# Some Weighted Estimates for Multilinear Fourier Multiplier Operators

Authors: Zengyan Si
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We first provide a weighted Fourier multiplier theorem for multilinear operators which extends Theorem 1.2 in Fujita and Tomita (2012) by usingLr-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).


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## Body

## 1. Introduction

During the last several years, considerable attention has been paid to the study of multilinear Fourier multiplier operators. Let𝒮(ℝd) be the Schwartz space of all rapidly decreasing smooth functions on ℝd, for some d∈ℤ+. The multilinear Fourier multiplier operator Tσ associated with a symbol σ is defined by
(1)Tσ(f1,…,fm)(x)=∫ℝmn‍e2πix·(ξ1+⋯+ξm)σ(ξ1,…,ξm)×f^1(ξ1)⋯f^m(ξm)dξ1⋯dξm
for fi∈𝒮(ℝn), i=1,…,m.Coifman and Meyer [1] proved that if σ is a bounded function on ℝmn∖{0} that satisfies
(2)|∂ξ1α1⋯∂ξmαmσ(ξ1,…,ξm)|≤Cα(|ξ1|+⋯+|ξm|)-(|α1|+⋯+|αm|)
away from the origin for all sufficiently large multi-indices αj, then Tσ is bounded from the product Lp1(ℝn)×⋯×Lpm(ℝn) to Lp(ℝn) for all 1<p1,…,pm,p<∞ satisfying 1/p1+⋯+1/pm=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 Lr-based Sobolev spaces (1<r≤2). Grafakos et al. [7] proved the L2-boundedness of  Tσ with multipliers of limited smoothness.In order to state other known results, we first introduce some notations. The Laplacian onℝd is Δg=∑j=1d‍∂2g/∂xj2, that is, the sum of the second partials of g in every variable. We define the operator (I-Δ)γ/2(g)=ℱ-1(wγℱ(g)), where wγ(ξ)=(1+4π2|ξ|2)γ/2 for γ>0. Let Lγr(ℝd) be the Lr-based Sobolev space with norm
(3)∥f∥Lγr=∥(I-Δ)γ/2f∥Lr(ℝd),
where 1≤r<∞.Lets→=(s1,…,sm) and let the product type Sobolev space Ws1,…,sm(ℝmn) consist of all functions F such that the following norm of F is finite:
(4)∥F∥Ws1,…,sm(ℝmn)=(∫ℝmn‍〈ξ1〉2s1⋯〈ξm〉2sm|F^(ξ)|2dξ)1/2,whereξ=(ξ1,…,ξm) and 〈ξk〉=(1+|ξk|2)1/2.Letψ∈𝒮(ℝmn) be such that suppψ⊂{ξ∈ℝmn:1/2≤|ξ|≤2} and ∑j∈ℤ‍ψ(2-jξ)=1 for ξ≠0.Let𝒮1(ℝd) be the set of all Schwartz functions Ψ on ℝd, whose Fourier transform is supported in an annulus of the form {ξ:c1<|ξ|<c2}, is nonvanishing in a smaller annulus {ξ:c1′≤|ξ|≤c2′} (for some choice of constants 0<c1<c1′<c2′<c2<∞), and satisfies
(5)∑j∈ℤ‍Ψ^(2-jξ)=constant,ξ∈ℝd∖{0}.The weighted estimate forTσ 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 Tσ under the Hörmander condition and classical Ap weights. For other works about the weighted estimates for Tσ, see [9, 10] and the references therein.Theorem A (see [8]).
Let1<p1,p2,…,pN<∞, 1/p1+⋯+1/pN=1/p, and Nn/2<s≤Nn. Assume (i)
minp1,…,pN>Nn/s and ω∈Aminp1s/Nn,…,pNs/Nn;(ii)
minp1,…,pN>(Nn/s)′ and 1<p<∞, ω1-p′∈Ap′s/(Nn).
Ifσ∈L∞ satisfies supk∈Z∥σ(2k·)ψ∥Ls2<∞, then Tσ is bounded from Lp1(ω)×⋯×Lpm(ω) to Lp(ω).An improvement of Theorem 1.2 is stated as follows.Theorem B (see [8]).
