Multilinear Singular Integral Operators on Generalized Weighted Morrey Spaces

Abstract

The purpose of this paper is to discuss the boundedness properties of multilinear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces.

1. Introduction and Results

Multilinear Calderón-Zygmund theory is a natural generalization of the linear case. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. In 2009, the weighted estimates of multilinear Calderón-Zygmund singular integral operator and its commutator were established in [5] by Lerner et al. In 2013, the results of [5] were extended to the weighted Morrey space (see [6, 7]). In this paper, we will discuss the boundedness properties of multilinear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces.

Let ℝ^𝑛 be the 𝑛-dimensional Euclidean space, and let (ℝ^𝑛)^𝑚=ℝ^𝑛×⋯×ℝ^𝑛 be the 𝑚-fold product space (𝑚∈ℕ). We denote by 𝒮(ℝ^𝑛) the space of all Schwartz functions on ℝ^𝑛 and denote by 𝒮^′(ℝ^𝑛) its dual space of the set of all tempered distributions on ℝ^𝑛.

We say that a locally integrable function 𝐾(𝑦₀,𝑦₁,…,𝑦_𝑚) defined away from the diagonal 𝑦₀=𝑦₁=⋯=𝑦_𝑚 in (ℝ^𝑛)^𝑚+1 is a kernel in the class 𝑚-𝐶𝑍𝐾(𝐴,𝜖) if it satisfies the size estimate 𝐾𝑦₀,𝑦₁,…,𝑦_𝑚≤𝐴∑^𝑚_𝑘,𝑙=0𝑦_𝑙−𝑦_𝑘^𝑚𝑛 (1) for some 𝐴>0 and all (𝑦₀,𝑦₁,…,𝑦_𝑚)∈(ℝ^𝑛)^𝑚+1 with 𝑦_𝑙≠𝑦_𝑘 for some 𝑙, 𝑘. Moreover, assume that for some 𝜖>0 and all 0≤𝑗≤𝑚 it satisfies the smoothness estimates 𝐾𝑦₀,…,𝑦_𝑗,…,𝑦_𝑚−𝐾𝑦₀,…,𝑦^′_𝑗,…,𝑦_𝑚≤𝐴𝑦_𝑗−𝑦^′_𝑗^𝜖∑^𝑚_𝑘,𝑙=0𝑦_𝑙−𝑦_𝑘^{𝑚𝑛+𝜖} (2) whenever 𝑦_𝑗−𝑦^′_𝑗≤12max_{0≤𝑘≤𝑚}𝑦_𝑗−𝑦_𝑘. (3)

Let 𝑇 be a multilinear operator initially defined on the 𝑚-fold product of Schwartz spaces, and, by taking values into the space of tempered distributions, 𝑇∶𝒮ℝ^𝑛×⋯×𝒮ℝ^𝑛⟶𝒮^′ℝ^𝑛. (4) Following [2], we say that 𝑇 is an 𝑚-linear Calderón-Zygmund operator if, for some 𝑞₁,…,𝑞_𝑚∈[1,∞) and 𝑞∈(0,∞) with 1/𝑞=∑^𝑚_𝑖=11/𝑞_𝑖, it extends to a bounded multilinear operator from 𝐿^𝑞1(ℝ^𝑛)×⋯𝐿^𝑞𝑚(ℝ^𝑛) into 𝐿^𝑞(ℝ^𝑛) and if there exists a kernel function 𝐾(𝑥,𝑦₁,…,𝑦_𝑚) in the class 𝑚-𝐶𝑍𝐾(𝐴,𝜖), defined away from the diagonal 𝑥=𝑦₁=⋯=𝑦_𝑚 in (ℝ^𝑛)^𝑚+1, satisfying 𝑇𝑓(𝑥)=𝑇𝑓₁,…,𝑓_𝑚(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑓𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚 (5) for all 𝑥∉⋂^𝑚_𝑗=1supp𝑓_𝑗 and 𝑓₁,…,𝑓_𝑚 are 𝐶^∞ functions with compact support.

Theorem 1 (see [5]). Let 𝑚≥2 and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. Suppose that 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, 𝜔=(𝜔₁,…,𝜔_𝑚) satisfies the 𝐴_𝑝 condition, and 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖.(1)If 1<𝑝_𝑖<∞, 𝑖=1,…,𝑚, then 𝑇𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖)}. (6)(2)If 1≤𝑝_𝑖<∞, 𝑖=1,…,𝑚, and at least one of the 𝑝_𝑖=1, then 𝑇𝑓_{𝑊𝐿𝑝(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖)}. (7)

The purpose of this paper is to discuss the boundedness properties of 𝑚-linear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces 𝑀^𝑝_𝜑(𝜔).

The classical Morrey spaces 𝐿^𝑝,𝜆 were originally introduced by Morrey in [8] to study the local behavior of solutions to the second-order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the readers to [9–11]. Moreover, various Morrey spaces are defined in the process of study. Mizuhara [12] introduced the generalized Morrey space 𝐿^𝑝,𝜑; Komori and Shirai [13] defined the weighted Morrey spaces 𝐿^𝑝,𝜅(𝜔); Guliyev [14] gave a concept of generalized weighted Morrey space 𝑀^𝑝_𝜑(𝜔) which could be viewed as extension of both 𝐿^𝑝,𝜑 and 𝐿^𝑝,𝜅(𝜔). The boundedness of some operators on these Morrey spaces can be seen in [12–16].

Our first result can be formulated as follows.

Theorem 2. Let 𝑚≥2, let 𝑇 be an 𝑚-linear Calderón-Zygmund operator, and let 𝜑_𝑘=(𝜑_𝑘1,…,𝜑_𝑘𝑚), 𝑘=1,2. Suppose that 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, 𝜔=(𝜔₁,…,𝜔_𝑚) satisfies the 𝐴_𝑝 condition with 𝜔₁,…,𝜔_𝑚∈𝐴_∞, and (𝜑₁,𝜑₂) satisfies the condition ^∞_𝑠essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑚_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑟))^1/𝑝𝑖𝑑𝑟𝑟≤𝐶𝜑₂(𝑥,𝑠). (8)(1)If 1<𝑝_𝑖<∞, 𝑖=1,…,𝑚, then 𝑇𝑓_{𝑀𝑝𝜑2(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}. (9)(2)If 1≤𝑝_𝑖<∞, 𝑖=1,…,𝑚, and at least one of the 𝑝_𝑖=1, then 𝑇𝑓_{𝑊𝑀𝑝𝜑2(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}, (10) where 𝜑₂=∏^𝑚_𝑖=1𝜑_2𝑖 and 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖.

