DISCRETE LITTLEWOOD-PALEY-STEIN THEORY AND MULTI-PARAMETER HARDY SPACES ASSOCIATED WITH FLAG SINGULAR INTEGRALS 8 0 Yongsheng Han 0 2 Department of Mathematics n Auburn University a Auburn, AL 36849, U.S.A J E-mail: [email protected] 1 1 ] A Guozhen Lu∗ C Department of Mathematics . h Wayne State University t a Detroit, MI 48202 m E-mail: [email protected] [ 1 v 1 Abstract. 0 7 The main purpose of this paper is to developa unified approach of multi-parameterHardy space 1 theory using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter . 1 structure. It is motivated by the goal to establish and develop the Hardy space theory for the flag 0 singular integral operators studied by Muller-Ricci-Stein [MRS] and Nagel-Ricci-Stein [NRS]. This 8 0 approach enables us to avoid the use of transference method of Coifman-Weiss [CW] as often used : in the Lp theory for p > 1 and establish the Hardy spaces Hp and its dual spaces associated with v F the flag singular integral operators for all 0 <p ≤1. We also prove the boundedness of flag singular i X integral operators on BMOF and HFp, and from HFp to Lp for all 0 < p ≤ 1 without using the r deep atomic decomposition. As a result, it bypasses the use of Journe’s type covering lemma in this a implicit multi-parameter structure. The method used here provides alternate approaches of those developedbyChang-R. Fefferman[CF1-3], Chang [Ch], R.Fefferman[F],Journe [J1-2], Pipher[P]in theirimportantwork inpure product setting. ACaldero´n-Zygmunddecompositionand interpolation theorem are also proved for the implicit multi-parameter Hardy spaces. Key words and phrases. Flagsingularintegrals,MultiparameterHardy spaces, DiscreteCaldero´nreproducing formulas, Discrete Littlewood-Paley-Stein analysis. (*) Research was supported partly by the U.S. NSF Grant DMS#0500853. Typeset by AMS-TEX 1 2 Y. HAN AND G. LU Table of Contents 1. Introduction and statement of results 2. Lp estimates for Littlewood-Paley-Stein square function: Proofs of Theorems 1.1 and 1.4 3. Test function spaces, almost orthogonality estimates, discrete Calder´on reproduc- ing formula: Proofs of Theorems 1.8 and 1.9 4. Discrete Littlewood-Paley-Stein square function and Hardy spaces, boundedness of flag singular integrals on Hardy spaces Hp, from Hp to Lp: Proofs of Theorems F F 1.10 and 1.11 5. Duality of Hardy spaces Hp and the boundedness of flag singular integrals on F BMO space: Proofs of Theorems 1.14, 1.16 and 1.18 F 6. Calder´on-Zgymund decomposition and interpolation on flag Hardy spaces Hp: F Proofs of Theorems 1.19 and 1.20 References 1. Introduction and statement of results The multi-parameter structures play a significant role in Fourier analysis. On the one hand, theclassicalCalder´on-Zygmund theorycanberegardedascentering aroundtheHardy-Littlewood maximal operator and certain singular integrals which commute with the usual dilations on Rn, given by δ ·x = (δx ,...,δx ) for δ > 0. On the other hand, if we consider the multi-parameter 1 n dilations on Rn, given by δ·x = (δ x ,...,δ x ), where δ = (δ ,...,δ ) ∈ Rn = (R )n, then these 1 1 n n 1 n + + n-parameter dilations are naturally associated with the strong maximal function ([JMZ]), given by 1 (1.1) M (f)(x) = sup |f(y)|dy, s |R| x∈R Z R where the supremum is taken over the family of all rectangles with sides parallel to the axes. This multi-parameter pure product theory has been developed by many authors over the past thirty years or so. For Calder´on-Zygmund theory in this setting, one considers operators of the form Tf = K ∗ f, where K is homogeneous, that is, δ ...