Some Weighted Estimates for Multilinear Fourier Multiplier Operators

Abstract

We first provide a weighted Fourier multiplier theorem for multilinear operators which extends Theorem 1.2 in Fujita and Tomita (2012) by using 𝐿^𝑟-based Sobolev spaces (1<𝑟≤2). Then, by using a different method, we obtain a result parallel to Theorem 6.2 which is an improvement of Theorem 1.2 under assumption (i) in Fujita and Tomita (2012).

1. Introduction

During the last several years, considerable attention has been paid to the study of multilinear Fourier multiplier operators. Let 𝒮(ℝ^𝑑) be the Schwartz space of all rapidly decreasing smooth functions on ℝ^𝑑, for some 𝑑∈ℤ⁺. The multilinear Fourier multiplier operator 𝑇_𝜎 associated with a symbol 𝜎 is defined by 𝑇_𝜎𝑓₁,…,𝑓_𝑚(𝑥)=_ℝ𝑚𝑛𝑒^{2𝜋𝑖𝑥⋅(𝜉1+⋯+𝜉𝑚)}𝜎𝜉₁,…,𝜉_𝑚×𝑓₁𝜉₁⋯𝑓_𝑚𝜉_𝑚𝑑𝜉₁⋯𝑑𝜉_𝑚 (1) for 𝑓_𝑖∈𝒮(ℝ^𝑛), 𝑖=1,…,𝑚.

Coifman and Meyer [1] proved that if 𝜎 is a bounded function on ℝ^𝑚𝑛⧵{0} that satisfies 𝜕^𝛼1_𝜉1⋯𝜕^𝛼𝑚_𝜉𝑚𝜎𝜉₁,…,𝜉_𝑚≤𝐶_𝛼𝜉₁+⋯+𝜉_𝑚^{−(|𝛼1|+⋯+|𝛼𝑚|)} (2) away from the origin for all sufficiently large multi-indices 𝛼_𝑗, then 𝑇_𝜎 is bounded from the product 𝐿^𝑝1(ℝ^𝑛)×⋯×𝐿^𝑝𝑚(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛) for all 1<𝑝₁,…,𝑝_𝑚,𝑝<∞ satisfying 1/𝑝₁+⋯+1/𝑝_𝑚=1/𝑝. The multiplier theorem of Coifman and Meyer was extended to indices 𝑝<1 (and larger than 1/𝑚) by Grafakos and Torres [2] and Kenig and Stein [3] (when 𝑚=2). Exploiting the idea of the proof of the Hörmander multiplier theorem in [4], Tomita [5] gave a Hörmander type theorem for multilinear Fourier multipliers with more weaker smoothness condition assumed on 𝜎 than (2). Grafakos and Si [6] gave similar results for 𝑝≤1 by using 𝐿^𝑟-based Sobolev spaces (1<𝑟≤2). Grafakos et al. [7] proved the 𝐿²-boundedness of  𝑇_𝜎 with multipliers of limited smoothness.

In order to state other known results, we first introduce some notations. The Laplacian on ℝ^𝑑 is Δ𝑔=∑^𝑑_𝑗=1𝜕²𝑔/𝜕𝑥²_𝑗, that is, the sum of the second partials of 𝑔 in every variable. We define the operator (𝐼−Δ)^𝛾/2(𝑔)=ℱ⁻¹(𝑤_𝛾ℱ(𝑔)), where 𝑤_𝛾(𝜉)=(1+4𝜋²|𝜉|²)^𝛾/2 for 𝛾>0. Let 𝐿^𝑟_𝛾(ℝ^𝑑) be the 𝐿^𝑟-based Sobolev space with norm 𝑓_𝐿𝑟𝛾=(𝐼−Δ)^𝛾/2𝑓_{𝐿𝑟(ℝ𝑑)}, (3) where 1≤𝑟<∞.

Let 𝑠⃗=(𝑠₁,…,𝑠_𝑚) and let the product type Sobolev space 𝑊^{𝑠1,…,𝑠𝑚}(ℝ^𝑚𝑛) consist of all functions 𝐹 such that the following norm of 𝐹 is finite: ‖𝐹‖_𝑊^𝑠_1𝑚^,…,𝑠_(ℝ^𝑚𝑛₎=_ℝ𝑚𝑛𝜉₁^2𝑠1⋯𝜉_𝑚^2𝑠𝑚𝐹(𝜉)²𝑑𝜉^1/2, (4)

where 𝜉=(𝜉₁,…,𝜉_𝑚) and ⟨𝜉_𝑘⟩=(1+|𝜉_𝑘|²)^1/2.

Let 𝜓∈𝒮(ℝ^𝑚𝑛) be such that supp𝜓⊂{𝜉∈ℝ^𝑚𝑛∶1/2≤|𝜉|≤2} and ∑_𝑗∈ℤ𝜓(2^−𝑗𝜉)=1 for 𝜉≠0.

Let 𝒮₁(ℝ^𝑑) be the set of all Schwartz functions Ψ on ℝ^𝑑, whose Fourier transform is supported in an annulus of the form {𝜉∶𝑐₁<|𝜉|<𝑐₂}, is nonvanishing in a smaller annulus {𝜉∶𝑐^′₁≤|𝜉|≤𝑐^′₂} (for some choice of constants 0<𝑐₁<𝑐^′₁<𝑐^′₂<𝑐₂<∞), and satisfies _𝑗∈ℤΨ2^−𝑗𝜉=constant,𝜉∈ℝ^𝑑⧵{0}. (5)

The weighted estimate for 𝑇_𝜎 is also an interesting topic in harmonic analysis. And it has attracted many authors in this area. Recently, Fujita and Tomita [8] established some weighted estimates of 𝑇_𝜎 under the Hörmander condition and classical 𝐴_𝑝 weights. For other works about the weighted estimates for 𝑇_𝜎, see [9, 10] and the references therein.

Theorem A (see [8]). Let 1<𝑝₁,𝑝₂,…,𝑝_𝑁<∞, 1/𝑝₁+⋯+1/𝑝_𝑁=1/𝑝, and 𝑁𝑛/2<𝑠≤𝑁𝑛. Assume (i)min𝑝₁,…,𝑝_𝑁>𝑁𝑛/𝑠 and 𝜔∈𝐴_{min𝑝1𝑠/𝑁𝑛,…,𝑝𝑁𝑠/𝑁𝑛};(ii)min𝑝₁,…,𝑝_𝑁>(𝑁𝑛/𝑠)^′ and 1<𝑝<∞, 𝜔^1−𝑝′∈𝐴_{𝑝′𝑠/(𝑁𝑛)}.
If 𝜎∈𝐿^∞ satisfies sup_𝑘∈𝑍‖𝜎(2^𝑘⋅)𝜓‖_𝐿2𝑠<∞, then 𝑇_𝜎 is bounded from 𝐿^𝑝1(𝜔)×⋯×𝐿^𝑝𝑚(𝜔) to 𝐿^𝑝(𝜔).

