 (R𝑛 ) (1 ≤ 𝑝 < ∞) spaces was fully studied by Ojanen in his doctoral dissertation [12] Let 𝐷𝑘 = {𝑥 ∈ R𝑛 : |𝑥| ≤ 2𝑘 } and let 𝐴 𝑘 = 𝐷𝑘 \ 𝐷𝑘−1 for 𝑘 ∈ 𝑍 Throughout this paper, we will denote by 𝜒𝐸 the characteristic function of the set 𝐸 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): −𝑛 TΩ 𝑓 (𝑥) ≤ 𝐶|𝑥| ∫ Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦, R𝑛 (7) when supp 𝑓 ⊆ 𝐴 𝑘 and |𝑥| ≥ 2𝑘+1 with 𝑘 ∈ Z and −𝑘𝑛 TΩ 𝑓 (𝑥) ≤ 𝐶2 ∫ Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦, 𝑛 R 𝐵 (8) ∫ 𝑤 (𝑥) 𝑑𝑥) |𝐵| 𝐵 𝐵 1/𝑝 } } } < ∞} , } } } (12) where 𝑤(𝐵) = ∫𝐵 𝑤(𝑥)𝑑𝑥 For 𝑤 ∈ 𝐴 𝑝 (1 ≤ 𝑝 < ∞), if 𝜆 = 0, then 𝑀𝑝,0 (𝑤) = 𝐿𝑝 (𝑤) while 𝜆 = implies 𝑀𝑝,1 (𝑤) = 𝐿∞ (𝑤) Now, we formulate our major results of this paper as follows Theorem Let < 𝜆 < 1, < 𝑟 ≤ ∞, and 𝑟 ≤ 𝑝 < ∞ and let a sublinear operator TΩ satisfy (7) and (8) If TΩ is bounded on 𝐿𝑝 (𝑤) with 𝑤 ∈ 𝐴 𝑝/𝑟 , then TΩ is bounded on 𝑀𝑝,𝜆 (𝑤) < 𝑝 < ∞, 𝑤 ({𝑥 ∈ 𝐵 : TΩ 𝑓 (𝑥) > 𝜇}) 1/𝑝 ×( ≤ 𝐶, ∫ 𝑤(𝑥)−𝑝 𝑑𝑥) |𝐵| 𝐵 < 𝑝, 𝑞 < ∞, (9) respectively, where 1/𝑝 + 1/𝑝 = For 𝑝 = 1, the 𝐴 and 𝐴 (1,𝑞) (1 < 𝑞 < ∞) weights are defined by 𝑀𝑤 (𝑥) ≤ 𝐶𝑤 (𝑥) , 𝐵 𝑤(𝐵)𝜆 𝑝 ∫ 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥) 𝐵 Theorem Let < 𝑟 < ∞ and 1/𝑟+𝜆 < and let TΩ satisfy (7) and (8) Then if TΩ is bounded from 𝐿1 (𝑤) to 𝐿1,∞ (𝑤) with 𝑤 ∈ 𝐴 , there exists a constant 𝐶 > such that for all 𝜇 > and all balls 𝐵, 1/𝑞 ∫ 𝑤(𝑥)𝑞 𝑑𝑥) |𝐵| 𝐵 𝐴 (1,𝑞) : sup( 𝐵 When 𝑝 = 1, we have the following theorem 𝑝−1 ×( ≤ 𝐶, ∫ 𝑤(𝑥)1−𝑝 𝑑𝑥) |𝐵| 𝐵 𝐴 (𝑝,𝑞) : sup( { { { 𝑀𝑝,𝜆 (𝑤) = {𝑓 : 𝑓𝑀𝑝,𝜆 (𝑤) { { { = sup( when supp 𝑓 ⊆ 𝐴 𝑘 and |𝑥| ≤ 2𝑘−1 with 𝑘 ∈ Z, respectively It is worth pointing out that 𝑀Ω satisfies conditions (7) and (8) Also, condition (5) implies the size conditions (7) and (8) since |𝑥 − 𝑦| > |𝑥|/2 when |𝑥| ≥ 2𝑘+1 and supp 𝑓 ⊆ 𝐴 𝑘 while supp 𝑓 ⊆ 𝐴 𝑘 and |𝑥| ≤ 2𝑘−1 imply |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 classes 𝐴 𝑝 