# The Boundedness on Mixed Hardy Spaces

Authors: Wei Ding; Meidi Qin; Yueping Zhu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The boundedness of operators on Hardy spaces is usually given by atomic decomposition. In this paper, we obtain the boundedness of singular integral operators in mixed Journé class on mixed Hardy spaces by a direct method.


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## Body

## 1. Introduction

Different from the classical harmonic analysis, more general nonisotropic groups of dilations are considered on multiparameter harmonic analysis. In the past decades, many multiparameter theories of harmonic analysis have been developed, such as [1–15].Discrete Littlewood–Paley theory is a useful tool to study multiparameter harmonic analysis, which is developed by Han, Lu, and Sawyer in [16] to study flag Hardy spaces. Recently, the authors in [17] introduced a new Hardy space as follows:Letψ01,ψ1∈Sℝn1 with(1)suppψ01^⊆ξ∈ℝn1:ξ≤2;ψ01^ξ=1,ifξ≤1,(2)suppψ1^⊆ξ∈ℝn1:12≤ξ≤2,(3)ψ01^ξ2+∑j=1∞ψ1^2−jξ2=1,forallξ∈ℝn1.Letψ2∈Sℝn2 with(4)suppψ2^⊆ξ∈ℝn2:12≤ξ≤2,(5)∑j∈ℤψ2^2−jξ2=1,forallξ∈ℝn20.Then, forj,k∈ℤ,j≥1, set ψj1x=2jn1ψ12jx, ψk2x=2kn2ψ22kx, ψj,kx,y=ψj1xψk2y, and ψ0,kx,y=ψ01xψk2y.DenoteS0ℝn1+n2=f∈Sℝn1+n2:∫ℝn2fxxαdx=0,∀α≥0, SNℝn=ϕ∈Sℝn:∫ℝnϕxxαdx=0,∀α≤N, and S∞ℝn=ϕ∈Sℝn:∫ℝnϕxxαdx=0,∀α>0. For i=1,2 and any j∈ℤ, denote Πjni=I:I are dyadic cubes in ℝni with the side length lI=2−j, and the left lower corners of I are xI=2−jℓ, ℓ∈ℤni, Dni=∪jΠjni and D=Dn1×Dn2.Then, the following multiparameter mixed Hardy space is introduced in [17].Definition 1.
Let0<p<∞, ψ01, ψ1∈Sℝn1, and ψ2∈Sℝn2 are functions satisfying conditions (1)–(5), respectively. The multiparameter mixed Hardy space Hmixp=Hmixpℝn1×ℝn2 is defined by(6)Hmixp=f∈S0′ℝn1+n2:fHmixp<∞,where(7)fHmixp=∑j∈ℕ,k∈ℤ∑I×J∈Πjn1×Πkn2ψj,k∗fxI,xJ2χIxχJy1/2Lpℝn1+n2.
The symbolℕ means the set of natural numbers including zero. The multiparameter mixed Hardy space is well defined, that is, Hmixp is independent of the choice of functions ψ01, ψ1, and ψ2, which has been proved in [17]. The main purpose of this paper is to establish the boundedness of operators on Hmixp.
Firstly, let us recall something about singular integrals of nonconvolution type. A locally integral functionKx,y defined away from the diagonal x=y in ℝn×ℝn is called a one-parameter nonconvolution Calderón–Zygmund kernel with a regularity exponent ε>0 if Kx,y satisfies, for some constant C>0, the size condition(8)Kx,y≤C1x−yn,forx≠y,and the regularity conditions(9)Kx,y−Kx,y′≤Cy−y′εx−yn+ε,whenever y−y′≤1/2x−y, and(10)Kx,y−Kx′,y≤Cx−x′εx−yn+ε,whenever x−x′≤1/2x−y. The smallest such constant C is denoted by KCZ. We call an operator T one-parameter nonconvolution Calderón–Zygmund operator if T is a singular integral operator associated with a one-parameter nonconvolution Calderón–Zygmund kernel Kx,y given by <Tf,g>=∫Kx,yfygxdxdy for all f,g∈Sℝn with disjoint supports. Define TCZ by TCZ=KCZ+TL2⟶L2. Then, T is bounded on Lpℝn1<p<+∞, if TCZ<+∞. However, by a classical result, Hp, 0<p≤1, boundedness of T implies that the operator must satisfy additional condition: ∫Tφxdx=0 for any φ∈S∞ℝn, named T∗1=0. On the contrary, to ensure the boundedness of T on local Hardy spaces hp, 0<p≤1, additional condition Kx,y≤C1/x−yn+δ, for some δ>0, should be included [18]. According to these, we extend this definition of singular integral operators in Journé class on ℝn1×ℝn2 suitable for our multiparameter mixed Hardy spaces, named mixed Journé class.Definition 2.
