Weighted Estimates for Maximal Commutators of Multilinear Singular Integrals

Abstract

This paper is concerned with the pointwise estimates for the sharp function of the maximal multilinear commutators 𝑇^∗_Σ𝑏 and maximal iterated commutator 𝑇^∗_Π𝑏, generalized by 𝑚-linear operator 𝑇 and a weighted Lipschitz function 𝑏. The (𝐿^𝑝1(𝜇)×⋯×𝐿^𝑝𝑚(𝜇),𝐿^𝑟(𝜇^1−𝑟)) boundedness and the (𝐿^𝑝1(𝜇)×⋯×𝐿^𝑝𝑚(𝜇),𝐿^𝑟(𝜇^1−𝑚𝑟)) boundedness are obtained for maximal multilinear commutator 𝑇^∗_Σ𝑏 and maximal iterated commutator 𝑇^∗_Π𝑏, respectively.

1. Introduction and Notation

The theory of multilinear Calderón-Zygmund singular integral operators,originated from the work of Coifman and Meyers’, has an important role in harmonic analysis. Its study has been attracting a lot of attention in the last few decades. So far, a number of properties for multilinear operators are parallel to those of the classical linear Calderón-Zygmund operators but new interesting phenomena have also been observed. A systematic analysis of many basic properties of such multilinear operators can be found in the articles by Coifman and Meyer [1], Grafakos and Torres [2–4], and Lerner et al. [5]. So we first recall the definition and results of multilinear Calderón-Zygmund operators as well as the corresponding maximal multilinear operators.

Definition 1.1. Let 𝑇 be a multilinear operator initially defined on the m-fold product of Schwartz space and taking values into the space of tempered distributions: 𝑇∶𝒮ℝ^𝑛×⋯×𝒮ℝ^𝑛→𝒮^′ℝ^𝑛. (1.1) Following [2], we say that 𝑇 is an m-linear Calderón-Zygmund operator if for some 1≤𝑞_𝑗<∞, it extends to a bounded multilinear operator from 𝐿^𝑞1×⋯×𝐿^𝑞𝑚 to 𝐿^𝑞, where 1/𝑞=(1/𝑞₁)+⋯+(1/𝑞_𝑚), and if there exists a function 𝐾, defined off the diagonal 𝑥=𝑦₁=⋯=𝑦_𝑚 in (ℝ^𝑛)^𝑚+1, satisfying 𝑇𝑓₁,…,𝑓_𝑚(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚, (1.2) for all 𝑥∉⋂^𝑚_𝑗=1supp𝑓_𝑗.
And 𝐾𝑦₀,𝑦₁,…,𝑦_𝑚≤𝐴∑^𝑚_𝑘,𝑙=0𝑦_𝑘−𝑦_𝑙^𝑚𝑛,𝐾𝑦₀,…,𝑦_𝑗,…,𝑦_𝑚−𝐾𝑦₀,…,𝑦^′_𝑗,…,𝑦_𝑚≤𝐴𝑦_𝑗−𝑦^′_𝑗^𝜀∑^𝑚_𝑘,𝑙=0𝑦_𝑘−𝑦_𝑙^{𝑚𝑛+𝜀}, (1.3)(1.4) for some 𝜀>0 and all 0≤𝑗≤𝑚, where |𝑦_𝑗−𝑦^′_𝑗|≤(1/2)max_{0≤𝑘≤𝑚}|𝑦_𝑗−𝑦_𝑘|.

The maximal multilinear singular integral operator was defined by 𝑇^∗→𝑓(𝑥)=sup_𝛿>0𝑇_𝛿𝑓₁,…,𝑓_𝑚(𝑥), (1.5) where 𝑇_𝛿 is the smooth truncation of 𝑇 given by 𝑇_𝛿𝑓₁,…,𝑓_𝑚(𝑥)=_{|𝑥−𝑦1|2+⋯+|𝑥−𝑦𝑚|2>𝛿2}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚. (1.6) As pointed in [4], 𝑇^∗(^→𝑓) is pointwise well defined when 𝑓_𝑗∈𝐿^𝑞𝑗(ℝ^𝑛) with 1≤𝑞_𝑗<∞.

The study for the multilinear singular integral operator and its maximal operators attracts many authors’ attention. For maximal multilinear operator 𝑇^∗, one can see [4] for details. We list some results for 𝑇^∗ as follows.

Theorem A (see [4]). Let 1≤𝑞_𝑗<∞ and q such that 1/𝑞=(1/𝑞₁)+⋯+(1/𝑞_𝑚) and 𝜔∈𝐴_𝑞1∩⋯∩𝐴_𝑞𝑚. Let 𝑇 be an m-linear Calderón-Zygmund operator. Then there exists a constant 𝐶_𝑛,𝑞<∞, such that for all ^→𝑓=(𝑓₁,…,𝑓_𝑚) satisfying 𝑇^∗→𝑓_{𝐿𝑞(𝜔)}≤𝐶_𝑛,𝑞(𝐴+𝑊)^𝑚_𝑗=1𝑓_{𝑗𝐿𝑞𝑗(𝜔)}, (1.7) where W is the norm of 𝑇 in the mapping 𝑇: 𝐿¹×⋯𝐿¹→𝐿^1/𝑚,∞.

Theorem B (see [4]). Let 𝑇 be an m -linear Calderón-Zygmund operator. Then, for all exponents 𝑝,𝑝₁,…,𝑝_𝑚, satisfying (1/𝑝₁)+⋯+(1/𝑝_𝑚)=1/𝑝, one has 𝑇^∗∶𝐿^𝑝1×⋯×𝐿^𝑝𝑚→𝐿^𝑝, (1.8) when 1<𝑝₁,…,𝑝_𝑚≤∞, one also has 𝑇^∗∶𝐿^𝑝1×⋯×𝐿^𝑝𝑚→𝐿^𝑝,∞, (1.9) when at least one 𝑝_𝑗 is equal one. In either cases the norm of 𝑇^∗ is controlled by a constant multiple of 𝐴+𝑊.

Definition 1.2 (see [5] (commutators in the 𝑗th entry)). Given a collection of locally integrable function ^→𝑏=(𝑏₁,…,𝑏_𝑚), we define the commutators of the 𝑚-linear Calderón-Zygmund operator 𝑇 to be 𝑇_Σ𝑏𝑓₁,…,𝑓_𝑚=^𝑚_𝑗=1𝑇^𝑗_𝑏𝑗^→𝑓, (1.10) where each term is the commutator of 𝑏_𝑗 and 𝑇 in the 𝑗th entry of 𝑇, that is 𝑇^𝑗_𝑏𝑗^→𝑓=𝑏_𝑗𝑇𝑓₁,…,𝑓_𝑗,…,𝑓_𝑚−𝑇𝑓₁,…,𝑏_𝑗𝑓_𝑗,…,𝑓_𝑚. (1.11)

In [6], the following more general iterated commutators of multilinear Calderón-Zygmund operators and pointwise multiplication with functions in BMO were defined and studied in products of Lebesgue spaces, including strong type and weak end-point estimates with multiple 𝐴^→_𝑝 weights. That is, 𝑇_Π𝑏^→𝑓(𝑥)=𝑏₁,𝑏₂,…,𝑏_𝑚−1,𝑏_𝑚,𝑇_{𝑚𝑚−121}=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚^𝑚_𝑗=1𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚. (1.12)

For the operator [𝑏,𝑇], when 𝑇 is the Calderón-Zygmund singular integral operator and 𝑏∈Λ̇_𝛽(ℝ^𝑛) (the homogeneous Lipschitz spaces), Paluszyński [7] established the (𝐿^𝑝,𝐿^𝑞) boundedness with 1<𝑝<1/𝛽 and 1/𝑞=1/𝑝+𝛽/𝑛. Hu and Gu [8] extended this results to the case: 𝑏∈Lip_𝛽,𝜇 with 𝜇∈𝐴₁.