δ K(δ · x) = K(x), or, more 1 n generally, K(x) satisfies the certain differential inequalities and cancellation conditions such that δ ...δ K(δ · x) also satisfy the same bounds. This type of operators has been the subject of 1 n extensive investigations in the literature, see for instances the fundamental works of Gundy-Stein ([GS]), R. Fefferman and Stein [FS1], R. Fefferman ([F]), Chang and R. Fefferman ([CF1], [CF2], [CF3]), Journe ([J1], [J2]), Pipher [P], etc. It is well-known that there is a basic obstacle to the pure product Hardy space theory and pure product BMO space. Indeed, the role of cubes in the classical atomic decomposition of Hp(Rn) was replaced by arbitrary open sets of finite measures in the product Hp(Rn ×Rm). Suggested by a counterexample constructed by L. Carleson [Car], the very deep product BMO(Rn ×Rm) and Hardy space Hp(Rn ×Rm) theory was developed by Chang and R. Fefferman ([Ch],[CF3]). Because of the complicated nature of atoms in product space, a certain geometric lemma, namely DISCRETE LITTLEWOOD-PALEY-STEIN MULTI-PARAMETER ANALYSIS 3 Journe’s covering lemma([J1], [J2] and [P]), played an important role in the study of the bound- edness of product singular integrals on Hp(Rn ×Rm) and BMO(Rn×Rm). While great progress has been made in the case of pure product structure for both Lp and Hp theory, multi-parameter analysis has only been developed in recent years for the Lp theory when the underlying multi -parameter structure is not explicit, but implicit, such as the flag multi-parameter structure studied in [MRS] and [NRS]. The main goal of this paper is to develop a theory of Hardy space in this setting. One of the main ideas of our program is to develop a discrete version of Calder´on reproducing formula associated with the given multiparameter structure, and thus prove a Plancherel-Pˆolya type inequality in this setting. This discrete scheme of Littlewood-Paley-Stein analysis is particularly useful in dealing with the Hardy spaces Hp for 0 < p ≤ 1. We now recall two instances of implicit multiparameter structures which are of interest to us in this paper. We begin with reviewing one of these cases first. In the work of Muller-Ricci-Stein [MRS], by considering an implicit multi-parameter structure on Heisenberg(-type) groups, the Marcinkiewicz multipliers on the Heisenberg groups yield a new class of flag singular integrals. To be more precise, let m(L,iT) be the Marcinkiewicz multiplier operator, where L is the sub- Laplacian, T is the central element of the Lie algebra on the Heisenberg group Hn, and m satisfies the Marcinkiewicz conditions. It was proved in [MRS] that the kernel of m(L,iT) satisfies the standard one-parameter Calder´on-Zygmund typeestimates associatedwith automorphic dilations intheregionwhere|t| < |z|2,andthemulti-parameterproductkernelintheregionwhere|t| ≥ |z|2 on the space Cn ×R. The proof of the Lp,1 < p < ∞, boundedness of m(L,iT) given in [MRS] requires lifting the operator to a larger group, Hn × R. This lifts K, the kernel of m(L,iT) on Hn, to a product kernel K on Hn ×R. The lifted kernel K is constructed so that it projects to K by ∞ e e K(z,t) = K(z,t−u,u)du Z −∞ e taken in the sense of distributions. The operator T corresponding to product kernel K can be dealt with in terms of tensor products of operators, and one can obtain their Lp,1 < p < ∞, boundedness by the known pure product theory. Fienally, the Lp,1 < p < ∞, boundedneses of operator