Boundedness of Sublinear Operators with Rough Kernels on Weighted Morrey Spaces

Abstract

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 function Ω over the unit sphere 𝑆^𝑛−1 of ℝ^𝑛(𝑛≥2) equipped with the normalized Lebesgue measure 𝑑𝜎 and 𝑥^′=𝑥/|𝑥|, a Calderón-Zygmund singular integral operator with rough kernel was given by 𝑇_Ω𝑓(𝑥)=p.v._ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^𝑛𝑓𝑦𝑑𝑦 (1) and a related maximal operator 𝑀_Ω𝑓(𝑥)=sup_𝑟>01𝑟_{𝑛𝐵(𝑥,𝑟)}Ω𝑥−𝑦𝑓𝑦𝑑𝑦, (2) where Ω is homogeneous of degree zero and satisfies Ω∈𝐿^𝑟𝑆^𝑛−1,1<𝑟≤∞_𝑆𝑛−1Ω𝑥^′𝑑𝑥^′=0. (3)(4) When Ω is a smooth kernel and 𝑇_Ω a standard Calderón-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∈ℝ^𝑛, 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 𝑇_Ω has no regularity, and so the operator 𝑇_Ω is called rough singular integral operator. In recent years, a variety of operators related to the singular integrals for Calderón-Zygmund, but lacking the smoothness required in the classical theory, have been studied. Duoandikoetxea [2] studied the norm inequalities for 𝑇_Ω in homogeneous case on weighted 𝐿^𝑝(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 𝒯_Ω𝑓(𝑥)≤𝐶_ℝ𝑛Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑛𝑑𝑦,𝑥∉supp𝑓 (5) for 𝑓∈𝐿¹(ℝ^𝑛) 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 𝑇_Ω (see [11]) and the oscillatory singular integral operator 𝑇_Ω𝑓(𝑥)=p.v._ℝ𝑛𝑒^{𝑖𝑃(𝑥,𝑦)}Ω𝑥−𝑦𝑥−𝑦^𝑛𝑓𝑦𝑑𝑦,𝑥∉supp𝑓, (6) where the phase is a polynomial. The boundedness of 𝑇_Ω on weighted 𝐿^𝑝(ℝ^𝑛)(1≤𝑝<∞) spaces was fully studied by Ojanen in his doctoral dissertation [12].

Let 𝐷_𝑘={𝑥∈ℝ^𝑛∶|𝑥|≤2^𝑘} and let 𝐴_𝑘=𝐷_𝑘⧵𝐷_𝑘−1 for 𝑘∈𝑍. Throughout this paper, we will denote by 𝜒_𝐸 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): 𝒯_Ω𝑓(𝑥)≤𝐶|𝑥|^−𝑛_ℝ𝑛Ω𝑥−𝑦𝑓𝑦𝑑𝑦, (7) when supp𝑓⊆𝐴_𝑘 and |𝑥|≥2^𝑘+1 with 𝑘∈ℤ and 𝒯_Ω𝑓(𝑥)≤𝐶2^−𝑘𝑛_ℝ𝑛Ω𝑥−𝑦𝑓𝑦𝑑𝑦, (8) when supp𝑓⊆𝐴_𝑘 and |𝑥|≤2^𝑘−1 with 𝑘∈ℤ, respectively. It is worth pointing out that 𝑀_Ω satisfies conditions (7) and (8). Also, condition (5) implies the size conditions (7) and (8) since |𝑥−𝑦|>|𝑥|/2 when |𝑥|≥2^𝑘+1 and supp𝑓⊆𝐴_𝑘 while supp𝑓⊆𝐴_𝑘 and |𝑥|≤2^𝑘−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_𝐵1|𝐵|_𝐵𝑤(𝑥)×1𝑑𝑥|𝐵|_𝐵𝑤(𝑥)^1−𝑝′𝑑𝑥^𝑝−1𝐴≤𝐶,1<𝑝<∞,_(𝑝,𝑞)∶sup_𝐵1|𝐵|_𝐵𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞×1|𝐵|_𝐵𝑤(𝑥)^−𝑝′𝑑𝑥^1/𝑝′≤𝐶,1<𝑝,𝑞<∞, (9) respectively, where 1/𝑝+1/𝑝^′=1. For 𝑝=1, the 𝐴₁ and 𝐴_(1,𝑞)(1<𝑞<∞) weights are defined by (𝑥)(𝑥),𝐴𝑀𝑤≤𝐶𝑤_(1,𝑞)∶sup_𝐵1|𝐵|_𝐵𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞×esssup_𝐵1𝑤(𝑥)≤𝐶, (10) respectively. Here esssup and the following essinf are the abbreviations of essential supremum and essential infimum, respectively. Clearly, 𝑤∈𝐴₁ if and only if there is a constant 𝐶>0 such that 1|𝐵|_𝐵𝑤(𝑥)𝑑𝑥≤𝐶essinf_𝐵𝑤(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. Let 1≤𝑝<∞,  0<𝜆<1 and let 𝑤 be a weight function. Then the weighted Morrey space 𝑀_𝑝,𝜆(𝑤) is defined by 𝑀_𝑝,𝜆(𝑤)=𝑓∶𝑓_{𝑀𝑝,𝜆(𝑤)}=sup_𝐵1𝑤(𝐵)^𝜆𝐵𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥^1/𝑝<∞, (12) where 𝑤(𝐵)=∫_𝐵𝑤(𝑥)𝑑𝑥. For 𝑤∈𝐴_𝑝(1≤𝑝<∞), if 𝜆=0, then 𝑀_𝑝,0(𝑤)=𝐿^𝑝(𝑤) while 𝜆=1 implies 𝑀_𝑝,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 operator 𝒯_Ω satisfy (7) and (8). If 𝒯_Ω is bounded on 𝐿^𝑝(𝑤) with 𝑤∈𝐴_𝑝/𝑟′, then 𝒯_Ω is bounded on 𝑀_𝑝,𝜆(𝑤).