Given a locally integrable vector function 𝑏=(𝑏₁,…,𝑏_𝑚)∈(𝐵𝑀𝑂)^𝑚, the commutator of 𝑏 and the 𝑚-linear Calderón-Zygmund operator 𝑇, denoted here by 𝑇_Σ𝑏, was introduced in [5] and is defined via 𝑇_Σ𝑏𝑓=^𝑚_𝑖=1𝑇^𝑖_𝑏𝑖𝑓, (11) where 𝑇^𝑖_𝑏𝑖𝑓=𝑏_𝑖𝑇𝑓−𝑇𝑓₁,…,𝑏_𝑖𝑓_𝑖,…,𝑓_𝑚. (12) And the iterated commutator 𝑇_Π𝑏 was introduced in [17] and is defined by 𝑇_Π𝑏𝑓=𝑏₁,…,𝑏_𝑚−1,𝑏_𝑚,𝑇_𝑚𝑚−1⋯₁𝑓. (13)

To clarify the notation, if 𝑇 is associated in the usual way with a Calderón-Zygmund kernel 𝐾, then, at a formal level, 𝑇_Σ𝑏𝑓(𝑥)=_{(ℝ𝑛)𝑚}^𝑚_𝑖=1𝑏_𝑖(𝑥)−𝑏_𝑖𝑦_𝑖𝐾𝑥,𝑦₁,…,𝑦_𝑚×𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚,𝑇_Π𝑏𝑓(𝑥)=_{(ℝ𝑛)𝑚}^𝑚_𝑖=1𝑏_𝑖(𝑥)−𝑏_𝑖𝑦_𝑖𝐾𝑥,𝑦₁,…,𝑦_𝑚×𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚. (14)

For 𝑇 is an 𝑚-linear Calderón-Zygmund operator, 𝜔∈𝐴_𝑝, and 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖 with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚 and 𝑏∈(𝐵𝑀𝑂)^𝑚, Lerner et al. [5] proved that 𝑇_Σ𝑏 bounded from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑚) to 𝐿^𝑝(𝜈_𝜔) and Pérez et al. [17] extended the result to 𝑇_Π𝑏. Our second result is to extend their results to the generalized weighted Morrey space.

Theorem 3. Let 𝑚≥2 and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. Suppose that 𝑝₁,…,𝑝_𝑚∈(1,∞), 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, 𝜔=(𝜔₁,…,𝜔_𝑚) satisfies the 𝐴_𝑝 condition with 𝜔₁,…,𝜔_𝑚∈𝐴_∞, and (𝜑₁,𝜑₂) satisfies the condition ^∞_𝑠𝑟1+ln𝑠^𝑚essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑚_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑟))^1/𝑝𝑖𝑑𝑟𝑟≤𝐶𝜑₂(𝑥,𝑠). (15) If𝑏_𝑖∈𝐵𝑀𝑂,𝑏=(𝑏₁,…,𝑏_𝑚), then there exist constants 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇_Σ𝑏𝑓_{𝑀𝑝𝜑2(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗⋅^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)},𝑇_Π𝑏𝑓_{𝑀𝑝𝜑2(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗⋅^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}, (16) where 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖 and 𝜑₂=∏^𝑚_𝑖=1𝜑_2𝑖.

Example 4. Let 𝜑_1𝑖(𝑥,𝑡)=𝜑_2𝑖(𝑥,𝑡)=𝜔_𝑖(𝐵(𝑥,𝑡))^{(𝜅−1)/𝑝𝑖} and 𝜔_𝑖∈𝐴_∞ for 𝑖=1,…,𝑚, 0<𝜅<1. Then, (𝜑₁,𝜑₂) satisfies conditions (8) and (15).

In fact, from (34), in Section 2, we have a constant 𝛿>0 such that 𝜔_𝑖𝐵𝑥,2^𝑘𝑠≥𝐶2^𝛿𝑛𝑘𝜔_𝑖(𝐵(𝑥,𝑠)) (17) for all 𝑖. Then, ^∞_𝑠essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑚_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑟))^1/𝑝𝑖𝑑𝑟𝑟≤^∞_𝑠𝑟1+ln𝑠^𝑚×essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑚_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑟))^1/𝑝𝑖𝑑𝑟𝑟=^∞_𝑘=0^2𝑘+1𝑠_2𝑘𝑠𝑟1+ln𝑠^𝑚𝑚_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑟))^{(𝜅−1)/𝑝𝑖}𝑑𝑟𝑟≤𝐶^∞_𝑘=0𝑘^𝑚𝑚_𝑖=1𝜔_𝑖𝐵𝑥,2^𝑘+1𝑠^{(𝜅−1)/𝑝𝑖}≤𝐶^∞_𝑘=0𝑘^𝑚2^{𝑚(𝑘+1)𝑛𝛿((𝜅−1)/𝑝)}_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑠))^{(𝜅−1)/𝑝𝑖}≤𝐶^𝑚_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑠))^{(𝜅−1)/𝑝𝑖}=𝐶^𝑚_𝑖=1𝜑_2𝑖(𝑥,𝑠). (18) Hence, we get the following corollaries.

Corollary 5 (see [6]). Let 𝑚≥2 and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. Suppose that 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, 0<𝜅<1, 𝜔=(𝜔₁,…,𝜔_𝑚) satisfies the 𝐴_𝑝 condition with 𝜔₁,…,𝜔_𝑚∈𝐴_∞, and 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖.(1)If 1<𝑝_𝑖<∞, 𝑖=1,…,𝑚, then 𝑇𝑓_{𝐿𝑝,𝜅(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝜔𝑖)}. (19)(2)If 1≤𝑝_𝑖<∞, 𝑖=1,…,𝑚, and at least one of the 𝑝_𝑖=1, then 𝑇𝑓_{𝑊𝐿𝑝,𝜅(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝜔𝑖)}. (20)

Corollary 6. Let 𝑚≥2 and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. Suppose that 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, 0<𝜅<1, 𝜔=(𝜔₁,…,𝜔_𝑚) satisfies the 𝐴_𝑝 condition with 𝜔₁,…,𝜔_𝑚∈𝐴_∞, and 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖. If 𝑏_𝑖∈𝐵𝑀𝑂, 𝑏=(𝑏₁,…,𝑏_𝑚), then there exist constants 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇_Σ𝑏𝑓_{𝐿𝑝,𝜅(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗⋅^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝜔𝑖)},𝑇_Π𝑏𝑓_{𝐿𝑝,𝜅(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗⋅^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝜔𝑖)}. (21)

2. Definitions and Preliminaries

A weight 𝜔 is a nonnegative, locally integrable function on ℝ^𝑛. Let 𝐵=𝐵(𝑥₀,𝑟_𝐵) denote the ball with the center 𝑥₀ and radius 𝑟_𝐵. For any ball 𝐵 and 𝜆>0, 𝜆𝐵 denotes the ball concentric with 𝐵 whose radius is 𝜆 times as long. For a given weight function 𝜔 and a measurable set 𝐸, we also denote the Lebesgue measure of 𝐸 by |𝐸| and set the weighted measure 𝜔(𝐸)=∫_𝐸𝜔(𝑥)𝑑𝑥.