An improvement of Theorem 1.2 is stated as follows.

Theorem B (see [8]). Let 1<𝑝₁,𝑝₂,…,𝑝_𝑁<∞, 1/𝑝₁+⋯+1/𝑝_𝑁=1/𝑝, and 𝑛/2<𝑠_𝑗≤𝑛, 𝑗=1,…,𝑁. Assume 𝑝_𝑗>𝑛/𝑠_𝑗 and 𝜔_𝑗∈𝐴_{𝑝𝑗𝑠𝑗/𝑛} for 1≤𝑗≤𝑁. If 𝜎∈𝐿^∞ satisfies sup_𝑘∈𝑍‖𝜎(2^𝑘⋅)𝜓‖_{𝑊1𝑁)(𝑠,⋯,𝑠(𝑅𝑁𝑛)}<∞, then 𝑇_𝜎 is bounded from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑁) to 𝐿^𝑝(𝜔), where 𝜔=𝜔^𝑝/𝑝1₁⋯𝜔^{𝑝/𝑝𝑁}_𝑁.

The first purpose of this paper is to improve Theorem A by using 𝐿^𝑟-based Sobolev spaces (1<𝑟≤2). The second purpose is to give a new proof of Theorem B. The following are the main results.

Theorem 1. For some 1<𝑟≤2, suppose that 𝜎∈𝐿^∞(ℝ^𝑚𝑛) and Ψ∈𝒮₁(ℝ^𝑚𝑛) satisfy, for some 𝑚𝑛/𝑟<𝛾≤𝑚𝑛, sup_𝑘∈ℤ𝜎(2^𝑘⋅)Ψ_{𝐿𝑟𝛾(ℝ𝑚𝑛)}=𝐾<∞. (6)
If 𝑝₁,…,𝑝_𝑚, 𝛾, and the weights 𝜔 satisfy one of the following two conditions: (i)min{𝑝₁,…,𝑝_𝑚}>𝑚𝑛/𝛾 and 𝜔∈𝐴_{min{𝑝1𝛾/(𝑚𝑛),…,𝑝𝑚𝛾/(𝑚𝑛)}},(ii)min{𝑝₁,…,𝑝_𝑚}>(𝑚𝑛/𝛾)^′, 1<𝑝<∞, and 𝜔^1−𝑝′∈𝐴_{𝑝′𝛾/(𝑚𝑛)},then there is a number 𝛿=𝛿(𝑚𝑛,𝛾,𝑟) satisfying 0<𝛿≤𝑟−1, such that the 𝑚-linear operator 𝑇_𝜎, associated with the multiplier 𝜎, is bounded from 𝐿^𝑝1(𝜔)×⋯×𝐿^𝑝𝑚(𝜔) to 𝐿^𝑝(𝜔), whenever 𝑟−𝛿<𝑝_𝑗<∞ for all 𝑗=1,…,𝑚, and 𝑝 is given by 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚.

Theorem 2. Let 1<𝑝₁,…,𝑝_𝑚<2 and let 𝑠₁>𝑛/𝑝₁,…,𝑠_𝑚>𝑛/𝑝_𝑚 and 𝑠₁+⋯+𝑠_𝑚 < 𝑛/𝑝₁+⋯+𝑛/𝑝_𝑚+1. If 𝜎∈𝐿^∞(ℝ^𝑚𝑛) satisfies sup_𝑘∈ℤ‖𝜎(2^𝑘⋅)𝜓‖_{𝑊𝑠1𝑚,…,𝑠(ℝ𝑚𝑛)}<∞, then 𝑇_𝜎 is bounded from 𝐿^𝑞1(𝑤^𝑞1₁)×⋯×𝐿^𝑞𝑚(𝑤^𝑞𝑚_𝑚) to 𝐿^𝑞(𝑤^𝑞), whenever 1<𝑞₁,…,𝑞_𝑚<∞, 1/𝑞₁+⋯+1/𝑞_𝑚=1/𝑞, and (𝑤^𝑞1₁,…,𝑤^𝑞𝑚_𝑚)∈(𝐴_𝑞1,…,𝐴_𝑞𝑚) with 𝑤=𝑤₁⋯𝑤_𝑚.

2. The Proof of Theorem 1

In this section we discuss the proof of Theorem 1. We begin with some definitions for maximal operators. Throughout the paper, 𝑀 denotes the Hardy-Littlewood maximal operator defined by 𝑀𝑓(𝑥)=sup_𝑥∈𝑄1|𝑄|_𝑄𝑓𝑦𝑑𝑦, (7) where 𝑄 moves over all cubes containing 𝑥. For 𝛿>0, 𝑀_𝛿 is the maximal function defined by 𝑀_𝛿𝑓(𝑥)=𝑀𝑓^𝛿1/𝛿(𝑥)=sup_𝑄∋𝑥1|𝑄|_𝑄𝑓𝑦^𝛿𝑑𝑦^1/𝛿. (8) In addition, 𝑀^♯ is the sharp maximal function of Fefferman and Stein: 𝑀^♯𝑓(𝑥)=sup_𝑄∋𝑥inf_𝑐1|𝑄|_𝑄𝑓𝑦−𝑐𝑑𝑦≈sup_𝑄∋𝑥1|𝑄|_𝑄𝑓𝑦−𝑓_𝑄𝑑𝑦, (9) where 𝑓_𝑄 denotes the average of 𝑓 over 𝑄 and a variant of 𝑀^♯ is given by 𝑀^♯_𝛿𝑓(𝑥)=𝑀^♯𝑓^𝛿1/𝛿(𝑥). (10)

We prepare some lemmas which will be used later.

Lemma 3 (see [11]). Let 1<𝑝<∞ and 𝜔∈𝐴_𝑝. Then (1) 𝜔^1−𝑝′∈𝐴_𝑝′; (2) there exists 𝑞<𝑝 such that 𝜔∈𝐴_𝑞.

Lemma 4 (see [12]). Let 1<𝑝,𝑞<∞, and 𝜔∈𝐴_𝑝. Then there exist positive finite constants 𝐶(𝑝,𝑞) such that _𝑘∈ℤ𝑀𝑓_𝑘^𝑞1/𝑞_{𝐿𝑝(𝜔)}≤𝐶𝑝,𝑞_𝑘∈ℤ𝑓_𝑘^𝑞1/𝑞_{𝐿𝑝(𝜔)} (11) for all sequences {𝑓_𝑘}_𝑘∈ℤ of locally integrable functions on ℝ^𝑛.