When 𝑝=1, we have the following theorem.

Theorem 2. Let 1<𝑟<∞and 1/𝑟+𝜆<1 and let 𝒯_Ω satisfy (7) and (8). Then if 𝒯_Ω is bounded from 𝐿¹(𝑤) to 𝐿^1,∞(𝑤) with 𝑤∈𝐴₁, there exists a constant 𝐶>0 such that for all 𝜇>0 and all balls 𝐵, 𝑤𝑥∈𝐵∶𝒯_Ω𝑓(𝑥)>𝜇≤𝐶𝜇⁻¹𝑓_{𝑀1,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]. Let 1≤𝑝<∞,  0<𝜆<1. For two weights 𝑤₁ and 𝑤₂, 𝑀_𝑝,𝜆𝑤₁,𝑤₂=𝑓∶𝑓_{𝑀𝑝,𝑘(𝑤1,𝑤2)}=sup_𝐵1𝑤₂(𝐵)^𝜆𝐵𝑓(𝑥)^𝑝𝑤₁(𝑥)𝑑𝑥^1/𝑝<∞. (14) If 𝑤₁=𝑤₂=𝑤, we write 𝑀_𝑝,𝜆(𝑤₁,𝑤₁)=𝑀_𝑝,𝜆(𝑤₂,𝑤₂)=𝑀_𝑝,𝜆(𝑤).

We can get similar results for fractional integrals following the line of Theorems 1 and 2.

Theorem 3. Let 0<𝛼<𝑛, 1≤𝑟<∞, and 0<𝜆<1. Suppose that a sublinear operator 𝒯_𝛼,Ω satisfies the size conditions 𝒯_𝛼,Ω𝑓(𝑥|𝑥|)≤𝐶^{−(𝑛−𝛼)}_ℝ𝑛Ω𝑥−𝑦𝑓𝑦𝑑𝑦 (15) when supp𝑓⊆𝐴_𝑘 and |𝑥|≥2^𝑘+1 with 𝑘∈𝑍 and 𝒯_𝛼,Ω𝑓(𝑥)≤𝐶2^{−𝑘(𝑛−𝛼)}_ℝ𝑛Ω𝑥−𝑦𝑓𝑦𝑑𝑦 (16) when supp𝑓⊆𝐴_𝑘 and |𝑥|≤2^𝑘−1 with 𝑘∈𝑍. Then one has the following.(a)If 𝒯_𝛼,Ω maps 𝐿^𝑝(𝑤^𝑝) into 𝐿^𝑞(𝑤^𝑞) with 𝑤∈𝐴_{(𝑝/𝑟′,𝑞)}, then 𝒯_𝛼,Ω is bounded from 𝑀_𝑝,𝜆(𝑤^𝑝,𝑤^𝑞) to 𝑀_{𝑞,𝑞𝜆/𝑝}(𝑤^𝑞), where 𝑟^′≤𝑝<𝑛/𝛼,1/𝑞=1/𝑝−𝛼/𝑛 and 𝑝≤𝑞≤∞.(b)If 𝒯_𝛼,Ω is bounded from 𝐿¹(𝑤) to 𝐿^𝑞,∞(𝑤^𝑞) 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, 𝑤𝑥∈𝐵∶𝒯_𝛼,Ω𝑓(𝑥)>𝜇^1/𝑞≤𝐶𝜇⁻¹𝑓_{𝑀1,𝜆(𝑤,𝑤𝑞)}𝑤(𝐵)^𝜆, (17)where 1<𝑞<∞.

We emphasize that (15) and (16) are weaker conditions than the following condition: 𝒯_𝛼,Ω𝑓(𝑥)≤𝐶_ℝ𝑛Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦,0<𝛼<𝑛 (18) for any integral function 𝑓 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]: 𝒯_𝛼,Ω𝑓(𝑥)=_ℝ𝑛Ω𝑥−𝑦𝑓𝑦𝑥−𝑦^𝑛−𝛼𝑑𝑦,0<𝛼<𝑛. (19) For some mapping properties of 𝒯_𝛼,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 2 contains the proofs of Theorems 1 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.