Now we present the definitions of two classes of maximal commutators of multilinear singular integral operators. One is 𝑇^∗_Σ𝑏^→𝑓(𝑥)=sup_𝛿>0^𝑚_{𝑗=1∑𝑚𝑖=1|𝑥−𝑦𝑖|2>𝛿2}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓_𝑗𝑦_𝑗⋯𝑓_𝑚𝑦_𝑚𝑑^→𝑦, (1.13) the other is 𝑇^∗_Π𝑏^→𝑓(𝑥)=sup_𝛿>0𝑏₁,𝑏₂,…,𝑏_𝑚−1,𝑏_𝑚,𝑇_{𝛿𝑚𝑚−1},…₂₁^→𝑓(𝑥)=sup_{𝛿>0∑𝑚𝑖=1|𝑥−𝑦𝑖|2>𝛿2}𝐾𝑥,𝑦₁,…,𝑦_𝑚^𝑚_𝑗=1𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑^→𝑦, (1.14) where 𝑑^→𝑦=𝑑𝑦₁⋯𝑑𝑦_𝑚. It is obvious to see that 𝑇^∗_Σ𝑏^→𝑓(𝑥)≤^𝑚_𝑗=1𝑇_𝑏^∗,𝑗_𝑗^→𝑓(𝑥). (1.15) The main purpose of this paper is to extend the results in [8] to the maximal commutators generated by multilinear singular integrals 𝑇 and Lip_𝛽,𝜇 functions ^→𝑏.

We can formulate our result as following.

Theorem 1.3. Assume that the kernel 𝐾 satisfies (1.3) and (1.4). Let 1<𝑞₁,…,𝑞_𝑚,𝑞<∞ be given numbers satisfying 1/𝑞=(1/𝑞₁)+⋯+(1/𝑞_𝑚). And assume that 𝑇 maps 𝐿^𝑞1(ℝ^𝑛)×⋯×𝐿^𝑞𝑚(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛). For 𝑗∈{1,…,𝑚} and let 1/𝑟=(1/𝑝)−(𝛽/𝑛), 1<𝑝<𝑟<∞, 0<𝛽<1, and 1/𝑝= 1/𝑝₁+⋯+1/𝑝_𝑚 with 1<𝑝_𝑖<∞, 𝑖=1,…,𝑚. Given 𝜇 such that 𝜇∈𝐴₁(ℝ^𝑛) and 𝑏_𝑗∈Lip_𝛽,𝜇(ℝ^𝑛)  (𝑗=1,…,𝑚), then one has 𝑇_𝑏^∗,𝑗_𝑗^→𝑓_{𝐿𝑟(𝜇1−𝑟)}≤𝐶𝑏_{𝑗Lip𝛽,𝜇}^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜇)},𝑗=1,…,𝑚. (1.16) From (1.15) and (1.16), one can get 𝑇^∗_Σ𝑏(^→𝑓)_{𝐿𝑟(𝜇1−𝑟)}≤𝐶^𝑚_𝑗=1𝑏_{𝑗Lip𝛽,𝜇}^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜇)}. (1.17)

If 𝜇=1, one can get the following.

Theorem 1.4. Assume that the kernel 𝐾 satisfies (1.3) and (1.4). Let 1<𝑞₁,…,𝑞_𝑚,𝑞<∞ be given numbers satisfying 1/𝑞=(1/𝑞₁)+⋯+(1/𝑞_𝑚). And assume that 𝑇 maps 𝐿^𝑞1(ℝ^𝑛)×⋯×𝐿^𝑞𝑚(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛). For 𝑗∈{1,…,𝑚} and let 1/𝑟=(1/𝑝)−(𝛽/𝑛), 1<𝑝<𝑟<∞, 0<𝛽<1 and 1/𝑝= (1/𝑝₁)+⋯+(1/𝑝_𝑚) with 1<𝑝_𝑖<∞, 𝑖=1,…,𝑚. Set 𝑏_𝑗∈Lip_𝛽(ℝ^𝑛)(𝑗=1,…,𝑚), then one has 𝑇_𝑏^∗,𝑗_𝑗^→𝑓_{𝐿𝑟(ℝ𝑛)}≤𝐶𝑏_𝑗Lip𝛽^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(ℝ𝑛)},𝑗=1,…,𝑚. (1.18) From (1.15), one can get 𝑇^∗_Σ𝑏(^→𝑓)_{𝐿𝑟(ℝ𝑛)}≤𝐶^𝑚_𝑗=1𝑏_𝑗Lip𝛽^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(ℝ𝑛)}. (1.19)

The following theorem states the weighted estimates with two different weights for maximal iterated commutator of multilinear singular integrals.

Theorem 1.5. Assume that the kernel 𝐾 satisfies (1.3) and (1.4). Let 1<𝑞₁,…,𝑞_𝑚,𝑞<∞ be given numbers satisfying 1/𝑞=(1/𝑞₁)+⋯+(1/𝑞_𝑚). And assume that 𝑇 maps 𝐿^𝑞1(ℝ^𝑛)×⋯×𝐿^𝑞𝑚(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛). Let 1/𝑟_𝑖=(1/𝑝_𝑖)−(𝛽_𝑖/𝑛), 1<𝑝_𝑖<𝑟_𝑖<∞, 0<𝛽_𝑖<1, 𝑖=1,…,𝑚 with 1/𝑝= (1/𝑝₁)+⋯+(1/𝑝_𝑚), 1/𝑟=(1/𝑟₁)+⋯+(1/𝑟_𝑚), and 𝛽=𝛽₁+⋯+𝛽_𝑚, 0<𝛽<1. Given 𝜇 such that 𝜇∈𝐴₁(ℝ^𝑛) and 𝑏_𝑖∈Lip_{𝛽𝑖,𝜇}(ℝ^𝑛)(𝑖=1,…,𝑚), then one has 𝑇^∗_Π𝑏^→𝑓_{𝐿𝑟(𝜇1−𝑚𝑟)}≤𝐶^𝑚_𝑖=1𝑏_{𝑖Lip𝛽𝑖,𝜇}𝑓_{𝑖𝐿𝑝𝑖(𝜇)}. (1.20)

Similarly as Theorem 1.4, one also obtains the unweighted estimates of maximal iterated commutators.

Theorem 1.6. Assume that the kernel 𝐾 satisfies (1.3) and (1.4). Let 1<𝑞₁,…,𝑞_𝑚,𝑞<∞ be given numbers satisfying 1/𝑞=(1/𝑞₁)+⋯+(1/𝑞_𝑚). And assume that 𝑇 maps 𝐿^𝑞1(ℝ^𝑛)×⋯×𝐿^𝑞𝑚(ℝ^𝑛) to 𝐿^𝑞(ℝ^𝑛). Let 1/𝑟_𝑖=(1/𝑝_𝑖)−(𝛽_𝑖/𝑛), 1<𝑝_𝑖<𝑟_𝑖<∞, 0<𝛽_𝑖<1, 𝑖=1,…,𝑚 with 1/𝑝=(1/𝑝₁)+⋯+(1/𝑝_𝑚), 1/𝑟=(1/𝑟₁)+⋯+(1/𝑟_𝑚), and 𝛽=𝛽₁+⋯+𝛽_𝑚, 0<𝛽<1. Set 𝑏_𝑖∈Lip_𝛽𝑖(ℝ^𝑛)(𝑖=1,…,𝑚), then one has 𝑇^∗_Π𝑏^→𝑓_{𝐿𝑟(ℝ𝑛)}≤𝐶^𝑚_𝑖=1𝑏_{𝑖Lip𝛽𝑖}𝑓_{𝑖𝐿𝑝𝑖(ℝ𝑛)}. (1.21)

The rest of this paper is organized as follows. In Section 2, we recall some standard definitions and lemmas. Section 3 is devoted to the proof of our theorems. Throughout this paper, we use the letter 𝐶 to denote a positive constant that varies line to line, but it is independent of the essential variable. For any 1≤𝑝≤∞, the 𝑝^′ is always used to denote the dual index such that (1/𝑝)+(1/𝑝^′)=1.

