# A Remark Related Pseudo-Differential Operators on Multi-Parameter Besov-Lipschitz and Triebel-Lizorkin Spaces

Authors: Chengdan Xu
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Let a symbolσx,ξ belongs to the class S1,1,δ1,δ20,0 with x,ξ∈ℝn1×ℝn2 and 0≤δ1,δ2<1. This paper mainly establish the boundedness of the class of all bi-parameter pseudo-differential operators associated with the symbol σx,ξ on the multi-parameter non-homogeneous Besov-Lipschitz and Triebel-Lizorkin spaces.


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## Body

## 1. Introduction

In this paper, we mainly consider the bi-parameter pseudo-differential operator(1)Tσfx1,x2=12πn1+n2∫ℝn1×ℝn2σx1,x2,ξ1,ξ2eix1·ξ1+x2·ξ2f^ξ1,ξ2dξ1dξ2for f∈Sℝn1×ℝn2, where the symbol σ∈BS1,δ0 with δ=δ1,δ2, x=x1,x2, ξ=ξ1,ξ2, and 0≤δi<1, xi∈ℝni, ξi∈ℝni for i=1,2.The pseudo-differential operator, which is formalised in Kohn and Nirenberg [1] and Hörmander [2] in 1960s, have been developed as a popular mathematical tool and widely applied in various fields of mathematics. Over the last few decades, the study of pseudo-differential operator have attracted many related mathematicians and many authors have been working on proving boundednesses of the one-parameter pseudo-differential operators on various classical spaces. It’s worth noting that the boundedness for the class S0,00 was first treated by Calderón-Vaillancourt in [3], which means that the boundedness of all the derivatives of a symbol ensures the L2-boundedness of the corresponding operator. Further, Laptev [4] proved that any pseudo-differential operator with symbols in S1,00 is a standard Calderón-Zygmund operator, hence the Lp1<p<∞-boundedness is obvious. Hörmander [5] pointed out that the pseudo-differential operators with symbols in Sρ,δm are bounded on Lp if and only if m≤nρ−11/p−1/2, where 0≤δ≤ρ≤1, δ<1 and 1<p<∞ (see also Fefferman [6] and Stein [7], Chapter VII, Section 5.12). Sugimoto [8] showed the pseudo-differential operators with symbols in Sρ,δm are bounded on Besov spaces Bp,qs when m≤nρ−11/p−1/2, where 0≤δ≤ρ≤1, δ<1, s∈ℝ and 1<p,q<∞. Furthermore, Jun Park [9] studied the boundedness of pseudo-differential operators with symbols in Sρ,ρm on Triebel-Lizorkin spaces and Besov spaces when m≤nρ−11/p−1/2+s1−s2, where 0<ρ<1, 0<p<∞ and 0<q≤∞. Some weighted results about boundednesses of pseudo-differential operators also have been verified, such as in [10–12] and so on.On the other hand, it’s well-known that there are two different directions being extended to developing the Calderón-Zygmund singular integral theory. One of the two extensions is non-convolution operators with a kernel, such as the Cauchy-operator on Lipschitz curves and the interested reader can see [13–16] for more reference. The other extension is due to R. Fefferman and E. M. Stein, they extended the classical singular integrals into the product situation in [17]. Later, Journé have unified these two different directions into a certain extent in [18], and studied a class of singular integral operators defined on a product of n Euclidean spaces, the reader can see [16, 19–23] for further information on this topic. For the theory of multi-parameter singular integrals and function spaces plays an important role in many aspects of harmonic analysis, many authors have studied the pure product theory (which is also called multi-parameter harmonic analysis) over the last decades. For instance, the Lp estimates for multi-parameter Coifman-Meyer Fourier multipliers in [24]; the Lp estimates and Calderón-Vaillancourt type theorem for multi-parameter and multi-linear Fourier multipliers and pseudo-differential operators in [25, 26]. In particular, Sugimoto [27] showed that any pseudo-differential operators with symbols in Besov space B∞,11/2,1/2ℝn×ℝn implies the L2-boundedness. Furthermore, Tomita [28] generalized the results in [27] and studied the Lp-boundedness of pseudo-differential operators with non-regular symbols, where 1<p<∞. As some applications, Tomita [28] also obtained the Lp-boundedness for symbols in Besov spaces and modulation spaces, respectively. Recently, the properties of multi-parameter Besov-Lipschitz and Triebel-Lizorkin spaces and the boundedness of multi-parameter singular integral operators on them have been established in [29–32].Generally speaking, the pseudo-differential operators can be regarded as an extension for Fourier multipliers. On the other hand, theα-modulation spaces Mp,qs,αℝn, which were introduced by Gröbner’s PH.D. thesis [33], function as a kind of intermediate space between modulation spaces and the well-known Besov spaces. Here the parameter α∈0,1 works as the “adjuster”, modulation spaces Mp,qsℝn are the special α-modulation spaces when α=0 and Besov spaces Bp,qsℝn can be treated as the limit case of α-modulation spaces when the parameter α tends to 1. The difference between modulation spaces Mp,qsℝn and Besov spaces Bp,qsℝn can be referred to Tomita [28]. We also remark that the boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces have been obtained in [34]. The author in this paper and others have introduced the bi-parameter α-modulation spaces with 0≤α<1 and bi-parameter modulation spaces in [35], and established the boundedness of bi-parameter pseudo-differential operators on these spaces. While not much is known about the boundedness of pseudo-differential operators on multi-parameter Besov-Lipschitz and Triebel-Lizorkin spaces:The main purpose of this paper is to investigate the boundedness of bi-parameter pseudo-differential operators on the bi-parameter non-homogeneous Besov spaces and the bi-parameter non-homogeneous Triebel spaces. Let us recall the definition of the bi-parameter pseudo-differential operatorsTσ as the following.Form=m1,m2,ρ=ρ1,ρ2, δ=δ1,δ2 be real pairs, where ρi,δi∈0,1 with i=1,2. A function σx1,x2,ξ1,ξ2∈C∞ℝ2n1×ℝ2n2 is said to belong to BSρ,δm if there exists a positive constant Cα1,α2,β1,β2 such that the following inequality
(2)∂x1α1∂x2α2∂ξ1β1∂ξ2β2σx1,x2,ξ1,ξ2≤Cα1,α2,β1,β21+ξ1m1+δ1α1−ρ1β11+ξ2m2+δ2α2−ρ2β2holds for all multi-indices α1,α2,β1,β2, where x=x1,x2∈ℝn1×ℝn2, ξ=ξ1,ξ2∈ℝn1×ℝn2.Denote the class of all bi-parameter pseudo-differential operatorsTσ with σ∈BSρ,δm by OPBSρ,δm,
(3)Tσfx1,x2=12πn1+n2∫ℝn1×ℝn2σx1,x2,ξ1,ξ2eix1·ξ1+x2·ξ2f^ξ1,ξ2dξ1dξ2for f∈Sℝn1×ℝn2, where x=x1,x2∈ℝn1×ℝn2, ξ=ξ1,ξ2∈ℝn1×ℝn2.Next we give the main results about the boundedness of bi-parameter pseudo-differential operators on the bi-parameter non-homogeneous Besov spaces and the bi-parameter non-homogeneous Triebel spaces.Theorem 1.