Let 1≤𝑝<∞, let 𝜑 be a positive measurable function on ℝ^𝑛×(0,∞), and let 𝜔 be a nonnegative measurable function on ℝ^𝑛. Following [14], we denote by 𝑀^𝑝_𝜑(𝜔) the generalized weighted Morrey space and the space of all functions 𝑓∈𝐿^𝑝_loc(𝜔) with finite norm 𝑓_𝑀^𝑝_𝜑(𝑤)=sup_{𝑥∈ℝ𝑛,𝑟>0}1𝜑()1𝑥,𝑟𝑓𝑤(𝐵(𝑥,𝑟))^{𝑝𝐿𝑝(𝜔,𝐵(𝑥,𝑟))1/𝑝}, (22) where 𝑓_{𝐿𝑝(𝜔,𝐵(𝑥,𝑟))}=_{𝐵(𝑥,𝑟)}𝑓𝑦^𝑝𝑤𝑦𝑑𝑦. (23) Furthermore, by 𝑊𝑀^𝑝_𝜑(𝜔) we denote the weak generalized weighted Morrey space of all functions 𝑓∈𝑊𝑀^𝑝_𝜑(𝜔) for which 𝑓_𝑊𝑀^𝑝_𝜑(𝑤)=sup_{𝑥∈ℝ𝑛,𝑟>0}1𝜑()1𝑥,𝑟𝑓𝑤(𝐵(𝑥,𝑟))^{𝑝𝑊𝐿𝑝(𝜔,𝐵(𝑥,𝑟))1/𝑝}, (24) where 𝑓_{𝑊𝐿𝑝(𝜔,𝐵(𝑥,𝑟))}=sup_𝑡>0𝑡𝜔({𝑦∈𝐵(𝑥,𝑟)∶|𝑓(𝑦)|>𝑡})^1/𝑝. (25)(1)If 𝜔=1 and 𝜑(𝑥,𝑟)=𝑟^{(𝜆−𝑛)/𝑝} with 0<𝜆<𝑛, then 𝑀^𝑝_𝜑(𝜔)=𝐿^𝑝,𝜆 is the classical Morrey space.(2)If 𝜑(𝑥,𝑟)=𝜔(𝐵(𝑥,𝑟))^{(𝜅−1)/𝑝}, then 𝑀^𝑝_𝜑(𝜔)=𝐿^𝑝,𝜅(𝜔) is the weighted Morrey space.(3)If 𝜑(𝑥,𝑟)=𝜈(𝐵(𝑥,𝑟))^𝜅/𝑝𝜔(𝐵(𝑥,𝑟))^−1/𝑝, then 𝑀^𝑝_𝜑(𝜔)=𝐿^𝑝,𝜅(𝜈,𝜔) is the two weighted Morrey space.(4)If 𝜔=1, then 𝑀^𝑝_𝜑(𝜔)=𝑀^𝑝_𝜑 is the generalized Morrey space.(5)If 𝜑(𝑥,𝑟)=𝜔(𝐵(𝑥,𝑟))^−1/𝑝, then 𝑀^𝑝_𝜑(𝜔)=𝐿^𝑝(𝜔).

A weight 𝜔 is said to belong to 𝐴_𝑝, for 1<𝑝<∞, if there exists a constant 𝐶 such that, for every ball 𝐵⊂ℝ^𝑛, 1|𝐵|_𝐵𝜔(𝑥)1𝑑𝑥|𝐵|_𝐵𝜔(𝑥)^1−𝑝′𝑑𝑥^𝑝−1≤𝐶, (26) where 𝑠^′ is the dual of 𝑠 such that 1/𝑠+1/𝑠^′=1. The class 𝐴₁ is defined by replacing the above inequality with 1|𝐵|_𝐵𝑤𝑦𝑑𝑦≤𝐶⋅essinf_𝑥∈𝐵𝑤(𝑥)foreveryball𝐵⊂ℝ^𝑛. (27)

A weight 𝜔 is said to belong to 𝐴_∞ if there are positive numbers 𝐶 and 0<𝛿<1 so that 𝜔(𝐸)𝜔(𝐵)|𝐸|≤𝐶|𝐵|^𝛿 (28) for all balls 𝐵 and all measurable 𝐸⊂𝐵. It is well known that 𝐴_∞=_1≤𝑝<∞𝐴_𝑝. (29)

By (26), we have _𝐵𝜔(𝑥)𝑑𝑥_𝐵𝜔(𝑥)^1−𝑝′𝑑𝑥^𝑝−1≤𝐶|𝐵|^𝑝 (30) for 1<𝑝<∞. Note that essinf_𝑥∈𝐸𝑓(𝑥)⁻¹=esssup_𝑥∈𝐸1𝑓(𝑥) (31) is true for any real-valued nonnegative function 𝑓 and measurable on 𝐸 (see [18], p.143) and, by (27), we get 𝜔⁻¹_{𝐿∞(𝐵)}=esssup_𝑥∈𝐵1𝑤(𝑥)=1essinf_𝑥∈𝐵𝑤(𝑥)≤𝐶|𝐵|𝜔(𝐵)⁻¹. (32)

The classical 𝐴_𝑝 weight theory was first introduced by Muckenhoupt in the study of weighted 𝐿^𝑝 boundedness of Hardy-Littlewood maximal function in [19].