2. Preliminaries

A nonnegative function 𝜇 defined on ℝ^𝑛 is called weight if it is locally integrable. A weight 𝜇 is said to belong to the Muckenhoupt class 𝐴_𝑝(ℝ^𝑛), 1<𝑝<∞, if there exists a constant 𝐶 such that sup_𝐵1|𝐵|_𝐵𝜇(𝑥)1𝑑𝑥|𝐵|_𝐵𝜇(𝑥)^{−1/(𝑝−1)}𝑑𝑥^𝑝−1≤𝐶<∞, (2.1) for every ball 𝐵⊂ℝ^𝑛. A weight 𝜇 is said to belong to class 𝐴₁(ℝ^𝑛) if 1|𝐵|_𝐵𝜇(𝑥)𝑑𝑥≤𝐶inf_𝑥∈𝐵𝜇(𝑥),almostall𝑥∈ℝ^𝑛, (2.2) for every ball 𝐵∋𝑥. The class 𝐴_∞(ℝ^𝑛) can be characterized as 𝐴_∞=⋃_1≤𝑝<∞𝐴_𝑝.

Many properties of weights can be found in the book [9], we only collect some of them in the following lemma which will be used bellow.

Lemma 2.1. (i)  𝐴_𝑝⊂𝐴_𝑞 for 1≤𝑝<𝑞≤∞;(ii) if 𝜇∈𝐴₁, then 𝜇^𝜃∈𝐴₁ for 0≤𝜃≤1;(iii)for 1<𝑝<∞, 𝜇∈𝐴_𝑝 if and only if 𝜇^1−𝑝′∈𝐴_𝑝′.

A locally integrable function 𝑓 belongs to the weighted Lipschitz space Lip^𝑝_𝛽,𝜇(ℝ^𝑛) for 1≤𝑝≤∞, 0<𝛽<1 and 𝜇∈𝐴_∞ if sup_𝐵∋𝑥1𝜇(𝐵)^𝛽/𝑛1𝐵_𝐵𝑓(𝑥)−𝑓_𝐵^𝑝𝜇(𝑥)^1−𝑝𝑑𝑥^1/𝑝≤𝐶<∞. (2.3) The smallest bound 𝐶 satisfying (1.19) is then taken to be the norm of 𝑓 denoted by ‖𝑓‖_Lip^𝑝_𝛽,𝜇. Put Lip_𝛽,𝜇=Lip¹_𝛽,𝜇.

If 𝜇∈𝐴₁,𝑏∈Lip_𝛽,𝜇(0<𝛽<1), from the definition of ‖𝑓‖_Lip𝛽,𝜇, it is obvious to see 𝑏_𝐵−𝑏₂^𝑘+1_𝐵≤𝐶𝑘𝜇(𝑥)𝜇𝐵𝑥,2^𝑘+1𝑅^𝛽/𝑛‖𝑏‖_Lip𝛽,𝜇, (2.4) where 𝑏_𝐵=(1/|𝐵|)∫_𝐵𝑏(𝑦)𝑑𝑦.

The important properties of the weights are the weighted estimates for the maximal function, the sharp maximal function and their variants. One first recalls the maximal function defined by 𝑀𝑓(𝑥)=sup_𝐵∋𝑥1|𝐵|_𝐵𝑓𝑦𝑑𝑦, (2.5) It is well known that for 1<𝑝<∞,  𝑀 maps 𝐿^𝑝(𝜇) into itself if and only if 𝜇∈𝐴_𝑝, see [10].

The sharp maximal function is defined by 𝑀^#𝑓(𝑥)=sup_𝐵∋𝑥1|𝐵|_𝐵𝑓(𝑥)−𝑓_𝐵𝑑𝑦≈sup_𝐵∋𝑥inf_𝑐1|𝐵|_𝐵𝑓(𝑥)−𝑐𝑑𝑦. (2.6) One also recalls the variants 𝑀_𝛿(𝑓)(𝑥)=(𝑀(|𝑓|^𝛿)(𝑥))^1/𝛿, and 𝑀^#_𝛿(𝑓)(𝑥)=(𝑀^#(|𝑓|^𝛿)(𝑥))^1/𝛿. We denote the weighted fractional maximal operators by 𝑀_{𝛼,𝜇,𝑠}𝑓(𝑥)=sup_𝐵∋𝑥1𝜇(𝐵)^{1−(𝑠𝛼/𝑛)𝐵}𝑓𝑦^𝑠𝜇𝑦𝑑𝑦^1/𝑠. (2.7) Recall that 𝑀_𝛼∶=𝑀_𝛼,1,1 is the weighted fractional maximal operators, that is 𝑀_𝛼𝑓(𝑥)=sup_𝐵∋𝑥1𝜇(𝐵)^{1−(𝛼/𝑛)𝐵}𝑓𝑦𝜇𝑦𝑑𝑦. (2.8) The following lemmas are all from [11].

Lemma 2.2 (Kolmogorov’s inequality). Let (𝑋,𝜇) be a probability measure space and let 0<𝑝<𝑞<∞ then there exists a constant 𝐶=𝐶_𝑝,𝑞 such that for any measurable function 𝑓𝑓_{𝐿𝑝(𝜇)}≤𝐶𝑓_{𝐿𝑞,∞(𝜇)}. (2.9)

Lemma 2.3. Let 0<𝑝,𝛿<∞ and 𝜇∈𝐴_∞(ℝ^𝑛), there exists 𝐶>0 depending on the 𝐴_∞(ℝ^𝑛) constant of 𝜇 such that 𝑀_𝛿𝑓_{𝐿𝑝(𝜇)}≤𝐶𝑀^#_𝛿𝑓_{𝐿𝑝(𝜇)}, (2.10) for any function 𝑓 for which the left side of the above inequality is finite.

Lemma 2.4. Suppose that 0<𝛼<𝑛, 0<𝑠<𝑝<𝛼/𝑛, 1/𝑞=(1/𝑝)−(𝛼/𝑛). If 𝜇∈𝐴_∞(ℝ^𝑛), then there exists a constant 𝐶=𝐶_𝑝,𝑞 such that for any measurable function 𝑓𝑀_{𝛼,𝜇,𝑠}𝑓_{𝐿𝑞(𝜇)}≤𝐶𝑓_{𝐿𝑝(𝜇)}. (2.11)

Lemma 2.5. Suppose that 0<𝛼<𝑛, 0<𝑠<𝑝<𝛼/𝑛, 1/𝑞=(1/𝑝)−(𝛼/𝑛). If 𝜇∈𝐴_{1+(𝑞/𝑝′)}(ℝ^𝑛), then there exists a constant 𝐶=𝐶_𝑝,𝑞 such that for any measurable function 𝑓𝑀_𝛼𝑓_{𝐿𝑞(𝜇)}≤𝐶𝑓_{𝐿𝑝(𝜇𝑝/𝑞)}. (2.12)

3. Two Estimates for Maximal Multilinear Commutators

We will prove our theorems in this section. To begin, we prepare another two iterated operators to control the commutators.

Let 𝜑,𝜓∈𝐶^∞([0,+∞)) such that |𝜑^′(𝑡)|≤𝐶/𝑡, |𝜓^′(𝑡)|≤𝐶/𝑡 and satisfying 𝜒_[2,∞)(𝑡)≤𝜑(𝑡)≤𝜒_[1,∞)(𝑡),𝜒_[1,2](𝑡)≤𝜑(𝑡)≤𝜒_[1/2,3](𝑡). (3.1) We define the maximal operators Φ^∗→𝑓(𝑥)=sup_{𝜂>0(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝜑𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚𝜂^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖𝑑^→𝑦,Ψ^∗→𝑓(𝑥)=sup_{𝜂>0(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝜓𝑥−𝑦₁+⋯+𝑥−𝑦_𝑚𝜂^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖𝑑^→𝑦. (3.2) For simplicity, we denote 𝐾_𝜑,𝜂(𝑥,𝑦₁,…,𝑦_𝑚)=𝐾(𝑥,𝑦₁,…,𝑦_𝑚)𝜑(|𝑥−𝑦₁|+…+|𝑥−𝑦_𝑚|/𝜂), 𝐾_𝜓,𝜂(𝑥,𝑦₁,…,𝑦_𝑚)=𝐾(𝑥,𝑦₁,…,𝑦_𝑚)𝜓(|𝑥−𝑦₁|+⋯+|𝑥−𝑦_𝑚|/𝜂) and Φ_𝜂^→𝑓(𝑥)=_{(ℝ𝑛)𝑚}𝐾_𝜑,𝜂𝑥,𝑦₁,…,𝑦_𝑚^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖𝑑^→𝑦,Ψ_𝜂^→𝑓(𝑥)=_{(ℝ𝑛)𝑚}𝐾_𝜓,𝜂𝑥,𝑦₁,…,𝑦_𝑚^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖𝑑^→𝑦. (3.3) The kernels of Φ_𝜂 and Ψ_𝜂 satisfy conditions (1.3) and (1.4) uniformly in 𝜂, respectively. And by the same argument in [4], both Φ^∗ and Ψ^∗ have the same weighted estimates to 𝑇^∗ that appeared in Theorems A and B.