and 𝐴 (𝑝,𝑞) [14] contain the functions 𝑤 which satisfy 𝐴 𝑝 : sup ( 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 Let ≤ 𝑝 < ∞, < 𝜆 < and let 𝑤 be a weight function Then the weighted Morrey space 𝑀𝑝,𝜆 (𝑤) is defined by 1/𝑞 ∫ 𝑤(𝑥)𝑞 𝑑𝑥) |𝐵| 𝐵 × ( ess sup 𝐵 (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] Let ≤ 𝑝 < ∞, < 𝜆 < For two weights 𝑤1 and 𝑤2 , 𝑀𝑝,𝜆 (𝑤1 , 𝑤2 ) = {𝑓 : 𝑓𝑀𝑝,𝑘 (𝑤1 ,𝑤2 ) (10) ) ≤ 𝐶, 𝑤 (𝑥) = sup( 𝐵 respectively Here ess sup and the following essinf are the abbreviations of essential supremum and essential infimum, respectively Clearly, 𝑤 ∈ 𝐴 if and only if there is a constant 𝐶 > such that ∫ 𝑤 (𝑥) 𝑑𝑥 ≤ Cess inf 𝑤 (𝑥) 𝐵 |𝐵| 𝐵 ≤ 𝐶𝜇−1 𝑓𝑀1,𝜆 (𝑤) 𝑤(𝐵)𝜆 (11) 𝑤2 (𝐵)𝜆 𝑝 ∫ 𝑓 (𝑥) 𝑤1 (𝑥) 𝑑𝑥) 𝐵 1/𝑝 < ∞} (14) If 𝑤1 = 𝑤2 = 𝑤, we write 𝑀𝑝,𝜆 (𝑤1 , 𝑤1 ) = 𝑀𝑝,𝜆 (𝑤2 , 𝑤2 ) = 𝑀𝑝,𝜆 (𝑤) We can get similar results for fractional integrals following the line of Theorems and Journal of Function Spaces and Applications Theorem Let < 𝛼 < 𝑛, ≤ 𝑟 < ∞, and < 𝜆 < Suppose that a sublinear operator T𝛼,Ω satisfies the size conditions T𝛼,Ω 𝑓 (𝑥) (15) ≤ 𝐶|𝑥|−(𝑛−𝛼) ∫ Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦 R𝑛 when supp 𝑓 ⊆ 𝐴 𝑘 and |𝑥| ≥ 2𝑘+1 with 𝑘 ∈ 𝑍 and ≤ 𝐶2 ∫ Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦 Proofs of Theorems and depend heavily on some properties of 𝐴 𝑝 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 of 𝐴 𝑝 weights without proofs, thus making our exposition self-contained Lemma Let ≤ 𝑝 < ∞ and 𝑤 ∈ 𝐴 𝑝 Then the following statements are true T𝛼,Ω 𝑓 (𝑥) −𝑘(𝑛−𝛼) Boundedness of Sublinear Operators (16) (a) There exists a constant 𝐶 such that 𝑤 (2𝐵) ≤ 𝐶𝑤 (𝐵) , R𝑛 when supp 𝑓 ⊆ 𝐴 𝑘 and |𝑥| ≤ 2𝑘−1 with 𝑘 ∈ 𝑍 Then one has the following (a) If T𝛼,Ω maps 𝐿𝑝 (𝑤𝑝 ) into 𝐿𝑞 (𝑤𝑞 ) with 𝑤 ∈ 𝐴 (𝑝/𝑟 ,𝑞) , then T𝛼,Ω is bounded from 𝑀𝑝,𝜆 (𝑤𝑝 , 𝑤𝑞 ) to 𝑀𝑞,𝑞𝜆/𝑝 (𝑤𝑞 ), where 𝑟 ≤ 𝑝 < 𝑛/𝛼, 1/𝑞 = 1/𝑝 − 𝛼/𝑛 and 𝑝 ≤ 𝑞 ≤ ∞ (b) If T𝛼,Ω is