with kernel K follows from transference method of Coifman and Weiss ([CW]), using the projection π : Hn × R → Hn by π((z,t),u) = (z,t+u). Anotherexampleofimplicitmulti-parameterstructureistheflagsingularintegralsonRn×Rm studied by Nagel-Ricci-Stein [NRS]. The simplest form of flag singular integral kernel K(x,y) on Rn ×Rm is defined through a projection of a product kernel K(x,y,z) defined on Rn+m ×Rm given by e (1.2) K(x,y) = K(x,y−z,z)dz. Z Rm e A more general definition of flag singular kernel was introduced in [NRS], see more details of definitions and applications of flag singular integrals there. We will also briefly recall them later in the introduction. Note that convolution with a flag singular kernel is a special case of product singularkernel. Asaconsequence, theLp,1 < p < ∞,boundedness offlagsingularintegralfollows directly from the product theory on Rn ×Rm. We note the regularity satisfied by flag singular 4 Y. HAN AND G. LU kernels is better than that of the product singular kernels. More precisely, the singularity of the standardpureproductkernel onRn×Rm,issets{(x,0)}∪{(0,y)}whilethesingularityofK(x,y), the flag singular kernel on Rn×Rm defined by (1.2), is a flag set given by {(0,0)} ⊆ {(0,y)}. For example, K (x,y) = 1 is a product kernel on R2 and K (x,y) = 1 is a flag kernel on R2. 1 xy 2 x(x+iy) The work of [NRS] suggests that a satisfactory Hardy space theory should be developed and boundedness of flag singular integrals on such spaces should be established. Thus some natural questionsarise. Fromnowon, wewillusethesubscript ”F”toexpressfunctionspacesorfunctions associated with the multi-parameter flag structure without further explanation. Question 1: What is the analogous estimate when p = 1? Namely, do we have a satisfactory flag Hardy space H1(Rn×Rm) theory associated with the flag singular integral operators? More F generally, can we develop the flag Hardy space Hp(Rn ×Rm) theory for all 0 < p ≤ 1 such that F the flag singular integral operators are bounded on such spaces? Question 2: Do we have a boundedness result on a certain type of BMO (Rn×Rm) space F for flag singular integral operators considered in [NRS]? Namely, does an endpoint estimate of the result by Nagel-Ricci-Stein hold when p = ∞? Question 3: What is the duality theory of so defined flag Hardy space? More precisely, do we have an analogue of BMO and Lipchitz type function spaces which are dual spaces of the flag Hardy spaces. Question 4: Is there a Calder´on-Zygmund decomposition in terms of functions in flag Hardy spaces Hp(Rn×Rm)? Furthermore, is there a satisfactory theory of interpolation on such spaces? F Question 5: What is the difference and relationship between the Hardy space Hp(Rn×Rm) in the pure product setting and Hp(Rn ×Rm) in flag multiparameter setting? F The original goal of our work is to address these questions. As in the Lp theory for p > 1 considered in [MRS], one is naturally tempted to establish the Hardy space theory under the implicit multi-parameter structure associated with the flag singular kernel by lifting method to the pure product setting together with the transference method in [CW]. However, this direct lifting method is not adaptable directly to the case of p ≤ 1 because the transference method is not known to be valid when p ≤ 1. This suggests that a different approach in dealing with the Hardy Hp(Rn ×Rm) space associated with this implicit multi-parameter structure is necessary. F This motivated our work in this paper. In fact, we will