# Dual Spaces of Multiparameter Local Hardy Spaces

Authors: Wei Ding; Feng Yu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study the duality theory of the multiparameter local Hardy spaceshpℝn1×ℝn2, and we prove that hpℝn1×ℝn2∗=cmopℝn1×ℝn2, where cmopℝn1×ℝn2 are defined by discrete Carleson measure. Moreover, we discuss the relationship among cmopℝn1×ℝn2, Lippℝn1×ℝn2, and rectangle cmorectpℝn1×ℝn2.


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## Body

## 1. Introduction

The classical theory of one-parameter harmonic analysis may be considered as centering around the Hardy-Littlewood maximal operator and its relationship with certain singular integral operators which commute with the usual one-parameter dilations onℝm, given by δx=δx1,⋯,δxm,δ>0. If this isotropic dilation is replaced by more general nonisotropic groups of dilations, then many nonisotropic variants of the classical theories can be produced, such as the strong maximal functions and multiparameter singular integral operators, corresponding to the multiparameter dilations δ:x⟶δ1x1,δ2x2,x=x1,x2∈ℝn×ℝm, and δ=δ1,δ2,δ1,δ2>0. Such a multiparameter theory has been developed extensively over the past decades. We refer the reader to the work in [1–18].SinceHpℝn space is well suited only to the pure Fourier analysis and not stable under multiplication by test functions which associated with PDE, in [19], Goldberg introduced the class of localizable Hardy spaces hpℝn,0<p<∞. Let φ∈Sℝn with ∫φ≠0 and
(1)mφfx=sup0<t<1φt∗fx.Then,hpℝn=f∈S′ℝn,mφf∈Lpℝn, where the symbol Sℝn has its usual meaning, the Schwartz class of rapidly decreasing test functions, and S′ℝn is the dual space of Sℝn. Goldberg proved that Sℝn is dense in hpℝn and hpℝn is preserved by pseudodifferential operators. Similar as Hardy space Hpℝn, local Hardy space hpℝn can be characterized by Poisson maximal function, Poisson area integral integral [19], and local Riesz transforms [20]. In 2001, Rychkov in [21] pointed out that this space has its continuous Littlewood-Paley characterization. Recently, its discrete Littlewood-Paley characterization has also been obtained in [22].Motivated by those, multiparameter local Hardy spaceshpℝn1×ℝn2 were discussed in [23, 24]. To be precise, let φ0i be functions defined on ℝni satisfying
(2)φ0i∈Dℝni,∫φ0i=1,where Dℝn denotes the set of all smooth functions with compact support on ℝn. Then, the multiparameter local Hardy spaces hpℝn1×ℝn2 are introduced in [23].Definition 1.
Let0<p<∞. For i=1,2, suppose that φ0i satisfies condition (2), and φix=φ0ix−2−niφ0ix/2. Then, the multiparameter local Hardy space hpℝn1×ℝn2 is defined to be the set of f∈S′ℝn1+n2 such that
(3)∑j,k∈ℕφj,k∗f·21/2Lpℝn1+n2<∞.In this paper, forj,k∈ℕ,j≥1, set ψj1x=2jn1ψ12jx, ψk2x=2kn2ψ22kx, ψj,kx,y=ψj1xψk2y, ψ0,kx,y=ψ01xψk2y, ψj,0x,y=ψj1xψ02y, and ψ0,0x,y=ψ01xψ02y. 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}, Πj,k=Πjn1×Πkn2, and Π=∪j,k∈ℕΠj,k, Π~=∪j,k∈ℤΠj,k.Similar to the classical one-parameter local Hardy space theory, it is proved in [23] that Sℝn1+n2 is dense in hpℝn1×ℝn2 and hpℝn1×ℝn2=Lpℝn1+n2, if p>1. Recently, we pointed out in [24] that this multiparameter local Hardy space can also be characterized by a discrete Littlewood-Paley norm. More precisely, for i=1,2, let ψ0i,ψi∈Sℝni with
(4)suppψ0i^⊆ξ∈ℝni:ξ≤2;ψ0i^ξ=1,ifξ≤1,(5)suppψi^⊆ξ∈ℝni:12≤ξ≤2,(6)ψ0i∧ξ2+∑j=1∞ψi∧2−jξ2=1,forallξ∈ℝni.Then, the following discrete multiparameter local Calderón identity is proved in [24]. For some other discrete Calderón’s identities, we refer the readers to [25–27].Theorem 2.
Fori=1,2, suppose that ψ0i and ψi∈Sℝni are functions satisfying conditions (4)-(6), respectively. Then,
(7)fx,y=∑j=0+∞∑k=0+∞∑I×J∈Πjn1×Πkn2IJψj,k∗fxI,xJ×ψj,kx−xI,y−xJ,where the series converges in L2ℝn1+n2, Sℝn1+n2, and S′ℝn1+n2.With the above discrete multiparameter local Calderón identity, multiparameter local Hardy spaceshpℝn1×ℝn2 can also be defined by discrete Littlewood-Paley norm [24]; that is, for every f∈S′ℝn1+n2, the hpℝn1×ℝn2 norm is equivalent to
(8)∑j,k∈ℕ∑I×J∈Πjn1×Πkn2ψj,k∗fxI,xJ2χI·χJ·1/2Lpℝn1+n2.The multiparameter Hardy space was introduced by Gundy and Stein in the 1970s in [11] and was developed by Chang and Fefferman in [1, 3]. Moreover, Chang and Fefferman in [1, 28] characterized duality of the product H1ℝn1×ℝn2 by the product Carleson measure. The generalized Carleson measure space CMOp was first introduced in [29]. It has been proved in [30] that the duality of multiparameter Hardy space Hpℝn1×ℝn2 is CMOpℝn1×ℝn2. We refer the readers to [26, 31] for the recent development of generalized Carleson measure spaces.The main goal of this paper is to identify the duality ofhpℝn1×ℝn2 with a new product Carleson measure space cmopℝn1×ℝn2.Definition 3.