Let1<p1,p2,…,pN<∞, 1/p1+⋯+1/pN=1/p, and n/2<sj≤n, j=1,…,N. Assume pj>n/sj and ωj∈Apjsj/nfor 1≤j≤N. If σ∈L∞ satisfies supk∈Z∥σ(2k·)ψ∥W(s1,⋯,sN)(RNn)<∞, then Tσ is bounded from Lp1(ω1)×⋯×Lpm(ωN) to Lp(ω), where ω=ω1p/p1⋯ωNp/pN.The first purpose of this paper is to improve Theorem A by usingLr-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 some1<r≤2, suppose that σ∈L∞(ℝmn) and Ψ∈𝒮1(ℝmn) satisfy, for some mn/r<γ≤mn,
(6)supk∈ℤ∥σ(2k·)Ψ^∥Lγr(ℝmn)=K<∞.
Ifp1,…,pm, γ, and the weights ω satisfy one of the following two conditions: (i)
min{p1,…,pm}>mn/γ and ω∈Amin{p1γ/(mn),…,pmγ/(mn)},(ii)
min{p1,…,pm}>(mn/γ)′, 1<p<∞, and ω1-p′∈Ap′γ/(mn),then there is a number δ=δ(mn,γ,r) satisfying 0<δ≤r-1, such that the m-linear operator Tσ, associated with the multiplier σ, is bounded from Lp1(ω)×⋯×Lpm(ω) to Lp(ω), whenever r-δ<pj<∞ for all j=1,…,m, and p is given by 1/p=1/p1+⋯+1/pm.Theorem 2.
Let1<p1,…,pm<2 and let s1>n/p1,…,sm>n/pm and s1+⋯+sm < n/p1+⋯+n/pm+1. If σ∈L∞(ℝmn) satisfies supk∈ℤ∥σ(2k·)ψ∥Ws1,…,sm(ℝmn)<∞, then Tσ is bounded from Lq1(w1q1)×⋯×Lqm(wmqm) to Lq(wq), whenever 1<q1,…,qm<∞, 1/q1+⋯+1/qm=1/q, and (w1q1,…,wmqm)∈(Aq1,…,Aqm) with w=w1⋯wm.


## 2. The Proof of Theorem1

In this section we discuss the proof of Theorem1. We begin with some definitions for maximal operators. Throughout the paper, M denotes the Hardy-Littlewood maximal operator defined by
(7)M(f)(x)=supx∈Q1|Q|∫Q‍|f(y)|dy,
where Q moves over all cubes containing x. For δ>0, Mδ is the maximal function defined by
(8)Mδf(x)=M(|f|δ)1/δ(x)=(supQ∋x1|Q|∫Q‍|f(y)|δdy)1/δ.
In addition, M♯ is the sharp maximal function of Fefferman and Stein:
(9)M♯f(x)=supQ∋xinfc1|Q|∫Q‍|f(y)-c|dy≈supQ∋x1|Q|∫Q‍|f(y)-fQ|dy,
where fQ denotes the average of f over Q and a variant of M♯ is given by
(10)Mδ♯f(x)=M♯(|f|δ)1/δ(x).We prepare some lemmas which will be used later.Lemma 3 (see [11]).
Let1<p<∞ and ω∈Ap. Then (1)ω1-p′∈Ap′; (2)there exists q<p such that ω∈Aq.Lemma 4 (see [12]).
Let1<p,q<∞, and ω∈Ap. Then there exist positive finite constants C(p,q) such that
(11)∥{∑k∈ℤ‍|M(fk)|q}1/q∥Lp(ω)≤C(p,q)∥{∑k∈ℤ‍|fk|q}1/q∥Lp(ω)
for all sequences {fk}k∈ℤ of locally integrable functions on ℝn.Lemma 5.
LetΔk be the Littlewood-Paley operator given by Δk(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|<2b}, 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 Lp(ω) functions f one has
(12)∥f∥Lp(ω)≤c∥(∑k∈ℤ‍|Δk(f)|2)1/2∥Lp(ω).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,
(13)|f(x)|=limt→0|Φt*f(x)|≤supt>0|Φt*f(x)|.
If ω∈A∞, the weighted Hardy space Hp(ω) coincides with the weighted Triebel-Lizorkin space F˙0p,2 for 0<p<∞. Hence, if ω∈A∞, we have
(14)∥f∥Lp(ω)≤∥supt>0|Φt*f(x)|∥Lp(ω)=c∥f∥Hp(ω)≈∥f∥F˙0p,2(ω)≤c∥(∑k∈ℤ‍|Δk(f)|2)1/2∥Lp(ω).