develop a unified approach to study multi- parameter Hardy space theory. Our approach will be carried out in the order of the following steps. (1) We first establish the theory of Littlewood-Paley-Stein square function g associated with F the implicit multi-parameter structure and the Lp estimates of g (1 < p < ∞). We then develop F a discrete Calder´on reproducing formula and a Plancherel-Polya type inequality in a test function space associated to this structure. As in the classical case of pure product setting, these Lp estimates can be used to provide a new proof of Nagel-Ricci-Stein’s Lp(1 < p < ∞) boundedenss of flag singular integral operators. p (2) We next develop the theory of Hardy spaces H associated to the multi-parameter flag F structures and the boundedness of flag singular integrals on these spaces; We then establish the boundedness of flag singular integrals from Hp to Lp. We refer to the reader the work of product F multi-parameter Hardy space theory by Chang-R. Fefferman [CF1-3], R. Fefferman [F1-3], Journe [J1-2] and Pipher [P]. DISCRETE LITTLEWOOD-PALEY-STEIN MULTI-PARAMETER ANALYSIS 5 p (3) We then establish the duality theory of the flag Hardy space H and introduce the dual F space CMOp, in particular, the duality of H1 and the space BMO . We then establish the F F F boundedness of flag singularintegrals onBMO . It isworthwhile to point out that inthe classical F one-parameter or pure product case, BMO(Rn) or BMO(Rn ×Rm) is related to the Carleson p p measure. The space CMO for all 0 < p ≤ 1, as the dual space of H introduced in this paper, F F is then defined by a generalized Carleson measure. (4) We further establish a Calder´on-Zygmund decomposition lemma for any Hp(Rn × Rm) F function (0 < p < ∞) in terms of functions in Hp1(Rn × Rm) and Hp2(Rn × Rm) with 0 < F F p < p < p < ∞. Then an interpolation theorem is established between Hp1(Rn × Rm) and 1 2 F Hp2(Rn × Rm) for any 0 < p < p < ∞ (it is noted that Hp(Rn × Rm) = Lp(Rn+m) for F 2 1 F 1 < p < ∞). In the present paper, we will use the above approach to study the Hardy space theory associ- ated with the implicit multi-parameter structures induced by the flag singular integrals. We now describe our approach and results in more details. We first introduce the continuous version of the Littlewood-Paley-Stein square function g . F Inspired by the idea of lifting method of proving the Lp(Rn ×Rm) boundedness given in [MRS], we will use a lifting method to construct a test function defined on Rn × Rm, given by the non-standard convolution ∗ on the second variable only: 2 (1.3) ψ(x,y) = ψ(1) ∗ ψ(2)(x,y) = ψ(1)(x,y−z)ψ(2)(z)dz, 2 Z Rm where ψ(1) ∈ S(Rn+m),ψ(2) ∈ S(Rm), and satisfy |ψ(1)(2−jξ ,2−jξ )|2 = 1 1 2 j X d for all (ξ ,ξ ) ∈ Rn ×Rm\{(0,0)}, and 1 2 |ψ(2)(2−kη)|2 = 1 k X b for all η ∈ Rm\{0}, and the moment conditions xαyβψ(1)(x,y)dxdy = zγψ(2)(z)dz = 0 Z Z Rn+m Rm forallmulti-indicesα,β,andγ.Weremarkherethatitisthissubtleconvolution∗ whichprovides 2 a rich theory for the implicit multi-parameter analysis. For f ∈ Lp,1 < p < ∞,g (f), the Littlewood-Paley-Stein square function of f, is defined by F 1 2 (1.4) g (f)(x,y) = |ψ ∗f(x,y)|2 F j,k   Xj Xk  where functions   (1) (2) ψ (x,y) = ψ ∗ ψ (x,y), j,k j 2 k 6 Y. HAN AND G. LU ψ(1)(x,y) = 2(n+m)jψ(1)(2jx,2jy) and ψ(2)(z) = 2mkψ(2)(2kz). j k We remark here that the terminology ”implicit multi-parameter structure” is clear from the fact that the dilation ψ (x,y) is not induced from ψ(x,y) explicitly. j,k By taking the Fourier transform, it