Let0<p≤1. For i=1,2, suppose that ψ0i and ψi∈Sℝni are functions satisfying conditions (4)-(6), respectively. The multiparameter local cmopℝn1×ℝn2 is defined to be the set of f∈S′ℝn1+n2 such that
(9)fcmopℝn1×ℝn2=supΩ1Ω2/p−1∫Ω∑j,k∈ℕ∑R∈Πj,k,R⊆Ωψj,k∗fxI,xJ2χRxdx1/2<∞,where Ω ranges over all open sets in ℝn1+n2 with finite measures.To see that the spacecmopℝn1×ℝn2 is well defined, one needs to show the following theorem.Theorem 4.
Let0<p≤1. Suppose that both ψj,k and φj,k satisfy the same conditions in Definition 3. Then, one has
(10)supΩ1Ω2/p−1∫Ω∑j,k∈ℕ∑R∈Πj,k,R⊆Ωψj,k∗fxI,xJ2χRxdx1/2≈supΩ1Ω2/p−1∫Ω∑j,k∈ℕ∑R∈Πj,k,R⊆Ωφj,k∗fxI,xJ2χRxdx1/2
for allf∈S′ℝn1+n2.This result can be obtained if we prove the following Theorem5. We will give a direct proof in Section 2.Theorem 5.
Let0<p≤1. Then,
(11)hpℝn1×ℝn2∗=cmopℝn1×ℝn2.
Namely, ifg∈cmopℝn1×ℝn2, the map lg, given by lgf=f,g, defined initially for f∈Sℝn1+n2, extends to a continuous linear functional on hpℝn1×ℝn2 with lg≲gcmopℝn1×ℝn2. Conversely, every l∈hpℝn1×ℝn2∗ satisfies l=lg for some g∈cmopℝn1×ℝn2 with lg≈gcmopℝn1×ℝn2.Certainly, the dual spaces of one-parameter local Hardy spaceshpℝn can be also characterized by the above form, namely, cmopℝn. Details can be seen in [22]. On the other hand, in [19], Goldberg showed that the duality of h1ℝn is bmoℝn, which is defined as the set of f∈Lloc1ℝn such that
(12)S1f=supQ<11Q∫Qfx−fQdx<∞,S2f=supQ≥11Q∫Qfxdx<∞,equipped with the norm fbmoℝn=maxS1f,S2f, where fQ is the mean of f over Q, i.e., fQ=1/Q∫Qfxdx. For 0<r<1, let Λr=f∈L∞ℝn:supx≠yfx−fy/x−yr<∞. Then, the duality of hpℝn is Λn1/p−1 for n/n+1<p<1. We refer the reader to [19] for more details of Λr when r≥1. It is convenient to denote Λ0=bmoℝn. Since cmopℝn and Λn1/p−1 are all dualities of hpℝn for n/n+1<p≤1, cmopℝn should be coincident with Λn1/p−1. In [32], we give a direct proof to identify them with equivalent norms. Moreover, we proved that, for 0<p<1, cmopℝn is coincident with the local Lipschitz space Lippℝn defined by
(13)Lippℝn=f∈S′ℝn:fLippℝn<∞,where
(14)fLippℝn=supj≥0,x∈ℝn2jn1/p−1ψj∗fx.A natural question, does this result hold in multiparameter setting? First, we introduce the following multiparameter local Lipschitz spaces.Definition 6.
Let0<p≤1. Suppose that ψj,k satisfy the conditions in Definition 3. The multiparameter local Lipschitz space Lippℝn1×ℝn2 is defined by
(15)Lippℝn1×ℝn2=f∈S′ℝn1+n2:fLippℝn1×ℝn2<∞,where
(16)fLippℝn1×ℝn2=supj,k≥0,x∈ℝn1+n22jn11/p−12kn21/p−1ψj,k∗fx.Some results about Lipschitz spaces associated with mixed homogeneities can be seen in [33]. In this paper, one of our main results is to discuss the relationship among Lippℝn1×ℝn2, cmopℝn1×ℝn2, and the following multiparameter local rectangle cmorectpℝn1×ℝn2.Definition 7.
Let0<p≤1. For i=1,2, suppose that ψ0i and ψi∈Sℝni are functions satisfying conditions (4)-(6), respectively. The multiparameter local rectangle cmorectpℝn1×ℝn2 is defined as the set of f∈S′ℝn1+n2 such that
(17)fcmorectpℝn1×ℝn2=supR01R02/p−1∫R0∑j,k∈ℕ∑R∈Πj,k,R⊆R0ψj,k∗fxI,xJ2χRxdx1/2<∞,where the supremum is taken over all possible dyadic rectangles R0∈Π~.It is easy to see that, for0<p≤1, cmorectpℝn1×ℝn2⊇cmopℝn1×ℝn2. Moreover, due to Carleson, there is a well-known counterexample to show that this inclusion is true inclusion when p=1 [4, 34].Theorem 8.