A singular integral operatorT is said to be in mixed Journé class on ℝn1×ℝn2 with regularity exponent ε∈0,1 and δ>0 if(11)Tfx1,x2=∫ℝn1×ℝn2Kx1,y1,x2,y2fy1,y2dy1dy2,where the kernel K satisfies the following conditions.
For eachx1,y1∈ℝn1, set K˜1x1,y1 to be the singular integral operator acting on functions on ℝn2 with the kernel(12)K˜1x1,y1x2,y2=Kx1,y1,x2,y2,and similarly,(13)K˜2x2,y2x1,y1=Kx1,y1,x2,y2.
Then there exists a constantC>0 such that(1)
T is bounded on L2ℝn1+n2(2)
K˜1x1,y1CZ≤Cmin1/x1−y1n1,1/x1−y1n1+δ,ifx1−y1>0, K˜1x1,y1−K˜1x1,y1′CZ≤Cy1−y1′ε/x1−y1n1+ε,ify1−y1′≤1/2x1−y1, and K˜1x1,y1−K˜1x1′,y1CZ≤Cx1−x1′ε/x1−y1n1+ε,ifx1−x1′≤1/2x1−y1(3)
K˜2x2,y2CZ≤C1/x2−y2n2,ifx2−y2>0, K˜2x2,y2−K˜2x2,y2′CZ≤Cy2−y2′ε/x2−y2n2+ε,ify2−y2′≤1/2x2−y2, and K˜2x2,y2−K˜2x2′,y2CZ≤Cx2−x2′ε/x2−y2n2+ε,ifx2−x2′≤1/2x2−y2
The smallest such constantC is denoted by KCZ. The norm of the operator T is defined by TCZ=TL2ℝn1+n2⟶L2ℝn1+n2+KCZ.Theorem 1.
Suppose thatT is a singular integral operator in mixed Journé class on ℝn1×ℝn2 with the regularity exponent ε∈0,1, δ>0. Then, T is bounded on Hmixp for maxn1/n1+δ,n1/n1+ε,n2/n2+ε<p≤1 if T11=T21=T1∗1=T2∗1=0.Remark 1.
The meaning ofT11=0 is understood as follows: for any fixed y2∈ℝn2,(14)∫Kx1,y1,x2,y2ψ1x1ψ2x2dx1dx2dy1=0,foranyψi∈SMℝni.
Similarly,T21=0, where M=Mp,n1,n2 is a large positive integer dependent only on p,n1,n2.
T
1
∗
1
=
0 is known as for any fixed x2∈ℝn2,(15)∫Kx1,y1,x2,y2ψ1y1ψ2y2dy1dy2dx1=0,foranyψi∈SMℝni,i=1,2,similarly for T2∗1=0.
The organization of this paper is as follows. In section2, we recall a discrete Calderón-type identity on L2ℝn1+n2∩Hmixp which is proved in [17]. In section 3, we prove Theorem 1.
Finally, we make some conventions. Throughout the paper,C denotes a positive constant that is independent of the main parameters involved, but whose value may vary from line to line. Constants with subscript, such as C1, do not change in different occurrences. We denote f≤Cg by f≲g. If f≲g≲f, we write f≈g.


