Characterizations of function spaces by the discrete Radon transform

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Characterizations of function spaces by the discrete Radon transform

A. Abouelaz ^a & T. Kawazoe ^b

a Department of Mathematics, Faculty of Sciences and

Informatics , University Hassan II , B.P. 5366, Maarif , Casablanca , Morocco

b Department of Mathematics , Keio University at Fujisawa , Endo, Fujisawa , Kanagawa , 252-8520 , Japan

Published online: 26 Sep 2011.

To cite this article: A. Abouelaz & T. Kawazoe (2012) Characterizations of function spaces by the discrete Radon transform, Integral Transforms and Special Functions, 23:9, 627-637, DOI:

10.1080/10652469.2011.618928

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Integral Transforms and Special Functions Vol. 23, No. 9, September 2012, 627–637

Characterizations of function spaces by the discrete Radon transform

A. Abouelaz^a* and T. Kawazoe^b

aDepartment of Mathematics, Faculty of Sciences and Informatics, University Hassan II, B.P. 5366 Maarif, Casablanca, Morocco;^bDepartment of Mathematics, Keio University at Fujisawa, Endo,

Fujisawa, Kanagawa 252-8520, Japan (Received 22 May 2011; final version received 27 August 2011)

LetZⁿbe the lattice inRⁿandGthe set of all discrete hyperplanes inZⁿ. Similarly, as in the Euclidean case, for a functionfonZⁿ, the discrete Radon transformRfis defined by the integral off over discrete hyperplanes, andRmaps functions onZⁿto functions onG. In this paper, we determine the Radon transform images of the Schwartz spaceS(Zⁿ), the space of compactly supported functions onZⁿ, and a discrete Hardy spaceH¹(Zⁿ).

Keywords: linear diophantine equations; discrete Radon transform; discrete Fourier transform; characterization of the discrete Radon transform images

2000 Mathematics Subject Classification: Primary: 44A12; 44A53; Secondary: 05C25; 05C65; 08H10

1. Introduction

The Radon transform is a fundamental topic in the integral geometry, which plays a preponderant role in various fields of mathematics, physics, medicine, and so on. We briefly recall the classical Radon transformRon the Euclidean spaceRⁿ, which is defined by integrating functionsf onRⁿ over hyperplanesξ inRⁿ

Rf (ξ )=

ξ

f (x)dξ(x),

where dξ(x) is then−1-dimensional Lebesgue measure on the hyperplaneξ. The fundamental problems of the integral geometry is to recoverf(x) fromRf (ξ )and to characterize the Radon transform images of some function spaces onRⁿ. We refer to Helgason [4–6] to the ranges of the Radon transform. The goal of this paper is to obtain analogous results for the discrete Radon transformRonZⁿ.

LetZⁿbe the lattice inRⁿ. Fora =(a₁, a₂, . . . , a_n)∈Zⁿ\{0}andk∈Z, the linear diophantine equationa·x=khas an infinity of solutions inZⁿif and only ifkis an integral multiple of the

*Corresponding author. Email: [email protected]

ISSN 1065-2469 print/ISSN 1476-8291 online

http://dx.doi.org/10.1080/10652469.2011.618928 http://www.tandfonline.com

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greatest common divisor d(a) ofa₁,a₂,. . .,an. Therefore, fora∈P = {z∈Zⁿ\{0} |d(z)=1}, the set of solutionsH (a, k)= {x∈Zⁿ|a·x=k}forms a discrete hyperplane inZⁿ. LetGbe the set of all discrete hyperplanes inZⁿ, which can be parametrized asP×Z/{±1}[1, §2]. As an analogue of the Euclidean case, the discrete Radon transformR, which maps functions onZⁿ to functions onG, is given by