Let0<p<1. Then, for every f∈S′ℝn1+n2, one has
(18)fcmorectpℝn1×ℝn2≲fLippℝn1×ℝn2≲fcmopℝn1×ℝn2.One may ask whetherfcmorectpℝn1×ℝn2≈fLippℝn1×ℝn2 or fcmopℝn1×ℝn2≈fLippℝn1×ℝn2 is true. It would be an interesting question.The organization of this paper is as follows. In Section2, we introduce the multiparameter ψ-transform Sψ and its inverse ψ-transform Tψ. These transforms correspond between hpℝn1×ℝn2cmopℝn1×ℝn2 and sequence spaces. Using these ψ-transform Sψ, we prove Theorem 4. In Section 3, we establish the duality of sequence spaces. The proof of Theorem 5 is placed in Section 4. In the last section, we prove Theorem 8.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. Multiparameterψ-Transform

In order to prove the duality theorems, following the idea of Frazier and Jawerth in [35], we first define and study the corresponding sequence spaces. For any R∈Πj,k, setting ψRx1,x2=R1/2ψj,kx1−xI,x2−xJ, then by (7), it is easy to have
(19)fx=∑R∈Πf,ψRψRx.Definition 9.
The multiparameterψ-transform Sψ is the map taking f∈S′ℝn1+n2 to the sequence Sψf=SψfRR∈Π, where SψfR=f,ψR. Define the inverse multiparameter ψ-transform Tψ as the map taking a sequence s=sRR∈Π to Tψs=∑R∈ΠsRψRx.By (19), for f∈Sℝn1+n2 and g∈S′ℝn1+n2, one has
(20)f,g=∑R∈ΠSψfRψRx,g=Sψf,Sψg.For a sequences=sR, one also has the following identity:
(21)Sψf,s=∑R∈Πf,ψRsR=f,∑R∈ΠsRψR=f,Tψs.Now, we define two discrete sequence spaces corresponding tohpℝn1×ℝn2 and cmopℝn1×ℝn2, respectively.Definition 10.
Let0<p≤1. Sequence space hpℝn1×ℝn2 is defined to be the collection of all complex-valued sequences s=sRR∈Π such that
(22)shpℝn1×ℝn2=∑R∈ΠsRχ~R·21/2Lp<∞,where χ~Rx=R−1/2χRx.Definition 11.
For0<p≤1, define Cpℝn1×ℝn2 to be the collection of all complex-valued sequences t=tRR∈Π such that
(23)tCpℝn1×ℝn2=supΩ1Ω2/p−1∫Ω∑R⊆Ω,R∈ΠtRχ~Rx2dx1/2,where Ω ranges over all open sets in ℝn1+n2 with finite measures, and χ~Rx is defined as in Definition 10.Then, we have the following generalization of the fundamental result of Theorem 2.2 in [35].Theorem 12.
Let0<p<∞. For i=1,2, suppose that ψ0i and ψi∈Sℝni satisfy conditions (4)-(6), respectively. Then, the operators Sψ:hpℝn1×ℝn2⟶hpℝn1×ℝn2 and Tψ:hpℝn1×ℝn2⟶hpℝn1×ℝn2 are all bounded; moreover, Tψ∘Sψ is the identity on hpℝn1×ℝn2.Proof.
The boundedness ofSψ is obvious since
(24)Sψfhpℝn1×ℝn2≈fhpℝn1×ℝn2by Definition 10 and (8).
We now prove the boundedness ofTψ. Similar proofs can be seen in [25, 31]. For a sequence s=sRR∈Π, denote fx=Tψs=∑R∈ΠsRψRx. Then, by almost orthogonality estimates [36], for any positive integers N,L, there exists a constant C such that
(25)ψj′,k′∗ψj,kxI′−xI,xJ′−xJ≤C2j∧j′n12−j−j′N1+2j∧j′xI′−xIL2k∧k′n22−∣k−k′∣N1+2k∧k′xJ′−xJL,where the notation j∧k means minj,k. Hence, for any u∈I′,v∈J′,(26)f∗ψj′,k′xI′,xJ′≲∑j,k∈ℕ2−j−j′L2−k−k′LC1Ms∑R∈Πj,kR−1sR2χIχJδ/2u,v1/δ,where δ>0, which can be sufficiently small if L is large enough, and Ms is the strong maximal operator. Summing over j′,k′ and R′∈Πj′n1×Πk′n2, one has
(27)∑j′,k′∈ℕ∑R′∈Πj′,k′ψj′,k′∗fxI′,xJ′2χI′x1′χJ′x2′1/2≤C∑j′,k′∈ℕ∑j,k∈ℕ2−j−j′L2−k−k′LC1Ms∑R∈Πj,kR−1sR2χIχJδ/2u,v1/δ21/2.
Then, by Cauchy’s inequality, we obtain(28)∑j′,k′∈ℕ∑R′∈Πj′,k′ψj′,k′∗fxI′,xJ′2χI′x1′χJ′x2′1/2≲∑j,k∈ℕMs∑R∈Πj,kR−1sR2χIχJδ/2u,v2/δ1/2.