## 2. Calderón-Type Identity

To show Theorem1, we need a new discrete Calderón-type identity on L2ℝn1+n2∩Hmixp. To do this, for any large integer D=Dp,n1,n2, let ϕ01,ϕ1∈Sℝn1 with(16)suppϕ01⊆x∈ℝn1:x≤1;∫ϕ01=1,(17)suppϕ1⊆x∈ℝn1:x≤1;∫ϕ1xxαdx=0,forallα≤D,(18)ϕ01^ξ2+∑j=1∞ϕ1^2−jξ2=1,forallξ∈ℝn1.Letϕ2∈Sℝn2 with(19)suppϕ2⊆x∈ℝn2:x≤1;∫ϕ2xxαdx=0,for allα≤D,(20)∑j∈ℤϕ2^2−jξ2=1,forallξ∈ℝn20.Setϕ0,k=ϕ01∗ϕk2 and ϕj,k=ϕj1∗ϕk2 if j>1. Then, we have the following discrete Calderón-type identity on L2ℝn1+n2∩Hmixp [17].Theorem 2.
For0<p≤1, let ϕ01, ϕ1 and ϕ2 satisfy conditions (16)–(20), respectively. Then, for any f∈L2ℝn1+n2∩Hmixp, there exists a positive integer N such that(21)fx=∑j∈ℕ,k∈ℤ∑I×J∈Πj+Nn1×Πk+Nn2IJϕj,k∗hxI,xJ×ϕj,kx1−xI,x2−xJ,where the series converges in L2ℝn1+n2 and Hmixp, and h∈L2ℝn1+n2∩Hmixp satisfying(22)hL2ℝn1+n2≈fL2ℝn1+n2,hHmixp≈fHmixp.Corollary 1 (see [17]).
Let0<p≤1. Suppose that ϕj,k satisfies the same conditions as in Theorem 2. Then, for f∈L2ℝn1+n2∩Hmixp,(23)fHmixp≈∑j∈ℕ,k∈ℤ∑I×J∈Πj+Nn1×Πk+Nn2ϕj,k∗fxI,xJ2χIχJ1/2Lpℝn1+n2,where N is the same as in Theorem 2.


## 3. Boundedness

Now, we are ready to prove Theorem1. For f∈L2ℝn1+n2∩Hmixp, by Theorem 2, one has(24)fx,y=∑j∈ℕ,k∈ℤ∑I×J∈Πj+Nn1×Πk+Nn2IJψj,k∗hxI,xJ×ψj,kx−xI,y−xJ,in L2ℝn1+n2, where ψj,k satisfy the same conditions as in Theorem 2. By the L2 boundedness of T,(25)Tfx,y=∑j∈ℕ,k∈ℤ∑I×J∈Πj+Nn1×Πk+Nn2IJψj,k∗hxI,xJ×Tψj,k⋅−xI,⋅−xJx,y.Then, Corollary1 gives that(26)TfHmixp≤C∑j′∈ℕ,k′∈ℤ∑I′×J′∈Πj′+Nn1×Πk′+Nn2ϕj′,k′∗TfxI′,xJ′2χI′xχJ′y1/2Lpℝn1+n2=C∑j′∈ℕ,k′∈ℤ∑I′×J′∈Πj′+Nn1×Πk′+Nn2∑j∈ℕ,k∈ℤ∑I×J∈Πj+Nn1×Πk+Nn2IJψj,k∗hxI,xJ×ϕj′,k′∗Tψj,k⋅−xI,⋅−xJxI′,xJ′2χI′xχJ′y1/2Lpℝn1+n2,where ϕj′,k′andψj,k satisfy the conditions in Theorem 2.Formally, one has(27)ϕj′,k′∗Tψj,k⋅−u1,⋅−u2x1,x2=∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψj1z1−u1ψk2z2−u2dy1dy2dz1dz2.To deal with the case thaty1−z1 or y2−z2 is small, we use the argument in [19]. For 0<η<1, let C0ηℝn denote the space of continuous functions f with compact supports such that(28)fLipηℝn=supx≠yfx−fyx−yηis finite. Note that if φ∈Sℝn with suppφ⊆Q for some cube Q in ℝn, then φ∈Lipηℝn for any 0<η<1, and(29)φLipηℝn<+∞,and for any j∈ℤ,(30)φjLipηℝn≤C2jn+jη.A linear one-parameter operatorT from C0ηℝn to C0ηℝn′, 0<η<1, is said to satisfy the weak boundedness property if(31)<φ,Tψ>≤CQ1+2η/nφLipηℝnψLipηℝn,for any φ,ψ∈C0ηℝn with supports in cube Q⊆ℝn. If φ∈C0ηℝn with support in Q, using φL2ℝn≤CQ1/2+η/nφLipηℝn, one can see that L2ℝn boundedness of T implies that T satisfies the weak boundedness property. This property can be generalized to multiparameter operators.Lemma 1.
