Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces

Abstract

We will study the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.

1. Introduction and Main 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]. 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 by 𝒮^′(ℝ^𝑛) its dual space, the set of all tempered distributions on ℝ^𝑛. Let 𝑚≥2 and 𝑇 be an 𝑚-linear operator initially defined on the 𝑚-fold product of Schwartz spaces, and taking values into the space of tempered distributions, 𝑇∶𝒮ℝ^𝑛×⋯×𝒮ℝ^𝑛⟶𝒮^′ℝ^𝑛. (1) Following [2], for given 𝑓=(𝑓₁,…,𝑓_𝑚), 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_𝑘=11/𝑞_𝑘, 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 such that 𝑇𝑓(𝑥)=𝑇𝑓₁,…,𝑓_𝑚(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚, (2) whenever 𝑓₁,…,𝑓_𝑚∈𝒮(ℝ^𝑛) and 𝑥∉⋂^𝑚_𝑘=1supp𝑓_𝑘. We say that 𝐾(𝑥,𝑦₁,…,𝑦_𝑚) is a kernel in the class 𝑚-𝐶𝑍𝐾(𝐴,𝜀) if it satisfies the size condition 𝐾𝑥,𝑦₁,…,𝑦_𝑚≤𝐴𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^𝑚𝑛, (3) for some 𝐴>0 and all (𝑥,𝑦₁,…,𝑦_𝑚)∈(ℝ^𝑛)^𝑚+1 with 𝑥≠𝑦_𝑘 for some 1≤𝑘≤𝑚. Moreover, for some 𝜀>0, it satisfies the regularity condition that 𝐾𝑥,𝑦₁,…,𝑦_𝑚−𝐾𝑥^′,𝑦₁,…,𝑦_𝑚≤𝐴⋅𝑥−𝑥^′𝜀𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^{𝑚𝑛+𝜀} (4) whenever |𝑥−𝑥^′|≤(1/2)max_{1≤𝑘≤𝑚}|𝑥−𝑦_𝑘| and also that, for each fixed 𝑘 with 1≤𝑘≤𝑚, 𝐾𝑥,𝑦₁,…,𝑦_𝑘,…,𝑦_𝑚−𝐾𝑥,𝑦₁,…,𝑦^′_𝑘,…,𝑦_𝑚≤𝐴⋅𝑦_𝑘−𝑦^′_𝑘^𝜀𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^{𝑚𝑛+𝜀} (5) whenever |𝑦_𝑘−𝑦^′_𝑘|≤(1/2)max_{1≤𝑖≤𝑚}|𝑥−𝑦_𝑖|. In recent years, many authors have been interested in studying the boundedness of these operators on function spaces; see, for example, [5–8]. In 2009, the weighted strong and weak type estimates of multilinear Calderón-Zygmund singular integral operators were established in [9] by Lerner et al. New more refined multilinear maximal function was defined and used in [9] to characterize the class of multiple 𝐴_𝑃 weights.

Theorem A (see [9]). Let 𝑚≥2 and 𝑇 be an 𝑚-linear Calderón-Zygmund operator. If 𝑝₁,…,𝑝_𝑚∈(1,∞) and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘 and 𝑤=(𝑤₁,…,𝑤_𝑚) satisfies the 𝐴_𝑃 condition, then there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇𝑓_{𝐿𝑝(𝜈𝑤)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝑤𝑖)}, (6) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖.

Theorem B (see [9]). Let 𝑚≥2, and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. If 𝑝₁,…,𝑝_𝑚∈[1,∞), min{𝑝₁,…,𝑝_𝑚}=1 and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, and 𝑤=(𝑤₁,…,𝑤_𝑚) satisfies the 𝐴_𝑃 condition, then there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇𝑓_{𝑊𝐿𝑝(𝜈𝑤)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝑤𝑖)}, (7) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖.

Let 𝑚≥2, and let 0<𝛼<𝑚𝑛. For given 𝑓=(𝑓₁,…,𝑓_𝑚), the 𝑚-linear fractional integral operator is defined by 𝐼_𝛼𝑓(𝑥)=𝐼_𝛼𝑓₁,…,𝑓_𝑚(𝑥)=_{(ℝ𝑛)𝑚}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁,…,𝑥−𝑦_𝑚^{𝑚𝑛−𝛼}𝑑𝑦₁⋯𝑑𝑦_𝑚. (8) For the boundedness properties of multilinear fractional integrals on various function spaces, we refer the reader to [10–16]. In 2009, Moen [17] considered the weighted norm inequalities for multilinear fractional integral operators and constructed the class of multiple 𝐴_𝑃,𝑞 weights (see also [18]).

Theorem C (see [17, 18]). Let 𝑚≥2, 0<𝛼<𝑚𝑛, and let 𝐼_𝛼 be an 𝑚-linear fractional integral operator. If 𝑝₁,…,𝑝_𝑚∈(1,∞), 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘 and 1/𝑞=1/𝑝−𝛼/𝑛, and 𝑤=(𝑤₁,…,𝑤_𝑚) satisfies the 𝐴_𝑃,𝑞 condition, then there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝐼_𝛼𝑓_{𝐿𝑞((𝜈𝑤)𝑞)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝑤}^𝑝_𝑖𝑖), (9) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖.

Theorem D (see [17, 18]). Let 𝑚≥2, 0<𝛼<𝑚𝑛, and let 𝐼_𝛼 be an 𝑚-linear fractional integral operator. If 𝑝₁,…,𝑝_𝑚∈[1,∞), min{𝑝₁,…,𝑝_𝑚}=1, 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘 and 1/𝑞=1/𝑝−𝛼/𝑛, and 𝑤=(𝑤₁,…,𝑤_𝑚) satisfies the 𝐴_𝑃,𝑞 condition, then there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝐼_𝛼𝑓_{𝑊𝐿𝑞((𝜈𝑤)𝑞)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝑤}^𝑝_𝑖𝑖), (10) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖.

On the other hand, the classical Morrey spaces ℒ^𝑝,𝜆 were originally introduced by Morrey in [19] to study the local behavior of solutions to 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 reader to [20–22]. For the properties and applications of classical Morrey spaces, one can see [23–25] and the references therein.

In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces 𝐿^𝑝,𝜅(𝑤) which could be viewed as an extension of weighted Lebesgue spaces and studied the boundedness of the above classical operators in Harmonic Analysis on these weighted spaces. Recently, in [27–34], we have established the continuity properties of some other operators and their commutators on the weighted Morrey spaces 𝐿^𝑝,𝜅(𝑤).

The main purpose of this paper is to establish the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights. We now formulate our main results as follows.