Lemma 7 (see [19, 20]). Suppose that 𝜔∈𝐴_𝑝 and the following statements hold.(i)For any 1≤𝑝<∞, there are positive numbers 𝐶 and 0<𝛿<1 such that 𝜔(2𝐵)≤𝐶2^𝑛𝑝𝜔(𝐵),𝜔(2𝐵)≥𝐶2^𝑛𝛿𝜔(𝐵); (33)(34)(ii)𝐴_𝑝1⊂𝐴_𝑝2 for any 1≤𝑝₁<𝑝₂≤∞;(iii)for any 1<𝑝<∞, one has 𝜔^1−𝑝′∈𝐴_𝑝′.

Now, let us recall the definition of multiple weights. For 𝑚 exponents 𝑝₁,…,𝑝_𝑚, we write 𝑝=(𝑝₁,…,𝑝_𝑚). Let 𝑝₁,…,𝑝_𝑚∈[1,∞), and let 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, 𝜔=(𝜔₁,…,𝜔_𝑚). Given 𝜔=(𝜔₁,…,𝜔_𝑚), set 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖. We say that 𝜔 satisfies the 𝐴_𝑝 condition if it satisfies sup_𝐵1|𝐵|_𝐵𝜈_𝜔(𝑥)𝑑𝑥^𝑚1/𝑝_𝑖=11|𝐵|_𝐵𝜔_𝑖(𝑥)^{1−𝑝′𝑖}(𝑥)𝑑𝑥^{1/𝑝′𝑖}<∞. (35) When 𝑝_𝑖=1, (1/|𝐵|)∫_𝐵𝜔_𝑖(𝑥)^{1−𝑝′𝑖}(𝑥)𝑑𝑥 is understood as (inf_𝑥∈𝐵𝜔_𝑖(𝑥))⁻¹.

Lemma 8 (see [5]). Let 𝑝₁,…,𝑝_𝑚∈[1,∞), and let 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘. Then, 𝜔=(𝜔₁,…,𝜔_𝑚)∈𝐴_𝑝 if and only if 𝜈_𝜔∈𝐴_𝑝𝑚,𝜔^{1−𝑝′𝑖}_𝑖∈𝐴_{𝑚𝑝′𝑖}𝑓𝑜𝑟𝑖=1,…,𝑚, (36) where 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖 and the condition 𝜔^{1−𝑝′𝑖}_𝑖∈𝐴_{𝑚𝑝′𝑖} in the case 𝑝_𝑖=1 is understood as 𝜔_𝑖^1/𝑚∈𝐴₁.

Lemma 9 (see [6]). Let 𝑝₁,…,𝑝_𝑚∈[1,∞), and let 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘. Assume that 𝜔₁,…,𝜔_𝑚∈𝐴_∞ and 𝜈_𝜔=∏^𝑚_𝑖=1𝜔^{𝑝/𝑝𝑖}_𝑖. Then, for any ball 𝐵, there exists a constant 𝐶>0 such that ^𝑚_𝑖=1𝐵𝜔_𝑖(𝑥)𝑑𝑥^{𝑝/𝑝𝑖}≤𝐶_𝐵𝜈_𝜔(𝑥)𝑑𝑥. (37)

Let us recall the definition and some properties of 𝐵𝑀𝑂. Following [21], a locally integrable function 𝑏 is said to be in 𝐵𝑀𝑂 if sup_{𝐵⊂ℝ𝑛}1|𝐵|_𝐵𝑏(𝑥)−𝑏_𝐵𝑑𝑥=‖𝑏‖_∗<∞, (38) where 𝑏_𝐵=|𝐵|⁻¹∫_𝐵𝑏(𝑦)𝑑𝑦.

Lemma 10 (see [22]). Suppose that 𝜔∈𝐴_∞ and 𝑏∈𝐵𝑀𝑂. Then, for any 𝑝≥1, one has 1𝜔(𝐵)_𝐵𝑏(𝑥)−𝑏_𝐵^𝑝𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶‖𝑏‖_∗. (39)

Lemma 11 (see [23]). Let 𝑏∈𝐵𝑀𝑂, 1≤𝑝<∞, and 𝑟₁,𝑟₂>0. Then, 1𝐵𝑥₀,𝑟_{1𝐵(𝑥0,𝑟1)}𝑏𝑦−𝑏_{𝐵(𝑥0,𝑟2)}^𝑝𝑑𝑦^1/𝑝≤𝐶‖𝑏‖_∗𝑟1+ln¹𝑟₂, (40) where 𝐶>0 is independent of 𝑓, 𝑥₀, 𝑟₁, and 𝑟₂.

By Lemmas 10 and 11, it is easy to prove the following results.

Lemma 12. Suppose that 𝜔∈𝐴_∞ and 𝑏∈𝐵𝑀𝑂. Then, for any 1≤𝑝<∞ and 𝑟₁,𝑟₂>0, one has 1𝜔(𝐵(𝑥₀,𝑟₁))_{𝐵(𝑥0,𝑟1)}𝑏(𝑥)−𝑏_{𝐵(𝑥0,𝑟2)}^𝑝𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶‖𝑏‖_∗𝑟1+ln¹𝑟₂. (41)

3. Proof of Theorem 2

We first prove the following conclusions.

Theorem 13. Let 𝑚≥2, let 𝑇 be an 𝑚-linear Calderón-Zygmund operator, and let 𝜔=(𝜔₁,…,𝜔_𝑚) satisfy the 𝐴_𝑝 condition with 𝜔₁,…,𝜔_𝑚∈𝐴_∞. If 𝑝₁,…,𝑝_𝑚∈(1,∞) and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, then, for any 𝑥₀∈ℝ^𝑛, 𝑇𝑓_{𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}≤𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (42) If 𝑝₁,…,𝑝_𝑚∈[1,∞), min{𝑝₁,…,𝑝_𝑚}=1, and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, then, for any 𝑥₀∈ℝ^𝑛, 𝑇𝑓_{𝑊𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}≤𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (43)