Rf (H (a, k))=

m∈H (a,k)

f (m)

for a suitable functionf onZⁿ. In their previous paper, Abouelaz and Ihsane [1] investigated the basic properties ofR. Especially, the Strichartz-type inversion formula and the support theorem forRwere obtained. Moreover, they showed that the discrete Radon transformRis a continuous linear mapping ofS(Zⁿ)intoS(G)(see Section 2 for the definitions of the Schwartz spaces on ZⁿandG). Our natural question is whether this mapR:S(Zⁿ)→S(G)is bijective or not. Let us suppose thatf belongs toS(Zⁿ). LetFf (t )=

m∈Zⁿf (m)e⁻^im^·^t,t∈Tⁿ, denote the Fourier (inverse) transform off. Then, it follows from [1, Corollary 3.10] that forθ∈T

F1Rf (H (a,·))(θ )=Ff (θ a),

whereF1 is the Fourier (inverse) transforms onZ. We note thatFf is aC^∞ function onTⁿ. Therefore, in order to show the surjectivity of the above mapping, we have to construct aC^∞ function onTⁿfrom (θ,a) variables inT×P. However, it is impossible, becauseθavaries in a dense subset ofTⁿ(see Remark 9). Hence, the mapRfromS(Zⁿ)intoS(G)is not bijective, and to characterize the Radon transform image ofS(Zⁿ), a condition corresponding to the above relation inS(G)is required.

This paper is organized as follows. In Section 3, a characterization of the local Schwartz space S(Zⁿa)is given. Here, we use a terminology ‘local’, when we fix ana∈P and we restrict our attention to a local areaZⁿainZⁿandGa = {H (a, k)|k∈Z}inG. A Paley–Wiener-type theorem is obtained in Section 4 (see Theorem 4). We determine the Radon transform image of the global Schwartz spaceS(Zⁿ)in Section 5 (see Corollary 7). We introduce discrete Hardy spaces onZⁿ andG, locally and globally, in Section 6 and we characterize their Radon transform images (see Theorem 13).

2. Notations

LetZⁿ be a lattice inRⁿ of alla=(a₁, a₂, . . . , an), ai ∈Z, equipped with the norm a²= a₁²+a₂²+ · · · +a_n² and the inner producta·b=a₁b₁+a₂b₂+ · · · +anbn. For 1≤p <∞, let l^p(Zⁿ)denote the space of all complex-valued functionsf onZⁿwith finite norm:

fp=

m∈Zⁿ

|f (m)|^p _1/p

<∞.

We introduce a spaceGof hyperplanes inZⁿ. Fora =(a₁, a₂, . . . , a_n)∈Zⁿ\{0}, let d(a) denote the greatest common divisor ofa1,a2,. . .,an and putP = {a∈Zⁿ\{0} |d(a)=1}. For each (a, k)∈P×Z, we define a discrete hyperplaneH(a,k) inZⁿby

H (a, k)= {m∈Zⁿ|am=k}.

We denote by G the set of all discrete hyperplanes H(a,k) with (a, k)∈P×Z, which is parametrized asP×Z/{±1}. For 1≤p <∞, letl^p(G)denote the space of all complex-valued

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Integral Transforms and Special Functions 629

functionsFonGwith finite norm:

Fa,p=

k∈Z

|F (H (a, k))|^p 1/p

<∞

for alla∈P. We introduce the Schwartz spaces onZⁿandGas follows. LetS(Zⁿ)denote the space of all complex-valued functionsf onZⁿsuch that for allN ∈N,

p_N(f )= sup

m∈Zⁿ(1+ m²)^N|f (m)|<∞

andS(G)the space of all complex-valued functionsFonGsuch that for allN ∈N,

qN(f )= sup

a∈P,k∈Z

1+k² 1+ a²

N

|F (H (a, k))|<∞.