2. Boundedness of Sublinear Operators

Proofs of Theorems 1 and 3 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 4. Let 1≤𝑝<∞ and 𝑤∈𝐴_𝑝. Then the following statements are true.(a) There exists a constant 𝐶 such that𝑤(2𝐵)≤𝐶𝑤(𝐵), (20) where 𝑤 satisfies this condition; one says 𝑤 satisfies the doubling condition.(b) There exists a constant 𝐶>1 such that𝑤(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 Hölder inequality holds for every ball 𝐵⊂ℝ^𝑛:1|𝐵|_𝐵𝑤(𝑥)^𝑟𝑑𝑥^1/𝑟1≤𝐶|𝐵|_𝐵𝑤(𝑥)𝑑𝑥. (22)(d) For all 𝜆>1, one has𝑤(𝜆𝐵)≤𝐶𝜆^𝑛𝑝𝑤(𝐵). (23)(e) There exist two constants 𝐶 and 𝛿>0 such that for any measurable set 𝑄⊂𝐵𝑤(𝑄)𝑤(𝐵)|𝑄|≤𝐶|𝐵|^𝛿, (24) if 𝑤 satisfies (24); one says 𝑤∈𝐴_∞.(f) For all 𝑝<𝑞<∞, one has𝐴_∞=∪_𝑝𝐴_𝑝,𝐴_𝑝⊂𝐴_𝑞. (25)

The following lemma about the rough kernel Ω is essential to our proofs. One can find its proof in [21].

Lemma 5. Let Ω∈𝐿^𝑟(𝑆^𝑛−1) with 1≤𝑟<∞. Then the following statements are true.(a)If  𝑥∈𝐴_𝑘 and 𝑗≥𝑘+1, then ∫_𝐴𝑗|Ω(𝑥−𝑦)|^𝑟𝑑𝑦≤𝐶2^𝑗𝑛.(b)If 𝑦∈𝐴_𝑘 and 𝑘≥𝑗+1, then ∫_𝐴𝑗|Ω(𝑥−𝑦)|^𝑟𝑑𝑥≤𝐶2^{𝑘(𝑛−1)+𝑗}.

Proof of Theorem 1. Let 1<𝑟^′≤𝑝<∞, 𝑤∈𝐴_𝑝/𝑟′, and 0<𝜆<1. Our task is to show 1𝑤(𝐵)^𝜆𝐵𝒯_Ω𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥≤𝐶𝑓^𝑝_{𝑀𝑝,𝜆(𝑤)}. (26) For a fixed ball 𝐵=𝐵(𝑥₀,𝑟), there is no loss of generality in assuming 𝑟=1. We decompose 𝑓=𝑓𝜒_2𝐵+𝑓𝜒_(2𝐵)^𝑐∶=𝑓₁+𝑓₂. Since 𝒯_Ω is a sublinear operator, so we get 1𝑤(𝐵)^𝜆𝐵𝒯_Ω𝑓(𝑥)^𝑝𝑤(𝑥)≤𝐶𝑑𝑥𝑤(𝐵)^𝜆𝐵𝒯_Ω𝑓₁(𝑥)^𝑝𝑤(𝑥)+𝐶𝑑𝑥𝑤(𝐵)^𝜆𝐵𝒯_Ω𝑓₂(𝑥)^𝑝𝑤(𝑥)𝑑𝑥∶=𝐼+𝐼𝐼. (27)
By the assumption on 𝒯_Ω and (25), we can obtain 𝐶𝐼≤𝑤(𝐵)^𝜆ℝ𝑛𝒯_Ω𝑓₁(𝑥)^𝑝𝑤(𝑥)≤𝐶𝑑𝑥𝑤(𝐵)^𝜆2𝐵𝑓(𝑥)^𝑝𝑤(𝑥)𝑓𝑑𝑥≤𝐶^𝑝_{𝑀𝑝,𝜆(𝑤)}. (28)
For the term 𝐼𝐼, by (8) we have 𝐶𝐼𝐼≤𝑤(𝐵)^𝜆𝐵∞_𝑘=12^−𝑘𝑛𝒯_Ω,𝑘𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥, (29) where 𝒯_Ω,𝑘𝑓(𝑥)=_𝐴𝑘+1Ω𝑥−𝑦𝑓𝑦𝑑𝑦. (30)
We distinguish two cases according to the size of 𝑝 and 𝑟 to get the estimates for 𝒯_Ω,𝑘.
Case 1   (𝑝>𝑟^′). In this case, 𝑤∈𝐴_𝑝/𝑟′ implies that _𝐵𝑤^{−𝑟′/(𝑝−𝑟′)}|𝐵|𝑑𝑦≤^{𝑝/(𝑝−𝑟′)}𝑤(𝐵)^{𝑟′/(𝑝−𝑟′)}. (31) By (31), Hölder’s inequality, and Lemma 5, we have 𝒯_Ω,𝑘𝑓(𝑥)≤𝐶_𝐴𝑘+1Ω𝑥−𝑦^𝑟𝑑𝑦^1/𝑟×_2𝑘+1𝐵𝑓𝑦^𝑟′𝑑𝑦^1/𝑟′≤𝐶2^{(𝑘+1)𝑛/𝑟}×_2𝑘+1𝐵𝑓𝑦^𝑟′𝑤𝑦^𝑟′/𝑝𝑤𝑦^{−𝑟′/𝑝}𝑑𝑦^1/𝑟′≤𝐶2^{(𝑘+1)𝑛/𝑟}_2𝑘+1𝐵𝑓𝑦^𝑝𝑤𝑦𝑑𝑦^1/𝑝×_2𝑘+1𝐵𝑤𝑦^{−𝑟′/(𝑝−𝑟′)}𝑑𝑦^{(𝑝−𝑟′)/(𝑝𝑟′)}≤𝐶𝑓_{𝑀𝑝,𝜆(𝑤)}2^{(𝑘+1)𝑛/𝑟}2^𝑘+1𝐵^1/𝑟′𝑤2^𝑘+1𝐵^{(1−𝜆)/𝑝}≤𝐶𝑓_{𝑀𝑝,𝜆(𝑤)}2^(𝑘+1)𝑛𝑤2^𝑘+1𝐵^{(1−𝜆)/𝑝}. (32)
Case 2  (𝑝=𝑟^′). In this case, 𝑤∈𝐴₁ implies that essinf_{𝑥∈2𝑘+1𝐵}𝑤(𝑥)⁻¹≤2^𝑘+1𝐵𝑤2^𝑘+1𝐵, (33) which in combination with the Hölder inequality and Lemma 5 yields that 𝒯_Ω,𝑘𝑓(𝑥)≤𝐶2^{(𝑘+1)𝑛/𝑟}×_2𝑘+1𝐵𝑓𝑦^𝑝𝑤𝑦𝑤𝑦⁻¹𝑑𝑦^1/𝑝≤𝐶𝑓_{𝑀𝑝,𝜆(𝑤)}2^(𝑘+1)𝑛𝑤2^𝑘+1𝐵^{(1−𝜆)/𝑝}. (34)
Substituting (32) and (34) into (29), we can assert that 𝐼𝐼≤𝐶𝑓^𝑝_{𝑀𝑝,𝜆(𝑤)}^∞_𝑘=1𝑤(𝐵)^{(1−𝜆)/𝑝}𝑤2^𝑘+1𝐵^{(1−𝜆)/𝑝𝑝}≤𝐶𝑓^𝑝_{𝑀𝑝,𝜆(𝑤)}, (35) where we have used (21) in the last inequality. Combining (28) and (29), we obtain the proof of Theorem 1.