Lemma 5. Let Δ_𝑘 be the Littlewood-Paley operator given by Δ_𝑘(𝑔)(̂𝜉)=𝑔(𝜉)Ψ(2^−𝑘𝜉), 𝑘∈ℤ, where Ψ is a Schwartz function whose Fourier transform is supported in the annulus {𝜉∶2^−𝑏<|𝜉|<2^𝑏}, for some 𝑏∈ℤ⁺, and satisfies ∑_𝑘∈ℤΨ(2^−𝑘𝜉)=𝑐₀, for some constant 𝑐₀. Let 0<𝑝<∞ and 𝜔∈𝐴_∞. Then there is a constant 𝑐=𝑐(𝑛,𝑝,𝑐₀,Ψ), such that for 𝐿^𝑝(𝜔) functions 𝑓 one has 𝑓_{𝐿𝑝(𝜔)}≤𝑐_𝑘∈ℤΔ_𝑘𝑓^21/2_{𝐿𝑝(𝜔)}. (12)

Proof. The proof follows from similar steps in Lemma 4 of [6] and combines the method used in Remark 2.6 of [8]. Let Φ be a Schwartz function with integral one. Then, 𝑓(𝑥)=lim_𝑡→0Φ_𝑡∗𝑓(𝑥)≤sup_𝑡>0Φ_𝑡∗𝑓(𝑥). (13) If 𝜔∈𝐴_∞, the weighted Hardy space 𝐻^𝑝(𝜔) coincides with the weighted Triebel-Lizorkin space 𝐹̇₀^𝑝,2 for 0<𝑝<∞. Hence, if 𝜔∈𝐴_∞, we have 𝑓_{𝐿𝑝(𝜔)}≤sup_𝑡>0Φ_𝑡∗𝑓(𝑥)_{𝐿𝑝(𝜔)}=𝑐𝑓_{𝐻𝑝(𝜔)}≈𝑓_̇𝐹0^𝑝,2_(𝜔)≤𝑐_𝑘∈ℤΔ_𝑘𝑓^21/2_{𝐿𝑝(𝜔)}. (14) The proof is complete.

Now we give the proof of Theorem 1.