Forf ∈l¹(Zⁿ), the discrete Radon transformRf onGis given by Rf (H (a, k))=

am=k

f (m). (1)

Then,Ris a continuous linear mapping ofl¹(Zⁿ)intol¹(G)[1, Remark 3.8] and the Strichartz-type inversion formula is given as follows:

f (m)= lim

j→∞Rf (H (a_j, a_j ·m)),

whereaj=(1,j,j²,. . .,jⁿ⁻¹) [1, Theorem 4.1]. Moreover,Ris a continuous linear mapping of S(Zⁿ)intoS(G)[1, Theorem 3.7]. Forf ∈S(Zⁿ)and a moderate growth functionφonZ

k∈Z

Rf (H (a, k))φ (k)=

m∈Zⁿ

f (m)φ (a·m)

for all a∈P [1, Proposition 3.9]. In particular, by takingφ(k)=e⁻^ikθ, it follows that for all (a, θ )∈P×T,

F1Rf (H (a,·))(θ )=Ff (θ a), (2) whereFandF1are Fourier (inverse) transforms onTⁿandT, respectively. Similarly, by taking φ(k)=k, it follows that for allp=0, 1, 2,. . .

k∈Z

Rf (H (a, k))k^p=

m∈Zⁿ

f (m)(a·m)^p. (3)

In particular, as a function ofa∈P,

k∈Z

Rf (H (a, k))k^pis a homogeneous polynomial inaof degreep. (4)

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3. Radon transform of local Schwartz spaces

As in the classical case (cf. [3, Theorem 2.4]), we shall consider local Schwartz spaces onZⁿand G, respectively. In what follows, we fix ana ∈P. For each hyperplaneH(a,k), letpa,k∈Zⁿbe the point inH(a,k) that is nearest from the origin. We set

Zⁿa = {pa,k|k∈Z}. We define a local Schwartz spaceS(Zⁿa)onZⁿby

S(Zⁿa)= {f ∈S(Zⁿ)|supp(f )⊂Zⁿa}. We denote byGathe set of all hyperplanes with directiona, that is,

Ga = {H (a, k)|k∈Z}.

SinceH (a, k)∩H (a, k)= ∅ifk=k and∪k∈ZH (a, k)=Zⁿ, it is clear thatGa∩Ga = ∅if a=a and∪a∈PGa =G(see [1, §2] for more details). We define a local Schwartz spaceS(Ga) onGby

S(Ga)= {F ∈S(G)|supp(F )⊂Ga}.

For a functionFonG, we denote byP_G_a(F )the function onGsuch thatP_G_a(F )(H )=F (H )if H ∈Gaand 0 otherwise.

Theorem1 For alla∈P,

P_G_a◦R(S(Zⁿa))=S(Ga).

Proof The argument used in the proof that R(S(Zⁿ))⊂S(G) [1, Theorem 3.7] yields that P_G_a◦R(S(Zⁿa))⊂S(G_a). We shall prove the converse inclusion. ForF ∈S(Ga), we put

f (m)=

k∈Z

F (H (a, k))χp_a,k(m),

whereχzdenotes the characteristic function of the pointz∈Zⁿ. Then,f is supported onZⁿaand it follows Proposition 3.4 in [1] that

Rf (H )=

k∈Z

F (H (a, k))χ_p^G

a,k(H ),

whereχ_z^Gis the characteristic function of the set of all hyperplanes containingz, that is,χ_z^G= R(χz). IfH=H (a, k)∈Ga, thenRf(H(a,k))=F(H(a,k)) andRf(H)=0 otherwise. Therefore, P_G_a◦R(f )=F. Hence, to complete the proof of the surjectivity, it is enough to prove that f ∈S(Zⁿa). Sincef is supported onZⁿaandpa,k ≤ |k|, it follows that for allm=pa,k,

(1+ m²)^N|f (m)| =(1+ pa,k²)^N|f (pa,k)|

≤(1+k²)^N|F (H (a, k))|.

SinceF ∈S(G), it follows thatpN(f ) <∞.