Proof. We represent 𝑓_𝑖 as 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_{𝐵(𝑥0,2𝑠)}, 𝑖=1,…,𝑚, and 𝜒_{𝐵(𝑥0,2𝑠)} denotes the characteristic function of 𝐵(𝑥₀,2𝑠). Then, ^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖=^𝑚_𝑖=1𝑓⁰_𝑖𝑦_𝑖+𝑓^∞_𝑖𝑦_𝑖=_{𝛼1,…,𝛼𝑚∈{0,∞}}𝑓^𝛼1₁𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑦_𝑚=^𝑚_𝑖=1𝑓⁰_𝑖𝑦_𝑖+Σ^′𝑓^𝛼1₁𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑦_𝑚, (44) where each term of Σ^′ contains at least one 𝛼_𝑖≠0. Since 𝑇 is an 𝑚-linear operator, then 𝑇𝑓_{𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}≤𝐶𝑇(𝑓⁰₁,…,𝑓⁰_𝑚)_{𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}+𝐶Σ^′𝑇(𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚)_{𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}=𝐼^0,…,0+Σ^′𝐼^{𝛼1,…,𝛼𝑚},𝑇𝑓_{𝑊𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}≤𝐶𝑇(𝑓⁰₁,…,𝑓⁰_𝑚)_{𝑊𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}+𝐶Σ^′𝑇(𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚)_{𝑊𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}=𝐽^0,…,0+Σ^′𝐽^{𝛼1,…,𝛼𝑚}. (45) Then, by Theorem 1, if 1<𝑝_𝑖<∞, 𝑖=1,…,𝑚, we get 𝐼^0,…,0≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑠))}. (46) If 1≤𝑝_𝑖<∞, min{𝑝₁,…,𝑝_𝑚}=1, then 𝐽^0,…,0≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑠))}. (47) Applying Hölder’s inequality, for 1≤𝑝_𝑖<∞, 𝑖=1,…,𝑚, we have 𝐵𝑥₀,2𝑠≤𝜔_𝑖𝐵𝑥₀,2𝑠^1/𝑝𝑖𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,2𝑠))}. (48) Thus, for 1≤𝑝_𝑖<∞, ^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑠))}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑠))}⋅𝐵𝑥₀,2𝑠^𝑚∞_2𝑠𝑑𝑟𝑟^𝑚𝑛+1≤𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,2𝑠^1/𝑝𝑖𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑠))}×𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,2𝑠))}^∞_2𝑠𝑑𝑟𝑟^𝑚𝑛+1≤𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,2𝑠^1/𝑝𝑖⋅^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,𝑟))}×𝑑𝑟𝑟^𝑚𝑛+1. (49) From (35) and Lemma 9, we get ^𝑚_𝑖=1𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,𝑟))}≤𝐶𝐵𝑥₀,𝑟^{1/𝑝+∑𝑚𝑖=1(1/𝑝′𝑖)}_{𝐵(𝑥0,𝑟)}𝜈_𝜔(𝑥)𝑑𝑥^−1/𝑝≤𝐶𝐵𝑥₀,𝑟^𝑚𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}. (50) Then, for 1≤𝑝_𝑖<∞, 𝑖=1,…,𝑚, ^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑠))}≤𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖⋅^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (51) This gives that 𝐼^0,…,0 and 𝐽^0,…,0 are majored by 𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖⋅^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (52)
For the other term, let us first consider the case when 𝛼₁=⋯=𝛼_𝑚=∞. By the size condition (1), for any 𝑥∈𝐵(𝑥₀,𝑠), we obtain 𝑇𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶_{(ℝ𝑛⧵𝐵(𝑥0,2𝑠))𝑚}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^𝑚𝑛𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^∞_{𝑗=1(𝐵(𝑥0,2𝑗+1𝑠)⧵𝐵(𝑥0,2𝑗𝑠))𝑚}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚×𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^𝑚𝑛−1𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^∞𝑚_{𝑗=1𝑖=1𝐵(𝑥0,2𝑗+1𝑠)⧵𝐵(𝑥0,2𝑗𝑠)}𝑓_𝑖𝑦_𝑖𝑥−𝑦_𝑖^𝑛𝑑𝑦_𝑖≤𝐶^∞𝑚_𝑗=1𝑖=12^𝑗+1𝑠^−𝑛_{𝐵(𝑥0,2𝑗+1𝑠)}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖. (53) Applying Hölder’s inequality, it can be found that sup_{𝑥∈𝐵(𝑥0,𝑠)}|𝑇(𝑓^∞₁,…,𝑓^∞_𝑚)(𝑥)| is less than 𝐶^∞𝑚_𝑗=1𝑖=12^𝑗+1𝑠^−𝑛𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑗+1𝑠))}𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,2𝑗+1𝑠))}. (54) Hence, sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶^∞_𝑗=1^2𝑗+2𝑠_2𝑗+1𝑠2^𝑗+2𝑠^{−𝑛𝑚−1}×^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑗+1𝑠))}𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,2𝑗+1𝑠))}𝑑𝑟≤𝐶^∞_𝑗=1^2𝑗+2𝑠_2𝑗+1𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑗+1𝑠))}×𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,2𝑗+1𝑠))}𝑑𝑟𝑟^𝑚𝑛+1≤𝐶^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑗+1𝑠))}𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,2𝑗+1𝑠))}𝑑𝑟𝑟^𝑚𝑛+1. (55) Substituting (50) into above, we obtain sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (56)
Using Hölder’s inequality, _{𝐵(𝑥0,𝑠)}𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖. (57) From (56) and (57), we know that 𝐼^∞,…,∞ and 𝐽^∞,…,∞ are not greater than (52) for 1≤𝑝_𝑖<∞, 𝑖=1,…,𝑚.
Now, we consider the case where exactly 𝜏 of the 𝛼_𝑖 are ∞ for some 1≤𝜏<𝑚. We only give the arguments for one of the cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Applying the size condition again, we deduce that, for any 𝑥∈𝐵(𝑥₀,𝑠), 𝑇𝑓^∞₁,…,𝑓^∞_𝜏,𝑓⁰_𝜏+1,…,𝑓⁰_𝑚(𝑥)≤𝐶_{(ℝ𝑛⧵𝐵(𝑥0,2𝑠))𝜏(𝐵(𝑥0,2𝑠))𝑚−𝜏}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚×𝑥−𝑦₁++⋯𝑥−𝑦_𝑚^𝑚𝑛−1𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^𝑚_{𝑖=𝜏+1𝐵(𝑥0,2𝑠)}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖×^∞_𝑗=11𝐵𝑥₀,2^𝑗+1𝑠^𝑚×_{(𝐵(𝑥0,2𝑗+1𝑠)⧵𝐵(𝑥0,2𝑗𝑠))𝜏}𝑓₁𝑦₁⋯𝑓_𝜏𝑦_𝜏𝑑𝑦₁⋯𝑑𝑦_𝜏≤𝐶^𝑚_{𝑖=𝜏+1𝐵(𝑥0,2𝑠)}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖⋅^∞_𝑗=11𝐵𝑥₀,2^𝑗+1𝑠^𝑚𝜏_𝑖=1^𝐵(𝑥₀^{,2𝑗+1𝑠)⧵𝐵(𝑥}₀^,2𝑗𝑠)𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶^∞𝑚_𝑗=1𝑖=11𝐵𝑥₀,2^𝑗+1𝑠_𝐵(𝑥0,2^𝑗+1_𝑠)𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖. (58) Similar to the estimates for 𝐼^∞,…,∞, we get sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓^∞₁,…,𝑓^∞_𝜏,𝑓⁰_𝜏+1,…,𝑓⁰_𝑚(𝑥)≤𝐶^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (59) Then, 𝐼^{∞,…,∞,0,…,0} and 𝐽^{∞,…,∞,0,…,0} are all less than 𝐶^𝑚_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖⋅^∞_2𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (60) Combining the above estimates, we complete the proof of Theorem 13.