Proof. Since the proof follows from similar steps in Theorem 1 in [6], we just give the different parts. For each 𝑗=1,…,𝑚, we let 𝑅_𝑗 be the set of points (𝜉₁,…,𝜉_𝑚) in (ℝ^𝑛)^𝑚 such that |𝜉_𝑗|=max{|𝜉₁|,…,|𝜉_𝑚|} and we introduce nonnegative smooth functions 𝜙_𝑗 on [0,∞)^𝑚−1 that are supported in [0,11/10]^𝑚−1 such that 1=^𝑚_𝑗=1𝜙_𝑗𝜉₁𝜉_𝑗𝜉,…,^𝑗𝜉_𝑗𝜉,…,^𝑚𝜉_𝑗 (15) for all (𝜉₁,…,𝜉_𝑚)≠0, with the understanding that the variable with the hat is missing. These functions introduce a partition of unity of (ℝ^𝑛)^𝑚⧵{0} subordinate to a conical neighborhood of the region 𝑅_𝑗.
Each region 𝑅_𝑗 can be written as the union of sets: 𝑅_𝑗,𝑘=𝜉₁,…,𝜉_𝑚∈𝑅_𝑗∶𝜉_𝑘≥𝜉_𝑠∀𝑠≠𝑗 (16) with 𝑘=1,…,𝑚. We need to work with a finer partition of unity, subordinate to each 𝑅_𝑗,𝑘. To achieve this, for each 𝑗, we introduce smooth functions 𝜙_𝑗,𝑘 on [0,∞)^𝑚−2 supported in [0,11/10]^𝑚−2 such that 1=^𝑚_{𝑘=1𝑘≠𝑗}𝜙_𝑗,𝑘𝜉₁𝜉_𝑘𝜉,…,^𝑘𝜉_𝑘𝜉,…,^𝑗𝜉_𝑘𝜉,…,^𝑚𝜉_𝑘 (17) for all (𝜉₁,…,𝜉_𝑚) in the support of 𝜙_𝑗 with 𝜉_𝑘≠0.
We now have obtained the following partition of unity of (ℝ^𝑛)^𝑚⧵{0}: 1=^𝑚𝑚_{𝑗=1𝑘=1𝑘≠𝑗}𝜙_𝑗(⋯)𝜙_𝑗,𝑘(⋯), (18) where the dots indicate the variables of each function.
We now introduce a nonnegative smooth bump 𝜓 supported in the interval [(10𝑚)⁻¹,2] and equal to 1 in the interval [(5𝑚)⁻¹,12/10] and we decompose 𝜎 into a finite number of multipliers: 𝜎=^𝑚𝑚_{𝑗=1𝑘=1𝑘≠𝑗}𝜎Φ_𝑗,𝑘+𝜎Ψ_𝑗,𝑘, (19) where Φ_𝑗,𝑘𝜉₁,…,𝜉_𝑚=𝜙_𝑗(⋯)𝜙_𝑗,𝑘(⋯𝜉)1−𝜓^𝑘𝜉_𝑗,Ψ_𝑗,𝑘𝜉₁,…,𝜉_𝑚=𝜙_𝑗(⋯)𝜙_𝑗,𝑘(⋯)𝜓𝜉_𝑘𝜉_𝑗. (20) We will prove the required assertion for each piece of this decomposition, that is, for the multipliers 𝜎Φ_𝑗,𝑘 and 𝜎Ψ_𝑗,𝑘 for each pair (𝑗,𝑘) in the previous sum. In view of the symmetry of the decomposition, it suffices to consider the case of a fixed pair (𝑗,𝑘) in the previous sum. To simplify notation, we fix the pair (𝑚,𝑚−1); thus, for the rest of the proof we fix 𝑗=𝑚 and 𝑘=𝑚−1 and we prove boundedness for the 𝑚-linear operators whose symbols are 𝜎₁=𝜎Φ_{𝑚,𝑚−1} and 𝜎₂=𝜎Ψ_{𝑚,𝑚−1}. These correspond to the 𝑚-linear operators 𝑇_𝜎1 and 𝑇_𝜎2, respectively.
We first prove Theorem 1 under assumption (i). Since 1≤𝑚𝑛/𝛾<min{𝑟,𝑝₁,…,𝑝_𝑚}, we can take 𝜌 such that 1≤𝑚𝑛/𝛾<𝜌<min{𝑟,𝑝₁,…,𝑝_𝑚} and 𝜔∈𝐴_{min{𝑝1/𝜌,…,𝑝𝑚/𝜌}}. We first consider 𝑇_𝜎1(𝑓₁,…,𝑓_𝑚), where 𝑓_𝑗 are fixed Schwartz functions. We fix a Schwartz radial function 𝜂 whose Fourier transform is supported in the annulus 1−(1/25)≤|𝜉|≤2 and satisfies _𝑗∈𝐙𝜂2^−𝑗𝜉=1,𝜉∈ℝ^𝑛⧵{0}. (21) Associated with 𝜂 we define the Littlewood-Paley operator Δ_𝑗(𝑓)=𝑓∗𝜂_2−𝑗, where 𝜂_𝑡(𝑥)=𝑡^−𝑛𝜂(𝑡⁻¹𝑥) for 𝑡>0. We also define an operator 𝑆_𝑗 by setting 𝑆_𝑗𝑔=𝑔∗𝜁₂^−𝑗, (22) where 𝜁 is a smooth function whose Fourier transform is equal to 1 on the ball |𝑧|<3/5𝑚 and vanishes outside the double of this ball. As in [6, page 143], by using Lemma 5 we get 𝑇_𝜎1𝑓₁,…,𝑓_{𝑚𝐿𝑝(𝜔)}≤𝐶_𝑗𝑇_𝜎1𝑆_𝑗𝑓₁,…,𝑆_𝑗𝑓_𝑚−1,Δ_𝑗𝑓_𝑚^21/2_{𝐿𝑝(𝜔)}. (23)
We will use the following estimate for 𝑇_𝜎1 (see [6, page 145]): 𝑇_𝜎1𝑆_𝑗𝑓₁,…,𝑆_𝑗𝑓_𝑚−1,Δ_𝑗𝑓_𝑚≤𝐶𝐾^𝑚−1_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌𝑀Δ_𝑗𝑓_𝑚^𝜌1/𝜌. (24)
We now square the previous expression, we sum over 𝑗∈ℤ, and we take square roots. Since 𝑟−𝛿=𝜌, the hypothesis 𝑝_𝑗>𝑟−𝛿 implies 𝑝_𝑗>𝜌, and thus each term (𝑀(𝑀(𝑓_𝑖)^𝜌))^1/𝜌 is bounded on 𝐿^𝑝𝑗(𝜔). We obtain 𝑇_𝜎1𝑓₁,…,𝑓_𝑚−1,𝑓_{𝑚𝐿𝑝(𝜔)}≤𝐶𝐾_𝑗𝑇_𝜎1𝑆_𝑗𝑓₁,…,𝑆_𝑗𝑓_𝑚−1,Δ_𝑗𝑓_𝑚^21/2_{𝐿𝑝(𝜔)}≤𝐶^′𝐾_𝑗𝑀Δ_𝑗𝑓_𝑚^{𝜌2/𝜌1/2}_{𝐿𝑝𝑚(𝜔)}×^𝑚−1_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌_{𝐿𝑝𝑖(𝜔)}≤𝐶^′′𝐾_𝑗𝑀Δ_𝑗𝑓_𝑚^{𝜌2/𝜌𝜌/2}_𝐿^1/𝜌_{𝑝𝑚/𝜌(𝜔)}^𝑚−1_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔)}≤𝐶^′′𝐾^𝑚_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔)}, (25) where the last step holds due to Lemma 4 with 𝑞=2/𝜌 and the weighted Littlewood-Paley theorem.
Next we deal with 𝜎₂. Following [6, page 146], we write 𝑇_𝜎2𝑓₁,…,𝑓_𝑚−1,𝑓_𝑚=_𝑗∈ℤ𝑇_𝜎2𝑆^′_𝑗𝑓₁,…,𝑆^′_𝑗𝑓_𝑚−2,Δ^′_𝑗𝑓_𝑚−1,Δ_𝑗𝑓_𝑚,𝑇_𝜎2𝑆^′_𝑗𝑓₁,…,𝑆^′_𝑗𝑓_𝑚−2,Δ^′_𝑗𝑓_𝑚−1,Δ_𝑗𝑓_𝑚≤𝐶𝐾^𝑚−2_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌𝑀Δ^′_𝑗𝑓_𝑚−1^𝜌1/𝜌×𝑀Δ_𝑗𝑓_𝑚^𝜌1/𝜌, (26) for some other Littlewood-Paley operator Δ^′_𝑗 which is given on the Fourier transform by multiplication with a bump Θ(2^−𝑗𝜉), where Θ is equal to one on the annulus {𝜉∈ℝ^𝑛∶(24/25)⋅(1/10𝑚)≤|𝜉|≤4} and vanishes on a larger annulus. Also, 𝑆^′_𝑗 is given by convolution with 𝜁^′_2−𝑗, where 𝜁^′ is a smooth function whose Fourier transform is equal to 1 on the ball |𝑧|<(22/10) and vanishes outside the double of this ball.
Summing over 𝑗 and taking 𝐿^𝑝(𝜔) norms yield 𝑇_𝜎2𝑓₁,…,𝑓_𝑚−1,𝑓_{𝑚𝐿𝑝(𝜔)}≤𝐶𝐾^𝑚−2_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌_𝑗∈ℤ𝑀Δ^′_𝑗𝑓_𝑚−1^𝜌1/𝜌×𝑀Δ_𝑗𝑓_𝑚^𝜌1/𝜌_{𝐿𝑝(𝜔)}≤𝐶𝐾^𝑚−2_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌×^𝑚_{𝑖=𝑚−1𝑗∈ℤ}𝑀Δ_𝑗𝑓_𝑖^{𝜌2/𝜌1/2}_{𝐿𝑝(𝜔)}, (27) where the last step holds due to the Cauchy-Schwarz inequality and we omitted the prime from the term with 𝑖=𝑚−1 for the matter of simplicity. Applying Hölder’s inequality and using that 𝜌<2 and Lemma 4 we obtain the conclusion that the expression above is bounded by 𝐶^′𝐾𝑓_{1𝐿𝑝1(𝜔)}⋯𝑓_{𝑚𝐿𝑝𝑚(𝜔)}. (28)
We next prove Theorem 1 under assumption (ii). It was proven in [6, page 136] that condition (6) is invariant under the adjoints; that is, it is also valid for the symbols of the dual operators 𝜎^∗𝑚(𝜉₁,…,𝜉_𝑚)=𝜎(𝜉₁,…,𝜉_𝑚−1,−(𝜉₁+⋯+𝜉_𝑚)). To prove the required assertion, by duality, it is enough to prove that 𝑇_𝜎1^∗𝑚 and 𝑇_𝜎2^∗𝑚 are bounded from 𝐿^𝑝1(𝜔)×⋯×𝐿^𝑝𝑚−1(𝜔)×𝐿^𝑝′(𝜔^1−𝑝′) to 𝐿^𝑝′𝑚(𝜔^{1−𝑝′𝑚}). We may assume that 𝑝_𝑚=min{𝑝₁,…,𝑝_𝑚}. Since 𝑝_𝑚<(𝑚𝑛/𝛾)^′, we see 1/𝑝^′, 1/𝑝_𝑘<1/𝑝^′+1/𝑝₂+⋯+1/𝑝_𝑚=1/𝑝^′_𝑚<𝛾/(𝑚𝑛). Hence, 𝑚𝑛/𝛾<min{𝑟,𝑝^′,𝑝₁,…,𝑝_𝑚−1}. Since 𝑝<𝑝_𝑚 and 𝜔^1−𝑝′∈𝐴_{𝑝′𝛾/(𝑚𝑛)}⊂𝐴_𝑝′, we deduce that 𝜔∈𝐴_𝑝⊂𝐴_𝑝𝑚; then 𝜔^{1−𝑝′𝑚}∈𝐴_𝑝′𝑚. It is obvious that 𝜔^{(1−𝑝′𝑚)/𝑝′𝑚}=𝜔^−1/𝑝𝜔^1/𝑝1⋯𝜔^1/𝑝𝑚. Since 𝑝_𝑚<𝑝_𝑘, 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑘+⋯+1/𝑝_𝑚−1>1/𝑝_𝑚+1/𝑝_𝑘≥2/𝑝_𝑘. That is, 𝑝<𝑝_𝑘/2; then 𝜔∈𝐴_𝑝⊂𝐴_𝑝𝑘/2⊂𝐴_{𝑝𝑘𝛾/(𝑚𝑛)}. Therefore, we take a positive number 𝜌 such that 1≤𝑚𝑛/𝛾<𝜌<min{𝑟,𝑝^′,𝑝₁,…,𝑝_𝑚−1}, and 𝛾>𝑚𝑛/𝜌 such that 𝜔^1−𝑝′∈𝐴_𝑝′/𝜌 and 𝜔∈𝐴_{𝑝𝑘/𝜌}. We have 𝑇_𝜎1^∗𝑚𝑓₁,…,𝑓_𝑚−1,𝑓_{𝑚𝐿𝑝′𝑚(𝜔′𝑚1−𝑝)}≤𝐶𝐾_𝑗𝑇_𝜎1𝑆_𝑗𝑓₁,…,𝑆_𝑗𝑓_𝑚−1,Δ_𝑗𝑓_𝑚^21/2_{𝐿𝑝′𝑚(𝜔′𝑚1−𝑝)}≤𝐶^′𝐾_𝑗𝑀Δ_𝑗𝑓_𝑚^{𝜌2/𝜌1/2}_{𝐿𝑝′(𝜔′1−𝑝)}×^𝑚−1_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌_{𝐿𝑝𝑖(𝜔)}≤𝐶^′′𝐾_𝑗𝑀Δ_𝑗𝑓_𝑚^{𝜌2/𝜌𝜌/2}_𝐿^1/𝜌_{𝑝′/𝜌𝜔′1−𝑝}×^𝑚−1_𝑖=1𝑓_{𝑖𝐿𝑝𝑖(𝜔)}≤𝐶^′𝐾𝑓_{1𝐿𝑝1(𝜔)}⋯𝑓_{𝑚𝐿𝑝𝑚(𝜔)}. (29) Similarly, we have 𝑇_𝜎2^∗𝑚𝑓₁,…,𝑓_𝑚−1,𝑓_{𝑚𝐿𝑝′𝑚(𝜔′𝑚1−𝑝)}≤𝐶𝐾^𝑚−2_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌𝜔^1/𝑝𝑖×_𝑗∈ℤ𝑀Δ^′_𝑗𝑓_𝑚−1^𝜌1/𝜌×𝑀Δ_𝑗𝑓_𝑚^𝜌1/𝜌𝜔^{−1/𝑝+1/𝑝𝑚−1}_{𝐿𝑝′𝑚(𝜔′𝑚1−𝑝)}≤𝐶𝐾^𝑚−2_𝑖=1𝑀𝑀𝑓_𝑖^𝜌1/𝜌_{𝐿𝑝𝑖(𝜔𝑖)}×_𝑗∈ℤ𝑀Δ_𝑗𝑓_𝑚−1^{𝜌2/𝜌𝜌/2}_{𝐿𝑝𝑚−1/𝜌(𝜔)}×_𝑗∈ℤ𝑀Δ_𝑗𝑓_𝑚^{𝜌2/𝜌𝜌/2}_{𝐿𝑝′/𝜌(𝜔′1−𝑝)}≤𝐶^′𝐾𝑓_{1𝐿𝑝1(𝜔)}⋯𝑓_{𝑚𝐿𝑝𝑚(𝜔)}. (30)
This concludes the proof of Theorem 1.