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4. A Paley–Wiener-type theorem

We shall consider a discrete Paley–Wiener theorem relative to the discrete Radon transform, which characterizes the Radon transform image of functions on Zⁿ with finite support. Let K= {x₁,x₂,. . .,xl}be a finite set inZⁿ. We denote byDK(Zⁿ)the subspace ofS(Zⁿ)consisting of all complex-valued functions onZⁿsuch that suppf ⊂K. LetGK = {H∈G|H∩K= ∅}. We denote byDK(G)the subspace ofS(G)consisting of all complex-valued functions onGsuch that suppF ⊂GK. Then, each functionf ∈DK(Zⁿ)is of the form

f =

z∈K

f (z)χz.

As shown in the support theorem [1, Theorem 5.3],Rf is of the form

Rf =

z∈K

f (z)χ_z^G (5)

and suppRf ⊂GK. Hence,Rfbelongs toDK(G). In what follows, we shall characterize functions of the form (5). We define a subspace ofDK(G)as

D∗,K(G)={F ∈DK(G)|F satisfies the moment condition (4) and for eachm∈Zⁿ, there existsjm∈Nfor which

F (H (a_j, a_j ·m))=F (H (a_j_m, a_j_m·m))for allj ≥j_m, (6) whereaj =(1, j, j², . . . , jⁿ⁻¹)}.

Lemma2 LetF ∈DK(G)anda∈P. Assume that

k∈Z

F (H (a, k))k^p =0 for allp=0,1,2, . . . .

Then, F(H(a, k))=0 for allk∈Z.

Proof We note that for a fixeda∈P,F(H(a,k))=0 except finitekand ∞

p=0

k∈Z

F (H (a, k))(iθ k)^p

p! =

k∈Z

F (H (a, k))e^{iθ k}=0

for all θ∈T. Therefore,F1F (H (a,·))(θ )=0 and thus F(H(a,k))=0 for all k∈Z. Hence,

F=0.

Lemma3 Letm∈Zⁿ.

(i) The case of m∈K: Letj_m,K=

z∈Kz−m². If j≥jm,K, then H (aj, aj ·m)∩K= ∅. (ii) The case of m∈K: Letj_K =

z∈Kz². If j>2jK, then H (aj, aj ·m)∩K= {m}.

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Proof (i) Letm∈K andj≥jm,K. We assume thatH (aj, aj ·m)∩K = ∅andz∈H (aj, aj · m)∩K. This implies that

(z₁−m₁)+(z₂−m₂)j+ · · · +(zn−mn)jⁿ⁻¹=0.

Hence,j|(z₁−m₁), and thus, there existsα∈Zsuch thatz₁−m₁=αj. Then, it follows that (αj )²=(z₁−m₁)²≤ z−m²≤j_m,K.

Therefore,αmust be 0 andz1=m1. We repeat the same argument forz2−m2and so on. Then, z=m. This contradictsm∈Kandz∈K. Hence,H (aj, aj ·m)∩K= ∅for allj≥jm,K.

(ii) Let m∈K and j>2jK. We assume that z∈H (aj, aj·m)∩K. Similarly, as above, z₁−m₁=αj. Then, it follows that

j_K≥ z²≥z²₁=m²₁+(αj )²+2αj m1

≥(αj )²−2|α|jjK = |α|j (|α|j−2jK).

Therefore,α=0 andz1=m1. We repeat the same argument forz2−m2and so on. Then,z=m.

Theorem4 R is a linear bijection ofDK(Zⁿ)ontoD∗,K(G).