Applying Fefferman-Stein’s vector-valued strong maximal inequality onLp/δℓ2/δ provided δ<p, we complete the proof.☐In the next theorem, we discuss the actions of the multiparameterψ-transform Sψ and its inverse ψ-transform Tψ on the space cmopℝn1×ℝn2 and discrete sequence space Cpℝn1×ℝn2, respectively. We will obtain that operators Sψ:cmopℝn1×ℝn2⟶Cpℝn1×ℝn2 and Tψ:Cpℝn1×ℝn2⟶cmopℝn1×ℝn2 are bounded, and Tψ∘Sψ is the identity on cmopℝn1×ℝn2.Theorem 13.
Let0<p<∞. For i=1,2, suppose that ψ0i and ψi∈Sℝni are functions satisfying conditions (4)-(6), respectively. Then, the operators Sψ:cmopℝn1×ℝn2⟶Cpℝn1×ℝn2 and Tψ:Cpℝn1×ℝn2⟶cmopℝn1×ℝn2 are bounded, and Tψ∘Sψ is the identity on cmopℝn1×ℝn2.Proof.
We only need to prove thatTψ is bounded. Let t=tR′R′∈Π∈Cpℝn1×ℝn2 and f=∑R′∈ΠtR′ψR′. For i=1,2, suppose that φ0i and φi∈Sℝni are functions satisfying conditions (4)-(6), respectively. We are going to prove
(29)supΩ1Ω1/p−1/2∑R=I×J⊆Ω,R∈Πφj,k∗fxI,xJ2R1/2≲supΩ1Ω1/p−1/2∑R′=I′×J′⊆Ω,R′∈ΠtR′R′−1/22R′1/2.
Given anyR′∈Πj′n1×Πk′n2, by classical almost orthogonality estimates, for any positive integers L1,L2,M1,M2, there exists a constant C such that
(30)φj,k∗ψR′xI,xJ≤CR′1/22j∧j′n12−j−j′L11+2j∧j′xI−xI′M12k∧k′n22−k−k′L21+2k∧k′xJ−xJ′M2,which implies that, for any positive integers L1,L2,M1,M2,
(31)∣φj,k∗ψR′xI,xJ∣≲CR′−1/22−∣j−j′∣L11+2j′∣xI−xI′∣M12−∣k−k′∣L21+2k′∣xJ−xJ′∣M2.
On the other hand, by conditions (4) and (5), one has that φj,k∗ψj′,k′∧ξ=0 if j′−j>4 or k′−k>4; it follows that
(32)φj,k∗fxI,xJ2≲∑R′∈ΠtR′2φj,k∗ψR′xI,xJ2.
Combining the above two estimates, one has(33)∑R=I×J⊆Ω,R∈Πφj,k∗fxI,xJ2R≲∑R=I×J⊆Ω,R∈Π∑R′=I′×J′∈ΠtR′2R′−1RII′∧I′I2L1/n111+distI,I′/ℓI′2M1JJ′∧J′J2L2/n211+distJ,J′/ℓJ′2M2=∑R=I×J⊆Ω,R∈Π∑R′=I′×J′∈ΠII′∧I′I2L1/n1JJ′∧J′J2L2/n2RR′11+distI,I′/ℓI′2M111+distJ,J′/ℓJ′2M2tR′R′−1/22R′,which gives that
(34)∑R=I×J⊆Ω,R∈Πφj,k∗fxI,xJ2R≲∑R=I×J⊆Ω,R∈Π∑R′=I′×J′∈ΠtR′R′−1/22R′II′∧I′ILJJ′∧J′JL11+distI,I′/ℓI′M11+distJ,J′/ℓJ′M
for any positive integersL,M.
Thus,(35)supΩ1Ω1/p−1/2∑R=I×J⊆Ω,R∈Πψj,k∗fxI,xJ2R1/2≲supΩ1Ω1/p−1/2∑R=I×J⊆Ω,R∈Π∑R′=I′×J′∈ΠrR,R′pR,R′tR′R′−1/22R′1/2,where
(36)rR,R′=II′∧I′ILJJ′∧J′JL,pR,R′=11+distI,I′/ℓI′M11+distJ,J′/ℓJ′M.
To obtain the inequality (29), by (35), we only need to prove
(37)supΩ1Ω1/p−1/2∑R=I×J⊆Ω,R∈Π∑R′=I′×J′∈ΠrR,R′pR,R′tR′R′−1/22R′1/2≲supΩ1Ω1/p−1/2∑R′=I′×J′⊆Ω,R′∈ΠtR′R′−1/22R′1/2.
Let(38)Ω0,0=⋃R=I×J⊆Ω,R∈Π3I×J.
Given anyR⊆Ω, let Ai,lR be the set of dyadic rectangles R′ defined as follows:
(39)A0,0R=R′=I′×J′∈Π:distI,I′≤ℓI∨ℓI′,distJ,J′≤ℓJ∨ℓJ′,and for i≥1,
(40)Ai,0R=R′=I′×J′∈Π:2i−1ℓI′∨ℓI<distI,I′≤2iℓI′∨ℓI,distJ,J′≤ℓJ∨ℓJ′,and for l≥1,
(41)A0,lR=R′=I′×J′∈Π:distI,I′≤ℓI∨ℓI′,2l−1ℓJ′∨ℓJ<distJ,J′≤2lℓJ′∨ℓJ,and for i,l≥1,
(42)Ai,lR=R′=I′×J′∈Π:2i−1ℓI′∨ℓI<distI,I′≤2iℓI′∨ℓI,2l−1ℓJ′∨ℓJ<distJ,J′≤2lℓJ′∨ℓJ,and i,l≥0,
(43)Ai,j=R′=I′×J′∈Π:32iI′×2lJ′∩Ω0,0≠∅.