bounded from 𝐿1 (𝑤) to 𝐿𝑞,∞ (𝑤𝑞 ) with 𝑤 ∈ 𝐴 (1,𝑞) and 1/𝑟 + 𝜆 < 1, then there exists a constant 𝐶 > such that for all 𝜇 > and all balls 𝐵, 1/𝑞 𝑤({𝑥 ∈ 𝐵 : T𝛼,Ω 𝑓 (𝑥) > 𝜇}) ≤ 𝐶𝜇−1 𝑓𝑀1,𝜆 (𝑤,𝑤𝑞 ) 𝑤(𝐵)𝜆 , (17) We emphasize that (15) and (16) are weaker conditions than the following condition: T𝛼,Ω 𝑓 (𝑥) Ω (𝑥 − 𝑦) 𝑓 (𝑦) ≤ 𝐶∫ 𝑑𝑦, 𝑥 − 𝑦𝑛−𝛼 R𝑛 0 such that the following reverse Hăolder inequality holds for every ball 𝐵 ⊂ R𝑛 : 1/𝑟 1 ∫ 𝑤(𝑥)𝑟 𝑑𝑥) ≤ 𝐶 ( ∫ 𝑤 (𝑥) 𝑑𝑥) |𝐵| 𝐵 |𝐵| 𝐵 (22) (d) For all 𝜆 > 1, one has 𝑤 (𝜆𝐵) ≤ 𝐶𝜆𝑛𝑝 𝑤 (𝐵) (23) (e) There exist two constants 𝐶 and 𝛿 > such that for any measurable set 𝑄 ⊂ 𝐵 𝛿 𝑤 (𝑄) |𝑄| ≤ 𝐶( ) , 𝑤 (𝐵) |𝐵| (24) if 𝑤 satisfies (24); one says 𝑤 ∈ 𝐴 ∞ (f) For all 𝑝 < 𝑞 < ∞, one has 𝐴 ∞ = ∪𝑝 𝐴 𝑝 , < 𝛼 < 𝑛 (21) where 𝑤 satisfies this condition; one says 𝑤 satisfies the reverse doubling condition ( where < 𝑞 < ∞ (20) 𝐴 𝑝 ⊂ 𝐴 𝑞 (25) (19) For some mapping properties of T𝛼,Ω on various kinds of function spaces, see [17–19] and the references therein We end this section with the outline of this paper Section contains the proofs of Theorems and 3; this part is partly motivated by the methods in [20] dealing with the case of the Lebesgue measure In Section 3, we extend the corresponding results to commutators of certain sublinear operators The following lemma about the rough kernel Ω is essential to our proofs One can find its proof in [21] Lemma Let Ω ∈ 𝐿𝑟 (𝑆𝑛−1 ) with ≤ 𝑟 < ∞ Then the following statements are true (a) If 𝑥 ∈ 𝐴 𝑘 and 𝑗 ≥ 𝑘+1, then ∫𝐴 |Ω(𝑥−𝑦)|𝑟 𝑑𝑦 ≤ 𝐶2𝑗𝑛 𝑗 (b) If 𝑦 ∈ 𝐴 𝑘 and 𝑘 ≥ 𝑗 + 1, then ∫𝐴 |Ω(𝑥 − 𝑦)|𝑟 𝑑𝑥 ≤ 𝐶2𝑘(𝑛−1)+𝑗 𝑗 Journal of Function Spaces and Applications Proof of Theorem Let < 𝑟 ≤ 𝑝 < ∞, 𝑤 ∈ 𝐴 𝑝/𝑟 , and < 𝜆 < Our task is to show By (31), Hăolders inequality, and Lemma 5, we have 𝑟 Ω (𝑥 − 𝑦) 𝑑𝑦) 𝐴k+1 TΩ,𝑘 𝑓 (𝑥) ≤ 𝐶(∫ 𝜆 𝑤(𝐵) 𝑝 𝑝 ∫ TΩ 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 𝐵 (26) For a fixed ball 𝐵 = 