Theorem 1. Let 𝑚≥2, and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. If 𝑝₁,…,𝑝_𝑚∈(1,∞) and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘 and 𝑤=(𝑤₁,…,𝑤_𝑚)∈𝐴_𝑃 with 𝑤₁,…,𝑤_𝑚∈𝐴_∞, then for any 0<𝜅<1, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇(𝑓)_{𝐿𝑝,𝜅(𝜈𝑤)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}, (11) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖.

Theorem 2. Let 𝑚≥2 and 𝑇 be an 𝑚-linear Calderón-Zygmund operator. If 𝑝₁,…,𝑝_𝑚∈[1,∞), min{𝑝₁,…,𝑝_𝑚}=1 and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, and 𝑤=(𝑤₁,…,𝑤_𝑚)∈𝐴_𝑃 with 𝑤₁,…,𝑤_𝑚∈𝐴_∞, then for any 0<𝜅<1, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇𝑓_{𝑊𝐿𝑝,𝜅(𝜈𝑤)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}, (12) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖.

Theorem 3. Let 𝑚≥2, let 0<𝛼<𝑚𝑛, and let 𝐼_𝛼 be an 𝑚-linear fractional integral operator. If 𝑝₁,…,𝑝_𝑚∈(1,∞), 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, 1/𝑞_𝑘=1/𝑝_𝑘−𝛼/𝑚𝑛 and 1/𝑞=∑^𝑚_𝑘=11/𝑞_𝑘=1/𝑝−𝛼/𝑛, and 𝑤=(𝑤₁,…,𝑤_𝑚)∈𝐴_𝑃,𝑞 with 𝑤^𝑞1₁,…,𝑤^𝑞𝑚_𝑚∈𝐴_∞, then for any 0<𝜅<𝑝/𝑞, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝐼_𝛼(𝑓)_{𝐿𝑞,𝜅𝑞/𝑝((𝜈𝑤)𝑞)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖), (13) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖.

Theorem 4. Let 𝑚≥2, let 0<𝛼<𝑚𝑛, and let 𝐼_𝛼 be an 𝑚-linear fractional integral operator. If 𝑝₁,…,𝑝_𝑚∈[1,∞), min{𝑝₁,…,𝑝_𝑚}=1, 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, 1/𝑞_𝑘=1/𝑝_𝑘−𝛼/𝑚𝑛 and 1/𝑞=∑^𝑚_𝑘=11/𝑞_𝑘=1/𝑝−𝛼/𝑛, and 𝑤=(𝑤₁,…,𝑤_𝑚)∈𝐴_𝑃,𝑞 with 𝑤^𝑞1₁,…,𝑤^𝑞𝑚_𝑚∈𝐴_∞, then, for any 0<𝜅<𝑝/𝑞, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝐼_𝛼(𝑓)_{𝑊𝐿𝑞,𝜅𝑞/𝑝((𝜈𝑤)𝑞)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖), (14) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖.

2. Notations and Definitions

The classical 𝐴_𝑝 weight theory was first introduced by Muckenhoupt in the study of weighted 𝐿^𝑝 boundedness of Hardy-Littlewood maximal functions in [35]. A weight 𝑤 is a nonnegative, locally integrable function on ℝ^𝑛; 𝐵=𝐵(𝑥₀,𝑟_𝐵) denotes the ball with the center 𝑥₀ and radius 𝑟_𝐵. For 1<𝑝<∞, a weight function 𝑤 is said to belong to 𝐴_𝑝 if there is a constant 𝐶>0 such that, for every ball 𝐵⊆ℝ^𝑛, 1|𝐵|_𝐵𝑤(𝑥)1𝑑𝑥|𝐵|_𝐵𝑤(𝑥)^{−1/(𝑝−1)}𝑑𝑥^𝑝−1≤𝐶, (15) where |𝐵| denotes the Lebesgue measure of 𝐵. For the case 𝑝=1, 𝑤∈𝐴₁, if there is a constant 𝐶>0 such that for every ball 𝐵⊆ℝ^𝑛, 1|𝐵|_𝐵𝑤(𝑥)𝑑𝑥≤𝐶⋅essinf_𝑥∈𝐵𝑤(𝑥). (16) A weight function 𝑤∈𝐴_∞ if it satisfies the 𝐴_𝑝 condition for some 1<𝑝<∞. We also need another weight class 𝐴_𝑝,𝑞 introduced by Muckenhoupt and Wheeden in [36]. A weight function 𝑤 belongs to 𝐴_𝑝,𝑞 for 1<𝑝<𝑞<∞ if there is a constant 𝐶>0 such that, for every ball 𝐵⊆ℝ^𝑛, 1|𝐵|_𝐵𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞1|𝐵|_𝐵𝑤(𝑥)^−𝑝′𝑑𝑥^1/𝑝′≤𝐶. (17) When 𝑝=1, 𝑤 is in the class 𝐴_1,𝑞 with 1<𝑞<∞ if there is a constant 𝐶>0 such that, for every ball 𝐵⊆ℝ^𝑛, 1|𝐵|_𝐵𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞esssup_𝑥∈𝐵1𝑤(𝑥)≤𝐶. (18)

Now let us recall the definitions of multiple weights. For 𝑚 exponents 𝑝₁,…,𝑝_𝑚, we will write 𝑃 for the vector 𝑃=(𝑝₁,…,𝑝_𝑚). Let 𝑝₁,…,𝑝_𝑚∈[1,∞), and let 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘. Given 𝑤=(𝑤₁,…,𝑤_𝑚), set 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖. We say that 𝑤 satisfies the 𝐴_𝑃 condition if it satisfies sup_𝐵1|𝐵|_𝐵𝜈_𝑤(𝑥)𝑑𝑥^𝑚1/𝑝_𝑖=11|𝐵|_𝐵𝑤_𝑖(𝑥)^{1−𝑝′𝑖}𝑑𝑥^{1/𝑝′𝑖}<∞. (19) When 𝑝_𝑖=1, ((1/|𝐵|)∫_𝐵𝑤_𝑖(𝑥)^{1−𝑝′𝑖}𝑑𝑥)^{1/𝑝′𝑖} is understood as (inf_𝑥∈𝐵𝑤_𝑖(𝑥))⁻¹.

Let 𝑝₁,…,𝑝_𝑚∈[1,∞), let 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, and let 𝑞>0. Given 𝑤=(𝑤₁,…,𝑤_𝑚), set 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖. We say that 𝑤 satisfies the 𝐴_𝑃,𝑞 condition if it satisfies sup_𝐵1|𝐵|_𝐵𝜈_𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞×^𝑚_𝑖=11|𝐵|_𝐵𝑤_𝑖(𝑥)^{−𝑝′𝑖}𝑑𝑥^{1/𝑝′𝑖}<∞. (20) When 𝑝_𝑖=1, ((1/|𝐵|)∫_𝐵𝑤_𝑖(𝑥)^{−𝑝′𝑖}𝑑𝑥)^{1/𝑝′𝑖} is understood as (inf_𝑥∈𝐵𝑤_𝑖(𝑥))⁻¹.