is easy to see the following continuous version of the Calder´on reproducing formula holds on L2(Rn+m), (1.5) f(x,y) = ψ ∗ψ ∗f(x,y). j,k j,k j k XX Note that if one considers the summation on the right hand side of (1.5) as an operator then, by the construction of function ψ, it is a flag singular integral and has the implicit multi-parameter structure as mentioned before. Using iteration and the vector-valued Littlewood-Paley-Stein estimate together with the Calder´on reproducing formula on L2 allows us to obtain the Lp,1 < p < ∞, estimates of g . F Theorem 1.1:. Let 1 < p < ∞. Then there exist constants C and C depending on p such that 1 2 for C kfk ≤ kg (f)k ≤ C kfk . 1 p F p 2 p In order to state our results for flag singular integrals, we need to recall some definitions given in [NRS]. Following closely from [NRS], we begin with the definitions of a class of distributions on an Euclidean space RN. A k −normalized bump function on a space RN is a Ck−function supported on the unit ball with Ck−norm bounded by 1. As pointed out in [NRS], the definitions given below are independent of the choices of k, and thus we will simply refer to ”normalized bump function” without specifying k. For the sake of simplicity of presentations, we will restrict our considerations to the case RN = Rn+m ×Rm. We will rephrase Definition 2.1.1 in [NRS] of product kernel in this case as follows: Definition1.2:. A productkernel on Rn+m×Rm is a distributionK on Rn+m+m which coincides with a C∞ function away from the coordinate subspaces (0,0,z) and (x,y,0), where (0,0) ∈ Rn+m and (x,y) ∈ Rn+m, and satisfies (1) (Differential Inequalities) For any multi-indices α = (α ,··· ,α ), β = (β ,··· ,β ) and 1 n 1 m γ = (γ ,··· ,γ ) m 1 m |∂α∂β∂γK(x,y,z)| ≤ C (|x|+|y|)−n−m−|α|−|β| ·|z|−m−|γ| x y z α,β,γ for all (x,y,z) ∈ Rn ×Rm ×Rm with |x|+|y| =6 0 and |z| =6 0. (2) (Cancellation Condition) | ∂α∂βK(x,y,z)φ (δz)dz| ≤ C (|x|+|y|)−n−m−|α|−|β| x y 1 α,β ZRm for all multi-indices α,β and every normalized bump function φ on Rm and every δ > 0; 1 | ∂γK(x,y,z)φ (δx,δy)dxdy| ≤ C |z|−m−|γ| z 2 γ ZRm DISCRETE LITTLEWOOD-PALEY-STEIN MULTI-PARAMETER ANALYSIS 7 for every multi-index γ and every normalized bump function φ on Rn+m and every δ > 0; and 2 | K(x,y,z)φ (δ x,δ y,δ z)dxdydz| ≤ C 3 1 1 2 ZRn+m+m for every normalized bump function φ on Rn+m+m and every δ > 0 and δ > 0. 3 1 2 Definition 1.3:. A flag kernel on Rn × Rm is a distribution on Rn+m which coincides with a C∞ function away from the coordinate subspace {(0,y)} ⊂ Rn+m, where 0 ∈ Rn and y ∈ Rm and satisfies (1) (Differential Inequalities) For any multi-indices α = (α ,··· ,α ), β = (β ,··· ,β ) 1 n 1 m |∂α∂βK(x,y)| ≤ C |x|−n−|α| ·(|x|+|y|)−m−|β| x y α,β for all (x,y) ∈ Rn ×Rm with |x| =6 0. (2) (Cancellation Condition) | ∂αK(x,y)φ (δy)dy| ≤ C |x|−n−|α| x 1 α ZRm for every multi-index α and every normalized bump function φ on Rm and every δ > 0; 1 | ∂βK(x,y)φ (δx)dx| ≤ C |y|−m−|β| y 2 γ ZRn for every multi-index β and every normalized bump function φ on Rn and every δ > 0; and 2 | K(x,y)φ (δ x,δ y)dxdy| ≤ C 3 1 2 ZRn+m for every normalized bump function φ on Rn+m and every δ > 0 and δ > 0. 