𝐵(𝑥0 , 𝑟), there is no loss of generality in assuming 𝑟 = We decompose 𝑓 = 𝑓𝜒2𝐵 + 𝑓𝜒(2𝐵)𝑐 := 𝑓1 + 𝑓2 Since TΩ is a sublinear operator, so we get 𝑤(𝐵)𝜆 + ≤ 𝐶2(𝑘+1)𝑛/𝑟 𝑟 /𝑝 −𝑟 /𝑝 𝑟 × (∫ 𝑑𝑦) 𝑓 (𝑦) 𝑤(𝑦) 𝑤(𝑦) 𝑘+1 𝐵 𝑝 𝑓 (𝑦) 𝑤 (𝑦) 𝑑𝑦) 𝑘+1 𝐵 𝐵 𝑤(𝐵)𝜆 𝑝 ∫ TΩ 𝑓1 (𝑥) 𝑤 (𝑥) 𝑑𝑥 𝐵 𝐶 𝑤(𝐵)𝜆 −𝑟 /(𝑝−𝑟 ) × (∫ (27) 𝑝 ∫ TΩ 𝑓2 (𝑥) 𝑤 (𝑥) 𝑑𝑥 𝐵 := 𝐼 + 𝐼𝐼 2𝑘+1 𝐵 𝑤(𝑦) 𝐼≤ 𝑤(𝐵)𝜆 ≤ 𝐶 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) (28) 𝑤(𝐵)𝜆 𝑝 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 1/𝑝 (𝑝−𝑟 )/(𝑝𝑟 ) 2(𝑘+1)𝑛 (1−𝜆)/𝑝 𝑤(2𝑘+1 𝐵) (32) 𝑝 ∫ TΩ 𝑓1 (𝑥) 𝑤 (𝑥) 𝑑𝑥 R𝑛 𝑝 ∫ 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥 2𝐵 𝑑𝑦) 1/𝑟 𝑘+1 1/𝑟 2 𝐵 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 2(𝑘+1)𝑛/𝑟 (1−𝜆)/𝑝 𝑤(2𝑘+1 𝐵) By the assumption on TΩ and (25), we can obtain 𝐶 1/𝑟 ≤ 𝐶2(𝑘+1)𝑛/𝑟 (∫ 𝑝 ∫ TΩ 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥 𝐶 ≤ 𝑟 × (∫ 𝑓 (𝑦) 𝑑𝑦) 𝑘+1 𝐵 1/𝑟 Case (𝑝 = 𝑟 ) In this case, 𝑤 ∈ 𝐴 implies that 2𝑘+1 𝐵 −1 (ess inf 𝑤 (𝑥)) ≤ 𝑘+1 , 𝑤 (2 𝐵) 𝑥∈2𝑘+1 𝐵 (33) which in combination with the Hăolder inequality and Lemma yields that TΩ,𝑘 𝑓 (𝑥) ≤ 𝐶2(𝑘+1)𝑛/𝑟 −1 𝑝 𝑓 (𝑦) 𝑤 (𝑦) 𝑤(𝑦) 𝑑𝑦) 2𝑘+1 𝐵 For the term 𝐼𝐼, by (8) we have × (∫ ∞ 𝑝 −𝑘𝑛 𝐼𝐼 ≤ ∫ ∑ TΩ,𝑘 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥, 𝜆 𝑤(𝐵) 𝐵 𝑘=1 𝐶 (29) ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) (𝑘+1)𝑛 (1−𝜆)/𝑝 𝑤(2𝑘+1 𝐵) 1/𝑝 (34) Substituting (32) and (34) into (29), we can assert that ∞ where 𝑝 𝐼𝐼 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) ( ∑ 𝑤(𝐵)(1−𝜆)/𝑝 𝑝 ) 𝑘+1 𝐵)(1−𝜆)/𝑝 𝑘=1 𝑤(2 TΩ,𝑘 𝑓 (𝑥) = ∫ Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦 𝑘+1 𝐴 (30) We distinguish two cases according to the size of 𝑝 and 𝑟 to get the estimates for TΩ,𝑘 Case (𝑝 > 𝑟 ) In this case, 𝑤 ∈ 𝐴 𝑝/𝑟 implies that where we have used (21) in the last inequality Combining (28) and (29), we obtain the proof of Theorem Proof of Theorem The task is now to show the following