Proof Letf ∈DK(Zⁿ). As mentioned in the beginning of this section,Rf ∈DK(G), and it satisfies (4), becauseDK(Zⁿ)⊂S(Zⁿ). Moreover, it follows from the proof of Theorem 4.1 in [1] that Rf satisfies (6). Therefore,Rf belongs toD∗,K(G). Since the inversion formula of R exists, it remains to prove the surjectivity ofR. LetF ∈D∗,K(G). We define a functiongonZⁿby g(m)=limj→∞F (H (a_j, a_j ·m)). Then,g∈DK(Zⁿ)by Lemma 3. SinceFsatisfies (6) andK is finite, there existsJ_K∈Nsuch thatg(m)=F(H(aj,aj·m)) for allm∈Kandj≥JK. Therefore, it follows from (3) and Lemma 3 that for allj > j₀ =max{2jK, J_K}andp=0, 1, 2,. . .,

k∈Z

Rg(H (a_j, k))k^p =

m∈Zⁿ

g(m)(a_j ·m)^p

=

m∈K

F (H (aj, aj ·m))(aj ·m)^p

=

k∈Z

⎛

⎝

m∈H (a_j,k)∩K

F (H (a_j, k))

⎞

⎠k^p

=

k∈Z

F (H (aj, k))k^p.

Hence,

k∈Z(Rg−F )(H (a_j, k))k^p=0 for allj>j₀. SinceRf andFsatisfies (4),

k∈Z(Rg− F )(H (a, k))k^pis a homogeneous polynomial inaof degreep, which is equal to 0 ata=ajfor allj>j0. Therefore,

k∈Z(Rg−F )(H (a, k))k^p=0 for alla∈P. Then, by Lemma 2, we see

thatF=Rg. This completes the proof of the theorem.

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We rewrite Theorem 4 in terms of trigonometric polynomials. LetPK(Tⁿ)=F(DK(Zⁿ)), that is, the set of trigonometric polynomials onTⁿof the form

p(x)=

m∈K

cme^im^·^x.

LetP_∗,K(P×T)denote the set of all functionsp(a,θ) onP×Twhose Fourier series p(a, θ )=

k∈Z

ca,ke^ikθ satisfies

(1) ca,k=0 ifk=a·m,m∈K, (2)

k∈Zca,kk^p is a homogeneous polynomial inaof degreep,

(3) for eachm∈Zⁿ, there existsjm∈Nsuch that for allj≥jm,caj,aj·m=ca_jm,a_jm·m. Then, it follow from Theorem 4 and (2) that

Corollary5 F1◦R◦F⁻¹is a linear bijection ofPK(Tⁿ)ontoP_∗,K(P×T). In particular, P_∗,K(P×T)= {p(θ a)|p(x)∈PK(Tⁿ)}.

5. Characterization of global spaces

We shall obtain a global characterization of the Radon transform images of subspaces ofl¹(Zⁿ) in terms of the Fourier series. For a function G(t )∈L¹(Tⁿ), let

m∈ZⁿG(m)eˆ ^im^·^t denote the Fourier series ofG. For a subspaceXofL¹(Tⁿ), letXˆ denote the space onZⁿconsisting of all Fourier coefficients ofG∈X, that is,

Xˆ = {f :Zⁿ→C| there existsG∈Xsuch thatf (m)= ˆG(m)}. LetX_Gdenote a subspace ofl¹(G)defined as

X_G = {F ∈l¹(G)| there exists aG∈Xfor whichF1F (H (a,·))(θ )

=G(θ a)for all(a, θ )∈P×T}. (7)

Theorem6 Suppose thatXˆ ⊂l¹(Zⁿ). Then, R is a bijection ofXˆ ontoX_G.

Proof SinceXˆ ⊂l¹(Zⁿ)andR(l¹(Zⁿ))⊂l¹(G),Ris defined onXˆ and eachRf,f ∈ ˆX, belongs tol¹(G). Moreover, it satisfies (7) withG=Ff ∈X(see (2)). Hence,R(X)ˆ ⊂X_G. Since the inversion formula ofRexists, it remains to prove the surjectivity ofR. LetF ∈X_Gand suppose thatF1F (H (a,·))(θ )=G(θ a)forG∈X. SinceXˆ ⊂l¹(Zⁿ), it follows that

G(θ a)=

m∈Zⁿ

G(m)eˆ ^−im^·^{θ a}=

m∈Zⁿ

G(m)eˆ ^{−iθ m}^·^a

=

k∈Z

⎛

⎝

m∈H (a,k)

G(m)ˆ

⎞

⎠e⁻^{iθ k}.