Let0<p,q<∞, s=s1,s2 and δ=δ1,δ2 with si∈ℝ and 0≤δi<1, i=1,2. Then the bi-parameter pseudo-differential operators Tσ defined in the form of (3) with symbols σ∈BS1,δ0 is bounded on the bi-parameter non-homogeneous Besov-Lipschitz Bp,qsℝn1×ℝn2. That is, there exists a positive constant C>0 such that
(4)TσfBp,qs≤CfBp,qsholds for all f∈Sℝn1×ℝn2 and σ∈BS1,δ0.Theorem 2.
Let0<p,q<∞, s=s1,s2 and δ=δ1,δ2 with si∈ℝ and 0≤δi<1, i=1,2. Then the bi-parameter pseudo-differential operators Tσ defined in the form of (3) with symbols σ∈BS1,δ0 is bounded on the bi-parameter non-homogeneous Triebel-Lizorkin spaces Fp,qsℝn1×ℝn2. That is, there exists a positive constant C>0 such that
(5)TσfFp,qs≤CfFp,qsholds for all f∈Sℝn1×ℝn2 and σ∈BS1,δ0.The contents of this paper are organized as follows: In Section2, we state some preliminary notation that will be used throughout this paper and introduce the bi-parameter non-homogeneous Besov-Lipschitz spaces and the bi-parameter non-homogeneous Triebel-Lizorkin spaces. Moveover, we give some necessary lemmas which will serve to prove the main theorems in Section 3. Since it is of importance for its own sake, we will give some self-contained proofs here. We give the detailed proof for Theorem 1 in Section 4 and the detailed proof for Theorem 2 in Section 5.As it was pointed out by one of the referees our work seems to have overlapping with our previous work [35]. The mainly thought to prove the main theorems maybe seem similar. However, we also give some new results, such as some norm estimates related to the bi-parameter pseudo-differential operators with different symbols (see lemma 16, lemma 17 and lemma 18). In particular, the method to prove Theorem 2 is slightly different from the idea to deal with Theorem 1, where we have applied the bi-parameter LΩplq-multiplier estimates (see lemma 19). These new lemmas are key tools in our proofs.


## 2. Preliminary Notation and Definition

In this section, we fix notations and provide some preliminary definitions that will be used in this paper.General notation. We will denote universal positive constants byC which can be different line to line. If A and B are two quantities (typically nonnegative), we use A≲B to denote the statement A≤C′B for some absolute constant C′>0. Also, we use A~B to denote the statement A≲B and B≲A.LetSℝn and S∞ℝn denote the Schwartz space of rapidly decreasing smooth functions and its dual space of tempered distributions, respectively, with their usual topologies. Write ℤ+n≔ℤ+×ℤ+×⋯×ℤ+, for any multi-index α=α1,α2,⋯,αn∈ℤ+n, denote ∂α=∂α1∂α2⋯∂αn and α≔α1+α2+⋯+αn.For0<p<∞, the bi-parameter Lebesgue space Lp=Lpℝn1×ℝn2 is equipped with the (quasi)-norm
(6)fLp=∫ℝn1×ℝn2fx1,x2pdx1dx21/p.Whenp=∞,
(7)fL∞=esssupx1∈ℝn1x2∈ℝn2fx1,x2.For0<q≤∞, we denote by lq=lqℤ+×ℤ+ the set of all complex number sequences aj1,j2j1∈ℤ+j2∈ℤ+ such that
(8)aj1,j2lq=∑j1=0∞∑j2=0∞aj1,j2q1/q,with the usual modification for q=∞.Through out the following paper, we denote·p=·Lpℝn1×ℝn2 and ·lq=·lqℤ+×ℤ+, ·Lplq=·lqLp and ·lqLp=·Lplq. We use F denote the Fourier transform and F−1 denote the inverse Fourier transform. That is,
(9)Ffξ1,ξ2≔f^ξ1,ξ2=∫ℝn1×ℝn2fx1,x2e−ix1·ξ1+x2·ξ2dx1dx2F−1fx1,x2≔f∨x1,x2=12πn1+n2∫ℝn1×ℝn2fξ1,ξ2eix1·ξ1+x2·ξ2dξ1dξ2for any x=x1,x2,ξ=ξ1,ξ2∈ℝn1×ℝn2.We also useF1,2 denote the partial Fourier transform of functions with respect to the x1-variable and the x2-variable, and F1,2−1 denote the inverse partial Fourier transform of functions with respect to the x1-variable and the x2-variable. That is,
(10)F1,2σy1,y2,ξ1,ξ2≔∫ℝn1×ℝn2σx1,x2,ξ1,ξ2e−ix1·y1+x2·y2dx1dx2F1,2−1σx1,x2,ξ1,ξ2≔12πn1+n2∫ℝn1×ℝn2σy1,y2,ξ1,ξ2eix1·y1+x2·y2dy1dy2for any x=x1,x2,y=y1,y2,ξ=ξ1,ξ2∈ℝn1×ℝn2.A class of Fourier multiplier operators can be stated as follow:(11)mDf=F−1mFf=F−1mFf.In order to introduce the definition of bi-parameter non-homogeneous Besov-Lipschitz spaces and non-homogeneous Triebel-Lizorkin spaces, we first recall the definition of the test function which is defined onℝn1×ℝn2.LetSℝn1×ℝn2 denote the space of Schwartz function and S∞ℝn1×ℝn2 be its dual. The test function ηx1,x2 defined on ℝn1×ℝn2 can be given by
(12)ηx1,x2=η1x1η2x2,where ηixi∈Aℝni, i=1,2. Set
(13)ηj1,j2x1,x2=ηj11x1ηj22x2.Definition 3.