Proof of Theorem 2. The task is now to show the following inequality: sup_𝜇>0𝜇𝑤(𝐵)^𝜆𝑤𝑥∈𝐵∶𝒯_Ω𝑓(𝑥𝑓)>𝜇≤𝐶^𝑝_{𝑀𝑝,𝜆(𝑤)}. (36) In order to get this inequality, it will be necessary to decompose 𝑓=𝑓𝜒_2𝐵+𝑓𝜒_(2𝐵)^𝑐∶=𝑓₁+𝑓₂ with 𝐵 as in Theorem 1. Since 𝒯_Ω is a sublinear operator, we can rewrite 𝑤𝑥∈𝐵∶𝒯_Ω𝑓(𝑥𝒯)>𝜇≤𝑤𝑥∈𝐵∶_Ω𝑓₁(𝑥>𝜇)2+𝑤𝑥∈𝐵∶𝒯_Ω𝑓₂(𝑥>𝜇)2∶=𝐽+𝐽𝐽. (37)
An application of (20) and the weighted weak type estimates for 𝒯_Ω yield that 𝐽≤𝐶𝜇⁻¹𝑓_𝑀1,𝜆(𝑤)𝑤(𝐵)^𝜇. (38)
To estimate the term 𝐽𝐽, we note that 𝐶𝐽𝐽≤𝜇_{{𝑥∈𝐵∶|𝒯Ω𝑓(𝑥)|>𝜇/2}}×^∞_𝑘=12^−𝑘𝑛𝒯_Ω,𝑘𝑓(𝑥)𝑤(𝑥)𝑑𝑥. (39) By (22), (33), and the Hölder inequality, we can estimate 𝐽𝐽 as 𝐶𝐽𝐽≤𝜇^∞_𝑘=12^−𝑘𝑛×_{2𝑘+1𝐵𝐵}Ω𝑤(𝑥)𝑓𝑦≤(𝐵)𝑥−𝑦𝑑𝑥𝑑𝑦𝐶𝑤𝜇^∞_𝑘=12^{−𝑘𝑛+𝑘(𝑛−1)/𝑟}×_2𝑘+1𝐵𝑓𝑦𝑤𝑦𝑤𝑦⁻¹≤(𝐵)𝑑𝑦𝐶𝑤𝜇𝑓_𝑀1,𝜆(𝑤)×^∞_𝑘=1𝑤2^𝑘+1𝐵^𝜆essinf_{𝑥∈2𝑘+1𝐵}𝑤⁻¹≤𝐶𝜇𝑓_𝑀1,𝜆(𝑤)×^∞_𝑘=12^{𝑘𝑛/𝑟−𝑘𝑛(1−𝜆)}𝑤(𝐵)^𝜆≤𝐶𝜇𝑓_𝑀1,𝜆(𝑤)𝑤(𝐵)^𝜆. (40) Combining these inequalities for 𝐽 and 𝐽𝐽, we have completed the proof of Theorem 2.

