CUBO A Mathematical Journal Vol.11, No¯ 04, (109–125). September 2009 A Characterization of the Product Hardy Space H1 Luiz Antonio Pereira Gomes Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-000 Maringa - PR, Brazil email: lapgomes@uem.br and Eduardo Brandani da Silva Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-000 Maringa - PR, Brazil email: ebsilva@wnet.com.br ABSTRACT A characterization of the product space H1 such as the two parameters space H 1,2 0 is obtained, where H 1,2 0 is a particular case of spaces H P,Q S , which are generalizations of spaces studied by J. Peetre and H. Triebel. RESUMEN Se obtiene una caracterización del espacio producto H1 como el espacio a dos parámet- ros H 1,2 0 , donde H 1,2 0 es un caso particular de los espacios H P,Q S , los cuales son gener- alizaciones de los espacios estudiados por J. Peetre y H. Triebel. Key words and phrases: Product Spaces, Singular Integral Operators, Vector-valued Operators. Math. Subj. Class.: 42B30, 46E40, 47B38. 110 Gomes and da Silva CUBO 11, 4 (2009) 1 Introduction Recent advances in the theory of product Hardy and BMO spaces (see [10], [11] and [18] for instance) have called the attention of many authors, which have achieve results about old and new problems of this rich area. One of these problems concern to the characterizations of Hardy spaces. In a fundamental work within the theory of Hardy spaces Hp over product of semi planes (or with two parameters), R. F. Gundy and E. M. Stein [13] proved that two parameters space H1 may be characterized by double and partial Hilbert transforms, using area integrals and maximal functions, with equivalent norms. After this initial work, several authors obtained other charac- terizations of the two parameters space H1. For more details see, for instance, [3], [4], [6], [7], [14], [19], [20] and [21]. In this work a characterization of the Hardy space H1 over product of semi-planes such as the two parameters space H 1,2 0 is obtained. This space is a particular case of the two parameters spaces H P,Q S (when S = (0, 0), P = (1, 1) and Q = (2, 2)). The H P,Q S spaces are generalizations of the one parameter spaces Hp,qs , studied by J. Peetre and H. Triebel. For the one parameter case, a characterization of space H1, such as that obtained in this work, was initially obtained by J. Peetre [15,16]. Later, H. Triebel obtained in [24] another proof by different arguments and after him, a new proof was achieved by J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea [17]. One of ingredients for the proof of the H1 characterization obtained in this work, consists of the theorems about singular integral vector operators contained in [12]. 2 Spaces H1(IR × IR) and HP,Q S (IR × IR) in the Product Case 2.1 Notations. The notations and basic results used through this work are introduced here. The letter C always denotes a constant which may assume different values in a sequence of inequalities. S(IR2) denotes the class of rapidly decreasing functions (at infinity). Let E be a Banach space. S′(IR2,E) is the class of all continuous linear maps T defined over S(IR2) with values in E (that is, if φj → φ in S(IR2) then T(φj) → T(φ)). If E is a Banach space in relation to the norm ||.