LetAℝni be the collection of all the systems ηi=ηjiξiji=0∞⊆SRni that satisfy the following conditions:
(i)
suppη0⊂ξi∈ℝni:ξi≤2, supp ηji⊂ξi∈ℝni:2ji−1≤ξi≤2ji+1 for ji=1,2,3,⋯, i=1,2;(ii)
∑ji=0∞ηjiξi=1 for all ξi∈ℝni, i=1,2;(iii)
There exists a positive numberCαi such that the inequality 2jiαi∂αiηjiξi≤Cαi holds for all non-negative integer ji and every multi-index αi and ξi∈ℝni, i=1,2.Definition 4.
Let0<p,q≤∞, s=s1,s2∈ℝ×ℝ. The bi-parameter non-homogeneous Besov-Lipschitz space Bp,qsℝn1×ℝn2 is defined by
(14)Bp,qsℝn1×ℝn2=f∈S∞ℝn1×ℝn2:fBp,qs<∞,where
(15)fBp,qs=∑j1≥0∑j2≥02j1s1q+j2s2qF−1ηj1,j2FfLpq1/q,with a slight modification in the case of p=∞ and q=∞. Here j1,j2 are non-negative integers.Definition 5.
Let0<p,q≤∞, s=s1,s2∈ℝ×ℝ. The bi-parameter non-homogeneous Triebel-Lizorkin space Fp,qsℝn1×ℝn2 is defined by
(16)Fp,qsℝn1×ℝn2=f∈S∞ℝn1×ℝn2:fFp,qs<∞,where
(17)fFp,qs=∑j1≥0∑j2≥02j1s1q+j2s2qF−1ηj1,j2Ffq1/qLp,with a slight modification in the case of p=∞ and q=∞. Here j1,j2 are non-negative integers.It should pointed out that the definition of bi-parameter non-homogeneous Besov-Lipschitz space and the definition of bi-parameter non-homogeneous Triebel-Lizorkin space are independent with the choosing ofη1 and η2. Definition 4 and Definition 5 are identical to definitions of weighted Besov-Lipschitz and Triebel-Lizorkin spaces defined in Sugimoto [7, 27] with the weight ρ=0. We also refer the reader to [29, 32] for the bi-parameter homogeneous cases.For simplicity and convenience, we will chooseη1=η2=η and denote the inverse Fourier transform of η by φ, that is, F−1η=φ. Denote Δj1,j2f=φj1,j2∗f and φtxi=t−niφt−1xi for all t>0, xi∈ℝni, i=1,2. Hence we can deduce F−1ηj1,j2=φj1,j2, φ2−jixi=2jiniφ2jixi for ji=1,2,3,⋯, and
(18)∑j1,j2Δj1,j2=I,where I denotes the identity operator. When the integer ji≥1, we denote φ2−jixi by φjixi for convenience. Hence we have
(19)φj1,j2x1,x2=φj1x1φj2x2(20)φ^j1,j2y1,y2=φ^2−j1y1φ^2−j2y2.For short, the bi-parameter non-homogeneous Besov-Lipschitz space will be called as the bi-parameter Besov space and the bi-parameter non-homogeneous Triebel-Lizorkin space called as the bi-parameter Triebel space in the next. According to the above, the bi-parameter Besov spaceBp,qsℝn1×ℝn2 and the bi-parameter Triebel space Fp,qsℝn1×ℝn2 can also be defined as follows:Definition 6.
Let0<p,q≤∞, s=s1,s2∈ℝ×ℝ. The bi-parameter Besov space Bp,qsℝn1×ℝn2 is defined by
(21)Bp,qsℝn1×ℝn2=f∈S∞ℝn1×ℝn2:fBp,qs<∞,where
(22)fBp,qs=∑j1≥0∑j2≥02j1s1q+j2s2qΔj1,j2fLpq1/q=∑j1≥0∑j2≥02j1s1q+j2s2qφ∧j1,j2Dx1,Dx2fLpq1/q=∑j1≥0∑j2≥02j1s1q+j2s2qφ∧2−j1Dx1φ∧2−j2Dx2fLpq1/q.Definition 7.