Now, we can give the proof of Theorem 2. From the definition of generalized Morrey space, the norm of 𝑇(𝑓) on 𝑀^𝑝_𝜑2(𝜈_𝜔) equals sup_{𝑥∈𝑅𝑛,𝑟>0}^𝑚_𝑖=1𝜑_2𝑖(𝑥,𝑠)⁻¹1𝜈_𝜔(𝐵(×𝑥,𝑠))_{𝐵(𝑥,𝑠)}𝑇𝑓𝑦^𝑝𝜈_𝜔(𝑦)𝑑𝑦^1/𝑝. (61) By (37), we get _{𝐵(𝑥,𝑠)}𝜈_𝜔(𝑥)𝑑𝑥^−1/𝑝≤𝐶^𝑚_{𝑖=1𝐵(𝑥,𝑠)}𝜔_𝑖(𝑥)𝑑𝑥^{−1/𝑝𝑖}. (62) Combining (42) and (62), 1𝜈_𝜔(𝐵(𝑥,𝑠))_{𝐵(𝑥,𝑠)}𝑇𝑓𝑦^𝑝𝜈_𝜔(𝑦)𝑑𝑦^1/𝑝≤𝐶^∞_𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}𝜔_𝑖(𝐵(𝑥,𝑟))^{−1/𝑝𝑖}𝑑𝑟𝑟. (63) Since 𝑓_𝑖∈𝑀^𝑝𝑖_𝜑1𝑖(𝜔_𝑖), then, by (31) and the fact that ‖𝑓_𝑖‖_{𝐿𝑝𝑖(𝐵(𝑥,𝑟))} are all nondecreasing function of 𝑟, we get ∏^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖≤esssup_{0<𝑟<𝑡<∞}∏^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖≤esssup_0<𝑡<∞∏^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑡))}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖≤𝐶^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}. (64) Then, ^∞_𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}𝜔_𝑖(𝐵(𝑥,𝑟))^{−1/𝑝𝑖}𝑑𝑟𝑟=^∞_𝑠∏^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖×essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑚_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖𝑑𝑟𝑟≤𝐶^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}×^∞_𝑠essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑛_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖𝑑𝑟𝑟. (65) By condition (8), we get ^∞_𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}𝜔_𝑖(𝐵(𝑥,𝑟))^1/𝑝𝑖𝑑𝑟𝑟≤𝐶^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}^𝑚_𝑖=1𝜑_2𝑖(𝑥,𝑠). (66) Combining (61), (63), and (66), the proof of the first part of Theorem 2 is completed.

Similarly, the norm of 𝑇(𝑓) on 𝑊𝑀^𝑝_𝜑2(𝜈_𝜔) equals sup_{𝑥∈𝑅𝑛,𝑟>0}^𝑚_𝑖=1𝜑_2𝑖(𝑥,𝑠)⁻¹×1𝜈_𝜔𝑇𝑓(𝐵(𝑥,𝑠))^{𝑝𝑊𝐿𝑝(𝜈𝜔,𝐵(𝑥,𝑠))1/𝑝}. (67) Combining (43) and (62), 1𝜈_𝜔𝑇𝑓(𝐵(𝑥,𝑠))^{𝑝𝑊𝐿𝑝(𝜈𝜔,𝐵(𝑥,𝑠))1/𝑝}≤𝐶^∞_𝑠^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}𝜔_𝑖(𝐵(𝑥,𝑟))^{−1/𝑝𝑖}𝑑𝑟𝑟. (68) Substituting (66) into (68), 1𝜈_𝜔𝑇𝑓(𝐵(𝑥,𝑠))^{𝑝𝑊𝐿𝑝(𝜈𝜔,𝐵(𝑥,𝑠))1/𝑝}≤𝐶𝜑₂(𝑥,𝑠)^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}. (69) Then, 𝑇𝑓_{𝑊𝑀𝑝𝜑2(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}. (70) This completes the proof of the second part of Theorem 2.

4. Proof of Theorem 3

Theorem 14. Let 𝑚≥2, let 𝑇 be an 𝑚-linear Calderón-Zygmund operator, and let 𝜔=(𝜔₁,…,𝜔_𝑚) satisfy the 𝐴_𝑝 condition with 𝜔₁,…,𝜔_𝑚∈𝐴_∞. If 𝑝₁,…,𝑝_𝑚∈(1,∞) and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑖=11/𝑝_𝑖, then, for any 𝑥₀∈ℝ^𝑛, 𝑇_Σ𝑏𝑓_{𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠𝑟1+ln𝑠^𝑚×^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟,𝑇_Π𝑏𝑓_{𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠𝑟1+ln𝑠^𝑚×^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (71)(72)