Given a ball 𝐵 and 𝜆>0, 𝜆𝐵 denotes the ball with the same center as 𝐵 whose radius is 𝜆 times that of 𝐵. For a given weight function 𝑤 and a measurable set 𝐸, we also denote the Lebesgue measure of 𝐸 by |𝐸| and the weighted measure of 𝐸 by 𝑤(𝐸), where 𝑤(𝐸)=∫_𝐸𝑤(𝑥)𝑑𝑥.

Lemma 5 (see [37]). Let 𝑤∈𝐴_𝑝 with 1≤𝑝<∞. Then, for any ball 𝐵, there exists an absolute constant 𝐶>0 such that 𝑤(2𝐵)≤𝐶𝑤(𝐵). (21)

Lemma 6 (see [38]). Let 𝑤∈𝐴_∞. Then for all balls 𝐵⊆ℝ^𝑛, the following reverse Jensen inequality holds: _𝐵𝑤(𝑥)|𝐵|1𝑑𝑥≤𝐶⋅exp|𝐵|_𝐵log𝑤(𝑥)𝑑𝑥. (22)

Lemma 7 (see [37]). Let 𝑤∈𝐴_∞. Then, for all balls 𝐵 and all measurable subsets 𝐸 of 𝐵, there exists 𝛿>0 such that 𝑤(𝐸)𝑤(𝐵)|𝐸|≤𝐶|𝐵|^𝛿. (23)

Lemma 8 (see [9]). Let 𝑝₁,…,𝑝_𝑚∈[1,∞), and let 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘. Then 𝑤=(𝑤₁,…,𝑤_𝑚)∈𝐴_𝑃 if and only if 𝜈_𝑤∈𝐴_𝑚𝑝,𝑤^{1−𝑝′𝑖}_𝑖∈𝐴_{𝑚𝑝′𝑖},𝑖=1,…,𝑚, (24) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖 and the condition 𝑤^{1−𝑝′𝑖}_𝑖∈𝐴_{𝑚𝑝′𝑖} in the case 𝑝_𝑖=1 is understood as 𝑤_𝑖^1/𝑚∈𝐴₁.

Lemma 9 (see [17, 18]). Let 0<𝛼<𝑚𝑛, and 𝑝₁,…,𝑝_𝑚∈[1,∞), let 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, and let 1/𝑞=1/𝑝−𝛼/𝑛. If 𝑤=(𝑤₁,…,𝑤_𝑚)∈𝐴_𝑃,𝑞, then 𝜈_𝑤^𝑞∈𝐴_𝑚𝑞,𝑤^{−𝑝′𝑖}_𝑖∈𝐴_{𝑚𝑝′𝑖},𝑖=1,…,𝑚, (25) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖.

Given a weight function 𝑤 on ℝ^𝑛, for 0<𝑝<∞, the weighted Lebesgue space 𝐿^𝑝(𝑤)   is defined as the set of all functions 𝑓 such that 𝑓_{𝐿𝑝(𝑤)}=_ℝ𝑛𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥^1/𝑝<∞. (26) We also denote by 𝑊𝐿^𝑝(𝑤) the weighted weak space consisting of all measurable functions 𝑓 such that 𝑓_{𝑊𝐿𝑝(𝑤)}=sup_𝜆>0𝜆⋅𝑤𝑥∈ℝ^𝑛∶𝑓(𝑥)>𝜆^1/𝑝<∞. (27)

In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces 𝐿^𝑝,𝜅(𝑤) for 1≤𝑝<∞. In order to deal with the multilinear case 𝑚≥2, we will define 𝐿^𝑝,𝜅(𝑤) for all 0<𝑝<∞.

Definition 10. Let 0<𝑝<∞, let 0<𝜅<1, and let 𝑤 be a weight function on ℝ^𝑛. Then the weighted Morrey space is defined by 𝐿^𝑝,𝜅(𝑤)=𝑓∈𝐿^𝑝_loc(𝑤)∶𝑓_{𝐿𝑝,𝜅(𝑤)}<∞, (28) where 𝑓_{𝐿𝑝,𝜅(𝑤)}=sup_𝐵1𝑤(𝐵)^𝜅_𝐵𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥^1/𝑝 (29) and the supremum is taken over all balls 𝐵 in ℝ^𝑛.

Definition 11. Let 0<𝑝<∞, let 0<𝜅<1, and let 𝑤 be a weight function on ℝ^𝑛. Then the weighted weak Morrey space is defined by 𝑊𝐿^𝑝,𝜅(𝑤)=𝑓measurable∶𝑓_{𝑊𝐿𝑝,𝜅(𝑤)}<∞, (30) where 𝑓_{𝑊𝐿𝑝,𝜅(𝑤)}=sup_𝐵sup_𝜆>01𝑤(𝐵)^𝜅/𝑝𝜆⋅𝑤𝑥∈𝐵∶𝑓(𝑥)>𝜆^1/𝑝. (31)

Furthermore, in order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weights.

Definition 12. Let 0<𝑝<∞ and 0<𝜅<1. Then for two weights 𝑢 and 𝑣, the weighted Morrey space is defined by 𝐿^𝑝,𝜅(𝑢,𝑣)=𝑓∈𝐿^𝑝_loc(𝑢)∶𝑓_{𝐿𝑝,𝜅(𝑢,𝑣)}<∞, (32) where 𝑓_{𝐿𝑝,𝜅(𝑢,𝑣)}=sup_𝐵1𝑣(𝐵)^𝜅_𝐵𝑓(𝑥)^𝑝𝑢(𝑥)𝑑𝑥^1/𝑝. (33)

Throughout this paper, we will use 𝐶 to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. Moreover, we will denote the conjugate exponent of 𝑝>1 by 𝑝^′=𝑝/(𝑝−1).

3. Proofs of Theorems 1 and 2

Before proving the main theorems of this section, we need to establish the following lemma.