It is easy to see that 𝑇^∗(^→𝑓)≤Φ^∗(^→𝑓)+Ψ^∗(^→𝑓). Moreover, 𝑇^∗_Σ𝑏^→𝑓≤Φ^∗_Σ𝑏^→𝑓+Ψ^∗_Σ𝑏^→𝑓,𝑇^∗_Π𝑏^→𝑓≤Φ^∗_Π𝑏^→𝑓+Ψ^∗_Π𝑏^→𝑓, (3.4) where Φ^∗_Σ𝑏^→𝑓(𝑥)=sup_𝜂>0^𝑚_{𝑗=1(ℝ𝑛)𝑚}𝐾_𝜑,𝜂𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓_𝑗𝑦_𝑗⋯𝑓_𝑚𝑦_𝑚𝑑^→𝑦≤^𝑚_𝑗=1sup_{𝜂>0(ℝ𝑛)𝑚}𝐾_𝜑,𝜂𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓_𝑗𝑦_𝑗⋯𝑓_𝑚𝑦_𝑚𝑑^→𝑦=^𝑚_𝑗=1Φ_𝑏^∗,𝑗_𝑗^→𝑓(𝑥),Ψ^∗_Σ𝑏^→𝑓(𝑥)=sup_𝜂>0^𝑚_{𝑗=1(ℝ𝑛)𝑚}𝐾_𝜓,𝜂𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓_𝑗𝑦_𝑗⋯𝑓_𝑚𝑦_𝑚𝑑^→𝑦≤^𝑚_𝑗=1sup_{𝜂>0(ℝ𝑛)𝑚}𝐾_𝜓,𝜂𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓_𝑗𝑦_𝑗⋯𝑓_𝑚𝑦_𝑚𝑑^→𝑦=^𝑚_𝑗=1Ψ_𝑏^∗,𝑗_𝑗^→𝑓(𝑥),Φ^∗_Π𝑏^→𝑓(𝑥)=sup_𝜂>0𝑏₁,𝑏₂,…,𝑏_𝑚−1,𝑏_𝑚,Φ_{𝜂𝑚𝑚−1}⋯₂₁^→𝑓(𝑥)=sup_{𝜂>0(ℝ𝑛)𝑚}𝐾_𝜑,𝜂𝑥,𝑦₁,…,𝑦_𝑚^𝑚_𝑗=1𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖𝑑^→𝑦,Ψ^∗_Π𝑏^→𝑓(𝑥)=sup_𝜂>0𝑏₁,𝑏₂,…,𝑏_𝑚−1,𝑏_𝑚,Ψ_{𝜂𝑚𝑚−1}…₂₁^→𝑓(𝑥)=sup_{𝜂>0(ℝ𝑛)𝑚}𝐾_𝜓,𝜂𝑥,𝑦₁,…,𝑦_𝑚^𝑚_𝑗=1𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗^𝑚_𝑖=1𝑓_𝑖𝑦_𝑖𝑑^→𝑦. (3.5) For simplicity, we will only prove for the case 𝑚=2. The arguments for the case 𝑚>2 are similar. For the similarity to the two commutators Φ^∗_Σ𝑏(^→𝑓) and Ψ^∗_Σ𝑏(^→𝑓), we might as well consider the former. We only consider the former. And we establish the following crucial lemma.

Lemma 3.1. Let 𝜇∈𝐴₁(ℝ^𝑛) and 𝑏_𝑗∈Lip_𝛽,𝜇, with 0<𝛽<1, 𝑗=1,2. Let 0<𝛿<1/2<1<𝑠<𝑛/𝛽. Then one has 𝑀^#_𝛿Φ_𝑏^∗,𝑗_𝑗𝑓₁,𝑓₂(𝑥)≤𝐶𝜇(𝑥)𝑏_{𝑗Lip𝛽,𝜇}𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓₂(𝑥)+𝐶𝜇(𝑥)𝑏_{𝑗Lip𝛽,𝜇}×𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀𝑓₂(𝑥)+𝑀𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂𝑥),𝑀_#𝛿(Ψ_𝑏^∗,𝑗_𝑗𝑓₁,𝑓₂(𝑥)≤𝐶𝜇(𝑥)𝑏_{𝑗Lip𝛽,𝜇}𝑀_{𝛽,𝜇,𝑠}Ψ^∗𝑓₁,𝑓₂(𝑥)+𝐶𝜇(𝑥)𝑏_{𝑗Lip𝛽,𝜇}×𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀𝑓₂(𝑥)+𝑀𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.6)(3.7)