Proof. We will give the proof for 𝑇_Π𝑏 because the proof for 𝑇_Σ𝑏 is very similar but easier. Moreover, for simplicity of the expansion, we only present the case 𝑚=2.
We represent 𝑓_𝑖 as 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_{𝐵(𝑥0,2𝑠)}, 𝑖=1,2, and 𝜒_{𝐵(𝑥0,2𝑠)} denotes the characteristic function of 𝐵(𝑥₀,2𝑠). Then, 𝑇_Π𝑏𝑓_{𝐿𝑝(𝜈𝜔,𝐵(𝑥0,𝑠))}≤𝐶_{𝐵(𝑥0,𝑠)}𝑇_Π𝑏(𝑓⁰₁,𝑓⁰₂)(𝑥)^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝+𝐶_{𝐵(𝑥0,𝑠)}𝑇_Π𝑏(𝑓⁰₁,𝑓^∞₂)(𝑥)^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝+𝐶_{𝐵(𝑥0,𝑠)}𝑇_Π𝑏(𝑓^∞₁,𝑓⁰₂)(𝑥)^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝+𝐶_{𝐵(𝑥0,𝑠)}𝑇_Π𝑏(𝑓^∞₁,𝑓^∞₂)(𝑥)^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝=𝐼+𝐼𝐼+𝐼𝐼𝐼+𝐼𝑉. (73) Since 𝑇_Π𝑏 bounded from 𝐿^𝑝1(𝜔₁)×𝐿^𝑝2(𝜔₂) to 𝐿^𝑝(𝜈_𝜔), we get _{𝐵(𝑥0,𝑠)}𝑇_Π𝑏(𝑓⁰₁,𝑓⁰₂)(𝑥)^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶^𝑚_𝑖=1𝑏_𝑖∗𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,2𝑠))}. (74) Then, by (51), we get 𝐼≤𝐶²_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖⋅^∞_2𝑠²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (75)
Owing to the symmetry of 𝐼𝐼 and 𝐼𝐼𝐼, we only estimate 𝐼𝐼. Taking 𝜆_𝑖=(𝑏_𝑖)_{𝐵(𝑥0,𝑠)}, then 𝑇_Π𝑏𝑓⁰₁,𝑓^∞₂(𝑥)=𝑏₁(𝑥)−𝜆₁𝑏₂(𝑥)−𝜆₂𝑇𝑓⁰₁,𝑓^∞₂(𝑥)−𝑏₁(𝑥)−𝜆₁𝑇𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)−𝑏₂(𝑥)−𝜆₂𝑇𝑏₁−𝜆₁𝑓⁰₁,𝑓^∞₂(𝑥)+𝑇𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)=𝐼𝐼₁+𝐼𝐼₂+𝐼𝐼₃+𝐼𝐼₄. (76) Similar to the estimate of (59), for any 𝑥∈𝐵(𝑥₀,𝑠), we can deduce sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓⁰₁,𝑓^∞₂(𝑥)≤𝐶^∞_2𝑠²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (77) By Lemma 8, we know that 𝜈_𝜔∈𝐴_∞. Applying Hölder’s inequality and Lemma 10, we have _{𝐵(𝑥0,𝑠)}𝑏₁(𝑥)−𝜆₁𝑏₂(𝑥)−𝜆₂^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶²_{𝑖=1𝐵(𝑥0,𝑠)}𝑏_𝑖(𝑥)−𝜆_𝑖^2𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/2𝑝≤𝐶²_𝑖=1𝑏_𝑖∗⋅𝜈_𝜔(𝐵(𝑥₀,𝑠))^1/𝑝. (78) Then, by (77), (78), and (57), we have _{𝐵(𝑥0,𝑠)}𝐼𝐼₁^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤_{𝐵(𝑥0,𝑠)}𝑏₁(𝑥)−𝜆₁𝑏₂(𝑥)−𝜆₂^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝×sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓⁰₁,𝑓^∞₂(𝑥)≤𝐶²_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖⋅^∞_2𝑠²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (79)
Applying the size condition, we deduce that, for any 𝑥∈𝐵(𝑥₀,𝑠), 𝑇𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)≤𝐶_{𝐵(𝑥0,2𝑠)ℝ𝑛⧵𝐵(𝑥0,2𝑠)}𝑓₁𝑦₁𝑏₂𝑦₂−𝜆₂𝑓₂𝑦₂×𝑥−𝑦₁+𝑥−𝑦₂^2𝑛−1𝑑𝑦₁𝑑𝑦₂≤𝐶^∞_𝑗=12^𝑗+1𝑠^−2𝑛_{𝐵(𝑥0,2𝑗+1𝑠)}𝑓₁𝑦₁𝑑𝑦₁×_{𝐵(𝑥0,2𝑗+1𝑠)}𝑏₂𝑦₂−𝜆₂𝑓₂𝑦₂𝑑𝑦₂. (80) Note that _{𝐵(𝑥0,2𝑗+1𝑠)}𝑓₁𝑦₁𝑑𝑦₁≤𝐶𝑓_{1𝐿𝑝1(𝜔1,𝐵(𝑥0,2𝑗+1𝑠))}𝜔^−1/𝑝1_{1𝐿𝑝′1(𝐵(𝑥0,2𝑗+1𝑠))},_{𝐵(𝑥0,2𝑗+1𝑠)}𝑏₂𝑦₂−𝜆₂𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝑓_{2𝐿𝑝2(𝜔2,𝐵(𝑥0,2𝑘+1𝑠))}𝑏₂(⋅)−𝜆_{2𝐿𝑝′2(𝜔′221−𝑝,𝐵(𝑥0,2𝑘+1𝑠))}. (81) Then, sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)≤𝐶^∞2_2𝑠𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}×𝜔^−1/𝑝1_{1𝐿𝑝′1(𝐵(𝑥0,𝑟))}𝑏₂(⋅)−𝜆_{2𝐿𝑝′2(𝜔′221−𝑝,𝐵(𝑥0,𝑟))}𝑑𝑟𝑟^2𝑛+1. (82) Since 𝜔^1−𝑝′2₂∈𝐴_2𝑝′2, then, from Lemma 12, we get 𝑏₂(⋅)−𝜆_{2𝐿𝑝′2(𝜔′221−𝑝,𝐵(𝑥0,𝑟))}≤𝐶_{𝐵(𝑥0,𝑟)}|𝑏₂(𝑧)−𝜆₂|^𝑝′2𝜔^1−𝑝′2₂(𝑧)𝑑𝑧^1/𝑝′2𝑟≤𝐶1+ln𝑠𝑏_2∗𝜔^1−𝑝′2₂𝐵𝑥₀,𝑟^1/𝑝′2𝑟=𝐶1+ln𝑠𝑏_2∗𝜔^−1/𝑝2_{2𝐿𝑝′2(𝐵(𝑥0,𝑟))}. (83) By (50), we have ²_𝑖=1𝜔^{−1/𝑝𝑖}_{𝑖𝐿𝑝′𝑖(𝐵(𝑥0,𝑟))}≤𝐶𝐵𝑥₀,𝑟²²_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}. (84) From (82), (83), and (84), we can deduce sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥𝑏)≤𝐶_2∗^∞_2𝑠𝑟1+ln𝑠ײ_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (85) Applying Lemma 10 and (57), we have _{𝐵(𝑥0,𝑠)}𝑏₁(𝑥)−𝜆₁^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶𝑏_1∗𝜈_𝜔(𝐵(𝑥₀,𝑠))^1/𝑝≤𝐶𝑏_1∗²_𝑖=1𝜔_𝑖𝐵𝑥₀,𝑟^1/𝑝𝑖. (86) Then, by (85), _{𝐵(𝑥0,𝑠)}𝐼𝐼₂^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤_{𝐵(𝑥0,𝑠)}𝑏₁(𝑥)−𝜆₁^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝×sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑇𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)≤𝐶²_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠𝑟1+ln𝑠ײ_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (87) Similarly, we also have that _{𝐵(𝑥0,𝑠)}𝐼𝐼₃^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶²_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠𝑟1+ln𝑠ײ_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (88)
Applying the size condition again, for any 𝑥∈𝐵(𝑥₀,𝑠), with the same method of estimate for (85), we have 𝑇𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)≤𝐶^∞_𝑗=12^𝑗+1𝑠^2−2𝑛_{𝑖=1𝐵(𝑥0,2𝑗+1𝑠)}𝑏_𝑖𝑦_𝑖−𝜆_𝑖𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶^∞_2𝑠²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}⋅𝑏_𝑖(⋅)−𝜆_{𝑖𝐿𝑝′𝑖(𝜔′𝑖𝑖1−𝑝,𝐵(𝑥0,𝑟))}𝑑𝑟𝑟^2𝑛+1≤𝐶²_𝑖=1𝑏_𝑖∗⋅^∞_2𝑠𝑟1+ln𝑠²×²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (89) Then, _{𝐵(𝑥0,𝑠)}𝐼𝐼₄^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶𝜈_𝜔𝐵𝑥₀,𝑠^1/𝑝×sup_{𝑥∈𝐵(𝑥0,𝑠)}𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)≤𝐶²_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖⋅^∞_2𝑠𝑟1+ln𝑠²×²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (90) Then, combining (79), (87), (88), and (90), we get _{𝐵(𝑥0,𝑠)}|𝐼𝐼|^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶²_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠𝑟1+ln𝑠²×²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (91)
Finally, we still decompose 𝑇_Π𝑏(𝑓^∞₁,𝑓^∞₂)(𝑥) as follows: 𝑇_Π𝑏𝑓^∞₁,𝑓^∞₂(𝑥)=𝑏₁(𝑥)−𝜆₁𝑏₂(𝑥)−𝜆₂𝑇𝑓^∞₁,𝑓^∞₂(𝑥)−𝑏₁(𝑥)−𝜆₁𝑇𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)−𝑏₂(𝑥)−𝜆₂𝑇𝑏₁−𝜆₁𝑓^∞₁,𝑓^∞₂𝑥)𝑏(+𝑇₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)=𝐼𝑉₁+𝐼𝑉₂+𝐼𝑉₃+𝐼𝑉₄. (92) Because each term 𝐼𝑉_𝑗 is completely analogous to 𝐼𝐼_𝑗, 𝑗=1,2,3,4 with a bit difference, we get the following estimate without details: _{𝐵(𝑥0,𝑠)}|𝐼𝑉|^𝑝𝜈_𝜔(𝑥)𝑑𝑥^1/𝑝≤𝐶²_𝑖=1𝑏_𝑖∗𝜔_𝑖𝐵𝑥₀,𝑠^1/𝑝𝑖×^∞_2𝑠𝑟1+ln𝑠²×²_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥0,𝑟))}𝜔_𝑖𝐵𝑥₀,𝑟^{−1/𝑝𝑖}𝑑𝑟𝑟. (93)
Summing up the above estimates, (72) is proved.