inequality: sup −𝑟 /(𝑝−𝑟 ) ∫ 𝑤 𝐵 𝜇>0 𝑑𝑦 ≤ |𝐵|𝑝/(𝑝−𝑟 ) 𝑤(𝐵) 𝑟 /(𝑝−𝑟 ) (31) (35) 𝑝 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) , 𝜇 𝑤(𝐵) 𝜆 𝑤 ({𝑥 ∈ 𝐵 : TΩ 𝑓 (𝑥) > 𝜇}) 𝑝 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) (36) Journal of Function Spaces and Applications In order to get this inequality, it will be necessary to decompose 𝑓 = 𝑓𝜒2𝐵 + 𝑓𝜒(2𝐵)𝑐 := 𝑓1 + 𝑓2 with 𝐵 as in Theorem Since TΩ is a sublinear operator, we can rewrite 𝑤 ({𝑥 ∈ 𝐵 : TΩ 𝑓 (𝑥) > 𝜇}) 𝜇 ≤ 𝑤 ({𝑥 ∈ 𝐵 : TΩ 𝑓1 (𝑥) > }) (37) 𝜇 + 𝑤 ({𝑥 ∈ 𝐵 : TΩ 𝑓2 (𝑥) > }) := 𝐽 + 𝐽𝐽 To estimate the term 𝐽𝐽, we note that 𝐶 𝐽𝐽 ≤ ∫ 𝜇 {𝑥∈𝐵:|TΩ 𝑓(𝑥)|>𝜇/2} (39) ∞ −𝑘𝑛 × ∑ TΩ,𝑘 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥 𝑘=1 By (22), (33), and the Hăolder inequality, we can estimate as −𝑘𝑛 ∑2 𝜇 𝑘=1 𝐶𝑤 (𝐵) (𝑤) 𝑓 𝜇 𝑀1,𝜆 −1 𝐵) (ess inf 𝑤) 𝑘𝑛/𝑟−𝑘𝑛(1−𝜆) × ∑2 𝑥∈2𝑘+1 𝐵 (43) 𝑞 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤𝑝 ,𝑤𝑞 ) 𝑤𝑞 (𝐵)𝑞𝜆/𝑝 , which implies that 𝐾 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤𝑝 ,𝑤𝑞 ) (44) For the term 𝐾𝐾, by the similar arguments as that of Theorem 1, we obtain 𝑞 𝑘 𝑞 (45) ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤𝑝 ,𝑤𝑞 ) ∞ × (∑ 𝑤𝑞 (𝐵)(1/𝑞−𝜆/𝑝) 𝑞 𝑘+1 𝐵) 𝑘=1 𝑤 (2 𝑞 ) (1/𝑞−𝜆/𝑝) 𝑞 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤𝑝 ,𝑤𝑞 ) 𝑤(𝐵) 𝜆 𝑘=1 ≤ 𝑞 ∫ T𝛼,Ω 𝑓1 (𝑥) 𝑤𝑞 (𝑥) 𝑑𝑥 𝐵 × 𝑤𝑞 (𝐵)1−𝑞𝜆/𝑝 𝐶 (𝑤) 𝑓 𝜇 𝑀1,𝜆 ∞ To estimate the term 𝐾, using the fact that T𝛼,Ω is bounded from 𝐿𝑝 (𝑤𝑝 ) to 𝐿𝑞 (𝑤𝑞 ) with 𝑤 ∈ 𝐴 (𝑝/𝑟 ,𝑞) , we can get 𝑘 (40) 𝜆 𝑘=1 ≤ 𝑞 +T𝛼,Ω 𝑓2 (𝑥) ) 𝑤𝑞 (𝑥) 𝑑𝑥 −1 𝑓 (𝑦) 𝑤 (𝑦) 𝑤(𝑦) 𝑑𝑦 2𝑘+1 𝐵 × ∑ 𝑤(2 (42) 1/𝑟 𝑟 ≤ 𝐶∑[2−𝑘(𝑛−𝛼)+𝑛𝑘/𝑟 (𝑓 (𝑦) 𝑑𝑦) ] ×∫ 𝑘+1 𝑞 ∫ (T𝛼,Ω 𝑓1 (𝑥) 𝐵 × 𝑤𝑞 (𝐵)1−𝑞𝜆/𝑝 𝐶𝑤 (B) ∞ −𝑘𝑛+𝑘(𝑛−1)/𝑟 ≤ ∑2 𝜇 𝑘=1 ∞ 𝑤𝑞 (𝐵)𝑞𝜆/𝑝 𝑘 2𝑘+1 𝐵 𝐵 ≤ ≤ 𝑞 ∫ T𝛼,Ω 𝑓(𝑥) 𝑤(𝑥)𝑞 𝑑𝑥 𝐵 𝐾𝐾 ≤ 𝐶∑(2−𝑘(𝑛−𝛼) ∫ Ω (𝑥 − 𝑦) 𝑓 (𝑦) 𝑑𝑦) 𝐴 ∫ Ω (𝑥 − 𝑦) 𝑤 (𝑥) 𝑑𝑥 𝑓 (𝑦) 𝑑𝑦 ×∫ 𝑤𝑞 (𝐵)𝑞𝜆/𝑝 := 𝐾 + 𝐾𝐾 An application of (20) and the weighted weak type estimates for TΩ yield