Proof of Theorem 3. We can use the similar arguments as in the proof of Theorem 1 and Theorem 2. For the proof of (𝑎), it suffices to show that 1𝑤^𝑞(𝐵)^{𝑞𝜆/𝑝𝐵}𝒯_𝛼,Ω𝑓(𝑥)^𝑞𝑤(𝑥)^𝑞𝑑𝑥≤𝐶𝑓^𝑞_{𝑀𝑝,𝜆(𝑤𝑝,𝑤𝑞)}. (41)
For a fixed ball 𝐵=𝐵(𝑥₀,1), we decompose 𝑓=𝑓𝜒_2𝐵+𝑓𝜒_(2𝐵)^𝑐∶=𝑓₁+𝑓₂. Since 𝒯_𝛼,Ω is a sublinear operator, we get 1𝑤^𝑞(𝐵)^{𝑞𝜆/𝑝𝐵}𝒯_𝛼,Ω𝑓(𝑥)^𝑞𝑤(𝑥)^𝑞≤1𝑑𝑥𝑤^𝑞(𝐵)^{𝑞𝜆/𝑝𝐵}𝒯_𝛼,Ω𝑓₁(𝑥)^𝑞+𝒯_𝛼,Ω𝑓₂(𝑥)^𝑞𝑤^𝑞(𝑥)𝑑𝑥∶=𝐾+𝐾𝐾. (42)
To estimate the term 𝐾, using the fact that 𝒯_𝛼,Ω is bounded from 𝐿^𝑝(𝑤^𝑝) to 𝐿^𝑞(𝑤^𝑞) with 𝑤∈𝐴_{(𝑝/𝑟′,𝑞)}, we can get _𝐵𝒯_𝛼,Ω𝑓₁(𝑥)^𝑞𝑤^𝑞(𝑥)𝑓𝑑𝑥≤𝐶^𝑞_{𝑀𝑝,𝜆(𝑤𝑝,𝑤𝑞)}𝑤^𝑞(𝐵)^{𝑞𝜆/𝑝}, (43) which implies that 𝐾≤𝐶𝑓_{𝑀𝑝,𝜆(𝑤𝑝,𝑤𝑞)}. (44)
For the term 𝐾𝐾, by the similar arguments as that of Theorem 1, we obtain 𝐾𝐾≤𝐶_𝑘2^{−𝑘(𝑛−𝛼)}_𝐴𝑘Ω𝑥−𝑦𝑓𝑦𝑑𝑦^𝑞×𝑤^𝑞(𝐵)^{1−𝑞𝜆/𝑝}≤𝐶_𝑘2^{−𝑘(𝑛−𝛼)+𝑛𝑘/𝑟}𝑓𝑦^𝑟′𝑑𝑦^{1/𝑟′𝑞}×𝑤^𝑞(𝐵)^{1−𝑞𝜆/𝑝}≤𝐶𝑓_{𝑀𝑝,𝜆(𝑤𝑝,𝑤𝑞)}×^∞_𝑘=1𝑤^𝑞(𝐵)^{(1/𝑞−𝜆/𝑝)}𝑤^𝑞2^𝑘+1𝐵^{(1/𝑞−𝜆/𝑝)𝑞}≤𝐶𝑓^𝑞_{𝑀𝑝,𝜆(𝑤𝑝,𝑤𝑞)}. (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 of 𝒯_𝛼,Ω similar to the proof of Theorem 2.

3. Boundedness of Commutators

We say that 𝑏 is a BMO(ℝ^𝑛) function if the following sharp maximal function is finite: 𝑏^#(𝑥)=sup_𝐵1|𝐵|_𝐵𝑏𝑦−𝑏_𝐵𝑑𝑦, (46) where the supreme is taken over all balls 𝐵⊂ℝ^𝑛 and 𝑓_𝐵=(1/|𝐵|)∫_𝐵𝑓(𝑦)𝑑𝑦. This means ‖𝑏‖_BMO(ℝ^𝑛₎=‖𝑏^#‖_𝐿^∞<+∞. An early work about BMO(ℝ^𝑛) space can be attributed to John and Nirenberg [22]. For 1<𝑝<∞, there is a close relation between BMO(ℝ^𝑛) and 𝐴_𝑝 weights: BMOℝ^𝑛=𝛼log𝑤∶𝑤∈𝐴_𝑝,𝛼≥0. (47)

Given an operator 𝑇 acting on a generic function 𝑓 and a function 𝑏, the commutator 𝑇_𝑏 is formally defined as 𝑇_𝑏𝑓=[𝑏,𝑇]𝑓=𝑏𝑇𝑓−𝑇𝑏𝑓. (48)

Since 𝐿^∞(ℝ^𝑛)⊊BMO(ℝ^𝑛), the boundedness of 𝑇_𝑏 is worse than 𝑇 (e.g., the singularity; see also [23]). Therefore, many authors want to know whether 𝑇_𝑏 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 𝒯_Ω and 𝒯_𝛼,Ω to 𝒯_Ω,𝑏 and 𝒯_𝛼,Ω,𝑏, respectively.

Theorem 6. Let 𝑟, 𝑝, 𝜆, and 𝑤 be as in Theorem 1. Suppose that the sublinear operator 𝒯_Ω satisfies condition (5) for any integral function 𝑓 with compact support. If 𝒯_Ω,𝑏 is bounded on 𝐿^𝑝(𝑤) with 𝑏∈𝐵𝑀𝑂(ℝ^𝑛), then 𝒯_Ω,𝑏 is bounded on 𝑀_𝑝,𝜆(𝑤).