Proof. Without loss of generality, we only consider the case 𝑗=1 and denote 𝑏₁ by 𝑏 for convenience. Fix 𝑥∈ℝ^𝑛 and let 𝐵=𝐵(𝑥,𝑅), 𝜆=𝑏_𝐵^∗ be the average of 𝑏 on 𝐵^∗, where 𝐵^∗=𝐵(𝑥,2𝑅). To proceed, we decompose 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_𝐵∗, 𝑖=1,2. Let 𝑐 be a constant to be fixed along the proof.
Since 0<𝛿<1, we have 1|𝐵|_𝐵Φ_𝑏^∗,1(𝑓₁,𝑓₂)(𝑦)^𝛿−|𝑐|^𝛿𝑑𝑦^1/𝛿≤1|𝐵|_𝐵Φ_𝑏^∗,1𝑓₁,𝑓₂𝑦−𝑐^𝛿𝑑𝑦^1/𝛿≤1|𝐵|_𝐵𝑏𝑦−𝜆Φ^∗𝑓₁,𝑓₂𝑦^𝛿𝑑𝑦^1/𝛿+1|𝐵|_𝐵Φ^∗(𝑏−𝜆)𝑓⁰₁,𝑓⁰₂𝑦^𝛿𝑑𝑦^1/𝛿+1|𝐵|_𝐵Φ^∗(𝑏−𝜆)𝑓⁰₁,𝑓^∞₂𝑦^𝛿𝑑𝑦^1/𝛿+1|𝐵|_𝐵Φ^∗(𝑏−𝜆)𝑓^∞₁,𝑓⁰₂𝑦^𝛿𝑑𝑦^1/𝛿+1|𝐵|_𝐵sup_𝜂>0Φ_𝜂(𝑏−𝜆)𝑓^∞₁,𝑓^∞₂𝑦−𝑐^𝛿𝑑𝑦^1/𝛿∶=𝐼+𝐼𝐼+𝐼𝐼𝐼+𝐼𝑉+𝑉. (3.8) For 𝐼, since 0<𝛿<1, 𝜇∈𝐴₁ and 𝑏∈Lip_𝛽,𝜇, by Hölder’ inequality, we have 1𝐼≤|𝐵|_𝐵𝑏𝑦−𝜆Φ^∗𝑓₁,𝑓₂𝑦≤𝐶𝑑𝑦|𝐵^∗|_𝐵^∗𝑏𝑦−𝑏_𝐵^∗𝜇𝑦^−1/𝑠Φ^∗𝑓₁,𝑓₂𝑦𝜇𝑦^1/𝑠≤𝐶𝑑𝑦|𝐵^∗|_𝐵^∗𝑏𝑦−𝑏_𝐵^∗𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′_𝐵∗Φ^∗𝑓₁,𝑓₂𝑦^𝑠𝜇𝑦𝑑𝑦^1/𝑠1≤𝐶𝜇(𝐵^∗)^𝛽/𝑛1𝜇(𝐵^∗)_𝐵^∗𝑏𝑦−𝑏_𝐵^∗𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′𝜇𝐵^{∗(𝛽/𝑛)+(1/𝑠′)}|𝐵^∗|×1𝜇(𝐵^∗)^{1−(𝑠𝛽/𝑛)𝐵∗}Φ^∗𝑓₁,𝑓₂𝑦^𝑠𝜇𝑦𝑑𝑦^1/𝑠𝜇𝐵^{∗(1/𝑠)−(𝛽/𝑛)}≤𝐶‖𝑏‖_Lip𝛽,𝜇𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓₂(𝑥)𝜇𝐵^∗|𝐵^∗|≤𝐶𝜇(𝑥)‖𝑏‖_Lip𝛽,𝜇𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓₂(𝑥). (3.9)
To estimate the second term 𝐼𝐼. Since 0<𝛿<1/2, using Kolmogorov’s inequality with 𝑝=𝛿, 𝑞=1/2, 𝑋=𝐵, 𝜔=𝑑𝑥/|𝐵| and the (𝐿¹(ℝ^𝑛)×𝐿¹(ℝ^𝑛),𝐿^1/2,∞(ℝ^𝑛))-boundedness of Φ^∗, we derive that 𝐼𝐼≤Φ^∗(𝑏−𝜆)𝑓⁰₁,𝑓⁰_{2𝐿1/2,∞(𝑑𝑦/|𝐵|)}1≤𝐶|𝐵|_𝐵^∗𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁1|𝐵|_𝐵^∗𝑓₂𝑦₂𝑑𝑦₂1≤𝐶|𝐵^∗|_𝐵^∗𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁1|𝐵^∗|_𝐵^∗𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝜇(𝑥)‖𝑏‖_Lip𝛽,𝜇𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀𝑓₂(𝑥), (3.10) where we have used the analogous technique in 𝐼 to get the last inequality.
For the term 𝐼𝐼𝐼, using the fact |𝑦−𝑦₂|∼|𝑦₂−𝑥| for any 𝑦₂∈(𝐵^∗)^𝑐, 𝑦∈𝐵, and note that 𝐾_𝜑,𝜂 satisfies (1.3) uniformly in 𝜂, we obtain 1𝐼𝐼𝐼≤|𝐵|_𝐵Φ^∗(𝑏−𝜆)𝑓⁰₁,𝑓^∞₂𝑦≤1𝑑𝑦|𝐵|_𝐵𝐵^∗_×(ℝ^𝑛_⧵𝐵^∗₎𝐴𝑏𝑦₁−𝜆𝑓₁𝑦₁𝑓₂𝑦₂𝑦−𝑦₁+𝑦−𝑦₂^2𝑛𝑑𝑦₁𝑑𝑦₂≤𝑑𝑦_𝐵∗𝑏𝑦₁−𝑏_𝐵^∗∣𝑓₁𝑦₁𝑑𝑦_{1ℝ𝑛⧵𝐵∗}𝑓₂𝑦₂𝑦₂−𝑥^2𝑛𝑑𝑦₂≤𝐶_𝐵∗𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁^∞_{𝑘=12𝑘𝐵∗⧵2𝑘−1𝐵∗}𝑓₂𝑦₂𝑦₂−𝑥^2𝑛𝑑𝑦₂1≤𝐶|𝐵^∗|_𝐵^∗𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁^∞_𝑘=12^−𝑘𝑛12^𝑘𝐵^{∗2𝑘𝐵∗}𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝜇(𝑥)‖𝑏‖_Lip𝛽,𝜇𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀𝑓₂(𝑥). (3.11) For the term 𝐼𝑉, using the fact |𝑦−𝑦₁|∼|𝑦₁−𝑥| for any 𝑦₁∈(𝐵^∗)^𝑐, 𝑦∈𝐵, and note that 𝐾_𝜑,𝜂 satisfies (1.3) uniformly in 𝜂, and using (2.4), we obtain 1𝐼𝑉≤|𝐵|_𝐵Φ^∗(𝑏−𝜆)𝑓^∞₁,𝑓⁰₂𝑦≤1𝑑𝑦|𝐵|_𝐵(ℝ^𝑛_⧵𝐵^∗_)×𝐵^∗𝐴𝑏𝑦₁−𝜆𝑓₁𝑦₁𝑓₂𝑦₂𝑦−𝑦₁+𝑦−𝑦₂^2𝑛𝑑𝑦₁𝑑𝑦₂𝑑𝑦≤𝐶_{ℝ𝑛⧵𝐵∗}𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑥−𝑦₁^2𝑛𝑑𝑦_1𝐵∗𝑓₂𝑦₂𝑑𝑦₂≤𝐶^∞_{𝑘=02𝑘+1𝐵∗⧵2𝑘𝐵∗}𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑥−𝑦₁^2𝑛𝑑𝑦_1𝐵∗𝑓₂𝑦₂𝑑𝑦₂≤𝐶^∞_𝑘=012^𝑘𝐵^{∗22𝑘+1𝐵∗}𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦_1𝐵∗𝑓₂𝑦₂𝑑𝑦₂≤𝐶^∞_𝑘=02^−𝑘𝑛12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁1|𝐵^∗|_𝐵^∗𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝑀𝑓₂(𝑥)^∞_𝑘=02^−𝑘𝑛12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑏𝑦₁−𝑏₂^𝑘+1_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁+𝑏_𝐵^∗−𝑏₂^𝑘+1_𝐵^∗_{2𝑘+1𝐵∗}𝑓₁𝑦₁𝑑𝑦₁≤𝐶𝑀𝑓₂(𝑥)^∞_𝑘=02^−𝑘𝑛𝜇(𝑥)‖𝑏‖_{𝐿𝑖𝑝𝛽,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)+𝑘𝜇(𝑥)‖𝑏‖_{𝐿𝑖𝑝𝛽,𝜇}𝜇𝐵𝑥,2^𝑘+1𝑅^𝛽/𝑛2^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑓₁𝑦₁𝑑𝑦₁≤𝐶𝜇(𝑥)‖𝑏‖_Lip𝛽,𝜇𝑀𝑓₂(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥). (3.12)
For 𝑉, fix the value of 𝑐 by taking 𝑐=Φ^∗((𝑏−𝜆)𝑓^∞₁,𝑓^∞₂)(𝑥), recall that 𝐾_𝜑,𝜂 satisfies (1.4) uniformly in 𝜂, then we can obtain 1𝑉≤|𝐵|_𝐵Φ^∗(𝑏−𝜆)𝑓^∞₁,𝑓^∞₂𝑦−Φ^∗(𝑏−𝜆)𝑓^∞₁,𝑓^∞₂(𝑥≤1)𝑑𝑦|𝐵|_𝐵sup_𝜂>0Φ_𝜂(𝑏−𝜆)𝑓^∞₁,𝑓^∞₂𝑦−Φ_𝜂(𝑏−𝜆)𝑓^∞₁,𝑓^∞₂(𝑥≤1)𝑑𝑦|𝐵|_𝐵(ℝ^𝑛_⧵𝐵^∗₎²sup_𝜂>0𝐾_𝜑,𝜂𝑦,𝑦₁,𝑦₂−𝐾_𝜑,𝜂𝑥,𝑦₁,𝑦₂𝑏𝑦₁−𝜆𝑓₁𝑦₁×𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂≤𝐶𝑑𝑦|𝐵|_𝐵(ℝ^𝑛_⧵𝐵^∗₎²𝑥−𝑦^𝜀𝑦−𝑦₁+𝑦−𝑦₂^2𝑛+𝜀𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂≤𝐶𝑑𝑦|𝐵|^∞_𝑘=02^𝑘+1_𝐵^∗_⧵2^𝑘_𝐵^∗𝑥−𝑦^𝜀𝑦₁−𝑥^2𝑛+𝜀𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂𝑑𝑦≤𝐶^∞_𝑘=0𝐵^∗𝜀/𝑛2^𝑘𝐵^{∗2+𝜀/𝑛2𝑘+1𝐵∗}𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦_{12𝑘+1𝐵∗}𝑓₂𝑦₂𝑑𝑦₂≤𝐶^∞_𝑘=02^−𝑘𝜀12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑏𝑦₁−𝑏_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝜇(𝑥)‖𝑏‖_Lip𝛽,𝜇𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀𝑓₂(𝑥), (3.13) where in the last inequality, we use the same computation in the 𝐼𝑉 term.
Consequently, combining the estimates of 𝐼,𝐼𝐼,𝐼𝐼𝐼, and 𝑉, we conclude the proof of Lemma 3.1.