Let0<p,q≤∞, s=s1,s2∈ℝ×ℝ. The bi-parameter Triebel space Fp,qsℝn1×ℝn2 is defined by
(23)Fp,qsℝn1×ℝn2=f∈S∞ℝn1×ℝn2:fFp,qs<∞,where
(24)fFp,qs=∑j1≥0∑j2≥02j1s1q+j2s2qΔj1,j2fq1/qLp=∑j1≥0∑j2≥02j1s1q+j2s2qφ∧j1,j2Dx1,Dx2fq1/qLp=∑j1≥0∑j2≥02j1s1q+j2s2qφ∧2−j1Dx1φ∧2−j2Dx2fq1/qLp.According to (18) and (20), the symbol σx1,x2,ξ1,ξ2 can be decomposed as
(25)σx1,x2,ξ1,ξ2=∑l1≥0∑l2≥0σx1,x2,ξ1,ξ2φ^2−l1ξ1φ^2−l2ξ2=∑l1≥0k1≥0∑l2≥0k2≥0φ^2−k1Dx1φ^2−k2Dx2σx1,x2,ξ1,ξ2φ^2−l1ξ1φ^2−l2ξ2,where the sequence φ^2−ki· and φ^2−li· are the decomposition constituting bi-parameter Besov spaces and Triebel space, and
(26)li−ji≤2by the support condition of ηji in Definition 3 for i=1,2. By using Definition 6 and Definition 7, we have
(27)TσfBp,qs=∑j1≥0∑j2≥02j1s1q+j2s2q∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥0φ∧2−j1Dx1φ∧2−j2Dx2Tσl1,l2,k1,k2fLpq1/qTσfFp,qs=∑j1≥0∑j2≥02j1s1q+j2s2q∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥0φ∧2−j1Dx1φ∧2−j2Dx2Tσl1,l2,k1,k2fq1/qLp,where
(28)σl1,l2,k1,k2x1,x2,ξ1,ξ2≔φ^2−k1Dx1φ^2−k2Dx2σx1,x2,ξ1,ξ2φ^2−l1ξ1φ^2−l2ξ2.


## 3. Necessary Lemmas

In this section, we show some necessary lemmas that will be used to prove the main theorem in this paper. LetΩ be a compact subset of ℝn1×ℝn2 and Ωι1,ι2ι1≥0,ι2≥0 be a sequence of compact subsets of ℝn1×ℝn2. Write
(29)LΩp≔f∈Lpℝn1×ℝn2∩S∞ℝn1×ℝn2:suppf^⊂ΩLΩplq≔gι1,ι2ι1≥0,ι2≥0∈S∞ℝn1×ℝn2:suppFgι1,ι2⊂Ωι1,ι2,where ιi=0,1,2 for i=1,2.The following lemmas (Lemma8 to Lemma 10) are requirements to prove the main theorem and they have been considered in our previous work [35], we will omit the detailed proof here.Lemma 8.
Let0<p≤1, R1,R2>0 and Ω=x1,x2∈ℝn1×ℝn2:x1−x1′≤R1,x2−x2′≤R2. Then we have the estimates
(30)g∗fp≲R1n11p−1R2n21p−1fpgpholds for all f,g∈LΩpℝn1×ℝn2, where the implicit constant is independent of x1′,x2′ and R1,R2.Lemma 9. (see [36]).
Let1<p,q<∞, then there exists a constant C>0 such that
(31)∑k∈ℤMsfkq1/qLp≤C∑k∈ℤfkq1/qLpfor all sequences fk of locally integrable functions on ℝn1×ℝn2, where Ms denotes the strong maximal operator. The strong maximal operator Ms is defined by
(32)Msfx=supr1>0r2>01r1n11r2n2∫Rfy1,y2dy1dy2for any x=x1,x2, where R=y1,y2∈ℝn1×ℝn2:y1−x1<r1,y2−x2<r2, and f is a locally integrable function on ℝn1×ℝn2.Lemma 10.
Let0<c,r<∞, for any t1,t2>0 and any function u∈C1ℝn1×ℝn2 whose Fourier transform is supported in ξ1≤ct1,ξ2≤ct2, assume that u^ is an integrable function. Then there exists two constants C1 and C2 (which depend only on n1,n2,c,r) such that
(33)supz1∈ℝn1z2∈ℝn21t1∇xux−z1,y−z21+t1z1n1/r1+t2z2n2/r+1t2∇yux−z1,y−z21+t1z1n1/r1+t2z2n2/r≤C1supz1∈ℝn1z2∈ℝn2ux1−z1,x2−z21+t1z1n1/r1+t2z2n2/r≤C2Msurx1/r,where Ms denotes the strong maximal operator.By Lemma9 and Lemma 10, we can deduce the following Lemma 11.Lemma 11.
Let0<p≤∞, R1,R2>0 and Ω=x1,x2∈ℝn1×ℝn2:x1−x1′≤R1,x2−x2′≤R2. If0<r<p, then we have
(34)supz1∈ℝn1z2∈ℝn2fx1−z1,x2−z21+R1z1n1/r1+R2z2n2/rLpℝx1n1×ℝx2n2≲fLpℝx1n1×ℝx2n2holds for all f∈LΩpℝn1×ℝn2, where the implicit constant is independent of x1′,x2′ and R1,R2.Lemma 12.
Let0<p≤∞. If 0<r<p, then we have the estimates
(35)supz1∈ℝn1z2∈ℝn2Δj1,j2fx1−z1,x2−z21+2j1z1n1/r1+2j2z2n2/rLpℝx1n1×ℝx2n2≲Δj1,j2fLpℝx1n1×ℝx2n2holds for all j1∈ℤ,j2∈ℤ.Proof.
We only need to proveΔj1,j2f∈LΩp by Lemma 11. Since
(36)suppFΔj1,j2f=suppφ^j1,j2·f^⊆suppφ^j1,j2.
By the definition of the sequenceφj1,j2, we get
(37)suppφ^j1,j2⊆y1,y2∈ℝn1×ℝn2:2ji−1≤yi≤2ji+1,i=1,2.
LetRi=2ji+1 and xi′=0 for i=1,2 in Lemma 11, hence we have Δj1,j2f∈LΩp.
By Lemma12, we can obtain the following lemma due to Triebel [37].Lemma 13.
Let0<p<∞ and 0<q≤∞. If 0<r<minp,q, then we have the estimates
(38)supz1∈ℝn1z2∈ℝn2Δj1,j2fx1−z1,x2−z21+2j1z1n1/r1+2j2z2n2/rLplq≲Δj1,j2fLplqholds for all j1∈ℤ,j2∈ℤ.For anyxi∈ℝni and ti∈ℝ with i=1,2, the bi-parameter Bessel potential can be denoted as I−Δt1/2,t2/2f=F−11+x12t1/21+x22t2/2f^. By adopting the main idea for Besov spaces and Triebel spaces in Triebel [37], we give the following lemma for the bi-parameter case.Lemma 14.