Lemma 13. Let 𝑚≥2, let 𝑝₁,…,𝑝_𝑚∈[1,∞), and let 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘. Assume that 𝑤₁,…,𝑤_𝑚∈𝐴_∞ and 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖; then, for any ball 𝐵, there exists a constant 𝐶>0 such that ^𝑚_𝑖=1𝐵𝑤_𝑖(𝑥)𝑑𝑥^{𝑝/𝑝𝑖}≤𝐶_𝐵𝜈_𝑤(𝑥)𝑑𝑥. (34)

Proof. Since 𝑤₁,…,𝑤_𝑚∈𝐴_∞, then, by using Lemma 6, we have ^𝑚_𝑖=1𝐵𝑤_𝑖(𝑥)𝑑𝑥^{𝑝/𝑝𝑖}≤𝐶^𝑚_𝑖=1|𝐵|1⋅exp|𝐵|_𝐵log𝑤_𝑖(𝑥)𝑑𝑥^{𝑝/𝑝𝑖}=𝐶^𝑚_𝑖=1|𝐵|^{𝑝/𝑝𝑖}1⋅exp|𝐵|_𝐵log𝑤_𝑖(𝑥)^{𝑝/𝑝𝑖}𝐵𝑑𝑥=𝐶⋅(||)^{∑𝑚𝑖=1𝑝/𝑝𝑖}⋅exp^𝑚_𝑖=11|𝐵|_𝐵log𝑤_𝑖(𝑥)^{𝑝/𝑝𝑖}𝑑𝑥. (35) Note that ∑^𝑚_𝑖=1𝑝/𝑝_𝑖=1 and 𝜈_𝑤(𝑥)=∏^𝑚_𝑖=1𝑤_𝑖(𝑥)^{𝑝/𝑝𝑖}. Then, by Jensen inequality, we obtain ^𝑚_𝑖=1𝐵𝑤_𝑖(𝑥)𝑑𝑥^{𝑝/𝑝𝑖}|𝐵|1≤𝐶⋅⋅exp|𝐵|_𝐵log𝜈_𝑤(𝑥)𝑑𝑥≤𝐶_𝐵𝜈_𝑤(𝑥)𝑑𝑥. (36) We are done.

Proof of Theorem 1. For any ball 𝐵=𝐵(𝑥₀,𝑟_𝐵)⊆ℝ^𝑛 and letting  𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_2𝐵, 𝑖=1,…,𝑚, and 𝜒_2𝐵 denotes the characteristic function of 2𝐵, then we write ^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+^′𝑓^𝛼1₁𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑦_𝑚, (37) where each term of ∑^′ contains at least one 𝛼_𝑖≠0. Since 𝑇 is an 𝑚-linear operator, then we have 1𝜈_𝑤(𝐵)^{𝜅/𝑝𝐵}𝑇𝑓₁,…,𝑓_𝑚(𝑥)^𝑝𝜈_𝑤(𝑥)𝑑𝑥^1/𝑝≤1𝜈_𝑤(𝐵)^{𝜅/𝑝𝐵}𝑇𝑓⁰₁,…,𝑓⁰_𝑚(𝑥)^𝑝𝜈_𝑤(𝑥)𝑑𝑥^1/𝑝+^′1𝜈_𝑤(𝐵)^{𝜅/𝑝𝐵}𝑇𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)^𝑝𝜈_𝑤(𝑥)𝑑𝑥^1/𝑝=𝐼⁰+^′𝐼^{𝛼1,…,𝛼𝑚}. (38) In view of Lemma 8, we have that 𝜈_𝑤∈𝐴_𝑚𝑝. Applying Theorem A and Lemmas 13 and 5, we get 𝐼⁰1≤𝐶⋅𝜈_𝑤(𝐵)^{𝑚𝜅/𝑝}_𝑖=1^2𝐵𝑓_𝑖(𝑥)^𝑝𝑖𝑤_𝑖(𝑥)𝑑𝑥^1/𝑝𝑖≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅∏^𝑚_𝑖=1𝑤_𝑖(2𝐵)^{𝜅/𝑝𝑖}𝜈_𝑤(𝐵)^𝜅/𝑝≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅𝜈_𝑤(2𝐵)^𝜅/𝑝𝜈_𝑤(𝐵)^𝜅/𝑝≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}. (39) For the other terms, let us first consider the case when 𝛼₁=⋯=𝛼_𝑚=∞. By the size condition, for any 𝑥∈𝐵, we obtain 𝑇𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶_{(ℝ𝑛)𝑚⧵(2𝐵)𝑚}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^𝑚𝑛𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^∞_{𝑗=1(2𝑗+1𝐵)𝑚⧵(2𝑗𝐵)𝑚}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^𝑚𝑛𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^∞𝑚_{𝑗=1𝑖=12𝑗+1𝐵⧵2𝑗𝐵}𝑓_𝑖𝑦_𝑖𝑥−𝑦_𝑖^𝑛𝑑𝑦_𝑖≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖, (40) where we have used the notation 𝐸^𝑚=𝐸×⋯×𝐸. Furthermore, by using Hölder’s inequality, the multiple 𝐴_𝑃 condition, and Lemma 13, we deduce that 𝑇𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖^𝑝𝑖𝑤_𝑖𝑦_𝑖𝑑𝑦_𝑖^1/𝑝𝑖×_2𝑗+1𝐵𝑤_𝑖𝑦_𝑖^{1−𝑝′𝑖}𝑑𝑦_𝑖^{1/𝑝′𝑖}≤𝐶^∞_𝑗=112^𝑗+1𝐵^𝑚⋅2^𝑗+1𝐵^{1/𝑝+∑𝑚𝑖=1(1−1/𝑝𝑖)}𝜈_𝑤2^𝑗+1𝐵^1/𝑝×^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}𝑤_𝑖2^𝑗+1𝐵^{𝜅/𝑝𝑖}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅^∞_𝑗=1∏^𝑚_𝑖=1𝑤_𝑖2^𝑗+1𝐵^{𝜅/𝑝𝑖}𝜈_𝑤2^𝑗+1𝐵^1/𝑝≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅^∞_𝑗=1𝜈_𝑤2^𝑗+1𝐵^{(𝜅−1)/𝑝}. (41) Since 𝜈_𝑤∈𝐴_𝑚𝑝⊂𝐴_∞, then it follows directly from Lemma 7 that 𝜈_𝑤(𝐵)𝜈_𝑤2^𝑗+1𝐵|𝐵|≤𝐶2^𝑗+1𝐵^𝛿. (42) Hence, 𝐼^∞,…,∞≤𝜈_𝑤(𝐵)^{(1−𝜅)/𝑝}𝑇𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅^∞_𝑗=1𝜈_𝑤(𝐵)^{(1−𝜅)/𝑝}𝜈_𝑤2^𝑗+1𝐵^{(1−𝜅)/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅^∞_𝑗=1|𝐵||2^𝑗+1𝐵|^{𝛿(1−𝜅)/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}, (43) where the last inequality holds since 0<𝜅<1 and 𝛿>0. We now consider the case where exactly ℓ of the 𝛼_𝑖 are ∞ for some 1≤ℓ<𝑚. We only give the arguments for one of these cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Using the size condition again, we deduce that, for any 𝑥∈𝐵, 𝑇𝑓^∞₁,…,𝑓^∞_ℓ,𝑓⁰_ℓ+1,…,𝑓⁰_𝑚(𝑥)≤𝐶_{(ℝ𝑛)ℓ⧵(2𝐵)ℓ(2𝐵)𝑚−ℓ}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^𝑚𝑛×𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^𝑚_{𝑖=ℓ+12𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖×^∞_𝑗=112^𝑗+1𝐵^𝑚×_{(2𝑗+1𝐵)ℓ⧵(2𝑗𝐵)ℓ}𝑓₁𝑦₁⋯𝑓_ℓ𝑦_ℓ𝑑𝑦₁⋯𝑑𝑦_ℓ≤𝐶^𝑚_{𝑖=ℓ+12𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖×^∞_𝑗=112^𝑗+1𝐵^𝑚ℓ_𝑖=1^{2𝑗+1𝐵⧵2𝑗𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖, (44) and we arrive at the expression considered in the previous case. So for any 𝑥∈𝐵, we also have 𝑇𝑓^∞₁,…,𝑓^∞_ℓ,𝑓⁰_ℓ+1,…,𝑓⁰_𝑚(𝑥)≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅^∞_𝑗=1𝜈_𝑤2^𝑗+1𝐵^{(𝜅−1)/𝑝}. (45) Therefore, by the inequality (42) and the above pointwise inequality, we have 𝐼^{𝛼1,…,𝛼𝑚}≤𝜈_𝑤(𝐵)^{(1−𝜅)/𝑝}𝑇𝑓^∞₁,…,𝑓^∞_ℓ,𝑓⁰_ℓ+1,…,𝑓⁰_𝑚(𝑥)≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅^∞_𝑗=1𝜈_𝑤(𝐵)^{(1−𝜅)/𝑝}𝜈_𝑤2^𝑗+1𝐵^{(1−𝜅)/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅^∞_𝑗=1|𝐵|2^𝑗+1𝐵^{𝛿(1−𝜅)/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}. (46) Combining the above estimates and then taking the supremum over all balls 𝐵⊆ℝ^𝑛, we complete the proof of Theorem 1.