||E and P = (p1,p2) with 0 < p1,p2 ≤ ∞, L P (IR 2 ,E) is the space of all functions f defined over IR2 with values in E, such that ‖f(x)‖E is Lebesgue measurable, and ‖f‖LP (IR2,E) = (∫ IR (∫ IR ‖f(x)‖p1E dx1 )p2/p1 dx2 )1/p2 < ∞ with usual modifications when some of pi are equal to ∞. We observe that if pi = p, for i = 1, 2, then LP (IR2,E) = Lp(IR2,E). CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 111 To avoid any confusion, we write LP (E) and ‖.‖LP (E) instead of LP (IR2,E) and ‖.‖LP (IR2,E), and when E = IC, the field of complex numbers, LP and ‖.‖LP are posited. Given a Banach space E and Q = (q1,q2) with 0 < q1,q2 ≤ ∞, the (multi-)sequence spaces ℓ Q (ZZ 2 ,E) (ℓQ(E) to avoid confusion) are defined in a analogous way. If E is a Banach space, the Fourier transform of a function f ∈ L1(IR2,E) is defined by Ff(x) = f̂(x) = ∫ ∫ IR2 e −2πix·y f(y) dy , where x · y = x1.y1 + x2.y2. The following notation is used, � = {(0, 0), (1, 0), (0, 1), (1, 1)}. 2.2 Definition. Let E be a Hilbert space and f ∈ L1(IR2,E). Their Hilbert transforms Hkf, k ∈ �, are the elements of S′(IR2,E) defined by : (1) F(H10f) = −i sgx F(f)(x,y), (2) F(H01f) = −i sgy F(f)(x,y), (3) F(H11f) = (−i sgx)(−i sgy)F(f)(x,y), (4) (H00f) = f. Spaces H1(IR×IR,E) and BMO(IR×IR,E) are defined, which generalize the product spaces H 1 (IR × IR) and BMO(IR × IR) for the vectorial case. 2.3 Definition. Let E be a Hilbert space. H1(IR × IR,E) is the vector space of function f in L1(IR2,E) such that their Hilbert transforms, Hkf, k ∈ � \ {(0, 0)}, belong to L1(IR2,E). We equipped space H1(IR × IR,E) with the norm: ‖f‖H1(IR×IR,E) = ∑ k∈� ‖Hkf‖L1(IR2,E) , where H00f = f. 2.4 Definition. Let E be a Hilbert space. A function g from IR2 to E belongs to BMO(IR× IR,E), if it may be represented as g = ∑ k∈� Hkgk , (1) 112 Gomes and da Silva CUBO 11, 4 (2009) where H00g00 = g00 and ∑ k∈� ||gk||L∞(IR2,E) < ∞. We equipped the space BMO(IR×IR,E) with the norm: ||g||BMO(IR×IR,E) = inf{ ∑ k∈� ||gk||L∞(IR2,E)} , where the infimum takes over all representations of g in the form (1). Chang-Fefferman proved in [6] that for real value functions, the product space BMO(IR×IR) is the dual of the product space H1(IR×IR). This result is valid also for spaces BMO(IR×IR,E), where E is a Hilbert space; therefore, the product space BMO(IR×IR,E) is the dual of the product space H1(IR × IR,E). Results on the action of singular vector integral operators with product kernel over the prod- uct spaces H1(IR × IR,E) and BMO(IR × IR,E) are given by the two following theorems. Proofs are provided in Gomes-Silva [12]. 