Suppose thatφ1,φ2∈C0ηℝn1 with supports in cube Q1⊆ℝn1, ψ1,ψ2∈C0ηℝn2 with supports in cube Q2⊆ℝn2. If linear two-parameter operator T is L2ℝn1+n2 bounded, then T satisfies the following weak boundedness property:(32)φ1ψ1,Tφ2ψ2≤CTL2ℝn1+n2⟶L2ℝn1+n2Q11+2η/n1Q21+2η/n2×φ1Lipηℝn1ψ1Lipηℝn2φ2Lipηℝn1ψ2Lipηℝn2.Next, we need some almost orthogonality estimates aboutϕj′,k′∗Tψj,k⋅−u1,⋅−u2x1,x2. From now on, ϕj′,k′andψj,k satisfy the conditions of Theorem 2, and T satisfies the conditions of Theorem 1. In Definition 2, without loss of generality, we may assume δ≥ε.Lemma 2.
For anyk,k′∈ℤ and any 0<ε′<ε, there exists a positive constant C such that(33)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2≤C2−k′−kε′11+x1−u1n1+ε2−k∧k′ε2−k∧k′+x2−u2n2+ε.Proof.
We divide the estimates into eight cases:Case I:k′>k,2kx2−u2≤10,x1−u1≤8Case II:k′>k,2kx2−u2>10,x1−u1≤8Case III:k′≤k,2k′x2−u2≤10,x1−u1≤8Case IV:k′≤k,2k′x2−u2>10,x1−u1≤8Case V:k′>k,2kx2−u2≤10,x1−u1>8Case VI:k′>k,2kx2−u2>10,x1−u1>8Case VII:k′≤k,2k′x2−u2≤10,x1−u1>8Case VIII:k′≤k,2k′x2−u2>10,x1−u1>8
Now, we deal with case I. SinceT21=0, we have(34)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψ01z1−u1ψk2z2−u2dy1dy2dz1dz2=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψ01z1−u1ψk2z2−u2−ψk2x2−u2dy1dy2dz1dz2.
Letρ0∈C∞ℝn2 be 1 on B0,2 and 0 outside its double. Then,(35)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψ01z1−u1ψk2z2−u2−ψk2x2−u2ρ02k′z2−x2dydz+∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψ01z1−u1ψk2z2−u2−ψk2x2−u21−ρ02k′z2−x2dydz=A+B.
One can check that(36)ϕk′2Lipηℝn2≤C2k′n2+k′η,(37)ψk2⋅−u2−ψk2x2−u2ρ02k′⋅−x2Lipηℝn2≤C2−k′−k2kn22k′η,and ϕ01Lipηℝn1+ψ01Lipηℝn1≤C [19]. Hence, by the weak boundedness of T on ℝn1+n2, we have(38)A≤C2−k′n2+2ηϕk′2Lipηℝn2ψk2⋅−u2−ψk2x2−u2ρ02k′⋅−x2Lipηℝn2≤C2−k′−k2kn2≤C2−k′−kε2kn2.