3. The Proof of Theorem 2

We begin with some lemmas which will be used in the proof of Theorem 2.

Lemma 6 (see [11]). Let 0<𝑝 and 𝛿<∞ and let 𝜔 be a weight in 𝐴_∞. Then, there exists 𝐶>0 (depending on the 𝐴_∞ constant of 𝜔) such that _𝑅𝑛𝑀_𝛿𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥≤𝐶_𝑅𝑛𝑀^♯_𝛿𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥, (31) for all function 𝑓 for which the left-hand side is finite.

Lemma 7 (see [13]). Let 0<𝑝₁,𝑝₂,𝑝≤∞, and 1/𝑝₁+1/𝑝₂=1/𝑝. Let 𝜎 be a multiplier satisfying sup_𝑘∈ℤ‖𝜎(2^𝑘⋅)𝜓‖_{𝑊𝑠12,𝑠(ℝ𝑚𝑛)}<∞ for 𝑠₁>max{𝑛/2,𝑛/𝑝₁−𝑛/2}, 𝑠₂>max{𝑛/2,𝑛/𝑝₂−𝑛/2}, and 𝑠₁+𝑠₂>𝑛/𝑝₁+𝑛/𝑝₂−𝑛/2; then 𝑇_𝜎 is bounded from 𝐻^𝑝1(ℝ^𝑛)×𝐻^𝑝2(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛).

Remark 8. It should be pointed out that Lemma 7 can be extended to the case 𝑚≥3.

Lemma 9 (see [8]). Let 𝑟>0, 𝑞₁,…,𝑞_𝑚∈[2,∞), and 𝑠₁,…,𝑠_𝑚≥0. Then there exists a constant 𝐶>0 such that _ℝ𝑛⋯_ℝ𝑛𝐹𝜉₁,…,𝜉_𝑚^𝑞1𝜉₁^𝑠1𝑑𝜉₁^𝑞2/𝑞1×𝜉₂^𝑠2𝑑𝜉₂^𝑞3/𝑞2⋯𝜉_𝑚^𝑠𝑚𝑑𝜉_𝑚^1/𝑞𝑚≤𝐶‖𝐹‖_𝑊^𝑠_11𝑚𝑚^{/𝑞,…,𝑠/𝑞}_(ℝ^𝑚𝑛₎ (32)

for all 𝐹∈𝑊^{𝑠1/𝑞1,…,𝑠𝑚/𝑞𝑚}(ℝ^𝑚𝑛) with supp𝐹⊂{|𝑥₁|²+⋯+|𝑥_𝑚|²≤𝑟}.

Next, we give a pointwise control of 𝑀^♯_𝛿𝑇_𝜎(𝑓) which becomes very useful in the proof of Theorem 2.