Theorem 7. Let 𝑝, 𝑟, 𝑞, 𝛼, 𝑤, and 𝜆 be as in Theorem 3(𝑎) and let the sublinear operator 𝒯_𝛼,Ω satisfy condition (18) for any integral function 𝑓 with compact support. If 𝒯_𝛼,Ω,𝑏 maps 𝐿^𝑝(𝑤^𝑝) into 𝐿^𝑞(𝑤^𝑞)with 𝑏∈𝐵𝑀𝑂(ℝ^𝑛), then 𝒯_𝛼,Ω,𝑏 is bounded from 𝑀_𝑝,𝜆(𝑤^𝑝,𝑤^𝑞) to 𝑀_{𝑞,𝑞𝜆/𝑝}(𝑤^𝑞).

The following lemmas about BMO(ℝ^𝑛) functions will help us to prove Theorems 6 and 7.

Lemma 8 (see [25, Theorem 3.8]). Let 1≤𝑝<∞ and 𝑏∈𝐵𝑀𝑂(ℝ^𝑛). Then for any ball 𝐵⊂ℝ^𝑛, the following statements are true.(a)There exist constants 𝐶₁ and 𝐶₂ such that for all 𝛼>0𝑥∈𝐵∶𝑏(𝑥)−𝑏_𝐵>𝛼≤𝐶₁|𝐵|𝑒^{−𝐶2𝛼/‖𝑏‖𝑛)𝐵𝑀𝑂(ℝ}. (49)(b)Inequality (49) is called John-Nirenberg inequality: 𝑏₂^𝜆_𝐵−𝑏_𝐵≤2^𝑛𝜆‖𝑏‖_{𝐵𝑀𝑂(ℝ}^𝑛₎. (50)

Lemma 9 ([1, Proposition 7.1.2] (see also [14, Theorem 5])). Let 𝑤∈𝐴_∞ and 1<𝑝<∞. Then the following statements are equivalent:(a)‖𝑏‖_{𝐵𝑀𝑂(ℝ}^𝑛₎∼sup_𝐵((1/|𝐵|)∫_𝐵|𝑏(𝑥)−𝑏_𝐵|^𝑝𝑑𝑥)^1/𝑝, (b)‖𝑏‖_{𝐵𝑀𝑂(ℝ}^𝑛₎∼sup_𝐵inf_𝑎∈ℝ(1/|𝐵|)∫_𝐵|𝑏(𝑥)−𝑎|𝑑𝑥, (c)‖𝑏‖_{𝐵𝑀𝑂(𝑤)}=sup_𝐵(1/𝑤(𝐵))∫_𝐵|𝑏(𝑥)−𝑏_𝐵,𝑤|𝑤(𝑥)𝑑𝑥, where 𝐵𝑀𝑂(𝑤)={𝑏∶‖𝑏‖_{𝐵𝑀𝑂(𝑤)}<∞} and 𝑏_𝐵,𝑤=(1/𝑤(𝐵))∫_𝐵𝑏(𝑦)𝑤(𝑦)𝑑𝑦.

Lemma 10. Let 𝑝, 𝑟, 𝑏, and 𝑤 be as in Theorem 6 and let 𝐵=𝐵(𝑥₀,1) be a generic fixed ball. Then the inequality _{|𝑥0−𝑦|>2}Ω𝑥₀−𝑦𝑓𝑦𝑥₀−𝑦^𝑛𝑏_𝐵,𝑤−𝑏𝑦𝑑𝑦^𝑝≤𝐶𝑓^𝑝_{𝑀𝑝,𝜆(𝑤)}𝑤(𝐵)^𝜆−1 (51) holds for every 𝑦∈(2𝐵)^𝑐, where (2𝐵)^𝑐=ℝ^𝑛⧵(2𝐵).