Let0<p,q≤∞, s=s1,s2, t=t1,t2 with si,ti∈ℝ, i=1,2. Then
(i)
The mappingI−Δt1/2,t2/2:Bp,qs↪Bp,qs−t is isomorphic(ii)
The mappingI−Δt1/2,t2/2:Fp,qs↪Fp,qs−t is isomorphicTo prove Theorem1 and Theorem 2, we will change the formulation of Tσl1,l2,m1,m2f slightly as follows. Choosing a function ψyi∈Sℝni and satisfy ψ^yi=1 on suppφ^yi, and ψ^yi=0 besides suppφ^yi with i=1,2.Set(39)ψl1,l2x1,x2=ψ2−l1x1ψ2−l2x2.Then, we have(40)ψl1,l2^ξ1,ξ2=ψ^2−l1ξ1ψ^2−l2ξ2≡1on the support of φ^2−l1ξ1φ^2−l2ξ2. Thus, we can deduce
(41)Tσl1,l2,k1,k2fx1,x2=2π−n1+n2∫ℝn1×ℝn2σl1,l2,k1,k2x1,x2,ξ1,ξ2eix1·ξ1+x2·ξ2f^ξ1,ξ2dξ1dξ2=2π−n1+n2∫ℝn1×ℝn2φ^2−k1Dx1φ^2−k2Dx2σx1,x2,ξ1,ξ2φ^2−l1ξ1φ^2−l2ξ2×ψ^2−l1ξ1ψ^2−l2ξ2eix1·ξ1+x2·ξ2f^ξ1,ξ2dξ1dξ2=2π−n1+n2∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2φ^2−l1ξ1φ^2−l2ξ2eix1·ξ1+x2·ξ2f^ξ1,ξ2dξ1dξ2=2π−n1+n2∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eix1·ξ1+x2·ξ2FF−1φ^2−l1·φ^2−l2·f^·,·ξ1,ξ2dξ1dξ2=2π−n1+n2∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eix1·ξ1+x2·ξ2Fφ^2−l1Dx1φ^2−l2Dx2fξ1,ξ2dξ1dξ2=Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fx1,x2,where
(42)σ~l1,l2,k1,k2x1,x2,ξ1,ξ2=φ^2−k1Dx1φ^2−k2Dx2σx1,x2,ξ1,ξ2·ψ^2−l1ξ1ψ^2−l2ξ2.Lemma 15.
Letδ=δ1,δ2 with 0≤δi<1, i=1,2. Suppose σ∈BS1,δ0. Then the estimates
(43)∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eiy1·ξ1+y2·ξ2dξ1dξ2≲2l1n12l2n21+2l1y1−N11+2l2y2−N2holds for any N1,N2≥0, where the implicit constant is independent of yi∈ℝni and li,ki∈ℤ.Proof.
Firstly, we deal with∂ξ1γ1∂ξ2γ2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2. For any M1,M2>0 and any multi-index γ1∈ℤ+n1,γ2∈ℤ+n2 with γ1=M1,γ2=M2, according to the Leibniz rule, we get
(44)∂ξ1γ1∂ξ2γ2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2=∂ξ1γ1∂ξ2γ2φ^2−k1Dx1φ^2−k2Dx2σx1,x2,ξ1,ξ2ψ^2−l1ξ1ψ^2−l2ξ2≤∑β1≤γ1∑β2≤γ2φ^2−k1Dx1φ^2−k2Dx2∂ξ1β1∂ξ2β2σx1,x2,ξ1,ξ2∂ξ1γ1−β1∂ξ2γ2−β2ψ^2−l1ξ1ψ^2−l2ξ2.
By the construction ofψ^2−liξi, i=1,2, we can deduce
(45)∂ξ1γ1−β1∂ξ2γ2−β2ψ^2−l1ξ1ψ^2−l2ξ2=∂ξ1γ1−β1ψ^2−l1ξ1∂ξ2γ2−β2ψ^2−l2ξ2=2−l1γ1−β1∂ξ1γ1−β1ψ^2−l1ξ12−l2γ2−β2∂ξ2γ2−β2ψ^2−l2ξ2=2−l1γ1−β12−l2γ2−β2∂ξ1γ1−β1ψ^2−l1ξ1∂ξ2γ2−β2ψ^2−l2ξ2.
Ifξ1∈supp∂ξ1γ1−β1ψ^2−l1ξ1 in the above inequality, then ξ1~2l1. Similarly, if ξ1∈supp∂ξ2γ2−β2ψ^2−l2ξ2, we have ξ2~2l2.
Now, we start to estimateφ^2−k1Dx12−k2Dx2∂ξ1β1∂ξ2β2σx1,x2,ξ1,ξ2. By using the Fubini-Tonelli theorem, we can deduce
(46)φ^2−k1Dx1φ^2−k2Dx2∂ξ1β1∂ξ2β2σx1,x2,ξ1,ξ2=F1,2−1φ^2−k1·φ^2−k2·F1,2∂ξ1β1∂ξ2β2σ·,·x1,x2,ξ1,ξ2=F1,2−1φ^2−k1·φ^2−k2·∗∂ξ1β1∂ξ2β2σx1,x2,ξ1,ξ2=∫ℝn1×ℝn2F−1φ^2−k1·φ^2−k2·x1−y1,x2−y2∂ξ1β1∂ξ2β2σy1,y2,ξ1,ξ2dy1dy2≤∫ℝn1×ℝn2φk1,k2x1−y1,x2−y2∂ξ1β1∂ξ2β2σy1,y2,ξ1,ξ2dy1dy2≤∫ℝn1×ℝn2φk1,k2x1−y1,x2−y21+ξ1−β11+ξ2−β2dy1dy2≤∫ℝn1×ℝn2φk1,k2x1−y1,x2−y21+ξ1−β11+ξ2−β2dy1dy2≲2−l1β12−l2β2.