The proof is complete.Now we give the proof of Theorem1.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 Rj be the set of points (ξ1,…,ξm) in (ℝn)m such that |ξj|=max{|ξ1|,…,|ξm|} and we introduce nonnegative smooth functions ϕj on [0,∞)m-1 that are supported in [0,11/10]m-1 such that
(15)1=∑j=1m‍ϕj(|ξ1||ξj|,…,|ξj||ξj|^,…,|ξm||ξj|)
for all (ξ1,…,ξm)≠0, with the understanding that the variable with the hat is missing. These functions introduce a partition of unity of (ℝn)m∖{0} subordinate to a conical neighborhood of the region Rj.
Each regionRj can be written as the union of sets:
(16)Rj,k={(ξ1,…,ξm)∈Rj:|ξk|≥|ξs|∀s≠j}
with k=1,…,m. We need to work with a finer partition of unity, subordinate to each Rj,k. To achieve this, for each j, we introduce smooth functions ϕj,k on [0,∞)m-2 supported in [0,11/10]m-2 such that
(17)1=∑k=1k≠jm‍ϕj,k(|ξ1||ξk|,…,|ξk||ξk|^,…,|ξj||ξk|^,…,|ξm||ξk|)
for all (ξ1,…,ξm) in the support of ϕj with ξk≠0.
We now have obtained the following partition of unity of(ℝn)m∖{0}:
(18)1=∑j=1m‍∑k=1k≠jm‍ϕj(⋯)ϕj,k(⋯),
where the dots indicate the variables of each function.
We now introduce a nonnegative smooth bumpψ 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:
(19)σ=∑j=1m‍∑k=1k≠jm‍[σΦj,k+σΨj,k],
where
(20)Φj,k(ξ1,…,ξm)=ϕj(⋯)ϕj,k(⋯)(1-ψ(|ξk||ξj|)),Ψj,k(ξ1,…,ξm)=ϕj(⋯)ϕj,k(⋯)ψ(|ξk||ξj|).
We will prove the required assertion for each piece of this decomposition, that is, for the multipliers σΦj,k and σΨj,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=σΦm,m-1 and σ2=σΨm,m-1. These correspond to the m-linear operators Tσ1 and Tσ2, respectively.
We first prove Theorem1 under assumption (i). Since 1≤mn/γ<min{r,p1,…,pm}, we can take ρ such that 1≤mn/γ<ρ<min{r,p1,…,pm} and ω∈Amin{p1/ρ,…,pm/ρ}. We first consider Tσ1(f1,…,fm), where fj 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
(21)∑j∈Z‍η^(2-jξ)=1,ξ∈ℝn∖{0}.
Associated with η we define the Littlewood-Paley operator Δj(f)=f*η2-j, where ηt(x)=t-nη(t-1x) for t>0. We also define an operator Sj by setting
(22)Sj(g)=g*ζ2-j,
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
(23)∥Tσ1(f1,…,fm)∥Lp(ω)≤C∥[∑j‍|Tσ1(Sj(f1),…,Sj(fm-1),Δj(fm))|2]1/2∥Lp(ω).
We will use the following estimate forTσ1 (see [6, page 145]):
(24)|Tσ1(Sj(f1),…,Sj(fm-1),Δj(fm))|≤CK∏i=1m-1(M(M(fi)ρ))1/ρ(M(|Δj(fm)|ρ))1/ρ.
We now square the previous expression, we sum overj∈ℤ, and we take square roots. Since r-δ=ρ, the hypothesis pj>r-δ implies pj>ρ, and thus each term (M(M(fi)ρ))1/ρ is bounded on Lpj(ω). We obtain
(25)∥Tσ1(f1,…,fm-1,fm)∥Lp(ω)≤CK∥{∑j‍|Tσ1(Δj(fm))Sj(f1),…,Sj(fm-1),Δj(fm))|2{∑j‍}1/2∥Lp(ω)≤C′K∥{∑jM(|Δj(fm)|ρ)2/ρ}1/2∥Lpm(ω)×∏i=1m-1∥(M(M(fi)ρ))1/ρ∥Lpi(ω)≤C′′K∥{∑jM(|Δj(fm)|ρ)2/ρ}ρ/2∥Lpm/ρ(ω)1/ρ∏i=1m-1∥fi∥Lpi(ω)≤C′′K∏i=1m∥fi∥Lpi(ω),
where the last step holds due to Lemma 4 with q=2/ρ and the weighted Littlewood-Paley theorem.