Proof. We will consider two cases.
Case 1  (𝑃>𝑟^′). In this case, 𝑤∈𝐴_𝑝/𝑟′. Using Hölder’s inequality and Lemma 5 to the left-hand side of (51), we have _{|𝑥0−𝑦|>2}Ω𝑥₀−𝑦𝑓𝑦𝑥₀−𝑦^𝑛𝑏_𝐵,𝑤𝑦≤−𝑏𝑑𝑦_{∞𝑗=12𝑗<|𝑥0−𝑦|<2𝑗+1}Ω𝑥₀−𝑦𝑓𝑦𝑥₀−𝑦^𝑛×𝑏_𝐵,𝑤𝑦≤−𝑏𝑑𝑦_∞𝑗=112^𝑗𝐵_𝐴𝑗Ω𝑥₀−𝑦𝑓𝑦𝑏_𝐵,𝑤𝑦𝑓−𝑏𝑑𝑦≤𝐶_{𝑀𝑝,𝜆}^∞_(𝑤)𝑗=12^{𝑛𝑗/𝑟}𝑤2^𝑗+1𝐵^𝜆/𝑝2^𝑗𝐵×_2𝑗+1𝐵𝑏_𝐵,𝑤−𝑏𝑦^{𝑝𝑟′/(𝑝−𝑟′)}×𝑤𝑦^{−𝑟′/(𝑝−𝑟′)}𝑑𝑦^{(𝑝−𝑟′)/𝑝𝑟′}. (52)
Set 𝐴=_2𝑗+1𝐵𝑏_𝐵,𝑤−𝑏𝑦^{𝑝′𝑟′}𝑤^1−𝑝′𝑑𝑦^{1/𝑝′𝑟′}, (53) where 𝑝=𝑝/𝑟^′>1. Thus 𝐴≤_2𝑗+1𝐵𝑏_{2𝑗+1𝐵,𝑤1−𝑝′}𝑦+𝑏−𝑏_{2𝑗+1𝐵,𝑤1−𝑝′}−𝑏_𝐵,𝑤^{𝑝′𝑟′}×𝑤𝑦^1−𝑝′𝑑𝑦^{1/𝑝′𝑟′}≤_2𝑗+1𝐵𝑏_{2𝑗+1𝐵,𝑤1−𝑝′}−𝑏𝑦^{𝑝′𝑟′}×𝑤𝑦^1−𝑝′𝑑𝑦^{1/𝑝′𝑟′}+𝑏_{2𝑗+1𝐵,𝑤1−𝑝′}−𝑏_𝐵,𝑤×𝑤^1−𝑝′2^𝑗+1𝐵^{1/𝑝′𝑟′}=∶𝐴₁+𝐴₂. (54)
Lemma 9 implies that 𝐴₁≤𝐶𝑤^1−𝑝′2^𝑗+1𝐵^{1/𝑝′𝑟′}. (55)
We are now in a position to deal with 𝐴₂; by (50), we have 𝑏_{2𝑗+1𝐵,𝑤1−𝑝′}−𝑏_𝐵,𝑤≤𝑏_{2𝑗+1𝐵,𝑤1−𝑝′}−𝑏₂^𝑗+1_𝐵+𝑏₂^𝑗+1_𝐵−𝑏_𝐵+𝑏_𝐵−𝑏_𝐵,𝑤≤1𝑤^1−𝑝′2^𝑗+1𝐵×_2𝑗+1𝐵𝑏𝑦−𝑏₂^𝑗+1_𝐵𝑤𝑦^1−𝑝′𝑑𝑦+2^𝑛𝑗+1‖𝑏‖_BMO(ℝ^𝑛₎+1𝑤(𝐵)_𝐵𝑏𝑦−𝑏_𝐵𝑤𝑦𝑑𝑦∶=𝐴₂₁+𝐴₂₂+𝐴₂₃. (56)
Combining (23) with (49), we have 𝐴₂₃=1𝑤(𝐵)×^∞₀𝑤𝑥∈𝐵∶𝑏𝑦−𝑏_𝐵>𝛼𝑑𝛼≤𝐶^∞₀𝑒^{−𝐶2𝛼𝛿/‖𝑏‖BMO𝑛)(ℝ}𝑑𝛼≤𝐶. (57)
In the same manner we can see that 𝐴₂₁≤𝐶. (58) It follows immediately that 𝐴₂≤𝐶2^𝑛𝑗+1+2𝑤^1−𝑝′2^𝑗+1𝐵^{1/𝑝′𝑟′}. (59) Therefore 𝐴≤𝐶𝑗+1𝑤^1−𝑝′2^𝑗+1𝐵^{1/𝑝′𝑟′}. (60)
A further use of (21) and 𝑤∈𝑝/𝑟^′ allow us to obtain ^∞_𝑗=12^{𝑛𝑗/𝑟}𝑤2^𝑗+1𝐵^𝜆/𝑝2^𝑗𝐵×_2𝑗+1𝐵𝑏𝑦−𝑏_𝐵,𝑤^{𝑝′𝑟′}𝑤𝑦^1−𝑝′𝑑𝑦^{1/𝑝′𝑟′}≤^∞_𝑗=12^{𝑛𝑗/𝑟}𝑤2^𝑗+1𝐵^𝜆/𝑝2^𝑗𝐵×𝑗+1𝑤𝑦^1−𝑝′2^𝑗+1𝐵^{1/𝑝′𝑟′}≤𝐶^∞_𝑗=12^{𝑛𝑗/𝑟}2^𝑗+1𝐵^1/𝑟′𝑗+12^𝑗𝐵×𝑤(𝐵)^{(1−𝜆)/𝑝}𝑤2^𝑗+1𝐵^{(1−𝜆)/𝑝}𝑤(𝐵)^{(𝜆−1)/𝑝}≤𝐶^∞_𝑗=1𝑗+1𝐷^{(𝑗+1)(1−𝜆)/𝑝}𝑤(𝐵)^{(𝜆−1)/𝑝}≤𝐶𝑤(𝐵)^{(𝜆−1)/𝑝}, (61) where 𝐷>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 of Theorem 1 (in the case 𝑃=𝑟^′) and Case 1.

Having disposed of the previous preliminary step, we can now return to the proofs of Theorems 6 and 7.