Obviously, for anyR⊆Ω,R∈Π, one has that ∪i,l≥0Ai,lR=Π; moreover, Ai,lR∩Ai′,l′R=∅ if i,l≠i′,l′ and Ai,lR⊆Ai,l. It is easy to see that, for R′∈Ai,jR,i,l≥0,
(44)1+distI,I′ℓI′≿2i,1+distJ,J′ℓJ′≿2l,which gives that
(45)pR,R≾2−i+lM.
Therefore,(46)supΩ1Ω1/p−1/2∑R=I×J⊆Ω,R∈Π∑R′=I′×J′∈ΠrR,R′pR,R′tR′R′−1/22R′1/2=supΩ1Ω1/p−1/2∑R=I×J⊆Ω,R∈Π∑i,l≥0∑R′∈Ai,lRrR,R′pR,R′tR′R′−1/22R′1/2=supΩ1Ω1/p−1/2∑R=I×J⊆Ω,R∈Π∑i,l≥0∑R′∈Ai,lχR′∈Ai,lRrR,R′pR,R′tR′R′−1/22R′1/2=supΩ1Ω1/p−1/2∑i,l≥0∑R′∈Ai,l∑R=I×J⊆Ω,R∈ΠχR′∈Ai,lRrR,R′pR,R′tR′R′−1/22R′1/2≲supΩ1Ω1/p−1/2∑i,l≥0∑R′∈Ai,l∑R=I×J⊆Ω,R∈ΠχR′∈Ai,lR2−i+lMrR,R′tR′R′−1/22R′1/2=supΩ1Ω1/p−1/2∑R′∈A0,0+∑i≥1∑R′∈Ai,0+∑l≥1∑R′∈A0,l+∑i,l≥1∑R′∈Ai,l2−i+lM∑R=I×J⊆Ω,R∈ΠχR′∈Ai,lRrR,R′tR′R′−1/22R′1/2=I1+I2+I3+I41/2.
We only estimateI4 since the other three terms can be considered by the same routine. Details can be seen in [26, 29].
For each integerh≥1, let Fhi,l=R′=I′×J′∈Ai,l:∣32iI′×2lJ′∩Ω0,0∣≥1/2h∣2iI′×2lJ′∣. Denote F0i,l=∅,
(47)Dhi,l=Fhi,l\Fh−1i,l,Ωhi,l=⋃R′∈Dhi,lR′.
Then,I4 can be rewritten as
(48)I4=1Ω2/p−1∑i,l≥1∑h≥1∑R′∈Dhi,l2−i+lM∑R⊆Ω,R∈ΠχR′∈Ai,lRrR,R′tR′R′−1/22R′.
To estimateI4, we first estimate the following:
(49)∑R⊆Ω,R∈ΠχR′∈Ai,lRrR,R′.
Note that ifR′∈Ai,lR, one has 3R∩32iI′×2jJ′≠∅. For R⊆Ω,R∈Π, we consider the following four cases:
Case 1.2iI′≥I, 2lJ′≥J.
Case 2.2iI′≥I, 2lJ′≤J.
Case 3.2iI′≤I, 2lJ′≥J.
Case 4.2iI′≤I, 2lJ′≤J.
From the definition ofAi,lR, one can see that, if R′∈ Case 2, then
(50)ℓJ=2l−1ℓJ′∨ℓJ<distJ,J′≤2lℓJ′∨ℓJ=ℓJ,which implies Case 2 is an empty set. For the same reason, Case 3 and Case 4 are also empty. Hence,
(51)I4=1Ω2/p−1∑i,l≥1∑h≥1∑R′∈Dhi,l2−i+lM∑R∈Case1χR′∈Ai,lRrR,R′tR′R′−1/22R′.
SinceR′∈Ai,lR and R′∈Dhi,l, one has
(52)R≤3R∩32iI′×2lJ′≤32iI′×2lJ′∩Ω0,0≤12h−132iI′×2lJ′.
To estimaterR,R′, we should compare the side length of R with the side length of R′. Hence, we divide R⊆Ω into four categories.
Category 1.I≤I′, J≤J′.
Category 2.I≤I′, J>J′.
Category 3.I>I′, J≤J′.
Category 4.I>I′, J>J′.
In Category 1, by (52), one has 2in1+ln2R′=2h−1−2n1+n2+ηR for some integer η>0, since I,I′ are all dyadic. Note that, for fixed i,l,h, these η should be bigger than some positive integer η0 to ensure I≤I′, J≤J′, that is, satisfying 2in1+ln2/2h−1−2n1+n2+η≤1,η≥η0. Moreover, for each fixed η>η0, the number of such R’s must be less than 7n1+n22h−1−2n1+n2+η since R⊆72iI′×2lJ′. Therefore,
(53)∑R∈Case1,I≤I′,J≤J′χR′∈Ai,lRrR,R′≤∑η≥η07n1+n22h−1−2n1+n2+η2in1+ln22h−1−2n1+n2+ηL≲2−hL−1+i+2n1L+l+2n2L.