Now we are ready to return to prove Theorem 1.3.

Proof. First, by Lemma 2.1, we have that 𝜇∈𝐴₁⊂𝐴_𝑟′, and hence 𝜇^1−𝑟∈𝐴_𝑟⊂𝐴_∞. Then by Lemma 2.3, we obtain Φ_𝑏^∗,𝑗_𝑗𝑓₁,𝑓₂(𝑥)_{𝐿𝑟(𝜇1−𝑟)}≤𝑀_𝛿Φ_𝑏^∗,𝑗_𝑗𝑓₁,𝑓₂(𝑥)_{𝐿𝑟(𝜇1−𝑟)}≤𝑀^#_𝛿Φ_𝑏^∗,𝑗_𝑗𝑓₁,𝑓₂(𝑥)_{𝐿𝑟(𝜇1−𝑟)}. (3.14) For 𝑗=1,2, by Lemma 3.1, we reduce to bound the ‖⋅‖_𝐿^𝑟_{(𝜇1−𝑟)} norm of the right-hand side of (3.6). For the first term, since 1/𝑟=(1/𝑝)−(𝛽/𝑛) and taking 𝑠 such that 1<𝑠<𝑝<𝑛/𝛽, by Lemma 2.4, and Theorem B(ii), we have 𝜇𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓_{2𝐿𝑟(𝜇1−𝑟)}=𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓_{2𝐿𝑟(𝜇)}≤𝐶Φ^∗𝑓₁,𝑓_{2𝐿𝑝(𝜇)}≤𝐶𝑓_{1𝐿𝑝1(𝜇)}𝑓_{2𝐿𝑝2(𝜇)}. (3.15)
For the second term, we let 1/𝑟=1/𝑝₂+1/𝑙, and 1/𝑙=1/𝑝₁−𝛽/𝑛. Then by Lemma 2.4 again, together with Hölder’s inequality, we obtain 𝜇𝑀_{𝛽,𝜇,𝑠}𝑓₁𝑀𝑓_{2𝐿𝑟(𝜇1−𝑟)}=𝑀_{𝛽,𝜇,𝑠}𝑓₁𝑀𝑓_{2𝐿𝑟(𝜇)}≤𝑀_{𝛽,𝜇,𝑠}𝑓_{1𝐿𝑙(𝜇)}𝑀𝑓_{2𝐿𝑝2(𝜇)}≤𝐶𝑓_1𝐿𝑝1𝜇𝑓_2𝐿𝑝2𝜇. (3.16) We can obtain that Φ_𝑏^∗,1𝑓₁,𝑓_{2𝐿𝑟(𝜇1−𝑟)}≤𝐶‖𝑏‖_Lip𝛽,𝜇𝑓_{1𝐿𝑝1(𝜇)}𝑓_{2𝐿𝑝2(𝜇)}. (3.17) Similarly, we have Ψ_𝑏^∗,1𝑓₁,𝑓_{2𝐿𝑟(𝜇1−𝑟)}≤𝐶‖𝑏‖_Lip𝛽,𝜇𝑓_{1𝐿𝑝1(𝜇)}𝑓_{2𝐿𝑝2(𝜇)}. (3.18) Consequently, by the above arguments, we conclude the proof of Theorem 1.3.

Similarly as the proof of Lemma 3.1 and that of Theorem 1.3, we only consider the case 𝑚=2 and establish the following sharp maximal function for Φ^∗_Π𝑏.

Lemma 3.2. Let 𝜇∈𝐴₁(ℝ^𝑛) and 𝑏_𝑖∈Lip_𝛽1,𝜇,𝑖=1,2; 𝛽=𝛽₁+𝛽₂, and 0<𝛽<1. And let 0<𝛿<1/3<1<𝑠_𝑖<𝑛/𝛽_𝑖, 𝑖=1,2. Then one has 𝑀^#_𝛿Φ^∗_Π𝑏𝑓₁,𝑓₂(𝑥)≤𝜇(𝑥)²²_𝑗=1𝑏_{𝑗Lip𝛽𝑗,𝜇}×𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓₂(𝑥)+𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥),𝑀^#_𝛿Ψ^∗_Π𝑏𝑓₁,𝑓₂(𝑥)≤𝜇(𝑥)²²_𝑗=1𝑏_{𝑗Lip𝛽𝑗,𝜇}×𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓₂(𝑥)+𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.19)(3.20)