Next we deal withσ2. Following [6, page 146], we write
(26)Tσ2(f1,…,fm-1,fm)=∑j∈ℤ‍Tσ2(Sj′(f1),…,Sj′(fm-2),Δj′(fm-1),Δj(fm)),|Tσ2(Sj′(f1),…,Sj′(fm-2),Δj′(fm-1),Δj(fm))|≤CK∏i=1m-2‍(M(M(fi)ρ))1/ρ(M(|Δj′(fm-1)|ρ))1/ρ×(M(|Δj(fm)|ρ))1/ρ,
for some other Littlewood-Paley operator Δj′ 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∈ℝn:(24/25)·(1/10m)≤|ξ|≤4} and vanishes on a larger annulus. Also, Sj′ is given by convolution with ζ2-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 overj and taking Lp(ω) norms yield
(27)∥Tσ2(f1,…,fm-1,fm)∥Lp(ω)≤CK∥∏i=1m-2‍(M(M(fi)ρ))1/ρ∑j∈ℤ‍(M(|Δj′(fm-1)|ρ))1/ρ×(M(|Δj(fm)|ρ))1/ρ∏i=1m-2‍(M(M(fi)ρ))1/ρ∑j∈ℤ‍(M(|Δj′(fm-1)|ρ))1/ρ∥Lp(ω)≤CK∥×{∏i=m-1m‍∑j∈ℤ‍|M(|Δj(fi)|ρ)|2/ρ}1/2∏i=1m-2‍(M(M(fi)ρ))1/ρ×{∏i=m-1m‍∑j∈ℤ‍|M(|Δj(fi)|ρ)|2/ρ}1/2∥Lp(ω),
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
(28)C′K∥f1∥Lp1(ω)⋯∥fm∥Lpm(ω).
We next prove Theorem1 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,…,ξm)=σ(ξ1,…,ξm-1,-(ξ1+⋯+ξm)). To prove the required assertion, by duality, it is enough to prove that Tσ1*m and Tσ2*m are bounded from Lp1(ω)×⋯×Lpm-1(ω)×Lp′(ω1-p′) to Lpm′(ω1-pm′). We may assume that pm=min{p1,…,pm}. Since pm<(mn/γ)′, we see 1/p′, 1/pk<1/p′+1/p2+⋯+1/pm=1/pm′<γ/(mn). Hence, mn/γ<min{r,p′,p1,…,pm-1}. Since p<pm and ω1-p′∈Ap′γ/(mn)⊂Ap′, we deduce that ω∈Ap⊂Apm; then ω1-pm′∈Apm′. It is obvious that ω(1-pm′)/pm′=ω-1/pω1/p1⋯ω1/pm. Since pm<pk, 1/p=1/p1+⋯+1/pk+⋯+1/pm-1>1/pm+1/pk≥2/pk. That is, p<pk/2; then ω∈Ap⊂Apk/2⊂Apkγ/(mn). Therefore, we take a positive number ρ such that 1≤mn/γ<ρ<min{r,p′,p1,…,pm-1}, and γ>mn/ρ such that ω1-p′∈Ap′/ρ and ω∈Apk/ρ. We have
(29)∥Tσ1*m(f1,…,fm-1,fm)∥Lpm′(ω1-pm′)≤CK∥{∑j|Tσ1(Δj(fm))Sj(f1),…,Sj(fm-1),‍Δj(fm))|2{∑j|Tσ1(Δj(fm))Sj(f1),…,Sj(fm-1),‍}1/2∥Lpm′(ω1-pm′)≤C′K∥{∑j‍M(|Δj(fm)|ρ)2/ρ}1/2∥Lp′(ω1-p′)×∏i=1m-1‍∥(M(M(fi)ρ))1/ρ∥Lpi(ω)≤C′′K∥{∑j‍M(|Δj(fm)|ρ)2/ρ}ρ/2∥Lp′/ρ(ω1-p′)1/ρ×∏i=1m-1‍∥fi∥Lpi(ω)≤C′K∥f1∥Lp1(ω)⋯∥fm∥Lpm(ω).