Hence,F (H (a, k))=

m∈H (a,k)G(m). Therefore, if we define a functionˆ gonZⁿbyg(m)= G(m),ˆ m∈Zⁿ, then F=Rg. Clearly, G∈X implies g∈ ˆX. This completes the proof of the

theorem.

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LetX=C^∞(Tⁿ). Then,Xˆ =S(Zⁿ)andX_Gequals to

S∗(G)= {F ∈S(G)| there exists aG∈C^∞(Tⁿ)for which (7) holds.} Corollary7 R is a bijection ofS(Zⁿ)ontoS∗(G).

Let X=A¹(Tⁿ) be the subspace of L¹(Tⁿ) consisting of all G∈L¹(Tⁿ) such that

m∈Zⁿ| ˆG(m)|<∞. Then,Xˆ =l¹(Zⁿ)andX_Gequals to

l_∗¹(G)= {F ∈l¹(G)| there exists aG∈A¹(Tⁿ)for which (7) holds.} Corollary8 R is a bijection ofl¹(Zⁿ)ontol_∗¹(G).

Remark 9 (i) In Corollaries 7 and 8,Ris continuous; however, the continuity ofR⁻¹is an open problem. (ii) The left-hand side of (7) is a function ofθwith 0≤θ≤2πandGin the right-hand side is a function onTⁿ. Therefore, for a fixeda ∈P,θin the right-hand side varies in the set of 0≤θ≤La/a, whereLais the length of the line segment with directionabetween the origin and the boundary ofTⁿ. Hence, if we rewrite (7) as

F1F (H (a,·)) θ

a

=G|

θ a a

, 0≤θ≤La, (8)

thena/a,a∈P, moves all rational points inSⁿ⁻¹∩Tⁿ. Therefore, condition (8) implies that the left-hand side defined on

{rω|ω∈Sⁿ⁻¹∩Tⁿand rational, 0≤r≤L_ω} can be extended to a functionGonTⁿ.

6. Discrete Hardy spaces

The latticeZⁿis a space of homogeneous type, becauseZⁿis equipped with a Euclidean distance and a counting measure. Hence, we can introduce real Hardy spaces onZⁿaccording to the process in [2]. We shall prove that the Radon transformRlocally mapsH¹(Zⁿ)toH¹(Z)and globally maps to an atomic Hardy space onG.

We briefly overview the definition ofH¹(Zⁿ)and its atomic decomposition. Forφ∈S(Rⁿ), we define a discrete dilationφ_t,t ∈N, ofφby

φt(x)=t⁻ⁿ

m∈Zⁿ

φ (t⁻¹m)χm(x), x∈Zⁿ.

In the Euclidean case,φtis an approximate identity ast→0. Hence, to keep this property, we put φ₀(x)=φ (0)χ₀(x).

In what follows, we suppose that

Rⁿφ (x)dx =0. For anyf ∈S(Zⁿ), we define a radial maximal functionMφf onZⁿby

M_φf (x)= sup

t=0,1,2,...|f ∗φ_t(x)|, x∈Zⁿ,

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where * is the discrete convolution onZⁿ. Then, the discrete Hardy spaceH¹(Zⁿ)is defined as H¹(Zⁿ)= {f ∈S(Zⁿ)| Mφf ∈L¹(Zⁿ)}.