that 𝐽 ≤ 𝐶𝜇−1 𝑓𝑀1,𝜆 (𝑤) 𝑤(𝐵)𝜇 (38) 𝐽𝐽 ≤ For a fixed ball 𝐵 = 𝐵(𝑥0 , 1), we decompose 𝑓 = 𝑓𝜒2𝐵 + 𝑓𝜒(2𝐵)𝑐 := 𝑓1 + 𝑓2 Since T𝛼,Ω is a sublinear operator, we get 𝐶 (𝑤) 𝑤(𝐵)𝜆 𝑓 𝜇 𝑀1,𝜆 Combining these inequalities for 𝐽 and 𝐽𝐽, we have completed the proof of Theorem Proof of Theorem We can use the similar arguments as in the proof of Theorem and Theorem For the proof of (𝑎), it suffices to show that 𝑞 𝑞 ∫ T𝛼,Ω 𝑓(𝑥) 𝑤(𝑥)𝑞 𝑑𝑥 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤𝑝 ,𝑤𝑞 ) (41) 𝑞 𝑤 (𝐵)𝑞𝜆/𝑝 𝐵 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 of T𝛼,Ω similar to the proof of Theorem Boundedness of Commutators We say that 𝑏 is a BMO(R𝑛 ) function if the following sharp maximal function is finite: 𝑏# (𝑥) = sup 𝐵 ∫ 𝑏 (𝑦) − 𝑏𝐵 𝑑𝑦, |𝐵| 𝐵 (46) where the supreme is taken over all balls 𝐵 ⊂ R𝑛 and 𝑓𝐵 = (1/|𝐵|) ∫𝐵 𝑓(𝑦)𝑑𝑦 This means ‖𝑏‖BMO(R𝑛 ) = ‖𝑏# ‖𝐿∞ < +∞ 6 Journal of Function Spaces and Applications An early work about BMO(R𝑛 ) space can be attributed to John and Nirenberg [22] For < 𝑝 < ∞, there is a close relation between BMO(R𝑛 ) and 𝐴 𝑝 weights: BMO (R𝑛 ) = {𝛼 log 𝑤 : 𝑤 ∈ 𝐴 𝑝 , 𝛼 ≥ 0} Lemma 10 Let 𝑝, 𝑟, 𝑏, and 𝑤 be as in Theorem and let 𝐵 = 𝐵(𝑥0 , 1) be a generic fixed ball Then the inequality (∫ (47) |𝑥0 −𝑦|>2 Given an operator 𝑇 acting on a generic function 𝑓 and a function 𝑏, the commutator 𝑇𝑏 is formally defined as 𝑇𝑏 𝑓 = [𝑏, 𝑇] 𝑓 = 𝑏𝑇 (𝑓) − 𝑇 (𝑏𝑓) (48) Since 𝐿∞ (R𝑛 ) ⊊ BMO(R𝑛 ), the boundedness of 𝑇𝑏 is worse than 𝑇 (e.g., the singularity; see also [23]) Therefore, many authors want to know whether 𝑇𝑏 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 of TΩ and T𝛼,Ω to TΩ,𝑏 and T𝛼,Ω,𝑏 , respectively holds for every 𝑦 ∈ (2𝐵)𝑐 , where (2𝐵)𝑐 = R𝑛 \ (2𝐵) Proof We will consider two cases Case (𝑃 > 𝑟 ) In this case, 𝑤 ∈ 𝐴 𝑝/𝑟 Using Hăolders inequality and Lemma to the left-hand side of (51), we have ∫ |𝑥0 −𝑦|>2 (49) 𝑗 𝑗+1 𝑗=1 𝛼} ≤ 𝐶1 |𝐵| 𝑒 