For Category 2,I≤I′, J>J′, from (52), one has
(54)IJ′≤R≤∣3R∩32iI′×2lJ′∣≤32iI′×2lJ′∩Ω0,0≤12h−132iI′×2lJ′,which implies that
(55)I≤12h−13n1+n22in1+ln2I′.
Hence,2in1+ln2I′=2h−1−2n1+n2+θI for some integer θ>0. Note that for each fixed θ>0, the number of such I’s must be less than 7n12h−1−2n1+n2−ln2+θ because I⊆72iI′. Moreover, from J′<J≤2lJ′, one has 2βJ′=J for some integer β with 1≤β≤l. For each fixed β>0, the number of such J’s must be less than 7n22l−βn2 since J⊆72lJ′. So
(56)∑R∈Case1,I≤I′,J>J′χR′∈Ai,lRrR,R′≤∑θ>0∑β=1l7n12h−1−2n1+n2−ln2+θ2in1+ln22h−1−2n1+n2+θL7n22l−βn212Ln2β≾l2−hL−1+i+2n1L+l+2n2L2l−Ln2≤2−hL−1+i+2n1L+l+4n2L.
Similar as Category 2, for Category 3, one can obtain the following estimate:(57)∑R∈Case1,I>I′,J≤J′χR′∈Ai,lRrR,R′≲2−hL−1+l+2n2L+i+4n1L.
For Category 4,I>I′, J>J′, from (52), one has
(58)R′≤12h−132iI′×2lJ′,which implies that 3n1+n22in1+ln2≥2h−1. Furthermore, by ∣R′∣≤∣R∣≤∣2iI′×2lJ′∣, one has 2in1+ln2∣R′∣=2λ∣R∣ for some integer 0≤λ≤in1+ln2 and for each fixed λ≥0, the number of such R’s must be less than 7n1+n22λ since R⊆72iI′×2lJ′. Hence,
(59)∑R∈Case1,∣I∣>∣I′∣,J>J′χR′∈Ai,lRrR,R′≤∑λ=0in1+ln27n1+n22λ2λ2in1+ln2L≲2in1+ln2.
Then,(60)I4=1Ω2/p−1∑i,l≥1∑h≥1∑R′∈Dhi,l2−i+lM∑R∈Case1χR′∈Ai,lRrR,R′tR′R′−1/22R′≲1Ω2/p−1∑i,l≥1∑h≥12−hL−1+l+4n2L+i+4n1L+∑3n1+n22in1+ln2≥2h−12in1+ln22−i+lMDhi,l2/p−11Dhi,l2/p−1∑R′∈Dhi,ltR′R′−1/22R′.
From the definition ofDhi,l, it is easy to see that Dhi,l⊆x∈ℝn1+n2:MsχΩ0,0x≥1/2h−1, which implies Dhi,l22hΩ0,022hΩ. Hence,
(61)I4≲∑i,l≥1∑h≥12−hL−1+l+4n2L+i+4n1L+∑3n1+n22in1+ln2≥2h−12in1+ln22−i+lM22h⋅supΩ1Ω2/p−1∑R′=I′×J′⊆Ω,R′∈ΠtR′R′−1/22R′≲∑i,l≥12−i+lM22l+4n2L+2i+4n1L⋅supΩ1Ω2/p−1∑R′=I′×J′⊆Ω,R′∈ΠtR′R′−1/22R′≲supΩ1Ω2/p−1∑R′=I′×J′⊆Ω,R′∈ΠtR′R′−1/22R′,by choosing proper M,L.
We complete the proof.☐Proof of Theorem 4.
Suppose thatψj,k,φj,k are functions satisfying Definition 3. We may assume that
(62)supΩ1Ω2/p−1∫Ω∑j,k∈ℕ∑R∈Πj,k,R⊆Ωφj,k∗fxI,xJ2χRxdx1/2<∞.
ForR∈Πj,k, set φRx=R1/2φj,kx1−xI,x2−xJ, and tR=f,φR=R1/2φj,k∗fxI,xJ. Then,
(63)supΩ1Ω2/p−1∫Ω∑j,k∈ℕ∑R∈Πj,k,R⊆Ωφj,k∗fxI,xJ2χRxdx1/2=supΩ1Ω2/p−1∫Ω∑R⊆Ω,R∈ΠtRχ~Rx2dx1/2,that is, t=tRR∈Π∈Cpℝn1×ℝn2.
By (19), fx=∑R∈ΠtRφRx. Hence, by Theorem 13,
(64)supΩ1Ω2/p−1∫Ω∑j,k∈ℕ∑R∈Πj,k,R⊆Ωψj,k∗fxI,xJ2χRxdx1/2supΩ1Ω2/p−1∫Ω∑R⊆Ω,R∈ΠtRχ~Rx2dx1/2.
Thus, we complete the proof.


## 3. Duality of Sequence Spaces

In this section, we establish the dual relationship betweenhpℝn1×ℝn2 and Cpℝn1×ℝn2.Theorem 14.
Let0<p≤1. Then,
(65)hpℝn1×ℝn2∗=Cpℝn1×ℝn2.Proof.
Lett∈Cpℝn1×ℝn2. For any s∈hpℝn1×ℝn2, denote
(66)Hsx=∑R∈ΠsRχ~Rx21/2,and for k∈ℤ,
(67)Ωk=x∈ℝn1+n2:Hsx>2k,Bk=R∈Π:R∩Ωk>12R,R∩Ωk+1≤12R.