We putfH¹= Mφf1. Since|f(x)| ≤cMφf(x), it follows thatH¹(Zⁿ)⊂l¹(Zⁿ). According to the process in [2, Chapter 3], we can obtain an atomic decomposition ofH¹(Zⁿ): we say that a functionbonZⁿis a(1,∞,0)-atom onZⁿif it satisfies

(i) suppb⊂B(m₀, r), (ii) b_∞≤r⁻ⁿ, (iii)

m∈Zⁿb(m)=0,

whereB(m₀,r) is a closed ball centred atm₀∈Zⁿand radiusr∈N, which depends onb. Then, f ∈H¹(Zⁿ)if and only if there exist a collection{bi}of(1,∞,0)-atom onZⁿand a sequence {λ_i}of complex numbers with

i|λ_i|<∞so that

f =

i

λ_ib_i,

and moreover,fH¹ ∼inf

i|λi|, where the infimum is taken over all atomic decomposition off. Similarly, we can defineH¹(Z)onZ. Forl∈N, we define a subspaceHl(Z)ofH (Z)as

H_l(Z)= {f ∈H (Z)| f has an atomic decomposition with(1,∞,0)-atoms onZsupported on intervals with length 2rl,r∈N}.

Now, we shall characterize the image RbonGof a (1,∞,0)-atomb onZⁿ. As pointed in Section 3, Rb=

z∈B(m₀,r)b(z)χ_z^G. Therefore, for each fixeda ∈P, as a function ofk∈Z, Rb(H(a,k)) is supported on{a·z|z∈B(m0,r)}. We denote byk0=a·m0the middle point of this support. Then, we can easily deduce that

suppRb(H (a,·))⊂ [k₀− ar, k₀+ ar],

because ifk=a·zis in the support ofRb(H(a,·)),|k−k0| = |a·(z−m0)| ≤ ar. We note that

|Rb(H (a, k))| ≤

m∈H (a,k)

|b(m)|

≤r⁻ⁿ|H (a, k)∩B(m₀, r)|

≤r⁻ⁿcrⁿ⁻¹

a =c(ar)⁻¹ (9)

and

k∈Z

Rb(H (a, k))=

m∈Zⁿ

b(m)=0. (10)

These properties imply thatc⁻¹Rb(H(a,k) is a(1,∞,0)-atom onZwith radiusar. Therefore, if we defineRaby

R_af (k)=Rf (H (a, k)), we can obtain the following.

Proposition10 For eacha∈P,Racontinuously mapsH¹(Zⁿ)intoH¹_a(Z).

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For the surjectivity, we shall prove the following lemmas.

Lemma11 Leta∈Pand Q be a cube inRⁿsuch that it is centred at the origin, each side length is2ar withr∈Nand a face parallels to H(a,0). Then, for each integer l∈[− ar,ar],

|Q∩H (a, l)|contains at least(2r)ⁿ⁻¹elements.

Proof Since Q is cubic, we may suppose l=0. When n=2, the assertion is clear.

Let a=(a1,a2,. . .,an), n≥3, and bi=(0,. . ., 0,ai,ai+1, 0, . . ., 0) for 1≤i≤n−1. Then, it follows from the case of n=2 that there exist at least 2r elements ni,j= (0,. . ., 0,mi,j,mi+1,j, 0,. . ., 0)∈Qfor whicha·nij=bi·nij=mi,jai+mi+1,jai+1=0, 1≤j≤2r.

Hence, eachni,jbelongs toQ∩H (a,0). Sinceni,j, 1≤i≤n−1, are linearly independent, the

desired result follows.

Lemma12 Leta∈P. For each(1,∞,0)-atom B onZwith radiusar, r∈N,there exist a (1,∞,0)-atom b onZⁿand a constant C for which B=CRab, where C depends only on n and a.