Hence, we have(47)∂ξ1γ1∂ξ2γ2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2≲∑β1≤γ1∑β2≤γ22−l1γ1−β12−l2γ2−β22−l1β12−l2β2χξ1,ξ2∈ℝn1×ℝn2:ξi~2li,i=1,2=2−l1M12−l2M2χξ1,ξ2∈ℝn1×ℝn2:ξi~2li,i=1,2.
Obviously, whenγ1=γ2=0, it gives hat
(48)σ~l1,l2,k1,k2x1,x2,ξ1,ξ2≲χξ1,ξ2∈ℝn1×ℝn2:ξi~2li,i=1,2.
Therefore,(49)∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eiy1·ξ1+y2·ξ2dξ1dξ2≲2l1n12l2n2.
On the other hand, by using theN1 times integration by parts with respect to the ξ1-variable and using the N2 times integration by parts with respect to the ξ2-variable, respectively, we can infer that
(50)∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eiy1·ξ1+y2·ξ2dξ1dξ2=y1−N1y2−N2∫ℝn1×ℝn2eiy1·ξ1+y2·ξ2∂ξ1N1∂ξ2N2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2dξ1dξ2≲y1−N1y2−N22−l1N12−l2N22l1n12l2n2=2l1y1−N12l2y2−N22l1n12l2n2for any y1≠0,y2≠0.
According to above, we can obtain(51)∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eiy1·ξ1+y2·ξ2dξ1dξ2≲2l1n12l2n21+2l1y1−N11+2l2y2−N2holds for any yi∈ℝni and li,mi∈ℤ with i=1,2. Thus, our desired result can be immediately obtained.
By applying Lemma15, we have the following lemma.Lemma 16.
Let0<r<p≤∞, δ=δ1,δ2 with 0≤δi<1, i=1,2. Suppose σ∈BS1,δ0. Then we have the estimates
(52)Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLpℝx1n1×ℝx2n2≲2−N1+n1/r+n12−N2+n2/r+n2φ^2−l1Dx1φ^2−l2Dx2fLpℝx1n1×ℝx2n2holds for any Ni≥ni+ni/r+1,i=1,2.Proof.
TakingN1≥n1+n1/r+1 and N2≥n2+n2/r+1, respectively. By Lemma 15 and the Fubini-Tonelli theorem, we have
(53)Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fx1,x2=2π−n1+n2∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eix1·ξ1+x2·ξ2Fφ^2−l1Dx1φ^2−l2Dx2fξ1,ξ2dξ1dξ2=2π−n1+n2∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eix1·ξ1+x2·ξ2×∫ℝn1×ℝn2φ^2−l1Dx1φ^2−l2Dx2fy1,y2e−iy1·ξ1+y2·ξ2dy1dy2dξ1dξ2≤2π−n1+n2∫ℝn1×ℝn2∫ℝn1×ℝn2σ~l1,l2,k1,k2x1,x2,ξ1,ξ2eix1−y1·ξ1eix2−y2·ξ2dξ1dξ2×φ^2−l1Dx1φ^2−l2Dx2fy1,y2dy1dy2≤2π−n1+n2∫ℝn1×ℝn22l1n12l2n21+2l1x1−y1N11+2l2x2−y2N2φ^2−l1Dx1φ^2−l2Dx2fy1,y2dy1dy2≲supy1∈ℝn1y2∈ℝn2φ^2−l1Dx1φ^2−l2Dx2fx1−y1,x2−y21+2l1y1n1/r1+2l2y2n2/r∫ℝn1×ℝn22l1n12l2n21+2l1y1N1−n1/r1+2l2y2N2−n2/rdy1dy2.
Set(54)I≔∫ℝn1×ℝn22l1n12l2n21+2l1y1N1−n1/r1+2l2y2N2−n2/rdy1dy2.
It remains to estimate the integration expression ofI. Next we will deal with I by dividing it into the following four parts
(55)I=∫2l1y1<12l2y2<12l1n12l2n21+2l1y1N1−n1/r1+2l2y2N2−n2/rdy1dy2+∫2l1y1<12l2y2≥12l1n12l2n21+2l1y1N1−n1/r1+2l2y2N2−n2/rdy1dy2+∫2l1y1≥12l2y2<12l1n12l2n21+2l1y1N1−n1/r1+2l2y2N2−n2/rdy1dy2+∫2l1y1≥12l2y2≥12l1n12l2n21+2l1y1N1−n1/r1+2l2y2N2−n2/rdy1dy2≔I1+I2+I3+I4.
Now we need to considerI1,I2,I3 and I4, respectively.Case 1.
When2l1y1<1 and 2l2y2<1, then
(56)I1≲∫2l1y1<12l2y2<12l1n12l2n2dy1dy2≲1.Case 2.
When2l1y1<1 and 2l2y2≥1, by transformation of spherical polar coordinates, we get
(57)I2≲∫2l2y2≥12l2n21+2l2y2N2−n2/rdy2≲∫Sn2−1∫1∞Rn2−11+Rl2N2−n2/rdRdθ≲2−N2+n2/r+n2.Case 3.
When2l1y1≥1 and 2l2y2<1, by symmetry with Case 2, we obtain
(58)I3≲2−N1+n1/r+n1.Case 4.
When2l1y1≥1 and 2l2y2≥1, by transformation of spherical polar coordinates, we have
(59)I4≲2−N1+n1/r+n12−N2+n2/r+n2.Combining above four cases together, we have(60)I≲2−N1+n1/r+n12−N2+n2/r+n2.Taking theLp (quasi) norm to the both sides and using Lemma 12, we can get the desired result.Lemma 17.
Let0<p<∞ and 0<q≤∞, δ=δ1,δ2 with 0≤δi<1, i=1,2. Suppose σ∈BS1,δ0. If 0<r<minp,q, then we have the estimates
(61)Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLplq≲2−N1+n1/r+n12−N2+n2/r+n2φ^2−l1Dx1φ^2−l2Dx2fLplqholds for any Ni≥ni+ni/r+1,i=1,2.Proof.