Similarly, we have
(30)∥Tσ2*m(f1,…,fm-1,fm)∥Lpm′(ω1-pm′)≤CK∥∏i=1m-2‍(M(M(fi)ρ))1/ρω1/pi×∑j∈ℤ(M(|Δj′(fm-1)|ρ))1/ρ‍×(M(|Δj(fm)|ρ))1/ρω-1/p+1/pm-1∥∏i=1m-2‍(M(M(fi)ρ))1/ρω1/pi∥Lpm′(ω1-pm′)≤CK∥∏i=1m-2‍(M(M(fi)ρ))1/ρ∥Lpi(ωi)×∥{∑j∈ℤ‍|M(|Δj(fm-1)|ρ)|2/ρ}ρ/2∥Lpm-1/ρ(ω)×∥{∑j∈ℤ‍|M(|Δj(fm)|ρ)|2/ρ}ρ/2∥Lp′/ρ(ω1-p′)≤C′K∥f1∥Lp1(ω)⋯∥fm∥Lpm(ω).
This concludes the proof of Theorem1.


## 3. The Proof of Theorem2

We begin with some lemmas which will be used in the proof of Theorem2.Lemma 6 (see [11]).
Let0<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
(31)∫Rn(Mδf(x))pω(x)dx≤C∫Rn(Mδ♯f(x))pω(x)dx,
for all function f for which the left-hand side is finite.Lemma 7 (see [13]).
Let0<p1,p2,p≤∞, and 1/p1+1/p2=1/p. Let σ be a multiplier satisfying supk∈ℤ∥σ(2k·)ψ∥Ws1,s2(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 Tσ is bounded from Hp1(ℝn)×Hp2(ℝn) to Lp(ℝn).Remark 8.
It should be pointed out that Lemma7 can be extended to the case m≥3.Lemma 9 (see [8]).
Letr>0, q1,…,qm∈[2,∞), and s1,…,sm≥0. Then there exists a constant C>0 such that
(32)(×〈ξ2〉s2dξ2(∫ℝn‍|F^(ξ1,…,ξm)|q1〈ξ1〉s1dξ1)q2/q1)q3/q2∫ℝn‍⋯((∫ℝn‍|F^(ξ1,…,ξm)|q1〈ξ1〉s1dξ1)q2/q1×〈ξ2〉s2dξ2(∫ℝn‍|F^(ξ1,…,ξm)|q1〈ξ1〉s1dξ1)q2/q1)q3/q2⋯〈ξm〉smdξm∫ℝn‍⋯)1/qm≤C∥F∥Ws1/q1,…,sm/qm(ℝmn)for allF∈Ws1/q1,…,sm/qm(ℝmn) with suppF⊂{|x1|2+⋯+|xm|2≤r}.Next, we give a pointwise control ofMδ♯Tσ(f→) which becomes very useful in the proof of Theorem 2.Lemma 10.
Let1<p1,…,pm<2. Assume that σ∈L∞(ℝmn) which satisfies supk∈ℤ∥σ(2k·)ψ∥<∞Ws1,…,sm(ℝmn) for s1>n/p1,…,sm>n/pm and s1+⋯+sm < n/p1 + ⋯ + n/pm + 1. For any 0<δ<1/m, one has Mδ♯Tσ(f→)(x)≤C∏j=1m‍Mpjfj(x).Proof.
For simplicity, we only prove for the casem=2, since there is no essential difference for the general case. Fix an x∈ℝn and a cube Q with side length l, such that x∈Q. Let fi=fi0+fi∞, where fi0=fiχQ*  and fi∞=fiχ(Q*)c for i=1,2 and Q*=4nQ. Since 0<δ<1/2, we have
(33)(1|Q|∫Q‍||Tσ(f1,f2)(z)|δ-|C|δ|dz)1/δ≤C(1|Q|∫Q‍|Tσ(f1,f2)(z)-C|δdz)1/δ≤C(1|Q|∫Q‍|Tσ(f10,f20)(z)+Tσ(f1∞,f2∞)(z)+Tσ(f10,f2∞)(z)+Tσ(f10,f20)(z)-C|δdz(1|Q|∫Q‍)1/δ≤C(1|Q|∫Q‍|Tσ(f10,f20)(z)|δdz)1/δ+(1|Q|∫Q‍|Tσ(f1∞,f2∞)(z)+Tσ(f10,f2∞)(z)+Tσ(f1∞,f20)(z)-C|δdz(1|Q|∫Q‍)1/δ=U1+U2.