Proof of Theorem 2. For any ball 𝐵=𝐵(𝑥₀,𝑟_𝐵)⊆ℝ^𝑛 and decomposing 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_2𝐵, 𝑖=1,…,𝑚, then, for any given 𝜆>0, we can write 𝜈_𝑤𝑥∈𝐵∶𝑇𝑓₁,…,𝑓_𝑚>𝜆^1/𝑝≤𝜈_𝑤𝑥∈𝐵∶𝑇𝑓⁰₁,…,𝑓⁰_𝑚>𝜆2^𝑚1/𝑝+^′𝜈_𝑤𝑥∈𝐵∶𝑇𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚>𝜆2^𝑚1/𝑝=𝐼⁰_∗+^′𝐼^{𝛼1,…,𝛼𝑚}_∗, (47) where each term of ∑^′ contains at least one 𝛼_𝑖≠0. By Lemma 8 again, we know that 𝜈_𝑤∈𝐴_𝑚𝑝 with 1≤𝑚𝑝<∞. Applying Theorem B and Lemmas 13 and 5, we have 𝐼⁰_∗≤𝐶𝜆^𝑚_𝑖=12𝐵𝑓_𝑖(𝑥)^𝑝𝑖𝑤_𝑖(𝑥)𝑑𝑥^1/𝑝𝑖≤𝐶⋅∏^𝑚_𝑖=1𝑤_𝑖(2𝐵)^{𝜅/𝑝𝑖}𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}≤𝐶⋅𝜈_𝑤(2𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}≤𝐶⋅𝜈_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}. (48) In the proof of Theorem 1, we have already showed the following pointwise estimate (see (40) and (44)). Consider 𝑇𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖. (49) Without loss of generality, we may assume that 𝑝₁=⋯=𝑝_ℓ=min{𝑝₁,…,𝑝_𝑚}=1 and 𝑝_ℓ+1,…,𝑝_𝑚>1. Using Hölder’s inequality, the multiple 𝐴_𝑃 condition, and Lemma 13, we obtain 𝑇𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)≤𝐶^∞ℓ_𝑗=1𝑖=112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖×^𝑚_𝑖=ℓ+112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶^∞ℓ_𝑗=1𝑖=112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖𝑤_𝑖𝑦_𝑖𝑑𝑦_𝑖×inf_{𝑦𝑖∈2𝑗+1𝐵}𝑤_𝑖𝑦_𝑖⁻¹×^𝑚_𝑖=ℓ+112^𝑗+1𝐵₂^𝑗+1_𝐵𝑓_𝑖𝑦_𝑖^𝑝𝑖𝑤_𝑖𝑦_𝑖𝑑𝑦_𝑖^1/𝑝𝑖×_2𝑗+1𝐵𝑤_𝑖𝑦_𝑖^{1−𝑝′𝑖}𝑑𝑦_𝑖^{1/𝑝′𝑖}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}^∞_𝑗=1𝜈_𝑤2^𝑗+1𝐵^{(𝜅−1)/𝑝}. (50) Observe that 𝜈_𝑤∈𝐴_𝑚𝑝 with 1≤𝑚𝑝<∞. Thus, it follows from the inequality (42) that, for any 𝑥∈𝐵, 𝑇𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)=𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅1𝜈_𝑤(𝐵)^{∞(1−𝜅)/𝑝}_𝑗=1𝜈_𝑤(𝐵)^{(1−𝜅)/𝑝}𝜈_𝑤2^𝑗+1𝐵^{(1−𝜅)/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅1𝜈_𝑤(𝐵)^{∞(1−𝜅)/𝑝}_𝑗=1|𝐵|2^𝑗+1𝐵^{𝛿(1−𝜅)/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅1𝜈_𝑤(𝐵)^{(1−𝜅)/𝑝}. (51) If {𝑥∈𝐵∶|𝑇(𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚)(𝑥)|>𝜆/2^𝑚}=∅, then the inequality 𝐼^{𝛼1,…,𝛼𝑚}_∗≤𝐶⋅𝜈_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)} (52) holds trivially. Now, if instead we suppose that {𝑥∈𝐵∶|𝑇(𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚)(𝑥)|>𝜆/2^𝑚}≠∅, then, by the pointwise inequality (51), we have 𝜆<𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}⋅1𝜈_𝑤(𝐵)^{(1−𝜅)/𝑝}, (53) which is equivalent to 𝜈_𝑤(𝐵)^1/𝑝≤𝐶⋅𝜈_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}. (54) Therefore, 𝐼^{𝛼1,…,𝛼𝑚}_∗≤𝜈_𝑤(𝐵)^1/𝑝≤𝐶⋅𝜈_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}. (55) Summing up all the above estimates and then taking the supremum over all balls 𝐵⊆ℝ^𝑛 and all 𝜆>0, we complete the proof of Theorem 2.