3 1 2 By a result in [MRS], we may assume first that a flag kernel K lies in L1(Rn+m). Thus, there exists a product kernel K♯ on Rn+m ×Rm such that K(x,y) = K♯(x,y−z,z)dz. ZRm Conversely, if a product kernel K♯ lies in L1(Rn+m×Rm), then K(x,y) defined as above is a flag kernel on Rn×Rm. As pointed out in [MRS], we may always assume that K(x,y), a flag kernel, is integrable on Rn ×Rm by using a smooth truncation argument. As a consequence of Theorem 1.1, we give a new proof of the Lp,1 < p < ∞, boundedness of flag singular integrals due to Nagel, Ricci and Stein in [NRS]. More precisely, let T(f)(x,y) = K ∗ f(x,y) be a flag singular integral on Rn ×Rm. Then K is a projection of a product kernel K♯ on Rn+m ×Rm. 8 Y. HAN AND G. LU Theorem 1.4:. Suppose that T is a flag singular integral defined on Rn × Rm with the flag kernel K(x,y) = K♯(x,y − z,z)dz, where the product kernel K♯ satisfies the conditions of Rm Definition 1.2 aboveR. Then T is bounded on Lp for 1 < p < ∞. Moreover, there exists a constant C depending on p such that for f ∈ Lp,1 < p < ∞, kT(f)k ≤ Ckfk . p p In order to use the Littlewood-Paley-Stein square function g to define the Hardy space, one F needstoextendtheLittlewood-Paley-Steinsquarefunctiontobedefinedonasuitabledistribution space. For this purpose, we first introduce the product test function space on Rn+m ×Rm. Definition 1.5:. A Schwartz test function f(x,y,z) defined on Rn ×Rm ×Rm is said to be a product test function on Rn+m ×Rm if (1.6) f(x,y,z)xαyβdxdy = f(x,y,z)zγdz = 0 Z Z for all multi-indices α,β,γ of nonnegative integers. If f is a product test function on Rn+m ×Rm we denote f ∈ S (Rn+m×Rm) and the norm ∞ of f is defined by the norm of Schwartz test function. We now define the test function space S on Rn ×Rm associated with the flag structure. F Definition 1.6:. A function f(x,y) defined on Rn×Rm is said to be a test function in S (Rn× F Rm) if there exists a function f♯ ∈ S (Rn+m ×Rm) such that ∞ (1.7) f(x,y) = f♯(x,y −z,z)dz. Z Rm If f ∈ S (Rn ×Rm), then the norm of f is defined by F kfk = inf{kf♯k : for all representations of f in (1.7)}. SF(Rn×Rm) S∞(Rn+m×Rm) We denote by (S )′ the dual space of S . F F We would like to point out that the implicit multi-parameter structure is involved in S . Since F thefunctions ψ constructed abovebelongtoS (Rn×Rm),sotheLittlewood-Paley-Steinsquare j,k F function g can be defined for all distributions in (S )′. Formally, we can define the flag Hardy F F space as follows. Definition 1.7:. Let 0 < p ≤ 1. Hp(Rn ×Rm) = {f ∈ (S )′ : g (f) ∈ Lp(Rn ×Rm)}. F F F If f ∈ Hp(Rn ×Rm), the norm of f is defined by F (1.8) kfkHp = kgF(f)kp. F A natural question arises whether this definition is independent of the choice of functions ψ . j,k p Moreover, to study the H -boundedness of flag singular integrals and establish the duality result F p p of H , this formal definition is not sufficiently good. We need to discretize the norm of H . In F F p order to obtain such a discrete H norm we will prove the Plancherel-Pˆolya-type inequalities. F The main tool to provide such inequalities is the Calder´on reproducing formula (1.5). To be more specific, we will prove that the formula (1.5) still holds on test function space S (Rn×Rm) and F its dual space (S )′ (see Theorem 3.6 below). Furthermore, using an approximation procedure F and the almost orthogonality argument, we prove the following discrete Calder´on reproducing formula. DISCRETE LITTLEWOOD-PALEY-STEIN MULTI-PARAMETER ANALYSIS 9 Theorem 1.8:. Suppose that ψ are the same as in (1.4). Then j,k (1.9) f(x,y) = |I||J|ψ (x,y,x ,y )ψ ∗f(x ,y ) j,k I J j,k I J j k J I XXXX e where ψ (x,y,x ,y ) ∈ S (Rn × Rm),I ⊂ Rn,J ⊂ Rm, are dyadic cubes with side-length j,k I J F ℓ(I) = 2−j−N and ℓ(J) = 2−k−N +2−j−N for a fixed large integer N,x ,y are any fixed points I J in I,J,erespectively, and the series in (1.9) converges in the