We first considerU1. By Kolmogorov’s inequality, Hölder’s inequality, and Lemma 7, we have
(34)(1|Q|∫Q‍|Tσ(f10,f20)(z)|δdz)1/δ≤C∥Tσ(f10,f20)∥Lp,∞(Q,dx/|Q|)≤C∏j=12‍(1|Q*|∫Q*‍|fj(z)|pjdz)1/pj≤C∏j=12‍Mpjfj(x),
where1/p=1/p1+1/p2 with p>δ and 1<p1, p2<∞.
Next we deal withU2. We choose C=∑i=13‍Ci, where
(35)C1=Tσ(f1∞,f2∞)(x),C2=Tσ(f10,f2∞)(x),C3=Tσ(f1∞,f20)(x).
We may splitU2 as U2≤U21+U22+U23, where
(36)U21=(1|Q|∫Q‍|Tσ(f1∞,f2∞)(z)-Tσ(f1∞,f2∞)(x)|δdz)1/δ,U22=(1|Q|∫Q‍|Tσ(f10,f2∞)(z)-Tσ(f10,f2∞)(x)|δdz)1/δ,U23=(1|Q|∫Q‍|Tσ(f1∞,f20)(z)-Tσ(f1∞,f20)(x)|δdz)1/δ.
Now we estimateU21 first. We decompose σ as
(37)σ=∑j∈ℤ‍σ(·)ψ(·/2j)≐∑j∈ℤ‍σj.
Letσj=σ(·)ψ(·/2j), where ψ∈𝒮(ℝ2n) with supp ψ⊂{ξ∈ℝ2n:1/2≤|ξ|≤2} and ∑j∈ℤ‍ψ(2-jξ)=1, ξ≠0. Thus, we have(38)|Tσ(f1∞,f2∞)(z)-Tσ(f1∞,f2∞)(x)|≤C∑j∈ℤ‍|Tσj(f1∞,f2∞)(z)-Tσj(f1∞,f2∞)(x)|≤C∑j∈ℤ‍∑k1=0∞‍∑k2=0∞‍∫2k2+1Q*∖2k2Q*‍∫2k1+1Q*∖2k1Q*|σj∨(z-y1,z-y2)-σj∨(x-y1,x-y2)||f1(y1)f2(y2)|dy1dy2‍≤C∑k1=0∞‍∑k2=0∞‍∑j∈ℤ‍∫2k2+1Q*∖2k2Q*‍∫2k1+1Q*∖2k1Q*|σj∨(z-y1,z-y2)-σj∨(x-y1,x-y2)|‍|f1(y1)f2(y2)|dy1dy2≐∑k1=0∞‍∑k2=0∞‍Ik1,k2.
Applying Hölder’s inequality we have(39)Ik1,k2=∑j∈ℤ‍∫2k2+1Q*∖2k2Q*‍∫2k1+1Q*∖2k1Q*|σj∨(z-y1,z-y2)-σj∨(x-y1,x-y2)|‍|f1(y1)f2(y2)|dy1dy2≤C∑j∈ℤ‍(∫2k2+1Q*∖2k2Q*‍(∫2k1+1Q*∖2k1Q*|σj∨(z-y1,z-y2)-σj∨(x-y1,x-y2)|p1′dy1)p2′/p1′dy2)1/p2′×(∫2k1+1Q*‍|f1(y1)|p1dy1)1/p1(∫2k2+1Q*‍|f2(y2)|p2dy2)1/p2≐C∑j∈ℤ‍Ik1,k2,j×(∫2k2+1Q*‍|f1(y1)|p1dy1)1/p1(∫2k2+1Q*‍|f2(y2)|p2dy2)1/p2.