Proof. Fix 𝑥∈ℝ^𝑛 and let 𝐵=𝐵(𝑥,𝑅) with 𝑛>0. Taking 𝜆_𝑖=(𝑏_𝑖)_𝐵^∗, the average of 𝑏_𝑖 on 𝐵^∗, 𝑖=1,2, where 𝐵^∗=𝐵(𝑥,2𝑅). Let 𝑐 be a constant to be fixed along the proof. We split Φ^∗_Π𝑏(𝑓₁,𝑓₂)(𝑦) in the following way: Φ^∗_Π𝑏𝑓₁,𝑓₂𝑦=sup_𝜂>0𝑏₁𝑦−𝜆₁𝑏₂𝑦−𝜆₂Φ_𝜂𝑓₁,𝑓₂𝑦−𝑏₁𝑦−𝜆₁Φ_𝜂𝑓₁,𝑏₂−𝜆₂𝑓₂𝑦−𝑏₂𝑦−𝜆₂Φ_𝜂×𝑏₁−𝜆₁𝑓₁,𝑓₂𝑦+Φ_𝜂𝑏₁−𝜆₁𝑓₁,𝑏₂−𝜆₂𝑓₂𝑦. (3.21) Since 0<𝛿<1/3, then we have 1|𝐵|_𝐵Φ^∗_Π𝑏𝑓₁,𝑓₂𝑦^𝛿−|𝑐|^𝛿𝑑𝑦^1/𝛿≤1|𝐵|_𝐵Φ^∗_Π𝑏𝑓₁,𝑓₂𝑦−𝑐^𝛿𝑑𝑦^1/𝛿≤1|𝐵|_𝐵𝑏₁𝑦−𝜆₁𝑏₂𝑦−𝜆₂Φ^∗𝑓₁,𝑓₂𝑦^𝛿𝑑𝑦^1/𝛿+1|𝐵|_𝐵sup_𝜂>0𝑏₁𝑦−𝜆₁Φ_𝜂𝑓₁,𝑏₂−𝜆₂𝑓₂𝑦^𝛿𝑑𝑦^1/𝛿+1|𝐵|_𝐵sup_𝜂>0𝑏₂𝑦−𝜆₂Φ_𝜂𝑏₁−𝜆₁𝑓₁,𝑓₂𝑦^𝛿𝑑𝑦^1/𝛿+1|𝐵|_𝐵sup_𝜂>0Φ_𝜂𝑏₁−𝜆₁𝑓₁,𝑏₂−𝜆₂𝑓₂−𝑐^𝛿𝑑𝑦^1/𝛿∶=𝑈₁+𝑈₂+𝑈₃+𝑈₄. (3.22) For the term 𝑈₁, since 0<𝛿<1/3, and 𝛽=𝛽₁+𝛽₂, then by Hölder’s inequality, we have 𝑈₁≤1|𝐵|_𝐵𝑏₁𝑦−𝜆₁^3𝛿𝑑𝑦^1/3𝛿1|𝐵|_𝐵𝑏₂𝑦−𝜆₂^3𝛿𝑑𝑦^1/3𝛿×1|𝐵|_𝐵Φ^∗𝑓₁,𝑓₂𝑦^3𝛿𝑑𝑦^1/3𝛿1≤𝐶|𝐵^∗|_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗1𝑑𝑦|𝐵^∗|_𝐵^∗𝑏₂𝑦−𝑏_2𝐵∗×1𝑑𝑦|𝐵^∗|_𝐵^∗Φ^∗𝑓₁,𝑓₂𝑦1𝑑𝑦≤𝐶|𝐵^∗|_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗^𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′1|𝐵^∗|_𝐵^∗𝜇𝑦𝑑𝑦^1/𝑠×1|𝐵^∗|_𝐵^∗𝑏₂𝑦−𝑏_2𝐵∗^𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′1|𝐵^∗|_𝐵^∗𝜇𝑦𝑑𝑦^1/𝑠×1|𝐵^∗|_𝐵^∗Φ^∗𝑓₁,𝑓₂𝑦^𝑠𝜇𝑦𝑑𝑦^1/𝑠1|𝐵^∗|_𝐵^∗𝜇𝑦^{−𝑠′/𝑠}𝑑𝑦^1/𝑠′1≤𝐶𝜇(𝐵^∗)^𝛽1/𝑛1𝜇(𝐵^∗)_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗^𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′𝜇𝐵^{∗(𝛽1/𝑛)+1}|𝐵^∗|×1𝜇(𝐵^∗)^𝛽2/𝑛1𝜇(𝐵^∗)_𝐵^∗𝑏₂𝑦−𝑏_2𝐵∗^𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′𝜇𝐵^{∗(𝛽2/𝑛)+1}|𝐵^∗|×1𝜇(𝐵^∗)^{1−(𝑠𝛽/𝑛)𝐵∗}Φ^∗𝑓₁,𝑓₂𝑦^𝑠𝜇𝑦𝑑𝑦^1/𝑠𝜇𝐵^{∗(1/𝑠)−(𝛽/𝑛)}𝜇𝐵^{∗(1/𝑠′)−1}|𝐵^∗|≤𝐶𝜇(𝑥)²𝑏_{1Lip𝛽1,𝜇}𝑏_{2Lip𝛽2,𝜇}𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓₂(𝑥). (3.23)
For the term 𝑈₂, noting that 0<𝛿<1/3, we use the facts 1=𝛿+(1−𝛿) and 0<𝛿/(1−𝛿)<1/2, then by Hölder’s inequality and Komolgorov’s inequality (Lemma 2.2) and Theorem B, we have 𝑈₂1≤𝐶|𝐵^∗|_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗1𝑑𝑦|𝐵^∗|_𝐵^∗sup_𝜂>0Φ_𝜂𝑓₁,𝑏₂−𝜆₂𝑓₂𝑦^{𝛿/(1−𝛿)}𝑑𝑦^{(1−𝛿)/𝛿}≤1|𝐵^∗|_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗1𝑑𝑦|𝐵^∗|_𝐵^∗Φ^∗𝑓₁,𝑏₂−𝜆₂𝑓₂𝑦^{𝛿/(1−𝛿)}𝑑𝑦^{1−𝛿/𝛿}≤1|𝐵^∗|_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗𝑑𝑦Φ^∗𝑓₁,𝑏₂−𝜆₂𝑓_{2𝐿1/2,∞(𝑑𝑦/|𝐵∗|)}≤1|𝐵^∗|_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗1𝑑𝑦|𝐵^∗|_𝐵^∗𝑓₁𝑦₁𝑑𝑦₁×1|𝐵^∗|_𝐵^∗𝑏₂𝑦₂−𝑏_2𝐵∗𝑓₂𝑦₂𝑑𝑦₂≤1|𝐵^∗|_𝐵^∗𝑏₁𝑦−𝑏_1𝐵∗^𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′_𝐵∗𝜇𝑦𝑑𝑦^1/𝑠×1|𝐵^∗|_𝐵^∗𝑓₁𝑦₁^𝑠𝜇𝑦₁𝑑𝑦₁^1/𝑠1|𝐵^∗|_𝐵^∗𝜇𝑦₁^{−𝑠′/𝑠}𝑑𝑦₁^1/𝑠′×1|𝐵^∗|_𝐵^∗𝑏₂𝑦₂−𝑏_2𝐵∗^𝑠′𝜇𝑦₂^1−𝑠′𝑑𝑦₂^1/𝑠′_𝐵∗𝑓₂𝑦₂^𝑠𝜇𝑦₂𝑑𝑦₂^1/𝑠≤1𝜇(𝐵^∗)^𝛽1/𝑛1𝜇(𝐵^∗)_𝐵^∗𝑏𝑦−𝑏_𝐵^∗𝑠′𝜇𝑦^1−𝑠′𝑑𝑦^1/𝑠′𝜇𝐵^{∗(𝛽1/𝑛)+1}|𝐵^∗|×1𝜇(𝐵^∗)^{1−(𝑠𝛽/𝑛)𝐵∗}𝑓₁𝑦₁^𝑠𝜇𝑦₁𝑑𝑦₁^1/𝑠𝜇𝐵^{∗(1/𝑠)−(𝛽/𝑛)}𝜇𝐵^∗−1/𝑠|𝐵^∗|×1𝜇(𝐵^∗)^𝛽2/𝑛1𝜇(𝐵^∗)_𝐵^∗𝑏₂𝑦−𝑏_2𝐵∗^𝑠′𝜇𝑦^1−𝑠′𝑑𝑦₂^1/𝑠′𝜇𝐵^{∗(𝛽2/𝑛)+(1/𝑠′)}|𝐵^∗|×1𝜇(𝐵^∗)^{1−(𝑠𝛽/𝑛)𝐵∗}𝑓₂𝑦₂^𝑠𝜇𝑦₂𝑑𝑦₂^1/𝑠𝜇𝐵^{∗(1/𝑠)−(𝛽/𝑛)}≤𝐶𝜇(𝑥)²𝑏_{1Lip𝛽1,𝜇}𝑏_{2Lip𝛽2,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.24) Similarly, for the term 𝑈₃, we have 𝑈₃≤𝐶𝜇(𝑥)²𝑏_{1Lip𝛽,𝜇}𝑏_{2Lip𝛽,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.25)
Now we turn to estimate the last term 𝑈₄. To proceed, we denote that 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_𝐵∗, 𝑖=1,2. Let 𝑐=𝑐₁+𝑐₂+𝑐₃, where 𝑐₁=Φ_𝜂𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥),𝑐₂=Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓⁰₂(𝑥),𝑐₃=Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥). (3.26) We split 𝐼𝑉 in the following way: 𝑈₄≤𝑈₄₁+𝑈₄₂+𝑈₄₃+𝑈₄₄, (3.27) where 𝑈₄₁=1|𝐵|_𝐵sup_𝜂>0Φ_𝜂𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓⁰₂𝑦^𝛿𝑑𝑦^1/𝛿,𝑈₄₂=1|𝐵|_𝐵sup_𝜂>0Φ_𝜂𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂𝑦−Φ_𝜂𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)^𝛿𝑑𝑦^1/𝛿,𝑈₄₃=1|𝐵|_𝐵sup_𝜂>0Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓⁰₂𝑦−Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓⁰₂(𝑥)^𝛿𝑑𝑦^1/𝛿,𝑈₄₄=1|𝐵|_𝐵sup_𝜂>0Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂𝑦−Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)^𝛿𝑑𝑦^1/𝛿. (3.28)
For the term 𝑈₄₁, we choose 1<𝑝₀<1/2𝛿 and use Kolmogorov’s inequality and Theorem B, then we use the same computation as 𝑈₂ to deduce that 𝑈₄₁≤1|𝐵|_𝐵Φ^∗𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓⁰₂𝑦^𝑝0𝛿𝑑𝑦^1/𝑝0𝛿≤Φ^∗𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓⁰_{2𝐿1/2,∞(𝑑𝑦/|𝐵|)}≤1|𝐵|_𝐵𝑏₁𝑦₁−𝜆₁𝑓⁰₁𝑦₁𝑑𝑦₁1|𝐵|_𝐵𝑏₂𝑦₂−𝜆₂𝑓⁰₂𝑦₂𝑑𝑦₂1≤𝐶|𝐵^∗|_𝐵^∗𝑏₁𝑦₁−𝑏_1𝐵∗𝑓₁𝑦₁𝑑𝑦₁1|𝐵^∗|_𝐵^∗𝑏₂𝑦₂−𝑏_2𝐵∗𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝜇(𝑥)²𝑏_{1Lip𝛽1,𝜇}𝑏_{2Lip𝛽2,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.29)
For 𝑈₄₂, by the fact |𝑦−𝑦₂|∼|𝑦₂−𝑥|, for any 𝑦₂∈(𝐵^∗)^𝑐, 𝑦∈𝐵, and note that 𝐾_𝜑,𝜂 satisfies (1.4) uniformly in 𝜂, then we get that 𝑈₄₂≤1|𝐵|_𝐵sup_𝜂>0Φ_𝜂𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂𝑦−Φ_𝜂𝑏₁−𝜆₁𝑓⁰₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥≤𝐶)𝑑𝑦|𝐵|_𝐵𝐵^∗𝑏₁𝑦₁−𝜆₁𝑓₁𝑦₁𝑑𝑦_{1(𝐵∗)𝑐}𝑦−𝑥^𝜀𝑏₂𝑦₂−𝜆₂𝑓₂𝑦₂𝑦−𝑦₁+𝑦−𝑦₂^2𝑛+𝜀𝑑𝑦₂≤𝑑𝑦_𝐵∗𝑏₁𝑦₁−𝑏_1𝐵∗𝑓₁𝑦₁𝑑𝑦₁×^∞_{𝑘=02𝑘+1𝐵∗⧵2𝑘𝐵∗}𝑦−𝑥^𝜀𝑏₂𝑦₂−𝑏_2𝐵∗𝑓₂𝑦₂𝑥−𝑦₂^2𝑛+𝜀𝑑𝑦₂≤_𝐵∗𝑏₁𝑦₁−𝑏_1𝐵∗𝑓₁𝑦₁𝑑𝑦₁×^∞_𝑘=0𝐵^∗𝜀/𝑛2^𝑘𝐵^{∗2+𝜀/𝑛2𝑘+1𝐵∗}𝑏₂𝑦₂−𝑏_2𝐵∗𝑓₂𝑦₂𝑑𝑦₂≤1|𝐵^∗|_𝐵^∗𝑏₁𝑦₁−𝑏_1𝐵∗𝑓₁𝑦₁𝑑𝑦₁×^∞_𝑘=02^−𝑘𝜀12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑏₂𝑦₂−𝑏_2𝐵∗𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝜇(𝑥)𝑏_{1Lip𝛽1,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)^∞_𝑘=12^−𝑘𝜀12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑏₂𝑦₂−𝑏_{22𝑘+1𝐵∗}𝑓₂𝑦₂𝑑𝑦₂+𝑏_2𝐵∗−𝑏_{22𝑘+1𝐵∗2𝑘+1𝐵∗}𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝜇(𝑥)²𝑏_{1Lip𝛽1,𝜇}𝑏_{2Lip𝛽2,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥), (3.30) where we have used the same computation of 𝐼𝑉 to gain the last inequality.
Similarly as 𝑈₄₂, we can get the estimates for 𝑈₄₃, 𝑈₄₃≤𝐶𝜇(𝑥)²𝑏_{1Lip𝛽,𝜇}𝑏_{2Lip𝛽,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.31)
Now we turn to 𝑈₄₄, by the fact |𝑦−𝑦₁|∼|𝑦₁−𝑥|, |𝑦−𝑦₂|∼|𝑦₂−𝑥| for any 𝑦₁,𝑦₂∈(𝐵^∗)^𝑐, 𝑦∈𝐵, and recalling that 𝐾_𝜑,𝜂 satisfies (1.4) uniformly in 𝜂, then we can obtain Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂𝑦−Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥)≤𝐶^∞_{𝑘=0(2𝑘+1𝐵∗⧵2𝑘𝐵∗)2}𝑥−𝑦^𝜀𝑦−𝑦₁+𝑦−𝑦₂^2𝑛+𝜀𝑏₁𝑦₁−𝜆₁𝑓₁𝑦₁𝑏₂𝑦−2−𝜆₂𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂≤𝐶^∞_{𝑘=0(2𝑘+1𝐵∗⧵2𝑘𝐵∗)2}𝑥−𝑦^𝜀𝑦−𝑦₁^2𝑛+𝜀𝑏₁𝑦₁−𝜆₁𝑓₁𝑦₁𝑏₂𝑦−2−𝜆₂𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂≤𝐶^∞_𝑘=0𝐵^∗𝜀/𝑛2^𝑘𝐵^{∗2+𝜀/𝑛2𝑘+1𝐵∗}𝑏₁𝑦₁−𝜆₁𝑓₁𝑦₁𝑑𝑦_{12𝑘+1𝐵∗}𝑏₂𝑦₂−𝜆₂𝑓₂𝑦₂𝑑𝑦₂≤𝐶^∞_𝑘=02^−𝑘𝜀12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑏₁𝑦₁−𝜆₁𝑓₁𝑦₁𝑑𝑦₁×12^𝑘+1𝐵^{∗2𝑘+1𝐵∗}𝑏₂𝑦₂−𝜆₂𝑓₂𝑦₂𝑑𝑦₂≤𝐶𝜇(𝑥)²𝑏_{1Lip𝛽1,𝜇}𝑏_{2Lip𝛽2,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.32) Therefore, 𝑈₄₄≤1|𝐵|_𝐵sup_𝜂>0Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂𝑦−Φ_𝜂𝑏₁−𝜆₁𝑓^∞₁,𝑏₂−𝜆₂𝑓^∞₂(𝑥(𝑥))𝑑𝑦≤𝐶𝜇²𝑏_{1Lip𝛽1,𝜇}𝑏_{2Lip𝛽2,𝜇}𝑀_{𝛽,𝜇,𝑠}𝑓₁(𝑥)𝑀_{𝛽,𝜇,𝑠}𝑓₂(𝑥). (3.33) Consequently, the estimate for 𝑈₁, together with those of 𝑈₂,𝑈₃, and 𝑈₄, can conclude the proof of Lemma 3.2.