By using Hölder’s inequality, it is easy to check that if each 𝑤_𝑖 is in 𝐴_𝑝𝑖, then ^𝑚_𝑖=1𝐴_𝑝𝑖⊂𝐴_𝑃, (56) and this inclusion is strict (see [9]). Thus, as direct consequences of Theorems 1 and 2, we immediately obtain the following.

Corollary 14. Let 𝑚≥2, and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. If 𝑝₁,…,𝑝_𝑚∈(1,∞) and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘 and 𝑤=(𝑤₁,…,𝑤_𝑚)∈∏^𝑚_𝑖=1𝐴_𝑝𝑖, then, for any 0<𝜅<1, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇𝑓_{𝐿𝑝,𝜅(𝜈𝑤)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}, (57) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖.

Corollary 15. Let 𝑚≥2 and let 𝑇 be an 𝑚-linear Calderón-Zygmund operator. If 𝑝₁,…,𝑝_𝑚∈[1,∞), min{𝑝₁,…,𝑝_𝑚}=1 and 𝑝∈(0,∞) with 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, and 𝑤=(𝑤₁,…,𝑤_𝑚)∈∏^𝑚_𝑖=1𝐴_𝑝𝑖, then, for any 0<𝜅<1, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝑇𝑓_{𝑊𝐿𝑝,𝜅(𝜈𝑤)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖,𝜅(𝑤𝑖)}, (58) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤^{𝑝/𝑝𝑖}_𝑖.

4. Proofs of Theorems 3 and 4

Following along the same lines as those of Lemma 13, we can also show the following result, which plays an important role in our proofs of Theorems 3 and 4.

Lemma 16. Let 𝑚≥2, 𝑞₁,…,𝑞_𝑚∈[1,∞) and 𝑞∈(0,∞) with 1/𝑞=∑^𝑚_𝑘=11/𝑞_𝑘. Assume that 𝑤^𝑞1₁,…,𝑤^𝑞𝑚_𝑚∈𝐴_∞ and 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖; then, for any ball 𝐵, there exists a constant 𝐶>0 such that ^𝑚_𝑖=1𝐵𝑤^𝑞𝑖_𝑖(𝑥)𝑑𝑥^{𝑞/𝑞𝑖}≤𝐶_𝐵𝜈_𝑤(𝑥)^𝑞𝑑𝑥. (59)