norm of S (Rn × Rm) and in the F dual space (S )′. F The discrete Calder´on reproducing formula (1.9) provides the following Plancherel-Pˆolya-type inequalities. We use the notation A ≈ B to denote that two quantities A and B are comparable independent of other substantial quantities involved in the context. Theorem 1.9:. Suppose ψ(1),φ(1) ∈ S(Rn+m),ψ(2),φ(2) ∈ S(Rm) and ψ(x,y) = ψ(1)(x,y−z)ψ(2)(z)dz, Z Rm φ(x,y) = φ(1)(x,y−z)ψ(2)(z)dz, Z Rm and ψ , φ satisfy the conditions in (1.4). Then for f ∈ (S )′ and 0 < p < ∞, jk jk F k{ sup |ψj,k ∗f(u,v)|2χI(x)χJ(y)}21kp u∈I,v∈J j k J I XXXX (1.10) ≈ k{ inf |φj,k ∗f(u,v)|2χI(x)χJ(y)}21||p u∈I,v∈J j k J I XXXX where ψ (x,y) and φ (x,y) are defined as in (1.4), I ⊂ Rn,J ⊂ Rm, are dyadic cubes with j,k j,k side-length ℓ(I) = 2−j−N and ℓ(J) = 2−k−N +2−j−N for a fixed large integer N,χ and χ are I J indicator functions of I and J, respectively. The Plancherel-Pˆolya-type inequalities in Theorem 1.9 give the discrete Littlewood-Paley- Stein square function 1 2 (1.11) gd(f)(x,y) = |ψ ∗f(x ,y )|2χ (x)χ (y) F  j,k I J I J  Xj Xk XJ XI    where I,J,x , and y are the same as in Theorem 1.8 and Theorem 1.9. I J p p From this it is easy to see that the Hardy space H in (1.8) is well defined and the H norm F F of f is equivalent to the Lp norm of gd. By use of the Plancherel-Pˆolya-type inequalities, we will F p prove the boundedness of flag singular integrals on H . F 10 Y. HAN AND G. LU Theorem 1.10:. Suppose that T is a flag singular integral with the kernel K(x,y) satisfying the p same conditions as in Theorem 1.4. Then T is bounded on H , for 0 < p ≤ 1. Namely, for all F 0 < p ≤ 1 there exists a constant C such that p kT(f)kHp ≤ CpkfkHp. F F To obtain the Hp → Lp boundedness of flag singular integrals, we prove the following general F result: Theorem 1.11. Let 0 < p ≤ 1. If T is a linear operator which is bounded on L2(Rn+m) and Hp(Rn×Rm), then T can be extended to a bounded operator from Hp(Rn×Rm) to Lp(Rn+m). F F From the proof, we can see that this general result holds in a very broad setting, which includes the classical one-parameter and product Hardy spaces and the Hardy spaces on spaces of homogeneous type. In particular, for flag singular integral we can deduce from this general result the following Corollary 1.12:. Let T be a flag singular integral as in Theorem 1.4. Then T is bounded from Hp(Rn ×Rm) to Lp(Rn+m) for 0 < p ≤ 1. F p p To study the duality of H , we introduce the space CMO . F F Definition 1.13:. Let ψ be the same as in (1.4). We say that f ∈ CMOp if f ∈ (S )′ and it j,k F F has the finite norm kfkCMOp defined by F 1 2 1 (1.12) sup |ψ ∗f(x,y)|2χ (x)χ (y)dxdy  2−1 j,k I J  Ω |Ω|p  Xj,k ΩZ I,J:XI×J⊆Ω  for all open setsΩ in Rn ×Rm with finite measures, and I ⊂ Rn,J ⊂ Rm, are dyadic cubes with side-length ℓ(I) = 2−j and ℓ(J) = 2−k +2−j respectively. Note that the Carleson measure condition is used and the implicit multi-parameter structure is involved in CMOp space. When p = 1, as usual, we denote by BMO the space CMO1. To F F F p p see the space CMO is well defined, one needs to show the definition of CMO is independent F F p of the choice of the functions ψ . This can be proved, again as in the Hardy space H , by the j,k F following Plancherel-Pˆolya-type inequality. Theorem 1.14:. Suppose ψ,φ satisfy the same conditions as in Theorem 1.9. Then for f ∈ (S )′, F 1 2 1 sup sup |ψ ∗f(u,v)|2|I||J| ≈ Ω |Ω|p2−1 u∈I,v∈J j,k   Xj Xk I×XJ⊆Ω    1 2 1 (1.13) sup inf |φ ∗f(u,v)|2|I||J| Ω |Ω|p2−1 u∈I,v∈J j,k   Xj Xk I×XJ⊆Ω   