2.5 Theorem. Let E, F and G be Banach spaces and k1 and k2 kernels in L 2 loc(IR 2 ,L(E,F)) and L2loc(IR 2 ,L(F,G)), respectively, satisfying ∫ |x−y′|>γ|y−y′| ‖kj(x,y) − kj(x,y′)‖Lj dx ≤ C · γ−δ , j = 1, 2, (1) for every γ ≥ 2 and some δ > 0, where L1 = L(E,F) and L2 = L(F,G). Let T1 and T2 be bounded linear operators from L2(IR,E) to L2(IR,F) and from L2(IR,F) to L2(IR,G), respectively, satisfying T1f(x) = ∫ IR k1(x,u) f(u) du , (2) for every f ∈ L2c(IR,E), and T2f(y) = ∫ IR k2(y,v) f(v) dv , (3) for every f ∈ L2c(IR,F). Let T be a linear operator from L2c(IR2,E) to M(IR2,G) satisfying Tf(x,y) = ∫ ∫ IR2 k2(y,v) k1(x,u) f(u,v) du dv , (4) for every f ∈ L2c(IR2,E) and (x,y) /∈ sup f. Suppose that T has a bounded extension from L 2 (IR 2 ,E) to L2(IR2,G). Then, T has a bounded extension from H1(IR × IR,E) to L1(IR2,G); that is, there exists a constant C > 0, such that ‖Tf‖L1(IR2,G) ≤ C ‖f‖H1(IR×IR,E) , CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 113 for all f ∈ H1(IR × IR,E). 2.6 Theorem. Let E be a Banach space, F and G Hilbert spaces and k1 and k2 kernels in L 1 loc(IR 2 ,L(E,F)) and L1loc(IR 2 ,L(F,G)), respectively, satisfying ∫ |x′−y|>γ|x−x′| ‖kj(x,y) − kj(x′,y)‖Lj dx ≤ C · γ−δ , j = 1, 2, (1) for every γ ≥ 2 and some δ > 0, where L1 = L(E,F) and L2 = L(F,G). Let T1 and T2 be bounded linear operators from L2(IR,E) to L2(IR,F) and from L2(IR,F) to L2(IR,G), respectively, satisfying T1f(x) = ∫ IR k1(x,u) f(u) du , (2) for every f ∈ L∞c (IR,E), and T2f(y) = ∫ IR k2(y,v) f(v) dv , (3) for every f ∈ L∞c (IR,F). Let T be a linear operator from L∞c (IR2,E) to M(IR2,G) satisfying Tf(x,y) = ∫ ∫ IR2 k2(y,v) k1(x,u) f(u,v) du dv , (4) for every f ∈ L∞c (IR2,E) and (x,y) /∈ sup f. Suppose that T has a bounded extension from L 2 (IR 2 ,E) to L2(IR2,G). Then, T is a bounded linear operator from L∞c (IR 2 ,E) to BMO(IR × IR,G); that is, there exists a constant C > 0, such that ‖Tf‖BMO(IR×IR,G) ≤ C ‖f‖L∞(IR2,E) , for all f ∈ L∞c (IR2,E). 2.7 Lemma. There exists ϕ ∈ S(IR), such that (1) sup Fϕ = {t ∈ IR : 2−1 ≤ |t| ≤ 2} ; (2) |Fϕ(t)| > 0 se 2−1 < |t| < 2 ; (3) ∑∞ i=−∞ Fϕ(2−it) = 1 se t 6= 0 . For the proof see Berg-Löfströn [2] 114 Gomes and da Silva CUBO 11, 4 (2009) 2.8 System of Test Functions. Let ϕ be given as in the Lemma 2.7 and for each i ∈ ZZ let ϕi be the function given by ϕi(t) = 2 i ϕ(2 i t). The family (ϕi)i∈ZZ is called a system of test functions over IR. Since Fϕi(t) = Fϕ(2−it) for each i ∈ ZZ, and from 2.7(1), 2.7(2) and 2.7(3), it follows that (1) sup Fϕi = {t ∈ IR : 2i−1 ≤ |t| ≤ 2i+1} ; i ∈ ZZ ; (2) |Fϕi(t)| > 0 se 2i−1 < |t| < 2i+1 ; (3) ∑∞ i=−∞ Fϕi(t) = 1 se t 6= 0 . 2.9 Definition. Let S = (s1,s2), P = (p1,p2) and Q = (q1,q2), such that sn ∈ IR, 0 < pn < ∞ and 0 < qn ≤ ∞, n = 1, 2. Let (ϕi)i∈ZZ and (ψj)j∈ZZ be systems of test func- tions as in 2.8. Then, H P,Q S (IR × IR) = H P,Q S (IR × IR,ϕ,ψ) is the vector space of all functions f in LP (IR2) ∩ S′(IR2) with real values, satisfying (2s1i+s2jϕiψj ∗ f)ij ∈ LP (ℓQ). Spaces H P,Q S (IR×IR) are equipped with the following quasi-norm (it is a norm if min (p1,p2,q1,q2) ≥ 1) : ‖f‖ϕ,ψ H P,Q S = ‖(2s1i+s2jϕiψj ∗ f)ij‖LP (ℓQ) . (1) To avoid any confusion, we simply denote ‖f‖ϕ,ψ H P,Q S by ‖f‖ H P,Q S . When S = (s,s), P = (p,p) and Q = (q,q), then ‖f‖ϕ,ψ H P,Q S = ‖(2s(i+j)ϕiψj ∗ f)ij‖Lp(ℓq ) and the space H P,Q S (IR × IR) is simply denoted by Hp,qs (IR × IR). The next result shows that the quasi-norm 2.9(1) is independent of the systems of test func- tions (ϕi)i∈ZZ and (ψj)j∈ZZ. 2.10 Theorem. Let (αi)i∈ZZ, (βj)j∈ZZ, (ϕk)k∈ZZ and (ψl)l∈ZZ be systems of test functions as in 2.8. Let S, P and Q, as in Definition 2.9. Then the quasi-norms ‖.