Proof of Theorem 3. Arguing as in the proof of Theorem 1, fix a ball 𝐵=𝐵(𝑥₀,𝑟_𝐵)⊆ℝ^𝑛 and decompose 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_2𝐵, 𝑖=1,…,𝑚. Since 𝐼_𝛼 is an 𝑚-linear operator, then we have 1𝜈^𝑞_𝑤(𝐵)^{𝜅/𝑝𝐵}𝐼_𝛼𝑓₁,…,𝑓_𝑚(𝑥)^𝑞𝜈_𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞≤1𝜈^𝑞_𝑤(𝐵)^{𝜅/𝑝𝐵}𝐼_𝛼𝑓⁰₁,…,𝑓⁰_𝑚(𝑥)^𝑞𝜈_𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞+^′1𝜈^𝑞_𝑤(𝐵)^{𝜅/𝑝𝐵}𝐼_𝛼𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)^𝑞𝜈_𝑤(𝑥)^𝑞𝑑𝑥^1/𝑞=𝐽⁰+^′𝐽^{𝛼1,…,𝛼𝑚}, (60) where each term of ∑^′ contains at least one 𝛼_𝑖≠0. In view of Lemma 9, we can see that (𝜈_𝑤)^𝑞∈𝐴_𝑚𝑞. Using Theorem C and Lemmas 16 and 5, we get 𝐽⁰1≤𝐶⋅𝜈^𝑞_𝑤(𝐵)^{𝑚𝜅/𝑝}_𝑖=1^2𝐵𝑓_𝑖(𝑥)^𝑝𝑖𝑤_𝑖(𝑥)^𝑝𝑖𝑑𝑥^1/𝑝𝑖≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅∏^𝑚_𝑖=1𝑤^𝑞𝑖_𝑖(2𝐵)^{𝜅𝑞/𝑝𝑞𝑖}𝜈^𝑞_𝑤(𝐵)^𝜅/𝑝=𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅∏^𝑚_𝑖=1𝑤^𝑞𝑖_𝑖(2𝐵)^{𝑞/𝑞𝑖𝜅/𝑝}𝜈^𝑞_𝑤(𝐵)^𝜅/𝑝≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅𝜈^𝑞_𝑤(2𝐵)^𝜅/𝑝𝜈^𝑞_𝑤(𝐵)^𝜅/𝑝≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖). (61) For the other terms, let us first deal with the case when 𝛼₁=⋯=𝛼_𝑚=∞. By the definition of 𝐼_𝛼, for any 𝑥∈𝐵, we obtain 𝐼_𝛼𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)=_{(ℝ𝑛)𝑚⧵(2𝐵)𝑚}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^{𝑚𝑛−𝛼}𝑑𝑦₁⋯𝑑𝑦_𝑚=^∞_{𝑗=1(2𝑗+1𝐵)𝑚⧵(2𝑗𝐵)𝑚}𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^{𝑚𝑛−𝛼}×𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^∞𝑚_{𝑗=1𝑖=12𝑗+1𝐵⧵2𝑗𝐵}𝑓_𝑖𝑦_𝑖𝑥−𝑦_𝑖^{𝑛−𝛼/𝑚}𝑑𝑦_𝑖≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵^{1−𝛼/𝑚𝑛2𝑗+1𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖. (62) Moreover, by using Hölder’s inequality, the multiple 𝐴_𝑃,𝑞 condition, and Lemma 16, we deduce that 𝐼_𝛼𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵^{1−𝛼/𝑚𝑛}×_2𝑗+1𝐵𝑓_𝑖𝑦_𝑖^𝑝𝑖𝑤_𝑖𝑦_𝑖^𝑝𝑖𝑑𝑦_𝑖^1/𝑝𝑖×_2𝑗+1𝐵𝑤_𝑖𝑦_𝑖^{−𝑝′𝑖}𝑑𝑦_𝑖^{1/𝑝′𝑖}≤𝐶^∞_𝑗=112^𝑗+1𝐵^{𝑚−𝛼/𝑛}⋅2^𝑗+1𝐵^{1/𝑞+∑𝑚𝑖=1(1−1/𝑝𝑖)}𝜈^𝑞_𝑤2^𝑗+1𝐵^1/𝑞×^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)𝑤^𝑞𝑖_𝑖2^𝑗+1𝐵^{𝜅𝑞/𝑝𝑞𝑖}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1∏^𝑚_𝑖=1𝑤^𝑞𝑖_𝑖2^𝑗+1𝐵^{𝑞/𝑞𝑖𝜅/𝑝}𝜈^𝑞_𝑤2^𝑗+1𝐵^1/𝑞≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1𝜈^𝑞_𝑤2^𝑗+1𝐵^{𝜅/𝑝−1/𝑞}. (63) Since (𝜈_𝑤)^𝑞∈𝐴_𝑚𝑞⊂𝐴_∞, then it follows immediately from Lemma 7 that 𝜈^𝑞_𝑤(𝐵)𝜈^𝑞_𝑤2^𝑗+1𝐵|𝐵|≤𝐶2^𝑗+1𝐵^𝛿′. (64) Hence, 𝐽^∞,…,∞≤𝜈^𝑞_𝑤(𝐵)^{1/𝑞−𝜅/𝑝}𝐼_𝛼𝑓^∞₁,…,𝑓^∞_𝑚(𝑥)≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1𝜈^𝑞_𝑤(𝐵)^{1/𝑞−𝜅/𝑝}𝜈^𝑞_𝑤2^𝑗+1𝐵^{1/𝑞−𝜅/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1|𝐵|2^𝑗+1𝐵^{𝛿′(1/𝑞−𝜅/𝑝)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖), (65) where in the last inequality we have used the fact that 0<𝜅<𝑝/𝑞 and 𝛿^′>0. We now consider the case where exactly ℓ of the 𝛼_𝑖 are ∞ for some 1≤ℓ<𝑚. We only give the arguments for one of these cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Using the definition of 𝐼_𝛼 again, we can see that, for any 𝑥∈𝐵, 𝐼_𝛼𝑓^∞₁,…,𝑓^∞_ℓ,𝑓⁰_ℓ+1,…,𝑓⁰_𝑚(𝑥=_{(ℝ𝑛)ℓ⧵(2𝐵)ℓ(2𝐵)𝑚−ℓ})𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚^{𝑚𝑛−𝛼}×𝑑𝑦₁⋯𝑑𝑦_𝑚≤𝐶^𝑚_{𝑖=ℓ+12𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖×^∞_𝑗=112^𝑗+1𝐵^{𝑚−𝛼/𝑛}×_{(2𝑗+1𝐵)ℓ⧵(2𝑗𝐵)ℓ}𝑓₁𝑦₁⋯𝑓_ℓ𝑦_ℓ𝑑𝑦₁⋯𝑑𝑦_ℓ≤𝐶^𝑚_{𝑖=ℓ+12𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖×^∞_𝑗=112^𝑗+1𝐵^{ℓ𝑚−𝛼/𝑛}_𝑖=1^{2𝑗+1𝐵⧵2𝑗𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵^{1−𝛼/𝑚𝑛2𝑗+1𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖, (66) and we arrive at the expression considered in the previous case. Thus, for any 𝑥∈𝐵, we also have 𝐼_𝛼𝑓^∞₁,…,𝑓^∞_ℓ,𝑓⁰_ℓ+1,…,𝑓⁰_𝑚(𝑥)≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1𝜈^𝑞_𝑤2^𝑗+1𝐵^{𝜅/𝑝−1/𝑞}. (67) Therefore, by the inequality (64) and the above pointwise inequality, we obtain 𝐽^{𝛼1,…,𝛼𝑚}≤𝜈^𝑞_𝑤(𝐵)^{1/𝑞−𝜅/𝑝}𝐼_𝛼𝑓^∞₁,…,𝑓^∞_ℓ,𝑓⁰_ℓ+1,…,𝑓⁰_𝑚(𝑥)≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1𝜈^𝑞_𝑤(𝐵)^{1/𝑞−𝜅/𝑝}𝜈^𝑞_𝑤2^𝑗+1𝐵^{1/𝑞−𝜅/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1|𝐵|2^𝑗+1𝐵^{𝛿′(1/𝑞−𝜅/𝑝)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖). (68) Summarizing the estimates derived above and then taking the supremum over all balls 𝐵⊆ℝ^𝑛, we finish the proof of Theorem 3.