Now we return to prove Theorem 1.5.

Proof. Similarly as the proof of Theorem 1.3, Φ^∗_Π𝑏𝑓₁,𝑓_{2𝐿𝑟(𝜇1−2𝑟)}≤𝑀_𝛿Φ^∗_Π𝑏𝑓₁,𝑓_{2𝐿𝑟(𝜇1−2𝑟)}≤𝑀^#_𝛿Φ^∗_Π𝑏𝑓₁,𝑓_{2𝐿𝑟(𝜇1−2𝑟)}. (3.34) We reduce to bound the ‖⋅‖_𝐿^𝑟_{(𝜇1−𝑟)} norm of the right-hand side of (3.19). We estimate each term as follows. For the first term, since 1/𝑟=(1/𝑝)−(𝛽/𝑛) and choosing 𝑠 such that 1<𝑠<𝑝<𝑛/𝛽, by Lemma 2.4 and Theorem A and observe that 𝜇∈𝐴₁, we obtain 𝜇²𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓_{2𝐿𝑟(𝜇1−2𝑟)}=𝑀_{𝛽,𝜇,𝑠}Φ^∗𝑓₁,𝑓_{2𝐿𝑟(𝜇)}≤𝐶Φ^∗𝑓₁,𝑓_{2𝐿𝑝(𝜇)}≤𝐶𝑓_{1𝐿𝑝1(𝜇)}𝑓_{2𝐿𝑝2(𝜇)}. (3.35) For the second term, since 1/𝑟=(1/𝑟₁)+(1/𝑟₂), by Hölder’s inequality and Lemma 2.4, we get 𝜇²𝑀_{𝛽,𝜇,𝑠}𝑓₁𝑀_{𝛽,𝜇,𝑠}𝑓_{2𝐿𝑟(𝜇1−2𝑟)}=𝑀_{𝛽,𝜇,𝑠}𝑓₁𝑀_{𝛽,𝜇,𝑠}𝑓_{2𝐿𝑟(𝜇)}≤𝑀_{𝛽,𝜇,𝑠}𝑓_{1𝐿𝑟1(𝜇)}𝑀_{𝛽,𝜇,𝑠}𝑓_{2𝐿𝑟2(𝜇)}≤𝐶𝑓_{1𝐿𝑝1(𝜇)}𝑓_{2𝐿𝑝2(𝜇)}. (3.36) Similarly, we also have Ψ^∗_Π𝑏𝑓₁,𝑓_{2𝐿𝑟(𝜇1−2𝑟)}≤𝐶𝑓_{1𝐿𝑝1(𝜇)}𝑓_{2𝐿𝑝2(𝜇)}. (3.37) This estimate together with that for Φ^∗_Π𝑏(𝑓₁,𝑓₂) finishes the proof of Theorem 1.5.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (1096015, 10871173, and 11261023) and the Natural Science Foundation of Jiangxiang Teacher’s Division (GJJ10397).