Leth=z-x and Q~=x-Q*. Then we have(40)Ik1,k2,j=(∫2k2+1Q*∖2k2Q*(∫2k1+1Q*∖2k1Q*|σj∨(z-y1,z-y2)-σj∨(x-y1,x-y2)|p1′dy1)p2′/p1′dy2)1/p2′≤C(∫2k2+1Q~∖2k2Q~(∫2k1+1Q~∖2k1Q~|σj∨(h+y1,h+y2)-σj∨(y1,y2)|p1′‍dy1)p2′/p1′‍dy2)1/p2′≤C(∫2k2+1Q~∖2k2Q~(∫2k1+1Q~∖2k1Q~|σj∨(y1,y2)|p1′‍dy1)p2′/p1′‍dy2)1/p2′≤C(2k1l)-s1(2k2l)-s2(×y1dy1∫ℝn‍)p2′/p1′)p2′/p1′∫2k2+1Q~∖2k2Q~‍(×s1p1′/22-jndy1∫ℝn‍)p2′/p1′∫2k1+1Q*∖2k1Q~‍|σj∨(y1,y2)|p1′×(1+|y1|2)s1p1′/2dy1×s1p1′/22-jndy1∫ℝn‍)p2′/p1′)p2′/p1′(1+|y2|2)s2p2′/2dy2)1/p2′≤C(2k1l)-s1(2k2l)-s2(×y1dy1∫ℝn‍)p2′/p1′)p2′/p1′∫ℝn‍(×s1p1′/22-jndy1∫ℝn‍)p2′/p1′∫ℝn‍|σj∨(2-jy1,2-jy2)|p1′×(1+|2-jy1|2)s1p1′/22-jndy1×y1dy1∫ℝn‍)p2′/p1′)p2′/p1′(1+|2-jy2|2)s2p2′/22-jndy2∫ℝn‍)1/p2′≤C(2k1l)-s1(2k2l)-s22-j(s1+s2)2jn((1/p1)+(1/p2))×(∫ℝn(∫ℝn‍|2-2jnσj∨(2-jy1,2-jy2)|p1′(1+|y1|2)s1p1′/2dy1)p2′/p1′(1+|y2|2)s2p2′/2dy2‍)1/p2′≤C(2k1l)-s1(2k2l)-s22-j(s1+s2)2jn((1/p1)+(1/p2))∥σ(2j·)ψ∥Ws1,s2,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
(41)∑j≥R‍Ik1,k2,j≤supj∥σ(2j·)ψ∥Ws1,s2×∑j≥R‍(2k1l)-s1(2k2l)-s22j(n/p1+n/p2-s1-s2)≤Csupj∥σ(2j·)ψ∥Ws1,s22-k1s12-k2s2l-n/p1l-n/p2.
On the other hand(42)Ik1,k2,j≤(∫2k2+1Q~∖2k2Q~‍(∫2k1+1Q~∖2k1Q~|σj∨(y1+h,y2+h)-σj∨(y1,y2)|p1′dy1∫∫2k2+1Q~∖2k2Q~‍2k2+1Q~∖2k2Q~‍)p2′/p1′dy2)1/p2′≤C(∫2k2+1Q~∖2k2Q~‍(∫2k1+1Q~∖2k1Q~‍(∫01‍|h→·∇(σj∨)(y1+θh,y2+θh)|dθ)p1′dy1)p2′/p1′dy2)1/p2′≤C∫01‍(dy1∫2k2+1Q~∖2k2Q~‍)(p2′/p1′)dy2)∫2k2+1Q~∖2k2Q~‍(∫2k1+1Q~∖2k1Q~‍|h→·∇(σj∨)(y1+θh,y2+θh)|p1′dy1)p2′/p1′dy2)1/p2′dθ≤C(∫2k2+1Q~∖2k2Q~(∫2k1+1Q~∖2k1Q~|h→·∇(σj∨)(y1,y2)|p1′dy1∫∫2k1+1Q~∖2k1Q~‍2k1+1Q~∖2k1Q~‍∫2k2+1Q~∖2k2Q~‍‍)p2′/p1′dy2)1/p2′,
whereh→=(h,h)∈ℝ2n. Since h→·∇(σj∨)(y1,y2)=∑r=12n‍hr∂r(σj∨)(y1,y2), we