Proof of Theorem 4. As before, fix a ball 𝐵=𝐵(𝑥₀,𝑟_𝐵)⊆ℝ^𝑛 and split 𝑓_𝑖 into 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_2𝐵, 𝑖=1,…,𝑚. Then for each fixed 𝜆>0, we can write 𝜈^𝑞_𝑤𝑥∈𝐵∶𝐼_𝛼𝑓₁,…,𝑓_𝑚>𝜆^1/𝑞≤𝜈^𝑞_𝑤𝑥∈𝐵∶𝐼_𝛼𝑓⁰₁,…,𝑓⁰_𝑚>𝜆2^𝑚1/𝑞+^′𝜈^𝑞_𝑤𝑥∈𝐵∶𝐼_𝛼𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚>𝜆2^𝑚1/𝑞=𝐽⁰_∗+^′𝐽^{𝛼1,…,𝛼𝑚}_∗, (69) where each term of ∑^′ contains at least one 𝛼_𝑖≠0. By Lemma 9 again, we know that (𝜈_𝑤)^𝑞∈𝐴_𝑚𝑞 with 1<𝑚𝑞<∞. Using Theorem D and Lemmas 16 and 5, we have 𝐽⁰_∗≤𝐶𝜆^𝑚_𝑖=12𝐵𝑓_𝑖(𝑥)^𝑝𝑖𝑤_𝑖(𝑥)^𝑝𝑖𝑑𝑥^1/𝑝𝑖≤𝐶⋅∏^𝑚_𝑖=1𝑤^𝑞𝑖_𝑖(2𝐵)^{𝜅𝑞/𝑝𝑞𝑖}𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)≤𝐶⋅𝜈^𝑞_𝑤(2𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)≤𝐶⋅𝜈^𝑞_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖). (70) In the proof of Theorem 3, we have already proved the following pointwise estimate (see (62) and (66)). Consider 𝐼_𝛼𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)≤𝐶^∞𝑚_𝑗=1𝑖=112^𝑗+1𝐵^{1−𝛼/𝑚𝑛2𝑗+1𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖. (71) Without loss of generality, we may assume that 𝑝₁=⋯=𝑝_ℓ=min{𝑝₁,…,𝑝_𝑚}=1 and 𝑝_ℓ+1,…,𝑝_𝑚>1. By using Hölder’s inequality, the multiple 𝐴_𝑃,𝑞 condition, and Lemma 16, we obtain 𝐼_𝛼𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)≤𝐶^∞ℓ_𝑗=1𝑖=112^𝑗+1𝐵^{1−𝛼/𝑚𝑛2𝑗+1𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖×^𝑚_𝑖=ℓ+112^𝑗+1𝐵^{1−𝛼/𝑚𝑛2𝑗+1𝐵}𝑓_𝑖𝑦_𝑖𝑑𝑦_𝑖≤𝐶^∞ℓ_𝑗=1𝑖=112^𝑗+1𝐵^{1−𝛼/𝑚𝑛}×_2𝑗+1𝐵𝑓_𝑖𝑦_𝑖𝑤_𝑖𝑦_𝑖𝑑𝑦_𝑖inf_{𝑦𝑖∈2𝑗+1𝐵}𝑤_𝑖𝑦_𝑖⁻¹×^𝑚_𝑖=ℓ+112^𝑗+1𝐵^{1−𝛼/𝑚𝑛2𝑗+1𝐵}𝑓_𝑖𝑦_𝑖^𝑝𝑖𝑤_𝑖𝑦_𝑖^𝑝𝑖𝑑𝑦_𝑖^1/𝑝𝑖×_2𝑗+1𝐵𝑤_𝑖𝑦_𝑖^{−𝑝′𝑖}𝑑𝑦_𝑖^{1/𝑝′𝑖}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅^∞_𝑗=1𝜈^𝑞_𝑤2^𝑗+1𝐵^{𝜅/𝑝−1/𝑞}. (72) Note that (𝜈_𝑤)^𝑞∈𝐴_𝑚𝑞 with 1<𝑚𝑞<∞. Hence, it follows from the inequality (64) that, for any 𝑥∈𝐵, 𝐼_𝛼𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥)=𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅1𝜈^𝑞_𝑤(𝐵)^{∞1/𝑞−𝜅/𝑝}_𝑗=1𝜈^𝑞_𝑤(𝐵)^{1/𝑞−𝜅/𝑝}𝜈^𝑞_𝑤2^𝑗+1𝐵^{1/𝑞−𝜅/𝑝}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅1𝜈^𝑞_𝑤(𝐵)^{∞1/𝑞−𝜅/𝑝}_𝑗=1|𝐵|2^𝑗+1𝐵^{𝛿′(1/𝑞−𝜅/𝑝)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅1𝜈^𝑞_𝑤(𝐵)^{1/𝑞−𝜅/𝑝}. (73) If {𝑥∈𝐵∶|𝐼_𝛼(𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚)(𝑥)|>𝜆/2^𝑚}=∅, then the inequality 𝐽^{𝛼1,…,𝛼𝑚}_∗≤𝐶⋅𝜈^𝑞_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖) (74) holds trivially. Now, if instead we assume that {𝑥∈𝐵∶|𝐼_𝛼(𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚)(𝑥)|>𝜆/2^𝑚}≠∅, then, by the pointwise inequality (73), we get 𝜆<𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖)⋅1𝜈^𝑞_𝑤(𝐵)^{1/𝑞−𝜅/𝑝}, (75) which in turn gives that 𝜈^𝑞_𝑤(𝐵)^1/𝑞≤𝐶⋅𝜈^𝑞_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖). (76) Therefore, 𝐽^{𝛼1,…,𝛼𝑚}_∗≤𝜈^𝑞_𝑤(𝐵)^1/𝑞≤𝐶⋅𝜈^𝑞_𝑤(𝐵)^𝜅/𝑝𝜆^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖). (77) Collecting all the above estimates and then taking the supremum over all balls 𝐵⊆ℝ^𝑛 and all 𝜆>0, we conclude the proof of Theorem 4.

By using Hölder’s inequality, it is easy to verify that if 1≤𝑝_𝑖<𝑞_𝑖, 1/𝑞=∑^𝑚_𝑘=11/𝑞_𝑘, and each 𝑤_𝑖 is in 𝐴_{𝑝𝑖,𝑞𝑖}, then we have ^𝑚_𝑖=1𝐴_{𝑝𝑖,𝑞𝑖}⊂𝐴_𝑃,𝑞, (78) and this inclusion is strict (see [17]). Also, recall that 𝑤∈𝐴_𝑝,𝑞 if and only if 𝑤^𝑞∈𝐴_{1+𝑞/𝑝′}⊂𝐴_∞ (see [36]). Thus, as straightforward consequences of Theorems 3 and 4, we finally obtain the following.

Corollary 17. Let 𝑚≥2, let 0<𝛼<𝑚𝑛, and let 𝐼_𝛼 be an 𝑚-linear fractional integral operator. If 𝑝₁,…,𝑝_𝑚∈(1,∞), 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, 1/𝑞_𝑘=1/𝑝_𝑘−𝛼/𝑚𝑛 and 1/𝑞=∑^𝑚_𝑘=11/𝑞_𝑘=1/𝑝−𝛼/𝑛, and 𝑤=(𝑤₁,…,𝑤_𝑚)∈∏^𝑚_𝑖=1𝐴_{𝑝𝑖,𝑞𝑖}, then, for any 0<𝜅<𝑝/𝑞, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝐼_𝛼(𝑓)_{𝐿𝑞,𝜅𝑞/𝑝((𝜈𝑤)𝑞)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖), (79) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖.

Corollary 18. Let 𝑚≥2, let 0<𝛼<𝑚𝑛, and let 𝐼_𝛼 be an 𝑚-linear fractional integral operator. If 𝑝₁,…,𝑝_𝑚∈[1,∞), min{𝑝₁,…,𝑝_𝑚}=1, 1/𝑝=∑^𝑚_𝑘=11/𝑝_𝑘, 1/𝑞_𝑘=1/𝑝_𝑘−𝛼/𝑚𝑛 and 1/𝑞=∑^𝑚_𝑘=11/𝑞_𝑘=1/𝑝−𝛼/𝑛, and 𝑤=(𝑤₁,…,𝑤_𝑚)∈∏^𝑚_𝑖=1𝐴_{𝑝𝑖,𝑞𝑖}, then, for any 0<𝜅<𝑝/𝑞, there exists a constant 𝐶>0 independent of 𝑓=(𝑓₁,…,𝑓_𝑚) such that 𝐼_𝛼(𝑓)_{𝑊𝐿𝑞,𝜅𝑞/𝑝((𝜈𝑤)𝑞)}≤𝐶^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖𝑖𝑖,𝜅𝑝𝑞/𝑝𝑞(𝑤}^𝑝_{𝑖𝑖,𝑤}^𝑞_𝑖𝑖), (80) where 𝜈_𝑤=∏^𝑚_𝑖=1𝑤_𝑖.