‖α,β H P,Q S and ‖.‖ϕ,ψ H P,Q S are equivalents, that is, there are positive constants C1 and C2, such that C1 · ‖f‖α,β H P,Q S ≤ ‖f‖ϕ,ψ H P,Q S ≤ C2 · ‖f‖α,β H P,Q S . (1) For the proof see Schmeisser-Triebel [22]. CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 115 2.11 Remark. From the proof of Theorem 2.10 it follows that condition 2.8(3) of the systems (αk)k and (βl)l is unnecessary, that is, ∞∑ k=−∞ Fαk(t) = ∞∑ l=−∞ Fβl(t) = 1 , t 6= 0 , to obtain inequalities of the type ‖f‖α,β H P,Q S ≤ C · ‖f‖ϕ,ψ H P,Q S . From systems (αk)k and (βl)l another kind of condition may be demanded, such as, ∞∑ k=−∞ [Fαk(t)]2 = ∞∑ l=−∞ [Fβl(t)]2 = 1 , t 6= 0 . This will be considered in the next section. 3 The Characterization H1(IR × IR) = H1,2 0 (IR × IR) 3.1 Lemma. Let ϕ ∈ S(IR) such that ϕ̂(0) = 0 and |ϕ̂(t)| > 0 if 2−1 < |t| < 2. Defining ϕj(x) = 2 j ϕ(2 j x), j ∈ ZZ, one has (1) ∑ j∈ZZ |ϕ̂j(t)|2 ≤ C ; (2) ∑ j∈ZZ |ϕj(x)|2 ≤ C · |x|−2; (3) ( ∑ j∈ZZ |ϕj(x − y) − ϕj(x)|2) 1 2 ≤ C · |y| |x|2 , if |x| > 2|y|. For the proof see Torrea [23]. 3.2 Theorem. Let ϕ and ψ be as in the Lemma 3.1. Then ‖(ϕiψj ∗ f)ij‖L1(IR2,ℓ2) ≤ C · ‖f‖H1(IR×IR) (1) for all f ∈ H1(IR × IR). Proof. Let us consider the linear operator defined on L2c(IR 2 ) by Tf = (ϕiψj ∗ f)ij ∈ M(IR2,ℓ2(ZZ2)). 116 Gomes and da Silva CUBO 11, 4 (2009) The operator T is well defined: if f ∈ L2c(IR2), then, by Plancherel’s Theorem and by 3.1(1), ∫ ∫ IR2 ∑ j∈ZZ ∑ i∈ZZ |ϕiψj ∗ f(x,y)|2dxdy = = ∑ j∈ZZ ∑ i∈ZZ ∫ ∫ IR2 |ϕiψj ∗ f(x,y)|2dxdy = ∑ j∈ZZ ∑ i∈ZZ ∫ ∫ IR2 |ϕ̂i(s)ψ̂j(t)f̂(s,t)|2dsdt = ∫ ∫ IR2 ( ∑ i∈ZZ |ϕ̂i(s)|2)( ∑ j∈ZZ |ψ̂j(t)|2)|f̂(s,t)|2dsdt ≤ C · ∫ ∫ IR2 |f̂(s,t)|2dsdt = C · ‖f‖L2(IR2) , (2) thus, it follows ∑ j∈ZZ ∑ i∈ZZ |ϕiψj ∗ f(x,y)|2 < ∞ for almost all (x,y); that is, Tf(x,y) ∈ ℓ2(ZZ2). To show that Tf is a measurable function, it is enough to verify that the map (x,y) −→ Tf(x,y).α is measurable for all α ∈ ℓ2(ZZ2), since ℓ2(ZZ2) is separable. If α = (αij)ij Tf(x,y).α = ∑ j∈ZZ ∑ i∈ZZ (ϕiψj ∗ f(x,y))αij = ∑ j∈ZZ ∑ i∈ZZ αijϕiψj ∗ f(x,y) , which is measurable. The inequality 3.2(2) shows that T is a bounded operator from L2(IR2) to L2(IR2,ℓ2(ZZ2)). For each n ∈ IN, let us consider the operators Tn, Tn 1 and Tn 2 defined in the following way: T n is defined on L2c(IR 2 ) by T n f = (ϕiψj ∗ f; −n ≤ i,j ≤ n) ∈ M(IR2,ℓ2(ZZ2)) ; T n 1 is defined on L2(IR) by T n 1 f = (ϕi ∗ f; −n ≤ i ≤ n) ∈ M(IR,ℓ2(ZZ)) ; CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 117 T n 2 is defined on L2c(IR,ℓ 2 (ZZ)) by T n 2 g = (ψj ∗ gi; −n ≤ i,j ≤ n) ∈ M(IR,ℓ2(ZZ2)) . Our next step it will be to show that these operators satisfy the hypothesis of Theorem 2.5. Analogously for operator T , it is easy to verify that for each n ∈ IN, Tn is well defined and Tnf is a measurable function. Moreover, from 3.2(2) it follows that operators Tn are all bounded from L 2 (IR 2 ) to L2(IR2,ℓ2(ZZ2)), with ‖Tn‖ bounded by a constant regardless of n. The operators Tn 1 are bounded from L2(IR) to L2(IR,ℓ2(ZZ)) with ‖Tn 1 ‖ bounded by a constant regardless n, since by the Plancherel’s Theorem and 3.1(1), ∫ IR n∑ i=−n |ϕi ∗ f(x)|2dx = n∑ i=−n ∫ IR |ϕ̂i(s)|2|f̂(s)|2ds ≤ C · ∫ IR |f̂(s)|2ds = C · ‖f‖L2(IR). Now, for each n ∈ IN, the kernel kn 1 defined by k n 1 (x) : λ ∈ IC −→ kn 1 (x).