Proof of Theorem 6. The task is now to find a constant 𝐶 such that for fixed ball 𝐵=𝐵(𝑥₀,1), we can obtain 1𝑤(𝐵)^𝜆𝐵𝒯_Ω,𝑏𝑓(𝑥)^𝑝𝑤(𝑥)𝑓𝑑𝑥≤𝐶^𝑝_{𝑀𝑝,𝜆(𝑤)}. (62)
We decompose 𝑓=𝑓𝜒_2𝐵+𝑓𝜒_(2𝐵)^𝑐∶=𝑓₁+𝑓₂ and consider the corresponding splitting _𝐵𝒯_Ω,𝑏𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥≤𝐶_𝐵𝒯_Ω,𝑏𝑓₁(𝑥)^𝑝𝑤(𝑥)𝑑𝑥+_𝐵𝒯_Ω,𝑏𝑓₂(𝑥)^𝑝𝑤(𝑥)𝑑𝑥=∶𝐿+𝐿𝐿. (63)
It follows from the 𝐿^𝑝(𝑤) boundedness of 𝒯_Ω,𝑏 and 𝑤∈𝐴_𝑝/𝑟′⊂𝐴_𝑝 that 𝐿≤𝐶_2𝐵𝑓(𝑥)^𝑝𝑤(𝑥)𝑓𝑑𝑥≤𝐶^𝑝_{𝑀𝑝,𝜆(𝑤)}𝑤(𝐵)^𝜆. (64) Then a further use of (5) derives that 𝒯_Ω,𝑏𝑓₂(𝑥)^𝑝≤𝐶_ℝ𝑛Ω𝑥−𝑦𝑓₂𝑦𝑏(𝑥)−𝑏𝑦𝑥−𝑦^𝑛𝑑𝑦^𝑝≤𝐶_{|𝑥0−𝑦|>2}Ω𝑥₀−𝑦𝑓𝑦𝑥₀−𝑦^𝑛×𝑏(𝑥)−𝑏_𝐵,𝑤+𝑏_𝐵,𝑤−𝑏𝑦𝑑𝑦^𝑝. (65) Therefore 𝐿𝐿≤𝐶_{|𝑥0−𝑦|>2}Ω𝑥₀−𝑦𝑓𝑦𝑥₀−𝑦^𝑛𝑑𝑦^𝑝×_𝐵𝑏(𝑥)−𝑏_𝐵,𝑤^𝑝𝑤(𝑥)𝑑𝑥+𝐶_{|𝑥0−𝑦|>2}Ω𝑥₀−𝑦𝑓𝑦𝑥₀−𝑦^𝑛×𝑏𝑦−𝑏_𝐵,𝑤𝑑𝑦^𝑝𝑤(𝐵)∶=𝐿𝐿₁+𝐿𝐿₂. (66)
By Lemma 10, we have 𝐿𝐿₂≤𝐶𝑓^𝑝_{𝑀𝑝,𝜆(𝑤)}𝑤(𝐵)^𝜆. (67)
We proceed to estimate 𝐿𝐿₁. Without loss of generality, we assume that 𝑝>𝑟^′. Taking into account (20), (22), and Lemma 9, we have 𝐿𝐿₁=^∞_{𝑗=12𝑗<|𝑥0−𝑦|<2𝑗+1}Ω𝑥₀−𝑦𝑓𝑦𝑥₀−𝑦^𝑛𝑑𝑦^𝑝×_𝐵𝑏(𝑥)−𝑏_𝐵,𝑤^𝑝𝑤(𝑥)≤𝑑𝑥_∞𝑗=112^𝑗𝐵_𝐴𝑗Ω𝑥₀−𝑦^𝑟𝑑𝑦^1/𝑟×_2𝑗+1𝐵𝑓𝑦^𝑟′𝑑𝑦^1/𝑟′×_𝐵𝑏(𝑥)−𝑏_𝐵,𝑤^𝑝𝑤(𝑥)𝑑𝑥≤𝐶^∞_𝑗=12^{𝑛𝑗/𝑟}2^𝑗𝐵×_2𝑗+1𝐵𝑓𝑦^𝑝𝑤𝑦𝑑𝑦^1/𝑝×_2𝑗+1𝐵𝑤𝑦^{−𝑟′/(𝑝−𝑟′)}𝑑𝑦^{(𝑝−𝑟′)/𝑝𝑟′𝑝}×_𝐵𝑏(𝑥)−𝑏_𝐵,𝑤^𝑝𝑤(𝑥)𝑓𝑑𝑥≤𝐶^𝑝_{𝑀𝑝,𝜆(𝑤)}×^∞_𝑗=1𝑤(𝐵)^{(1−𝜆)/𝑝}𝑤2^𝑗+1𝐵^{(1−𝜆)/𝑝𝑝}𝑤(𝐵)^𝜆≤𝐶𝑓^𝑝_{𝑀𝑝,𝜆(𝑤)}𝑤(𝐵)^𝜆. (68) Hence 𝐿𝐿≤𝐶𝑓^𝑝_{𝑀𝑝,𝜆(𝑤)}𝑤(𝐵)^𝜆. (69)
According to (64) and (69), we have completed the proof of Theorem 6.

Proof of Theorem 7. The proof of Theorem 7 is similar to that of Theorem 6, except using 𝑤∈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 supported by NSF of China (Grants nos. 11271175, 10901076, and 11171345) and NSF of Shandong Province (Grant no. ZR2012AQ026).