On the contrary, using∫ϕ2=0, one has(39)B=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2−Ky1,z1,x2,z2ψ01z1−u1ψk2z2−u2−ψk2x2−u21−ρ02k′z2−x2dydz,which implies that(40)B≤C∬ϕk′2x2−y2K˜2y2,z2−K˜2x2,z2CZψk2z2−u2−ψk2x2−u21−ρ02k′z2−x2dz2dy2,by the weak boundedness of operator K˜2y2,z2−K˜2x2,z2 on ℝn1. Combing the smooth condition of K˜2 with(41)ψk2z2−u2−ψk2x2−u2≤Cz2−x22−k+z2−x2ε2kn2,we have(42)B≤C∬x2−z2≥2−k′+1ψk′2x2−y2y2−x2εz2−x2n2+εz2−x22−k+z2−x2ε2kn2dy2dz2≤C2kn2∬x2−z2≥2−k′+1ψk′2x2−y22−k′εz2−x2n2dy212−k+z2−x2εdz2≤C2kn2∫x2−z2≥2−k′+12−k′εz2−x2n2+ε12−k+z2−x2εdz2≤C2−k′ε2kn2∫x2−z2>2−k1z2−x2n2+εdz2+C2−k′−kε2kn2∫2−k′+1≤x2−z2≤2−k1z2−x2n2dz2≤C2−k′ε2kn22kε+C2−k′−kε2kn2k′−k≤C2−k′−kε′2kn2.
Sincex2−u2≤10⋅2−k and x1−u1≤8, one has 2−k≈2−k+x2−u2 and 1+x1−u1≈1, which give that(43)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2≤C2−k′−kε′2kn2≈C2−k′−kε′2kn22−k2−k+x2−u2ε≈C2−k′−kε′11+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.
Now, we deal with case II. Using∫ϕ2=0, one has(44)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψ01z1−u1ψk2z2−u2dy1dy2dz1dz2=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2−Ky1,z1,x2,z2×ψ01z1−u1ψk2z2−u2dy1dy2dz1dz2.
By the weak boundedness ofK˜2y2,z2−K˜2x2,z2, it follows that(45)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2≤C∫ϕk′2x2−y2K˜2y2,z2−K˜2x2,z2CZψk2z2−u2dz2dy2.
In the above integral, one can check thaty2−z2≈x2−u2 and x2−u2≈2−k+x2−u2, which together with the smooth condition of K˜2 yield that(46)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2≤C∫ϕk′2x2−y2y2−x2εy2−z2n2+εψk2z2−u2dz2dy2≤C2−k′ε2−k+x2−u2n2+ε∫ϕk′2x2−y2ψk2z2−u2dz2dy2≤C2−k′−kε11+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.
By symmetry, with a similar proof, one can discuss case III and case IV sinceT2∗1=0. Now, we turn to case V. Firstly, proceeding as case I, one can divide ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2 into A and B, that is,(47)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2=A+B.
Then, by the weak boundedness of operatorK˜1y1,z1, one has(48)A=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψ01z1−u1ψk2z2−u2−ψk2x2−u2ρ02k′z2−x2dydz≤C∫ϕ01x1−y1K˜1y1,z1CZψ01z1−u1dy1dz1×2−k′n2+2ηϕk′2Lipηψk2⋅−u2−ψk2x2−u2ρ02k′⋅−x2Lipη≤C2−k′−kε2kn2∫ϕ01x1−y1K˜1y1,z1CZψ01z1−u1dy1dz1.
Using the support condition ofϕ01 and ψ01 and the size condition of K˜1y1,z1 when y1−z1>1, we can obtain that(49)A≤C2−k′−kε2kn2∫ϕ01x1−y11y1−z1n1+εψ01z1−u1dy1dz1≤C1x1−u1n1+ε2−k′−kε2kn2≈C2−k′−kε11+x1−u1n1+ε2−kε2−k+x2−u2n2+ε,since 2−k≈2−k+x2−u2 in this case.
With similar estimates in (42), using ∫ϕ2=0, one has(50)B=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2−Ky1,z1,x2,z2×ψ01z1−u1ψk2z2−u2−ψk2x2−u21−ρ02k′z2−x2dydz≤C∫ϕ01x1−y1ϕk′2x2−y2K˜1y1,z1CZy2−x2εz2−x2n2+ε×ψ01z1−u1ψk2z2−u2−ψk2x2−u21−ρ02k′z2−x2dydz≤C∫ϕ01x1−y11y1−z1n1+εψ01z1−u1dy1dz1×∫ϕk′2x2−y2y2−x2εz2−x2n2+εψk2z2−u2−ψk2x2−u21−ρ02k′z2−x2dy2dz2≤C11+x1−u1n1+ε2−k′−kε′2kn2≈C2−k′−kε′11+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.