have(43)Ik1,k2,j≤C∑r=12n‍l(∫2k2+1Q~∖2k2Q~‍(∫2k1+1Q~∖2k1Q~‍|∂r(σj∨)(y1,y2)|p1′dy1)p2′/p1′dy2)1/p2′Ik1,k2,j≤C∑r=12n‍l(2k1l)-s1(2k2l)-s2(∫ℝn‍(∫ℝn|∂r(σj∨)(2-jy1,2-jy2)|p1′(1+|2-jy2|2)s1p1′/22-jndy1‍)p2′/p1′Ik1,k2,j×(1+|2-jy2|2)s2p2′/22-jndy2(∫ℝn|∂r(σj∨)(2-jy1,2-jy2)|p1′)p2′/p1′)1/p2′≤C∑r=12n‍l(2k1l)-s1(2k2l)-s2(∫ℝn(∫ℝn|2-2jn∂r(σj∨)(2-jy1,2-jy2)|p1′(1+|y1|2)s1p1′/2dy1)p2′/p1′×(1+|y2|2)s2p2′/2dy2(∫ℝn|∂r(σj∨)(2-jy1,2-jy2)|p1′)p2′/p1′)1/p2′2-j(s1+s2)2jn((1/p1)+(1/p2))≤C∑r=12n‍l(2k1l)-s1(2k2l)-s22-j(s1+s2)2jn((1/p1)+(1/p2))2j∥σ(2j·)ψ∥Ws1,s2,where in the last inequality Lemma 9 was used again and hence
(44)∑j<R‍Ik1,k2,j≤Csupj∥σ(2j·)ψ∥Ws1,s22-k1s12-k2s2l-np1l-n/p2.
Combining the above arguments we have(45)|Tσ(f1∞,f2∞)(z)-Tσ(f1∞,f2∞)(x)|≤C∑k1=0∞‍∑k2=0∞‍2-k1s12-k2s2l-n/p1l-n/p2×(∫2k1+1Q*‍|f1(y1)|p1dy1)1/p1×(∫2k2+1Q*‍|f2(y2)|p2dy2)1/p2≤C∑k1=0∞‍2-k1(s1-n/p1)×∑k2=0∞‍2-k2(s2-n/p2)Mp1f1Mp2f2≤CMp1f1Mp2f2.
Thus, we obtainU21≤CMp1f1Mp2f2. What remain to be considered are U22 and U23. We just estimate U22 since the same arguments can be applied to U23:(46)|Tσ(f10,f2∞)(z)-Tσ(f10,f2∞)(x)|≤C∑j‍|Tσj(f10,f2∞)(z)-Tσj(f10,f2∞)(x)|≤C∑j‍∑k2=0∞‍∫2k2+1Q*‍|f2(y2)|∫Q*‍|σj∨(z-y1,z-y2)-σj∨(x-y1,x-y2)||f1(y1)|dy1dy2≤C∑j∈ℤ‍∑k2=0∞‍(∫2k2+1Q*∖2k2Q*(∫Q*‍|σj∨(z-y1,z-y2)-σj∨(x-y1,x-y2)|p1′dy1)p2′/p1′dy2)1/p2′×(∫Q*‍|f1(y1)|p1dy1)1/p1(∫2k2+1Q*‍|f2(y2)|p2dy2)1/p2.
Then by similar arguments as the above mentioned we get that(47)|Tσ(f10,f2∞)(z)-Tσ(f10,f2∞)(x)|≤C∑k2=0∞‍2-k2s2l-n/p1l-n/p2(∫Q*‍|f1(y1)|p1dy1)1/p1×(∫2k2+1Q*‍|f2(y2)|p2dy2)1/p2≤C∑k2=0∞‍2-k2(s2-p2)Mp1f1Mp2f2≤CMp1f1Mp2f2.
The proof of Lemma10 is complete.Now we are ready to give the proof of Theorem2.Proof.
By Lemma3, we can choose 1<p1<q1 and 1<p2<q2 such that ω1q1∈Aq1/p1 and ω2q2∈Aq2/p2. Then by the Hölder inequality, Lemma 10, and the weighted boundedness of M, we deduce that
(48)∥Tσ(f1,f2)∥Lq(wq)≤∥MδTσ(f1,f2)∥Lq(wq)≤C∥Mδ♯Tσ(f1,f2)∥Lq(wq)≤C∥Mp1f1Mp2f2∥Lq(wq)≤C∥Mp1f1∥Lq1(w1q1)∥Mp1f1∥Lq2(w2q2)≤C∥f1∥Lq1(w1q1)∥f2∥Lq2(w2q2).
The proof of Theorem2 is complete.



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*Source: 987205-2013-12-08.xml*