By the estimates in the proof of Lemma16, we get
(62)Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fx1,x2≲2−N1+n1/r+n12−N2+n2/r+n2supy1∈ℝn1y2∈ℝn2φ^2−l1Dx1φ^2−l2Dx2fx1−y1,x2−y21+2l1y1n1/r1+2l2y2n2/r.
Taking thelq (quasi) norm on l1 and l2 and Lp (quasi) norm on x1 and x2 both sides, respectively. We can obtain the desired result by using Lemma 13.
At last, we need the following lemma for the preparation of proving Theorem1.Lemma 18.
Let0<p≤∞ and l~i=maxli,ki+4 for i=1,2. Then we have
(63)Δj1,j2Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp≲2n1l~1+n2l~21min1,p−1Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp,where the implicit constant is independent of ji,li,ki, i=1,2.Proof.
When1≤p≤∞, by the Young inequality, we get
(64)Δj1,j2Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp=F−1φ^j1,j2∗Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp≤φj1,j2L1Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp≲Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp.
When0<p<1, by the decomposition of the symbol σl1,l2,k1,k2x1,x2,ξ1,ξ2 and the Fubini-Tonelli theorem, we can deduce
(65)FTσl1,l2,k1,k2fy1,y2=∫ℝn1×ℝn2e−ix1·y1+x2·y2Tσl1,l2,k1,k2fx1,x2dx1dx2=2π−n1+n2∫ℝn1×ℝn2∫ℝn1×ℝn2e−ix1·y1−ξ1+x2·y2−ξ1σl1,l2,k1,k2x1,x2,ξ1,ξ2f^ξ1,ξ2dξ1dξ2dx1dx2=2π−n1+n2∫ℝn1×ℝn2∫ℝn1×ℝn2e−ix1·y1−ξ1+x2·y2−ξ1φ^2−l1ξ1φ^2−l2ξ2f^ξ1,ξ2×φ^2−k1Dx1φ^2−k2Dx2σx1,x2,ξ1,ξ2dξ1dξ2dx1dx2=2π−n1+n2∫ℝn1×ℝn2φ^2−l1ξ1φ^2−l2ξ2f^ξ1,ξ2∫ℝn1×ℝn2e−ix1·y1−ξ1+x2·y2−ξ1×φ^2−k1Dx1φ^2−k2Dx2σx1,x2,ξ1,ξ2dx1dx2dξ1dξ2=2π−n1+n2∫ℝn1×ℝn2F1,2φ^2−k1Dx1φ^2−k2Dx2σy1−ξ1,y2−ξ2,ξ1,ξ2×φ^2−l1ξ1φ^2−l2ξ2f^ξ1,ξ2dξ1dξ2=2π−n1+n2∫ℝn1×ℝn2F1,2σy1−ξ1,y2−ξ2,ξ1,ξ2φ^2−l1ξ1φ^2−l2ξ2f^ξ1,ξ2×φ^2−k1y1−ξ1φ^2−k2y2−ξ2dξ1dξ2.
According to the facts that(66)suppφ^2−liξi⊂ξi∈ℝni:2li−1≤ξi≤2li+1,i=1,2,suppφ^2−kiyi−ξi⊂yi−ξi∈ℝni:2ki−1≤yi−ξi≤2ki+1,i=1,2.
It follows that(67)suppFTσl1,l2,k1,k2f⊂suppφ^2−l1ξ1φ^2−l2ξ2∩suppφ^2−k1y1−ξ1φ^2−k2y2−ξ2⊂y1,y2∈ℝn1×ℝn2:2maxli,ki−1−2minli,ki+1≤yi≤2li+1+2ki+1,i=1,2.
Besides, by the support ofφ^2−ji· defined as in the definition of the bi-parameter Besov space, we have
(68)suppφj1,j2^⊂yi∈ℝni:ξi∈ℝni:2ji−1≤yi≤2ji+1,i=1,2.
According to the above, if(69)suppφj1,j2^∩suppFTσl1,l2,k1,k2f≠∅for any non-negative integer ji,li,ki, i=1,2, we can get the following four cases:
Case 1:2maxl1,k1−1−2minl1,k1+1≤2j1−1≤2l1+1+2k1+1;2maxl2,k2−1−2minl2,k2+1≤2j2−1≤2l2+1+2k2+1.
Case 2:2maxl1,k1−1−2minl1,k1+1≤2j1−1≤2l1+1+2k1+1;2j2−1≤2maxl2,k2−1−2minl2,k2+1≤2j2+1.
Case 3:2maxl2,k2−1−2minl2,k2+1≤2j2−1≤2l2+1+2k2+1;2j1−1≤2maxl1,k1−1−2minl1,k1+1≤2j1+1.
Case 4:2j1−1≤2maxl1,k1−1−2minl1,k1+1≤2j1+1;2j2−1≤2maxl2,k2−1−2minl2,k2+1≤2j2+1.
Therefore, the estimates(70)Δj1,j2Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2falways vanishes unless the above four cases. From above four cases, it implies that
(71)suppFTσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2f⊂y1,y2∈ℝn1×ℝn2:yi≤2maxl1,k1+4,i=1,2suppφ^j1,j2⊂y1,y2∈ℝn1×ℝn2:yi≤2maxl1,k1+4,i=1,2.
Letl~i=maxli,ki+4 and set
(72)Ω=y1,y2∈ℝn1×ℝn2:yi≤2l~i,i=1,2.
Then, by Lemma8, we have
(73)suppFTσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2f⊂Ωandsuppφ^j1,j2⊂Ω.
Hence, it follows that(74)Δj1,j2Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp=F−1φ^j1,j2·FTσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp≲2n1l~1+n2l~21p−1Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLp.
According to above, we can get the desired result.