Then,(68)t,s=∑R∈ΠsRtR≤∑k∑R∈BktRsR≤∑k∑R∈BktR2p/2∑R∈BksR2p/21/p.
LetΩ~k=x∈ℝn1+n2,MsχΩkx>1/2, where Msf is the strong maximal function. Then, obviously, Ω~kΩk by the boundedness of Ms, and ∪R∈BkR⊆Ω~k. Hence,
(69)∑R∈BktR2=Ω~k2/p−11Ω~k2/p−1∑R∈BktR2≤tCpℝn1×ℝn22Ω~k2/p−1.
On the other hand,x∈R∈Bk implies MsχR∩Ω~k\Ωk+1x>1/2. Therefore,
(70)∑R∈BksR2=∑R∈Bk∫sR2R−1χRxdx≲∫∑R∈BkMs2sRR−1/2χR∩Ω~k\Ωk+1xdx≲∫Ω~k\Ωk+1∑R∈BksR2R−1χRxdx≤∫Ω~k\Ωk+1Hsx2dx22kΩk.
The above two estimates yield that(71)t,stCpℝn1×ℝn2∑k2kpΩk1/ptCpℝn1×ℝn2shpℝn1×ℝn2,which implies
(72)Cpℝn1×ℝn2⊆hpℝn1×ℝn2∗.
Conversely, letℓ∈hpℝn1×ℝn2∗; then, there exists some t=tRR∈Π, such that for every s=sRR∈Π∈hpℝn1×ℝn2, ℓs=∑R∈ΠsRtR, and
(73)∑R∈ΠsRtR≤ℓshpℝn1×ℝn2.
To complete the proof, it suffices to show thattCpℝn1×ℝn2≤ℓ. To do this, for any open set Ω⊆ℝn1+n2 with finite measure, denote X=R∈Π:R⊆Ω, and let μ be a measure on X such that μR=R if R∈X, and μR=0 when R∉X. Furthermore, let l2μ be the sequence space such that if sRR∈Π∈l2μ, ∑R∈ΠsR2μR=∑R∈XsR2∣R∣<∞.
Note that(74)∑R⊆Ω,R∈ΠtR21/2=tRR−1/2R∈Πl2μ=supsl2μ≤1∑R∈ΠtRR−1/2sRμR≤ℓsupsl2μ≤1R−1/2sRμRR∈Πhpℝn1×ℝn2.
On the other hand,(75)R−1/2sRμRR∈Πhpℝn1×ℝn2=∑R⊆Ω,R∈ΠR−1/2sRRχ~Rx21/2Lpℝn1+n2=∑R⊆Ω,R∈ΠsRχRx21/2Lpℝn1+n2≤∫Ω∑R⊆Ω,R∈DsRχRx2dx1/2Ω1/p−1/2=sl2μΩ1/p−1/2,by Hölder’s inequality. It gives that
(76)∑R⊆Ω,R∈ΠtR21/2≤ℓΩ1/p−1/2,that is,
(77)1Ω1/p−1/2∑R⊆Ω,R∈ΠtR21/2≤ℓ,for any open set Ω⊆ℝn1+n2 with finite measure.
We thus complete the proof.☐


## 4. Duality ofhpℝn1×ℝn2

Once the duality of sequence spaces are established, we can obtain the dual relationship betweenhpℝn1×ℝn2 and cmopℝn1×ℝn2 by multiparameter ψ-transform.Proof of Theorem 5.
It is known thatSℝn1+n2 is dense in hpℝn1×ℝn2 ([23]). Let g∈cmopℝn1×ℝn2, f∈Sℝn1+n2. Then, by (20), we have
(78)f,g=Sψf,Sψg.
By Theorem14, Theorem 12, and Theorem 13,
(79)f,g≤Sψfhpℝn1×ℝn2SψgCpℝn1×ℝn2fhpℝn1×ℝn2gcmopℝn1×ℝn2.
This implies that for anyg∈cmopℝn1×ℝn2, the maplg, given bylgf=f,g, is a continuous linear functional onSℝn1+n2, a dense subset ofhpℝn1×ℝn2, and hence can be extended to a continuous linear functional onhpℝn1×ℝn2withlggcmopℝn1×ℝn2.
Conversely, suppose thatl∈hpℝn1×ℝn2∗. Then, l1≡l∘Tψ∈hpℝn1×ℝn2∗ by Theorem 12. So there exists t=tRR∈Π∈Cpℝn1×ℝn2 by Theorem 14 such that
(80)l1s=s,t=∑R∈ΠsRt¯Rfor all s=sRR∈Π∈hpℝn1×ℝn2, and tCpℝn1×ℝn2≈l1l for the boundedness of Tψ. Obviously, l1∘Sψ=l∘Tψ∘Sψ=lsinceTψ∘Sψis identified by Theorem 12. If we denote g=Tψt, then for f∈Sℝn1+n2, one has
(81)lf=l1Sψf=Sψf,t=f,Tψt=f,gby (21), which implies that l=lg. At last, by Theorem 13, we have
(82)gcmopℝn1×ℝn2=Tψtcmopℝn1×ℝn2tCpℝn1×ℝn2l.
Thus, we complete the proof of Theorem5.