Ω (𝑥0 − 𝑦) 𝑓 (𝑦) 𝑏𝐵,𝑤 − 𝑏 (𝑦) 𝑑𝑦 𝑛 𝑥0 − 𝑦 ≤ ∑∫ Theorem 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 maps 𝐿𝑝 (𝑤𝑝 ) into 𝐿𝑞 (𝑤𝑞 )with 𝑏 ∈ BMO(R𝑛 ), then T𝛼,Ω,𝑏 is bounded from 𝑀𝑝,𝜆 (𝑤𝑝 , 𝑤𝑞 ) to 𝑀𝑞,𝑞𝜆/𝑝 (𝑤𝑞 ) (a) There exist constants 𝐶1 and 𝐶2 such that for all 𝛼 > (51) 𝑝 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 𝑤(𝐵)𝜆−1 Theorem Let 𝑟 , 𝑝, 𝜆, and 𝑤 be as in Theorem Suppose that the sublinear operator TΩ satisfies condition (5) for any integral function 𝑓 with compact support If TΩ,𝑏 is bounded on 𝐿𝑝 (𝑤) with 𝑏 ∈ BMO(R𝑛 ), then TΩ,𝑏 is bounded on 𝑀𝑝,𝜆 (𝑤) Lemma (see [25, Theorem 3.8]) Let ≤ 𝑝 < ∞ and 𝑏 ∈ BMO(R𝑛 ) Then for any ball 𝐵 ⊂ R𝑛 , the following statements are true 𝑝 Ω (𝑥0 − 𝑦) 𝑓 (𝑦) 𝑏 𝑑𝑦) − 𝑏 (𝑦) 𝑛 𝐵,𝑤 𝑥0 − 𝑦 ≤ (∫ 2𝑗+1 𝐵 𝑑𝑦) 1/𝑝 𝑟 𝑝 𝑟 𝑏 𝑗+1 1−𝑝 − 𝑏 (𝑦) 𝐵,𝑤 1−𝑝 × 𝑤(𝑦) 𝑑𝑦) 1/𝑝 𝑟 1/𝑝 𝑟 , (53) Journal of Function Spaces and Applications + 𝑏2𝑗+1 𝐵,𝑤1−𝑝 − 𝑏𝐵,𝑤 2𝑛𝑗/𝑟 𝑤(2𝑗+1 𝐵) ≤∑ 𝑗 2 𝐵 𝑗=1 ∞ 1/𝑝 𝑟 × 𝑤1−𝑝 (2𝑗+1 𝐵) × (𝑗 + 1) 𝑤(𝑦) =: 𝐴 + 𝐴 (54) Lemma implies that 𝐴 ≤ 𝐶w1−𝑝 (2𝑗+1 𝐵) 1/𝑝 𝑟 1−𝑝 (2𝑗+1 𝐵) 1/𝑝 𝑟 1/𝑟 2𝑛𝑗/𝑟 2𝑗+1 𝐵 (𝑗 + 1) ≤ 𝐶∑ 𝑗 2 𝐵 𝑗=1 ∞ (55) We are now in a position to deal with 𝐴 ; by (50), we have 𝑏2𝑗+1 𝐵,𝑤1−𝑝 − 𝑏𝐵,𝑤 ≤ 𝑏2𝑗+1 𝐵,𝑤1−𝑝 − 𝑏2𝑗+1 𝐵 + 𝑏2𝑗+1 𝐵 − 𝑏𝐵 + 𝑏𝐵 − 𝑏𝐵,𝑤 𝜆/𝑝 × 𝑤(𝐵)(1−𝜆)/𝑝 𝑤(2𝑗+1 𝐵) ∞ ≤ 𝐶∑ 𝑗=1 (1−𝜆)/𝑝 𝑗+1 𝐷(𝑗+1)(1−𝜆)/𝑝 𝑤(𝐵)(𝜆−1)/𝑝 𝑤(𝐵)(𝜆−1)/𝑝 ≤ 𝐶𝑤(𝐵)(𝜆−1)/𝑝 , (61) ≤ 𝑤1−𝑝 (2𝑗+1 𝐵) ×∫ 2𝑗+1 𝐵 (56) 1−𝑝 𝑏 (𝑦) − 𝑏2𝑗+1 𝐵 𝑤(𝑦) 𝑑𝑦 Case (𝑃 = 𝑟 ) In this case, 𝑤 ∈ 𝐴 We can prove (51) by a similar analysis as in the proof of Theorem (in the case 𝑃 = 𝑟 ) and Case + 2𝑛 (𝑗 + 1) ‖𝑏‖BMO(R𝑛 ) + where 𝐷 > is a constant that appeared in (21) ∫ 𝑏 (𝑦) − 𝑏𝐵 𝑤 (𝑦) 𝑑𝑦 𝑤 (𝐵) 𝐵 Having disposed of the previous preliminary step, we can now return to the proofs of Theorems and := 𝐴 21 + 𝐴 22 + 𝐴 23 Proof of Theorem The task is now to find a constant 𝐶 such that for fixed ball 𝐵 = 𝐵(𝑥0 , 1), we can obtain Combining (23) with (49), we