Lemma 10. Let 1<𝑝₁,…,𝑝_𝑚<2. Assume that 𝜎∈𝐿^∞(ℝ^𝑚𝑛) which satisfies sup_𝑘∈ℤ‖𝜎(2^𝑘⋅)𝜓‖<∞_{𝑊𝑠1𝑚,…,𝑠(ℝ𝑚𝑛)} for 𝑠₁>𝑛/𝑝₁,…,𝑠_𝑚>𝑛/𝑝_𝑚 and 𝑠₁+⋯+𝑠_𝑚 < 𝑛/𝑝₁ + ⋯ + 𝑛/𝑝_𝑚 + 1. For any 0<𝛿<1/𝑚, one has 𝑀^♯_𝛿𝑇_𝜎(𝑓)(𝑥)≤𝐶∏^𝑚_𝑗=1𝑀_𝑝𝑗𝑓_𝑗(𝑥).

Proof. For simplicity, we only prove for the case 𝑚=2, since there is no essential difference for the general case. Fix an 𝑥∈ℝ^𝑛 and a cube 𝑄 with side length 𝑙, such that 𝑥∈𝑄. Let 𝑓_𝑖=𝑓⁰_𝑖+𝑓^∞_𝑖, where 𝑓⁰_𝑖=𝑓_𝑖𝜒_𝑄∗  and 𝑓^∞_𝑖=𝑓_𝑖𝜒_(𝑄∗)^𝑐 for 𝑖=1,2 and 𝑄^∗=4√𝑛𝑄. Since 0<𝛿<1/2, we have 1|𝑄|_𝑄𝑇_𝜎𝑓₁,𝑓₂(𝑧)^𝛿−|𝐶|^𝛿𝑑𝑧^1/𝛿1≤𝐶|𝑄|_𝑄𝑇_𝜎𝑓₁,𝑓₂(𝑧)−𝐶^𝛿𝑑𝑧^1/𝛿1≤𝐶|𝑄|_𝑄𝑇_𝜎𝑓⁰₁,𝑓⁰₂(𝑧)+𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑧)+𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑧)+𝑇_𝜎𝑓⁰₁,𝑓⁰₂(𝑧)−𝐶^𝛿𝑑𝑧^1/𝛿1≤𝐶|𝑄|_𝑄𝑇_𝜎𝑓⁰₁,𝑓⁰₂(𝑧)^𝛿𝑑𝑧^1/𝛿+1|𝑄|_𝑄𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑧)+𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑧)+𝑇_𝜎𝑓^∞₁,𝑓⁰₂(𝑧)−𝐶^𝛿𝑑𝑧^1/𝛿=𝑈₁+𝑈₂. (33)
We first consider 𝑈₁. By Kolmogorov’s inequality, Hölder’s inequality, and Lemma 7, we have 1|𝑄|_𝑄𝑇_𝜎𝑓⁰₁,𝑓⁰₂(𝑧)^𝛿𝑑𝑧^1/𝛿≤𝐶𝑇_𝜎𝑓⁰₁,𝑓⁰_{2𝐿𝑝,∞(𝑄,𝑑𝑥/|𝑄|)}≤𝐶²_𝑗=11𝑄^∗_𝑄^∗𝑓_𝑗(𝑧)^𝑝𝑗𝑑𝑧^1/𝑝𝑗≤𝐶²_𝑗=1𝑀_𝑝𝑗𝑓_𝑗(𝑥), (34)
where 1/𝑝=1/𝑝₁+1/𝑝₂ with 𝑝>𝛿 and 1<𝑝₁, 𝑝₂<∞.
Next we deal with 𝑈₂. We choose 𝐶=∑³_𝑖=1𝐶_𝑖, where 𝐶₁=𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑥),𝐶₂=𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑥),𝐶₃=𝑇_𝜎𝑓^∞₁,𝑓⁰₂(𝑥). (35)
We may split 𝑈₂ as 𝑈₂≤𝑈₂₁+𝑈₂₂+𝑈₂₃, where 𝑈₂₁=1|𝑄|_𝑄𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑥)^𝛿𝑑𝑧^1/𝛿,𝑈₂₂=1|𝑄|_𝑄𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑥)^𝛿𝑑𝑧^1/𝛿,𝑈₂₃=1|𝑄|_𝑄𝑇_𝜎𝑓^∞₁,𝑓⁰₂(𝑧)−𝑇_𝜎𝑓^∞₁,𝑓⁰₂(𝑥)^𝛿𝑑𝑧^1/𝛿. (36)
Now we estimate 𝑈₂₁ first. We decompose 𝜎 as 𝜎=_𝑗∈ℤ𝜎(⋅)𝜓⋅/2^𝑗≐_𝑗∈ℤ𝜎_𝑗. (37)
Let 𝜎_𝑗=𝜎(⋅)𝜓(⋅/2^𝑗), where 𝜓∈𝒮(ℝ^2𝑛) with supp 𝜓⊂{𝜉∈ℝ^2𝑛∶1/2≤|𝜉|≤2} and ∑_𝑗∈ℤ𝜓(2^−𝑗𝜉)=1, 𝜉≠0. Thus, we have𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑥)≤𝐶∑_𝑗∈ℤ𝑇_𝜎𝑗𝑓^∞₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑗𝑓^∞₁,𝑓^∞₂(𝑥)≤𝐶∑^∞_𝑗∈ℤ∑_𝑘1^∞₌₀∑_𝑘2=0∫_{2𝑘2+1𝑄∗⧵2𝑘2𝑄∗}∫_{2𝑘1+1𝑄∗⧵2𝑘1𝑄∗}𝜎^∨_𝑗𝑧−𝑦₁,𝑧−𝑦₂−𝜎^∨_𝑗𝑥−𝑦₁,𝑥−𝑦₂𝑓₁𝑦₁𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂≤𝐶∑^∞_𝑘1^∞₌₀∑_𝑘2=0∑_𝑗∈ℤ∫_{2𝑘2+1𝑄∗⧵2𝑘2𝑄∗}∫_{2𝑘1+1𝑄∗⧵2𝑘1𝑄∗}𝜎^∨_𝑗𝑧−𝑦₁,𝑧−𝑦₂−𝜎^∨_𝑗𝑥−𝑦₁,𝑥−𝑦₂𝑓₁𝑦₁𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂≐∑^∞_𝑘1^∞₌₀∑_𝑘2=0𝐼_𝑘1,𝑘2. (38)
Applying Hölder’s inequality we have𝐼_𝑘1,𝑘2=_{𝑗∈ℤ2𝑘2+1𝑄∗⧵2𝑘2𝑄∗2𝑘1+1𝑄∗⧵2𝑘1𝑄∗}𝜎^∨_𝑗𝑧−𝑦₁,𝑧−𝑦₂−𝜎^∨_𝑗𝑥−𝑦₁,𝑥−𝑦₂𝑓₁𝑦₁𝑓₂𝑦₂𝑑𝑦₁𝑑𝑦₂≤𝐶_{𝑗∈ℤ2𝑘2+1𝑄∗⧵2𝑘2𝑄∗2𝑘1+1𝑄∗⧵2𝑘1𝑄∗}𝜎^∨_𝑗𝑧−𝑦₁,𝑧−𝑦₂−𝜎^∨_𝑗𝑥−𝑦₁,𝑥−𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2×_{2𝑘1+1𝑄∗}𝑓₁𝑦₁^𝑝1𝑑𝑦₁^1/𝑝1_{2𝑘2+1𝑄∗}𝑓₂𝑦₂^𝑝2𝑑𝑦₂^1/𝑝2≐𝐶_𝑗∈ℤ𝐼_{𝑘1,𝑘2,𝑗}×_{2𝑘2+1𝑄∗}𝑓₁𝑦₁^𝑝1𝑑𝑦₁^1/𝑝1_{2𝑘2+1𝑄∗}𝑓₂𝑦₂^𝑝2𝑑𝑦₂^1/𝑝2. (39)
Let ℎ=𝑧−𝑥 and 𝑄=𝑥−𝑄^∗. Then we