Proof We may suppose thatBis supported on [− ar,ar]. LetQbe a cube inRⁿsuch thatQis centred at the origin, each side length is 2arand a face parallelsH(a, 0). By Lemma 11, for each integerl∈[− ar,ar],Q∩H (a, l)contains (2r)ⁿ⁻¹integral points, say{m^q_l}, 1≤q≤(2r)ⁿ⁻¹. We define a functionbonZⁿas

b(m^q_l)=(2r)⁻⁽ⁿ⁻¹⁾B(l)

for− ar≤l≤ ar, 1≤q≤(2r)ⁿ⁻¹and 0 otherwise. Thenbis supported onQ⊂B(0, ([√ n] + 1)ar), where [x] is the greatest integer not exceedingx. We note that

m∈Zⁿ

b(m)=

l,q

b(m^q_l)=

l

B(l)=0

by the moment condition ofB. Moreover, it follows that

|b(m^q_l)| ≤(2r)⁻⁽ⁿ⁻¹⁾|B(l)| ≤2a⁻¹(2r)⁻ⁿ. Therefore, it is easy to see that 2ⁿ⁻¹([√

n] +1)⁻ⁿa⁻ⁿ⁺¹bis a(1,∞,0)-atom onZⁿ. Last we note that

Rab(l)=

m∈H (a,l)

b(m)=

q

b(m^q_l)=B(l).

Hence,B=R_ab=C_n,aR_a(C_n,a⁻¹b), whereC_n,a =2⁻ⁿ⁺¹([√

n] +1)ⁿaⁿ⁻¹. Let F ∈H¹_a(Z) and F =

iλ_iB_i an atomic decomposition of F, where each atom Bi

is supported on an interval with length 2ar. Then, it follows from Lemma 12 that there exists a(1,∞,0)-atombi onZⁿ so that Bi=Cn,aRabi. Therefore,F =R_a(

iλ_iC_n,ab_i)and

i|λ_i|C_n,a ≤C_n,aFH¹. Hence,

iλ_iC_n,ab_i ∈H¹(Zⁿ)andF ∈R_a(H¹(Zⁿ)).

Theorem13 For eacha∈P,Racontinuously mapsH¹(Zⁿ)ontoH¹_a(Z).

Now, we shall introduce an atomic Hardy space onG. LetB(m, r)⊂Zⁿbe a closed ball centred atm∈Zⁿwith radiusr∈N. We use the notation in Section 3 and we recall Theorem 4 and (9).

(13)

We say that a functionBonGis a(1,∞,0)-atom onGif it satisfies (i) B∈D_∗,B(m,r)(G),

(ii) |B(H (a, k))| ≤ |H (a, k)∩B(m, r)|r⁻ⁿ for allH (a, k)∈G, (11) (iii)

k∈Z

B(H (a, k))=0 for alla∈P,

whereB(m,r) depends onB. Clearly, if b is a(1,∞,0)-atom on Zⁿ, thenRb is a(1,∞,0)- atom onG(see Theorem 4, (9), (10)). LetB be a(1,∞,0)-atom onG. Then, by Theorem 4 and its proof, R⁻¹B is supported on B(m,r), and if x∈B(m,r), then for a sufficiently large j, H (a_j, a_j ·x)∩B(m, r)= {x} (see Lemma 3). Hence, it follows from (11) that

|R⁻¹B(x)| = |B(H(aj,aj·x))| ≤r⁻ⁿ, and thus,R⁻¹B_∞≤r⁻ⁿ. Moreover,

m∈Zⁿ

R⁻¹B(m)=

k∈Z

m∈H (a,k)

R⁻¹B(m)

=

k∈Z

B(H (a, k))=0.

Therefore, it is easy to see thatR⁻¹Bis a(1,∞,0)-atom onZⁿ.

Finally, we define the atomic Hardy spaceH_∞¹_,0(G)onGasF ∈H_∞¹_,0(G)if and only if there exist a collection{Bi}of (1,∞,0)-atoms onGand a sequence{λi}of complex numbers with

i|λi|<∞so thatF =

iλiBi. We putFH_∞,0¹ =inf

i|λi|, where the infimum is taken over all atomic decomposition ofF. Then, the previous argument yields the following.

Theorem14 R is an isomorphism ofH¹(Zⁿ)ontoH_∞¹_,0(G).

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