## 4. Proof of Theorem1

In this section, we will give a detailed proof for Theorem1.Note that for all0<p,q≤∞, we have
(75)∑m1∈ℤm2∈ℤam1,m2x1,x2Lpℝx1n1×ℝx2n2≤∑m1∈ℤm2∈ℤam1,m2x1,x2Lpℝx1n1×ℝx2n2min1,p1/min1,p∑m1∈ℤm2∈ℤbm1,m2,w1,w2lqℤw1×ℤw2≤∑m1∈ℤm2∈ℤbm1,m2,w1,w2lqℤw1×ℤw2min1,q1/min1,q.Now we give the detailed proof of Theorem1.Proof of Theorem 1.
SetJs1,s2=I−Δs1/2,s2/2. If all pseudo-differential operators Tσ with symbols σ∈BS1,δ0 are bounded on Bp,qs, by using Lemma 14, we can deduce
(76)TσfBp,qs~Js1,s2TσJ−s1,−s2Js1,s2fBp,q0≲Js1,s2fBp,q0~fBp,qs.
On the other hand, if allTσ are pseudo-differential operators with symbols σ∈BS1,δ0, then the operators Js1,s2TσJ−s1,−s2 are pseudo-differential operators with symbols σ∈BS1,δ0. Hence, we only need to prove the case that s1=s2=0. That is, we need to prove
(77)TσfBp,q0fBp,q0.
By Definition6, we obtian
(78)TσfBp,q0=∑j1≥0∑j2≥0Δj1,j2TσfLpq1/q.
For anyj1≥0,j2≥0, by Lemma 16 and Lemma 18, we have
(79)Δj1,j2TσfLp≲∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥0Δj1,j2Tσl1,l2,k1,k2fLp≲∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥0Δj1,j2Tσl1,l2,k1,k2fLpmin1,p1/min1,p=∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥0Δj1,j2Tσ~l1,l2,k1,k2φ∧2−l1Dx1φ∧2−l2Dx2fLpmin1,p1/min1,p≲∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥02n1l~11min1,p−1min1,p2n2l~21min1,p−1min1,p×Tσ~l1,l2,k1,k2φ∧2−l1Dx1φ∧2−l2Dx2fLpmin1,p1/min1,p≲∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥02n1l~11min1,p−1−N1+n1r+n1min1,p×2n2l~21min1,p−1−N2+n2r+n2min1,pφ∧2−l1Dx1φ∧2−l2Dx2fLpmin1,p1/min1,p.
Takingq-th power and summing the both sides over j1≥0 and j2≥0, we get
(80)TσfBp,q0≲∑j1≥0∑j2≥0∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥02n1l~11min1,p−1−N1+n1r+n1min1,p×2n2l~21min1,p−1−N2+n2r+n2min1,pφ∧2−l1Dx1φ∧2−l2Dx2fLpmin1,pq/min1,p1/q≲∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥02n1l~11min1,p−1−N1+n1r+n1min1,p,q×2n2l~21min1,p−1−N2+n2r+n2min1,p,qφ∧2−l1Dx1φ∧2−l2Dx2fLplqℤj1+×ℤj2+min1,p,q1/min1,p,q≲fBp,q0∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥02n1l~11min1,p−1−N1+n1r+n1min1,p,q2n2l~21min1,p−1−N2+n2r+n2min1,p,q1/min1,p,q.
ChoosingN1≥n1/r+n1+1 and N2≥n1/r+n1+1 such that
(81)n1l~11min1,p−1−N1+n1r+n1<andn2l~21min1,p−1−N2+n2r+n2<0.
Hence, we can obtain(82)TσfBp,q0≲fBp,q0.
Then the proof of Theorem1 is accomplished.


## 5. Proof of Theorem2

Ifb=b1,b2∈ℝ×ℝ, the bi-parameter Bessel-potential space H2bℝn1×ℝn2 is defined by
(83)H2bℝn1×ℝn2=f∈S∞ℝn1×ℝn2:fH2bℝn1×ℝn2<∞,where fH2bℝn1×ℝn2=1+x12b1/21+b22s2/2FfL2ℝn1×ℝn2.Next, we show the bi-parameterLΩplq-multiplier estimates, which will be used to prove Theorem 2. The main idea of proving this estimate is enlightened by Triebel’s book [37], we will omit the proof here.Lemma 19 (Bi-parameterLΩplq-multiplier estimates).
Let0<p<∞ and 0<q≤∞, b=b1,b2∈ℝ×ℝ, and Ωι1,ι2ι1≥0,ι2≥0 be a sequence of compact subsets of ℝn1×ℝn2. If bi>ni/2+ni/minp,q for i=1,2. Then there exists a constant C>0 such that
(84)F−1hι1,ι2Ffι1,ι2Lplq≤Csupι1,ι2hι1,ι2H2b1,b2fι1,ι2Lplqholds for all systems fι1,ι2∈LΩplq and all function spaces hι1,ι2∈H2b1,b2.Finally, we give the detailed proof of Theorem2.Proof of Theorem 2.
Similar to the analysis in the proof of Theorem1, we only need to prove the case for s1=s2=0. That is, it is enough to prove
(85)TσfFp,q0≲fFp,q0.
By Definition7, for any j1≥0,j2≥0,
(86)TσfFp,q0=∑j1≥0∑j2≥0Δj1,j2Tσfq1/qp≲∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥0Δj1,j2Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLplq.
By Lemma19 and Lemma 17, one can obtain
(87)Δj1,j2Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLplq=F−1φj1,j2FTσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLplq≲φj1,j2H2b1,b2Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLplq≲Tσ~l1,l2,k1,k2φ^2−l1Dx1φ^2−l2Dx2fLplq≲2−N1+n1/r+n12−N2+n2/r+n2φ^2−l1Dx1φ^2−l2Dx2fLplq.
ChoosingN1≥n1/r+n1+1 and N2≥n1/r+n1+1, hence we have
(88)TσfFp,q0≲∑l1≥0l1−j1≤2∑l2≥0l2−j2≤2∑k1≥0∑k2≥02−N1+n1/r+n12−N2+n2/r+n2φ^2−l1Dx1φ^2−l2Dx2fLplq≲fBp,q0.
Then the proof of Theorem2 is accomplished.



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*Source: 7953457-2022-11-07.xml*