have𝐼_{𝑘1,𝑘2,𝑗}=_{2𝑘2+1𝑄∗⧵2𝑘2𝑄∗2𝑘1+1𝑄∗⧵2𝑘1𝑄∗}𝜎^∨_𝑗𝑧−𝑦₁,𝑧−𝑦₂−𝜎^∨_𝑗𝑥−𝑦₁,𝑥−𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2≤𝐶_2𝑘2+1𝑄_⧵2^𝑘2𝑄₂^𝑘1+1𝑄^⧵2𝑘1𝑄𝜎^∨_𝑗ℎ+𝑦₁,ℎ+𝑦₂−𝜎^∨_𝑗𝑦₁,𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2≤𝐶_2𝑘2+1𝑄_⧵2^𝑘2𝑄₂^𝑘1+1𝑄_⧵2^𝑘1𝑄𝜎^∨_𝑗𝑦₁,𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2≤𝐶2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠2_2𝑘2+1𝑄_⧵2^𝑘2𝑄₂^𝑘1+1_𝑄^∗_⧵2^𝑘1𝑄𝜎^∨_𝑗𝑦₁,𝑦₂^𝑝′1×1+𝑦₁^{2𝑠1𝑝′1/2}𝑑𝑦₁^{𝑝′2/𝑝′1}1+𝑦₂^{2𝑠2𝑝′2/2}𝑑𝑦₂^1/𝑝′2≤𝐶2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠2_{ℝ𝑛ℝ𝑛}𝜎^∨_𝑗2^−𝑗𝑦₁,2^−𝑗𝑦₂^𝑝′1×1+2^−𝑗𝑦₁^{2𝑠1𝑝′1/2}2^−𝑗𝑛𝑑𝑦₁×𝑦₁^{𝑝′2/𝑝′1}1+2^−𝑗𝑦₂^{2𝑠2𝑝′2/2}2^−𝑗𝑛𝑑𝑦₂^1/𝑝′2≤𝐶2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠22^{−𝑗(𝑠1+𝑠2)}2^{𝑗𝑛((1/𝑝1)+(1/𝑝2))}×_{ℝ𝑛ℝ𝑛}2^−2𝑗𝑛𝜎^∨_𝑗2^−𝑗𝑦₁,2^−𝑗𝑦₂^𝑝′11+𝑦₁^{2𝑠1𝑝′1/2}𝑑𝑦₁^{𝑝′2/𝑝′1}1+𝑦₂^{2𝑠2𝑝′2/2}𝑑𝑦₂^1/𝑝′2≤𝐶2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠22^{−𝑗(𝑠1+𝑠2)}2^{𝑗𝑛((1/𝑝1)+(1/𝑝2))}𝜎2^𝑗⋅𝜓_{𝑊𝑠12,𝑠}, (40)where the last inequality holds due to Lemma 9. Suppose that 2^−𝑅≤𝑙<2^−𝑅+1. Since 𝑛/𝑝₁+𝑛/𝑝₂−𝑠₁−𝑠₂<0, we have _𝑗≥𝑅𝐼_{𝑘1,𝑘2,𝑗}≤sup_𝑗𝜎2^𝑗⋅𝜓_{𝑊𝑠12,𝑠}×_𝑗≥𝑅2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠22^{𝑗(𝑛/𝑝1+𝑛/𝑝2−𝑠1−𝑠2)}≤𝐶sup_𝑗𝜎2^𝑗⋅𝜓_{𝑊𝑠12,𝑠}2^{−𝑘1𝑠1}2^{−𝑘2𝑠2}𝑙^{−𝑛/𝑝1}𝑙^{−𝑛/𝑝2}. (41)
On the other hand𝐼_{𝑘1,𝑘2,𝑗}≤_2𝑘2+1𝑄_⧵2^𝑘2𝑄₂^𝑘1+1𝑄_⧵2^𝑘1𝑄𝜎^∨_𝑗𝑦₁+ℎ,𝑦₂+ℎ−𝜎^∨_𝑗𝑦₁,𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2≤𝐶_2𝑘2+1𝑄^⧵2𝑘2𝑄^2𝑘1+1𝑄^⧵2𝑘1𝑄¹⁰ℎ⋅∇𝜎^∨_𝑗𝑦₁+𝜃ℎ,𝑦₂+𝜃ℎ𝑑𝜃^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2≤𝐶¹_02𝑘2+1𝑄_⧵2^𝑘2𝑄₂^𝑘1+1𝑄_⧵2^𝑘1𝑄ℎ⋅∇𝜎^∨_𝑗𝑦₁+𝜃ℎ,𝑦₂+𝜃ℎ^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2𝑑𝜃≤𝐶_2𝑘2+1𝑄_⧵2^𝑘2𝑄₂^𝑘1+1𝑄_⧵2^𝑘1𝑄ℎ⋅∇𝜎^∨_𝑗𝑦₁,𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2, (42)
where ℎ=(ℎ,ℎ)∈ℝ^2𝑛. Since ℎ⋅∇(𝜎^∨_𝑗)(𝑦₁,𝑦₂)=∑^2𝑛_𝑟=1ℎ_𝑟𝜕_𝑟(𝜎^∨_𝑗)(𝑦₁,𝑦₂), we have𝐼_{𝑘1,𝑘2,𝑗}≤𝐶^2𝑛_𝑟=1𝑙_2𝑘2+1𝑄_⧵2^𝑘2𝑄₂^𝑘1+1𝑄_⧵2^𝑘1𝑄𝜕_𝑟𝜎^∨_𝑗𝑦₁,𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2≤𝐶^2𝑛_𝑟=1𝑙2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠2_{ℝ𝑛ℝ𝑛}𝜕_𝑟𝜎^∨_𝑗2^−𝑗𝑦₁,2^−𝑗𝑦₂^𝑝′11+2^−𝑗𝑦₂^{2𝑠1𝑝′1/2}2^−𝑗𝑛𝑑𝑦₁^{𝑝′2/𝑝′1}×1+2^−𝑗𝑦₂^{2𝑠2𝑝′2/2}2^−𝑗𝑛𝑑𝑦₂^1/𝑝′2≤𝐶^2𝑛_𝑟=1𝑙2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠2_{ℝ𝑛ℝ𝑛}2^−2𝑗𝑛𝜕_𝑟𝜎^∨_𝑗2^−𝑗𝑦₁,2^−𝑗𝑦₂^𝑝′11+𝑦₁^{2𝑠1𝑝′1/2}𝑑𝑦₁^{𝑝′2/𝑝′1}×1+𝑦₂^{2𝑠2𝑝′2/2}𝑑𝑦₂^1/𝑝′22^{−𝑗(𝑠1+𝑠2)}2^{𝑗𝑛((1/𝑝1)+(1/𝑝2))}≤𝐶^2𝑛_𝑟=1𝑙2^𝑘1𝑙^−𝑠12^𝑘2𝑙^−𝑠22^{−𝑗(𝑠1+𝑠2)}2^{𝑗𝑛((1/𝑝1)+(1/𝑝2))}2^𝑗𝜎2^𝑗⋅𝜓_{𝑊𝑠12,𝑠}, (43)where in the last inequality Lemma 9 was used again and hence _𝑗<𝑅𝐼_{𝑘1,𝑘2,𝑗}≤𝐶sup_𝑗𝜎2^𝑗⋅𝜓_{𝑊𝑠12,𝑠}2^{−𝑘1𝑠1}2^{−𝑘2𝑠2}𝑙^−𝑛𝑝1𝑙^{−𝑛/𝑝2}. (44)
Combining the above arguments we have𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑓^∞₁,𝑓^∞₂(𝑥)≤𝐶^∞_𝑘1^∞_=0𝑘2=02^{−𝑘1𝑠1}2^{−𝑘2𝑠2}𝑙^{−𝑛/𝑝1}𝑙^{−𝑛/𝑝2}×_{2𝑘1+1𝑄∗}𝑓₁𝑦₁^𝑝1𝑑𝑦₁^1/𝑝1×_{2𝑘2+1𝑄∗}𝑓₂𝑦₂^𝑝2𝑑𝑦₂^1/𝑝2≤𝐶^∞_𝑘1=02^{−𝑘1(𝑠1−𝑛/𝑝1)}×^∞_𝑘2=02^{−𝑘2(𝑠2−𝑛/𝑝2)}𝑀_𝑝1𝑓₁𝑀_𝑝2𝑓₂≤𝐶𝑀_𝑝1𝑓₁𝑀_𝑝2𝑓₂. (45)
Thus, we obtain 𝑈₂₁≤𝐶𝑀_𝑝1𝑓₁𝑀_𝑝2𝑓₂. What remain to be considered are 𝑈₂₂ and 𝑈₂₃. We just estimate 𝑈₂₂ since the same arguments can be applied to 𝑈₂₃:𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑥)≤𝐶_𝑗𝑇_𝜎𝑗𝑓⁰₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑗𝑓⁰₁,𝑓^∞₂(𝑥)≤𝐶_𝑗^∞_{𝑘2=02𝑘2+1𝑄∗}𝑓₂𝑦_2𝑄∗𝜎^∨_𝑗𝑧−𝑦₁,𝑧−𝑦₂−𝜎^∨_𝑗𝑥−𝑦₁,𝑥−𝑦₂𝑓₁𝑦₁𝑑𝑦₁𝑑𝑦₂≤𝐶^∞_{𝑗∈ℤ𝑘2=02𝑘2+1𝑄∗⧵2𝑘2𝑄∗𝑄∗}𝜎^∨_𝑗𝑧−𝑦₁,𝑧−𝑦₂−𝜎^∨_𝑗𝑥−𝑦₁,𝑥−𝑦₂^𝑝′1𝑑𝑦₁^{𝑝′2/𝑝′1}𝑑𝑦₂^1/𝑝′2×_𝑄∗𝑓₁𝑦₁^𝑝1𝑑𝑦₁^1/𝑝1_{2𝑘2+1𝑄∗}𝑓₂𝑦₂^𝑝2𝑑𝑦₂^1/𝑝2. (46)
Then by similar arguments as the above mentioned we get that 𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑧)−𝑇_𝜎𝑓⁰₁,𝑓^∞₂(𝑥)≤𝐶^∞_𝑘2=02^{−𝑘2𝑠2}𝑙^{−𝑛/𝑝1}𝑙^{−𝑛/𝑝2}_𝑄∗𝑓₁𝑦₁^𝑝1𝑑𝑦₁^1/𝑝1×_{2𝑘2+1𝑄∗}𝑓₂𝑦₂^𝑝2𝑑𝑦₂^1/𝑝2≤𝐶^∞_𝑘2=02^{−𝑘2(𝑠2−𝑝2)}𝑀_𝑝1𝑓₁𝑀_𝑝2𝑓₂≤𝐶𝑀_𝑝1𝑓₁𝑀_𝑝2𝑓₂. (47)
The proof of Lemma 10 is complete.