λ = (ϕi(x)λ; −n ≤ i ≤ n) ∈ ℓ2(ZZ) is well defined, belongs to L2loc(IR,L(IC,ℓ 2 (ZZ))), verifies the condition 2.5(1) with L1 = L(IC,ℓ 2 (ZZ)) and for all f ∈ L2c(IR), T n 1 f(x) = ∫ IR k n 1 (x − y)f(y)dy. (3) Indeed, kn 1 is well defined: if ϕ ∈ S(IR), then ‖kn 1 (x).λ‖ℓ2(ZZ) = ( n∑ i=−n |ϕi(x)|2) 1 2 |λ| ≤ C(n)|λ| (4) for all λ ∈ IC and all x ∈ IR. On the other hand, since L(IC,ℓ2(ZZ)) is isometric in ℓ2(ZZ), and the map x ∈ IR −→ n∑ i=−n αiϕi(x) is measurable for all α = (αi)i ∈ ℓ2(ZZ), it follows that kn1 is measurable. Now, if A ⊂ IR is a compact set, then by 3.2(4) 118 Gomes and da Silva CUBO 11, 4 (2009) ∫ A ‖kn 1 (x)‖2L1dx ≤ C(n)|A| < ∞ , where L1 = L(IC,ℓ 2 (ZZ)). This shows that kn 1 belongs to L2loc(IR,L1). To prove 3.2(3), since the map u ∈ IR −→ (ϕi(u)f(u); −n ≤ i ≤ n) ∈ ℓ2(ZZ) is integrable, we have T n 1 f(x) = (ϕi ∗ f(x); −n ≤ i ≤ n) = ( ∫ IR ϕi(x − u)f(u)du; −n ≤ i ≤ n) = ∫ IR (ϕi(x − u)f(u); −n ≤ i ≤ n)du = ∫ IR k n 1 (x − u)f(u)du. Finally, if |x − u′| > γ|y − u′|, with γ ≥ 2, then by 3.1(3) we obtain ‖kn 1 (x − u) − kn 1 (x − u′)‖L1 = ( n∑ i=−n |ϕi(x − u) − ϕi(x − u′)|2) 1 2 = ( n∑ i=−n |ϕi(x − u′ − (u − u′)) − ϕi(x − u′)|2) 1 2 ≤ C · |u − u ′| |x − u|2 , where C is a constant regardless of n. Therefore, the kernel kn 1 verifies the condition 2.5(1) with L1 = L(IC,ℓ 2 (ZZ)) and constant C regardless of n. The boundness of the operators Tn 2 from L2(IR,ℓ2(ZZ)) to L2(IR,ℓ2(ZZ2)), with ‖Tn 2 ‖ bounded by a constant regardless of n, follows from 3.1(1) using an analogous reasoning which was done for Tn 1 . Now, let us verify that for each n ∈ IN, there exists a kernel kn 2 in L2loc(IR,L(ℓ 2 (ZZ),ℓ 2 (ZZ 2 ))), satisfying 2.5(1) with L2 = L(ℓ 2 (ZZ),ℓ 2 (ZZ 2 )), such that T n 2 g(y) = ∫ IR k n 2 (y − v)g(v)dv, (5) for all g ∈ L2c(IR,ℓ2(ZZ)). Indeed, let kn2 be defined by k n 2 (y) : α = (αi)i ∈ ℓ2(ZZ) −→ kn2 (y).α = (ψj(y).αi; −n ≤ i,j ≤ n) ∈ ℓ2(ZZ2). CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 119 This function is well defined: if ψ ∈ S(IR), it follows that ‖kn 2 (y).α‖ℓ2(ZZ2) = ( n∑ j=−n n∑ i=−n |ψj(y)αi|2) 1 2 (6) ≤ ( n∑ j=−n |ψj(y)|2) 1 2 ( ∑ i∈ZZ |αi|2) 1 2 ≤ C(n).‖α‖ℓ2(ZZ) for all α = (αi)i ∈ ℓ2(ZZ) and all y ∈ IR. The measurability of kn2 follows from kn2 = ∑n i,j=−n k n 2,ij, where each kn 2,ij is defined by k n 2,ij(y).α = (....., 0,ψj(y)αi, 0, .....) and it is measurable, since each ψj is measurable. The fact that ‖kn2 (y)‖2L2 is locally integrable, where L2 = L(ℓ 2 (ZZ),ℓ 2 (ZZ 2 )), follows from 3.2(6). The verification of 3.2(5) is analougous to 3.2(3). As in case of the kernel kn 1 , it follows from 3.1(3) that ‖kn 2 (y − v) − kn 2 (y − v′)‖L2 ≤ C · |v − v′| |y − v′|2 where C is a constant regardless of n. Then, get kn 2 satisfies 2.5(1) with constant regardless of n. To complete the proof