## 5. The SpacesLippℝn1×ℝn2 and cmorectpℝn1×ℝn2

Proof of Theorem 8.
Given a dyadic rectanglesR0, obviously,
(83)1R02/p−1∫Ω∑j,k∈ℕ∑R∈Πjn1×Πkn2,R⊆Ωψj,k∗fxI,xJ2χRxdx=1R02/p−1∑j=0∞∑k=0∞∑R∈Πjn1×Πkn2,R⊆ΩRψj,k∗fxI,xJ2.
Setj0,k0∈ℤ such that R0=I0×J0∈Πj0,k0 and ∣R0∣=2−j0n12−k0n2.
Hence,(84)1R02/p−1∫R0∑j,k∈ℕ∑R∈Πjn1×Πkn2,R⊆R0ψj,k∗fxI,xJ2χRxdx≤fLippℝn1×ℝn221R02/p−1∑j=j0∨0∞∑k=k0∨0∞∑R∈Πjn1×Πkn2,R⊆R0R22jn11−1/p22kn21−1/p≤fLippℝn1×ℝn22R02−2/p∑j=j0∨0∞∑k=k0∨0∞22jn11−1/p22kn21−1/p≲fLippℝn1×ℝn22R02−2/p22j0∨0n11−1/p22k0∨0n21−1/p≤fLippℝn1×ℝn22R02−2/p22j0n11−1/p22k0n21−1/p=fLippℝn1×ℝn22,since 0<p≤1, where the notation j∨k means maxj,k.
To prove the second inequality in Theorem8, we first claim that
(85)ψj′,k′∗fx≲fcmopℝn1×ℝn2ψj′,k′x−·hpℝn1×ℝn2.
Then, if we obtain(86)ψj′,k′x−·hpℝn1×ℝn2≲2j′n11−1/p2k′n21−1/p,one can complete the proof. Now, we prove (85). Set ht=ψj′,k′x−t, and define
(87)Shx=∑j,k∈ℕ∑I×J∈Πjn1×Πkn2ψj,k∗hxI,xJ2χIxχJy1/2.
For anyi∈ℤ, denote
(88)Ωi=x∈ℝn1+n2:Shx>2i,Ω~i=x∈ℝn1+n2:MsχΩi>110n1+n2,Bi=R:R∈Π,R∩Ωi>12R,R∩Ωi+1≤12R.
Then,∪R∈BiR⊆Ω~i. By locally reproducing formula (7), one has
(89)ψj′,k′∗fx=∑j=0+∞∑k=0+∞∑I×J∈Πjn1×Πkn2IJψj,k∗fxI,xJψj′,k′∗ψj,kx−xI,y−xJ=∑j=0+∞∑k=0+∞∑I×J∈Πjn1×Πkn2IJψj,k∗fxI,xJh∗ψj,kxI,xJ=∑i=−∞+∞∑R∈BiRψR∗fxI,xJψR∗hxI,xJ,where ψj,k is denoted by ψR if R=I×J∈Πjn1×Πkn2.
Hence,(90)ψj′,k′∗fx≤∑i=−∞+∞∑R∈BiRψR∗fxI,xJψR∗gxI,xJ≤∑i=−∞+∞∑R∈BiRψR∗fxI,xJ2p/2∑R∈BiRψR∗hxI,xJ2p/21/p=∑i=−∞+∞Ω~i2/p−1∑R∈BiRψR∗hxI,xJ2p/21Ω~i2/p−1∑R∈BiRψR∗fxI,xJ2p/21/p≤fcmopℝn1×ℝn2∑i=−∞+∞Ω~i1−p/2∑R∈BiRψR∗hxI,xJ2p/21/p.
Note thatx∈R∈Bi implies MsχR∩Ω~i\Ωi+1x>1/2; then,
(91)χRx=χR2x≤22Ms2χR∩Ω~i\Ωi+1x,which follows that
(92)∑R∈BiψR∗hxI,xJχRx21/2L2ℝn1+n22≲∑R∈BiMs2ψR∗hxI,xJχR∩Ω~i\Ωi+1x1/2L2ℝn1+n22≲∫Ω~i\Ωi+1∑R∈BiψR∗hxI,xJ2χRxdxby the Fefferman-Stein vector-valued strong maximal inequality. Since Shx≤2i+1 in the domain Ω~i\Ωi+1, one has
(93)∑R∈BiψR∗hxI,xJχRx21/2L2ℝn1+n2222iΩ~i.
Hence,(94)ψj′,k′∗fx≲gcmopℝn1×ℝn2∑i=−∞+∞Ω~i1−p/222iΩ~ip/21/p≲fcmopℝn1×ℝn2hhpℝn1×ℝn2,which implies (85).
At last, to prove (86), using the continuous-type norm of hpℝn1×ℝn2, by almost orthogonality estimates, one has
(95)ψj′,k′x−·hpℝn1×ℝn2p=∫∑j.k∈ℕψj,k∗ψj′,k′x−·y2p/2dy≲∫∑j,k∈ℕ2j∧j′n1p2−j−j′Np1+2j∧j′y1Lp2k∧k′n2p2−k−k′Np1+2k∧k′y2Lpdy1dy2≲∑j,k∈ℕ2j∧j′n1p−n1·2−j−j′Np2k∧k′n2p−n2·2−k−k′Np≲2j′n1p−12k′n2p−1.
Thus, we have completed the proof of Theorem8.



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