have 𝐴 23 = 𝑤 (𝐵) ∞ × ∫ 𝑤 ({𝑥 ∈ 𝐵 : 𝑏 (𝑦) − 𝑏𝐵 > 𝛼}) 𝑑𝛼 (57) ∞ ≤ 𝐶 ∫ 𝑒−𝐶2 𝛼𝛿/‖𝑏‖BMO(R𝑛 ) 𝑑𝛼 In the same manner we can see that 𝐴 21 ≤ 𝐶 (58) It follows immediately that 1/𝑝 𝑟 𝐴 ≤ 𝐶 (2𝑛 (𝑗 + 1) + 2) 𝑤1−𝑝 (2𝑗+1 𝐵) (59) Therefore 𝐴 ≤ 𝐶 (𝑗 + 1) 𝑤1−𝑝 (2𝑗+1 𝐵) 1/𝑝 𝑟 (60) A further use of (21) and 𝑤 ∈ 𝑝/𝑟 allow us to obtain 𝜆/𝑝 2𝑛𝑗/𝑟 𝑤(2𝑗+1 𝐵) ∑ 𝑗 2 𝐵 𝑗=1 ∞ × (∫ 2𝑗+1 𝐵 (62) We decompose 𝑓 = 𝑓𝜒2𝐵 + 𝑓𝜒(2𝐵)𝑐 := 𝑓1 + 𝑓2 and consider the corresponding splitting ≤ 𝐶 𝑝 ∫ TΩ,𝑏 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥 𝑤(𝐵)𝜆 𝐵 𝑝 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 1−𝑝 𝑝 𝑟 𝑏 (𝑦) − 𝑏𝐵,𝑤 𝑤(𝑦) 𝑑𝑦) 1/𝑝 𝑟 𝑝 ∫ TΩ,𝑏 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥 𝐵 𝑝 ≤ 𝐶 (∫ TΩ,𝑏 𝑓1 (𝑥) 𝑤 (𝑥) 𝑑𝑥 𝐵 (63) 𝑝 + ∫ TΩ,𝑏 𝑓2 (𝑥) 𝑤 (𝑥) 𝑑𝑥) 𝐵 =: 𝐿 + 𝐿𝐿 It follows from the 𝐿𝑝 (𝑤) boundedness of TΩ,𝑏 and 𝑤 ∈ 𝐴 𝑝/𝑟 ⊂ 𝐴 𝑝 that 𝑝 𝐿 ≤ 𝐶 ∫ 𝑓 (𝑥) 𝑤 (𝑥) 𝑑𝑥 2𝐵 𝑝 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 𝑤(𝐵)𝜆 (64) Journal of Function Spaces and Applications ∞ 𝑛𝑗/𝑟 ≤ 𝐶 [ ∑ 𝑗 𝐵 [𝑗=1 Then a further use of (5) derives that 𝑝 TΩ,𝑏 𝑓2 (𝑥) 𝑝 Ω (𝑥 − 𝑦) 𝑓2 (𝑦) 𝑏 (𝑥) − 𝑏 (𝑦) ≤ 𝐶(∫ 𝑑𝑦) 𝑛 R𝑛 𝑥 − 𝑦 Ω (𝑥0 − 𝑦) 𝑓 (𝑦) ≤ 𝐶 (∫ 𝑛 |𝑥0 −𝑦|>2 𝑥0 − 𝑦 𝑝 𝑓 (𝑦) 𝑤 (𝑦) 𝑑𝑦) 𝑗+1 𝐵 × (∫ ×(∫ (65) 2𝑗+1 𝐵 ∞ 𝑤(𝐵)(1−𝜆)/𝑝 𝑗=1 (1−𝜆)/𝑝 𝑤(2𝑗+1 𝐵) × (∑ 𝑝 Ω (𝑥0 − 𝑦) 𝑓 (𝑦) 𝑑𝑦) 𝑛 𝑥0 − 𝑦 |𝑥0 −𝑦|>2 (69) According to (64) and (69), we have completed the proof of Theorem Proof of Theorem The proof of Theorem is similar to that of Theorem 6, except using 𝑤 ∈ 𝐴 (𝑝,𝑞) We omit its proof here Acknowledgments By Lemma 10, we have 𝑝 𝐿𝐿 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 𝑤(𝐵)𝜆 (67) We proceed to estimate 𝐿𝐿 Without loss of generality, we assume that 𝑝 > 𝑟 Taking into account (20), (22), and Lemma 9, we have Ω (𝑥0 − 𝑦) 𝑓 (𝑦) 𝑑𝑦) 𝑛 𝑥0 − 𝑦 𝑝 × ∫ 𝑏 (𝑥) − 𝑏𝐵,𝑤 𝑤 (𝑥) 𝑑𝑥 𝐵 ∞ 𝑟 ≤ ( ∑ 𝑗 ∫ Ω (𝑥0 − 𝑦) 𝑑𝑦) 𝐵 𝐴 𝑗 𝑗=1 𝑟 𝑓 (𝑦) 𝑑𝑦) 2𝑗+1 𝐵 × (∫ ) 𝑤(𝐵)𝜆 𝑝 𝐿𝐿 ≤ 𝐶𝑓𝑀𝑝,𝜆 (𝑤) 𝑤(𝐵)𝜆 (66) = 𝐿𝐿 + 𝐿𝐿 𝑗 𝑗+1 𝑗=1