that operators Tn, Tn 1 and Tn 2 satisfy the hypothesis of Theorem 2.5, we observe that the map (u,v) ∈ IR2 −→ (ϕi(u)ψj(v)f(u,v); −n ≤ i,j ≤ n) ∈ ℓ2(ZZ2) is integrable when f ∈ L2c(IR2); then we have T n f(x,y) = ∫ ∫ IR2 k n 2 (y − v)kn 1 (x − u)f(u,v)dudv for all f ∈ L2c(IR2). Therefore, by Theorem 2.5, ‖Tnf‖L1(IR2,ℓ2(ZZ2)) ≤ C · ‖f‖H1(IR×IR) (7) for all n and all f ∈ H1(IR × IR), where C is a constant regardless of n. Finally, applying the theorem of monotone convergence in 3.2(7), 3.2(1) is obtained as requested. As a consequence of Theorem 3.2, the following result gives us a part of the characterization H 1 (IR × IR) = H1,2 0 (IR × IR). 120 Gomes and da Silva CUBO 11, 4 (2009) 3.3 Corollary. The space H1(IR × IR) is continuously embedded in H1,2 0 (IR × IR), that is, there is a positive constant C, such that ‖f‖ H 1,2 0 (IR×IR) ≤ C · ‖f‖H1(IR×IR) for all f ∈ H1(IR × IR). Proof. It is enough to observe the test functions used to define the space H 1,2 0 (IR×IR) satisfies the hypothesis of the Theorem 3.2. The next theorem will be fundamental to prove the contrary immersion in the Corollary 3.3; that is, the space H 1,2 0 (IR × IR) is continuously embedded in the space H1(IR × IR). 3.4 Theorem. Let ϕ and ψ be given as in the Lemma 3.1. Then ‖(ϕiψj ∗ f)ij‖BMO(IR×IR,ℓ2) ≤ C · ‖f‖L∞(IR2) , for all f ∈ L∞c (IR2), where BMO(IR×IR,ℓ2) is the topological dual of the space H1(IR×IR,ℓ2). Proof. It is enough to follow the proof of Theorem 3.2, using in this case Theorem 2.6. 3.5 Theorem. Let O be the space of the real functions f ∈ S(IR2) with real values, such that (1) f̂ ∈ C∞c (IR2), (2) sup f̂ ∩ [(IR × {0}) ∪ ({0} × IR)] = ∅. Then O is a dense subspace of H1,2 0 (IR × IR). Proof. It is enough to adapt the arguments used by H. Sato in [19] to obtain a dense subspace of H1(IR × IR). 3.6 Theorem. A function f in L1(IR2) belongs to H1(IR × IR) if, and only if f belongs to H 1,2 0 (IR × IR). Moreover, there is a constant C > 0, such that C −1 .‖f‖H1(IR×IR) ≤ ‖f‖H1,20 (IR×IR) ≤ C · ‖f‖H1(IR×IR). (1) CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 121 Proof. It is enough to prove the first inequality in 3.6(1), since the second was proved in Corollary 3.3. Let f ∈ O and g ∈ L∞c (IR2) such that ‖g‖L∞(IR2) ≤ 1. Let α = (αi)i∈ZZ e β = (βj)j∈ZZ systems of test functions as given in 2.8, but with the condition 2.8(3) replaced by∑∞ i=−∞[α̂i(s)] 2 = 1 , s 6= 0 and ∑∞ j=−∞[β̂j(t)] 2 = 1, t 6= 0 (see remark 2.11). Thus, using the polarization formula and Plancherel’s Theorem, ∫ ∫ IR2 f(x,y)g(x,y)dxdy = (2) = 1 4 [ ∫ ∫ IR2 |f + g|2dxdy − ∫ ∫ IR2 |f − g|2dxdy] = 1 4 [ ∫ ∫ IR2 ∞∑ i=−∞ [α̂i(s)] 2 ∞∑ j=−∞ [β̂j(t)] 2|F(f + g)|2dsdt − − ∫ ∫ IR2 ∞∑ i=−∞ [α̂i(s)] 2 ∞∑ j=−∞ [β̂j(t)] 2|F(f − g)|2dsdt] = 1 4 [ ∞∑ j=−∞ ∞∑ i=−∞ ∫ ∫ IR2 |F(αiβj ∗ (f + g))|2dsdt − − ∞∑ j=−∞ ∞∑ i=−∞ ∫ ∫ IR2 |F(αiβj ∗ (f − g))|2dsdt] = ∫ ∫ IR2 1 4 [ ∞∑ j=−∞ ∞∑ i=−∞ (|αiβj ∗ (f + g)|2 − |αiβj ∗ (f − g)|2)]dxdy = ∫ ∫ IR2 ∞∑ j=−∞ ∞∑ i=−∞ (αiβj ∗ f)(αiβj ∗ g)dxdy. Now, by Theorem 3.4, ‖(αiβj ∗ g)ij‖BMO(IR×IR,ℓ2) ≤ C · ‖g‖L∞(IR2) , (3) for all g ∈ L∞c (IR2). This shows that (αiβj ∗ g)ij ∈ BMO(IR × IR,ℓ2(ZZ2)). On the other hand, if we denote by H the Hilbert transform in one variable and with the convention H0ϕ = ϕ and H1ϕ = Hϕ, then for each k = (l,m) ∈ �, we have Hk(αiβj ∗ f)(x,y) = (Hlαi Hmβj ∗ f)(x,y) . (4) It is enough to prove to k = (1, 0), since the another cases are similar. Indeed, by Definition 2.2, F[H10(αiβj ∗ f)](s,t) = −i sgs F(αiβj ∗ f)(s,t) = −i sgs α̂i(s)β̂j(t)f̂(s,t) = F(Hαi)(s) β̂j(t) f̂(s,t) = F[Hαi.