In case VI, similar as estimates (46), using ∫ϕ2=0, one has(51)ϕ0,k′∗Tψ0,k⋅−u1,⋅−u2x1,x2=∫ϕ01x1−y1ϕk′2x2−y2Ky1,z1,y2,z2−Ky1,z1,x2,z2×ψ01z1−u1ψk2z2−u2dy1dy2dz1dz2≤C∫ϕ01x1−y1ϕk′2x2−y2K˜1y1,z1CZy2−x2εy2−z2n2+ε×ψ01z1−u1ψk2z2−u2dydz≤C∫ϕ01x1−y11y1−z1n1+εψ01z1−u1dy1dz1×∫ϕk′2x2−y2y2−x2εy2−z2n2+εψk2z2−u2dz2dy2≤C2−k′−kε11+x1−u1n1+ε2−kε2−k+x2−u2n2+ε,since x1−u1>8 which implies that z1−y1>1.
By symmetry, we complete the proof.Lemma 3.
Suppose thatj,j′,k,k′∈ℤ, j′≥j≥0,j′>0, and k′≥k, then for any 0<ε′<ε, there exists a constant C>0 such that(52)ϕj′,k′∗Tψj,k⋅−u1,⋅−u2x1,x2≤C2−j′−jε′2−k′−kε′2−jε2−j+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.Proof.
We divide the estimates into four cases:2jx1−u1≤10, 2kx2−u2≤10; 2jx1−u1≤10,2kx2−u2>10; 2jx1−u1>10,2kx2−u2≤10; 2jx1−u1>10, 2kx2−u2>10.
We now discuss the first case:2jx1−u1≤10, 2kx2−u2≤10. Since T11=T21=0,(53)ϕj′,k′∗Tψj,k⋅−u1,⋅−u2x1,x2=∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψj1z1−u1ψk2z2−u2dy1dy2dz1dz2=∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2×ψj1z1−u1−ψj1x1−u1ψk2z2−u2−ψk2x2−u2dydz.
Letτ0,ρ0∈C∞ℝn2 be 1 on B0,2 and 0 outside its double. Set ρ1=1−ρ0,τ1=1−τ0. Then,(54)Tψj,k⋅−u1,⋅−u2x1,x2=∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψj1z1−u1−ψj1x1−u1×ψk2z2−u2−ψk2x2−u2τ02j′z1−x1ρ02k′z2−x2dydz+∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψj1z1−u1−ψj1x1−u1×ψk2z2−u2−ψk2x2−u2τ12j′z1−x1ρ02k′z2−x2dydz+∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψj1z1−u1−ψj1x1−u1×ψk2z2−u2−ψk2x2−u2τ02j′z1−x1ρ12k′z2−x2dydz+∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2ψj1z1−u1−ψj1x1−u1×ψk2z2−u2−ψk2x2−u2τ12j′z1−x1ρ12k′z2−x2dydz=A+B+D+E.
By (36) and (37), we have(55)A≤C2−k′n2+2η2−j′n1+2ηϕk′2Lipηψk2⋅−u2−ψk2x2−u2ρ02k′⋅−x2Lipη×ϕj′1Lipηψj1⋅−u1−ψj1x1−u1τ02j′⋅−x1Lipη≤C2−k′−k2kn22−j′−j2jn1≤C2−j′−jε2jn12−k′−kε2kn2≈C2−j′−jε2−k′−kε2−jε2−j+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.
Using∫ϕ1=0, one has(56)B=∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2−Kx1,z1,y2,z2×ψj1z1−u1−ψj1x1−u1×ψk2z2−u2−ψk2x2−u2×τ12j′z1−x1ρ02k′z2−x2dydz.