We give the proof of Theorem 3 in the following. From (62) and (72), 1𝜈_𝜔(𝐵(𝑥,𝑠))_{𝐵(𝑥,𝑠)}𝑇_Π𝑏𝑓𝑦^𝑝𝜈_𝜔(𝑦)𝑑𝑦^1/𝑝≤𝐶^𝑚_𝑖=1‖𝑏‖_∗⋅^∞_𝑠𝑟1+ln𝑠^𝑚×^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}𝜔_𝑖(𝐵(𝑥,𝑟))^{−1/𝑝𝑖}𝑑𝑟𝑟. (94) Since (𝜑₁,𝜑₂) satisfies condition (15), ‖𝑓_𝑖‖_{𝐿𝑝𝑖(𝐵(𝑥,𝑟))} are all nondecreasing function of 𝑟, and 𝑓_𝑖∈𝑀^𝑝𝑖_𝜑1𝑖(𝜔_𝑖), then ^∞_𝑠𝑟1+ln𝑠^𝑚𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}𝜔_𝑖(𝐵(𝑥,𝑟))^{−1/𝑝𝑖}𝑑𝑟𝑟=^∞_𝑠∏^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔𝑖,𝐵(𝑥,𝑟))}essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖×𝑟1+ln𝑠^𝑚×essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑛_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖𝑑𝑟𝑟≤𝐶^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}×^∞_𝑠𝑟1+ln𝑠^𝑚×essinf_{𝑟<𝑡<∞}∏^𝑚_𝑖=1𝜑_1𝑖(𝑥,𝑡)𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖∏^𝑛_𝑖=1𝜔_𝑖(𝐵(𝑥,𝑡))^1/𝑝𝑖𝑑𝑟𝑟≤𝐶𝜑₂(𝑥,𝑠)^𝑚_𝑖=1𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}. (95) Combining (94) and (95), we have proved that 𝑇_Π𝑏𝑓_{𝑀𝑝𝜑2(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑏_𝑖∗𝑓_𝑖𝑀^𝑝_{𝑖𝜑1𝑖(𝜔𝑖)}. (96)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous reviewers and the editor for their several comments and suggestions which contributed to the improvement of this paper. The project is supported by the National Natural Science Foundation of China (no. 61340014).

Supplementary Materials

Careful analysis of the proof in “Multilinear Commutators of Caldern-Zygmund Operator on Generalized Weighted Morrey Spaces”, Journal of Function Spaces, Volume 2014 (2014), Article ID 710542, by Guliyev, Vagif S. and Alizadeh Farida Ch., we found that there exist a fatal error in their proof: Under the condition of not satisfy Hardy inequality, they still use the Hardy inequality. So, in our paper, the proof of main theorem has adopted a completely new one. Our method avoid the application of Hardy inequality.

1. Supplementary Material