βj ∗ f](s,t) , 122 Gomes and da Silva CUBO 11, 4 (2009) and 3.6(4) is obtained to k = (1, 0). Moreover, the sequences (Hαi)i∈ZZ and (Hβj)j∈ZZ are systems of test functions satisfying 2.8(1) and 2.8(2), since F[Hαi](s) = −i sgs α̂i(s) and F[Hβj](t) = −i sgt β̂j(t). Thus, taking into account the Remark 2.11, Theorem 2.10 is applied to obtain ‖(αiβj ∗ f)ij‖H1(IR×IR,ℓ2) = ∑ k∈� ‖(Hk(αiβj ∗ f))ij‖L1(IR2,ℓ2) (5) = ∑ (l,m)∈� ‖(Hlαi Hmβj ∗ f)ij‖L1(IR2,ℓ2) ≤ C · ‖f‖ H 1,2 0 (IR×IR) . This shows that (αiβj ∗ f)ij ∈ H1(IR × IR,ℓ2(ZZ2)). Using 3.6(2), 3.6(3) and 3.6(5) and using the fact that BMO(IR × IR,ℓ2(ZZ2)) is the dual of H1(IR × IR,ℓ2(ZZ2)), | ∫ ∫ IR2 f.g dxdy| ≤ C · ‖(αiβj ∗ f)ij‖H1(IR×IR,ℓ2) ‖(αiβj ∗ g)ij‖BMO(IR×IR,ℓ2) ≤ C · ‖f‖ H 1,2 0 (IR×IR) . Taking the supremum over all functions g in L∞(IR2), such that ‖g‖L∞(IR2) ≤ 1, ‖f‖L1(IR2) ≤ C · ‖f‖H1,20 (IR×IR) (6) for all f ∈ O. If f belongs to O, then Hkf belongs too, for each k = (l,m) ∈ �. Therefore, 3.6(6) and Theorem 2.10 implies ‖f‖H1(IR×IR) = ∑ k∈� ‖Hkf‖L1(IR2) (7) ≤ C · ∑ k∈� ‖Hkf‖H1,20 (IR×IR) = C · ∑ k∈� ‖(ϕiψj ∗ Hkf)ij‖L1(IR2,ℓ2) = C · ∑ (l,m)∈� ‖(Hlϕi Hmψj ∗ f)ij‖L1(IR2,ℓ2) ≤ C · ‖(ϕiψj ∗ f)ij‖L1(IR2,ℓ2) = C · ‖f‖ H 1,2 0 (IR×IR) , for all f ∈ O. Finally, we may prove the inequality 3.6(7) is true for all f ∈ H1,2 0 (IR×IR). Let f ∈ H1,2 0 (IR×IR). By Theorem 3.5, O is dense in H1,2 0 (IR×IR); then there exists a sequence (fn)n of elements of O such that fn → f in the norm of H1,20 (IR × IR), from which it follows (fn)n is a CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 123 Cauchy sequence in H 1,2 0 (IR × IR). From 3.6(7) (fn)n is a Cauchy sequence in H1(IR × IR), from which results there exists an element g ∈ H1(IR×IR) such that fn → g in the norm of H1(IR×IR), since H1(IR × IR) is a complete space. By Corollary 3.3, H1(IR × IR) is continuously embedded in H 1,2 0 (IR × IR); then fn → g in the norm of H1,20 (IR × IR) and hence g = f. Thus, for all ε > 0, there is n ∈ IN, such that ‖fn −f‖H1(IR×IR) < ε and ‖fn −f‖H1,20 (IR×IR) < ε. Therefore, by 3.6(7), ‖f‖H1(IR×IR) ≤ ‖f − fn‖H1(IR×IR) + ‖fn‖H1(IR×IR) < ε + C · ‖fn − f‖H1,20 (IR×IR) + C · ‖f‖H1,20 (IR×IR) < (C + 1)ε + C · ‖f‖ H 1,2 0 (IR×IR) , for all ε > 0 and f ∈ H1,2 0 (IR × IR). This implies that 3.6(7) is true for all f ∈ H1,2 0 (IR × IR). Proof is complete. Acknowledgment. The authors would like to thank Professor Dicesar Lass Fernandez of Campinas State University for many valuable discussions and insights in this research. Received: April 2008. Revised: October 2008. References [1] A. Benedek and R. Panzone, The spaces LP with mixed norm, Duke Math. J. 28 (1961), 301–324. [2] J. Berg and J. Löfströn, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin- Heidelberg-New York, 1976. [3] N. V. P. Bertolo, On the Hardy space H1 on products of half-spaces, Pacific J. Math. 138 (1989), 347–356. [4] B. Bordin and D. L. Fernandez, Over the spaces Hp on products of semi-spaces, Anais Acad. Brasil. Ciências. 