Now we are ready to give the proof of Theorem 2.

Proof. By Lemma 3, we can choose 1<𝑝₁<𝑞₁ and 1<𝑝₂<𝑞₂ such that 𝜔^𝑞1₁∈𝐴_𝑞1/𝑝1 and 𝜔^𝑞2₂∈𝐴_𝑞2/𝑝2. Then by the Hölder inequality, Lemma 10, and the weighted boundedness of 𝑀, we deduce that 𝑇_𝜎𝑓₁,𝑓_{2𝐿𝑞(𝑤𝑞)}≤𝑀_𝛿𝑇_𝜎𝑓₁,𝑓_{2𝐿𝑞(𝑤𝑞)}≤𝐶𝑀^♯_𝛿𝑇_𝜎𝑓₁,𝑓_{2𝐿𝑞(𝑤𝑞)}≤𝐶𝑀_𝑝1𝑓₁𝑀_𝑝2𝑓_{2𝐿𝑞(𝑤𝑞)}≤𝐶𝑀_𝑝1𝑓_{1𝐿𝑞1(𝑤𝑞11)}𝑀_𝑝1𝑓_{1𝐿𝑞2(𝑤𝑞22)}≤𝐶𝑓_{1𝐿𝑞1(𝑤}^𝑞₁₁₎𝑓_{2𝐿𝑞2(𝑤}^𝑞₂₂₎. (48)
The proof of Theorem 2 is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (nos. 11226102 and 11226103) and Doctor Foundation of Henan Polytechnic University (no. B2012-055).