Hence, by (42)(57)B≤C2−k′n2+2ηϕk′2Lipηψk2⋅−u2−ψk2x2−u2ρ02k′⋅−x2Lipη×∫ϕj′1x1−y1K˜1y1,z1−K˜1x1,z1CZψj1z1−u1−ψj1x1−u1×τ12j′z1−x1dy1dz1≤C2−k′−k2kn2∫ϕj′1x1−y1y1−x1εz1−x1n1+εψj1z1−u1−ψj1x1−u1×τ12j′z1−x1dy1dz1≤C2−k′−k2kn22−j′−jε′2jn1,which implies that(58)B≤C2−j′−jε′2−k′−kε′2−jε2−j+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.
Similarly, one has(59)D≤C2−j′−jε′2−k′−kε′2−jε2−j+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.
ForE, since ∫ϕ1=∫ϕ2=0,(60)E=∫ϕj′1x1−y1ϕk′2x2−y2Ky1,z1,y2,z2−Kx1,z1,y2,z2−Ky1,z1,x2,z2−Kx1,z1,x2,z2ψj1z1−u1−ψj1x1−u1×ψk2z2−u2−ψk2x2−u2τ12j′z1−x1ρ12k′z2−x2dydz.
So,(61)E≤C∫ϕj′1x1−y1K1y1,z1−K1x1,z1CZy2−x2εz2−x2n2+εψj1z1−u1−ψj1x1−u1ψk2z2−u2−ψk2x2−u2×τ12j′z1−x1ρ12k′z2−x2dydz≤C∫ϕj′1x1−y1y1−x1εz1−x1n1+εy2−x2εz2−x2n2+εψj1z1−u1−ψj1x1−u1ψk2z2−u2−ψk2x2−u2×τ12j′z1−x1ρ12k′z2−x2dydz≤C2−j′−jε′2−k′−kε′2−jε2−j+x1−u1n1+ε2−kε2−k+x2−u2n2+ε.
The remaining cases are similar but easier. Thus, we complete the proof.
By symmetry, from Lemma2 and Lemma 3, one can obtain the following result.Corollary 2.
For anyj,j′∈ℕ,k,k′∈ℤ, then for any 0<ε′<ε, there exists a constant C>0 such that(62)ϕj′,k′∗Tψj,k⋅−u1,⋅−u2x1,x2≤C2−j′−jε′2−k′−kε′2−j∧j′ε2−j∧j′+x1−u1n1+ε2−k∧k′ε2−k∧k′+x2−u2n2+ε.Proof of Theorem 1.
Substituting the inequality in Corollary2 to (26) and by a standard estimate, we have(63)∑j′∈ℕ,k′∈ℤ∑I′×J′∈Πj′+Nn1×Πk′+Nn2ϕj′,k′∗TfxI′,xJ′2χI′xχJ′y≤∑j′∈ℕ,k′∈ℤ∑I′×J′∈Πj′+Nn1×Πk′+Nn2∑j∈ℕ,k∈ℤ∑I×J∈Πj+Nn1×Πk+Nn2IJψj,k∗hxI,xJ×ϕj′,k′∗Tψj,k⋅−xI,⋅−xJxI′,xJ′2χI′xχJ′y≤C∑j′,k′∈ℤ,j′≥0∑j,k∈ℤ,j≥02−j′−jε′2−k′−kε′C1ℳs∑R∈Πj+Nn1×Πk+Nn2ψj,k∗hxI,xJ2χIχJr/2u,v1/r2,where maxn1/n1+ε,n2/n2+ε<r≤1 and C1=C21−1/rn1j′∧j−j+n2k′∧k−k. It is easy to have(64)2−j′−jε′2−k′−kε′C1≤C2−j′−jε′−1/r−1n12−k′−kε′−1/r−1n2,since j′∧j−j≤j′−j,k′∧k−k≤k′−k. One can choose ε′<ε close to ε sufficiently such that ε′−1/r−1n1,ε′−1/r−1n2>0. Hence, there exists some ε″>0 such that(65)2−j′−jε′2−k′−kε′C1≤C2−j′−jε″2−k′−kε″.
Hence, we obtain(66)TfHmixp≤ChHmixp≈fHmixp.
SinceL2ℝn1+n2∩Hmixp is dense in Hmixp, we complete the Proof of Theorem 1.



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