55 (1983), 319–321 (in Portuguese). [5] B. Bordin and D. L. Fernandez, On a Littlewood-Paley theorem and connections between some non-isotropic distributions spaces. Rev. Real Acad. Cien. Ex. Fis. Nat. de Madrid, 80 (1986), 133–138. [6] S. Y. A. Chang and R. Fefferman, A continuous version of the duality of H1 and BMO on the bi-disc, Ann. of Math. 112 (1980), 179–201. [7] S. Y. A. Chang and R. Fefferman, The Calderon-Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), 445–468. 124 Gomes and da Silva CUBO 11, 4 (2009) [8] S. Y. A. Chang and R. Fefferman, Some recent developments in Fourier Analysis and H p-theory in product domains, Bull. Amer. Math. Soc. 12 (1985), 1–43. [9] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645. [10] S. H. Ferguson and M. T. Lacey, A characterization of product BMO by commutators, Acta Math., 189(2) (2002), 143–160. [11] S. H. Ferguson and C. Sadoski, Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures, J. D’Analyse Math., 81 (2000), 239–267. [12] L. A. P. Gomes and E. B. Silva, Vector-valued singular integral operators on the product spaces H1 and BMO, Int. J. Pure Appl. Math., 41(4) (2007), 577–595. [13] R. Gundy and E. M. Stein, Hp theory for the polydisc, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 1026–1029. [14] K. Merryfield, On the area integral, Carleson measures and Hp in the polydisc, Indiana Univ. Math. J. 34 (1985), 663–686. [15] J. Peetre, Hp Spaces, Lecture Notes, Lund, 1974. [16] J. Peetre, On spaces of Triebel-Lizorkin type, Ark. Mat. 13 (1975), 123–130. [17] J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. Math. 62 (1986), 7–48. [18] C. Sadosky, The BMO extended family in product spaces, Harmonic Analysis, Contemp. Math., 411 (2006), 63–78. [19] H. Sato, Caractérisation par les transformations de Riesz de la classe de Hardy H1 de fonc- tions bi-harmoniques sur IRm+1 + × IRn+1 + , These de Doctorat de Troisieme Cycle de Univ. Grenoble, 1979. [20] H. Sato, La classe de Hardy H1 de fonctions bi-harmoniques sur IRm+1 + × IRn+1 + ; su carac- terisation par les transformations de Riesz, C. R. Acad. Sc. Paris 291 (1980), 91–94. [21] S. Sato, Lusin functions and nontangential maximal functions in the Hp theory on the product of upper half-spaces, Tohoku Math. J. 37 (1985), 1–13. [22] H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Birkhauser Verlag, Basel, Boston and Stuttgart, 1987. [23] J. L. Torrea, Integrales Singulares Vectoriales, Notas de Algebra y Analisis no ¯ 12, Univ. Nac. del Sur, Bahia Blanca, 1984. CUBO 11, 4 (2009) A Characterization of the Product Hardy Space H1 125 [24] H. Triebel, Spaces of Besov-Hardy-Sobolev Type, Teubner-Texte Math. 15, Leipzig, Teub- ner, 1978. [25] H. Triebel, Theory of Function Spaces, Birkhauser Verlag, Basel, Boston and Stuttgart, 1983. Articulo 8