DISCRETE LITTLEWOOD-PALEY-STEIN THEORY AND MULTI-PARAMETER HARDY SPACES ASSOCIATED WITH FLAG SINGULAR INTEGRALS

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1 DSCRETE LTTLEWOOD-PALEY-STEN THEORY AND MULT-PARAMETER HARDY SPACES ASSOCATED WTH LAG SNGULAR NTEGRALS Yongsheng Han Department of Mathematics Auburn University Auburn, AL 36849, U.S.A Guozhen Lu Department of Mathematics Wayne State University Detroit, M gzlu@math.wayne.edu Abstract. The main purpose of this paper is to develop a unified approach of multi-parameter Hardy space theory using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. t is motivated by the goal to establish and develop the Hardy space theory for the flag singular integral operators studied by Muller-Ricci-Stein [MRS] and Nagel-Ricci-Stein [NRS]. This approach enables us to avoid the use of transference method of Coifman-Weiss [CW] as often used in the L p theory for p > and establish the Hardy spaces H p and its dual spaces associated with the flag singular integral operators for all 0 < p. We also prove the boundedness of flag singular integral operators on BMO and H p, and from Hp to Lp for all 0 < p without using the deep atomic decomposition. As a result, it bypasses the use of ourne s type covering lemma in this implicit multi-parameter structure. The method used here provides alternate approaches of those developed by Chang-R. efferman [C-3], Chang [Ch], R. efferman [], ourne [-], Pipher [P] in their important wor in pure product setting. A Calderón-Zygmund decomposition and interpolation theorem are also proved for the implicit multi-parameter Hardy spaces. Key words and phrases. lag singular integrals, Multiparameter Hardy spaces, Discrete Calderón reproducing formulas, Discrete Littlewood-Paley-Stein analysis. *) Research was supported partly by the U.S. NS Grant DMS# Typeset by AMS-TEX

2 Y. HAN AND G. LU. ntroduction and statement of results Table of Contents. L p estimates for Littlewood-Paley-Stein square function: Proofs of Theorems. and.4 3. Test function spaces, almost orthogonality estimates, discrete Calderón reproducing formula: Proofs of Theorems.8 and.9 4. Discrete Littlewood-Paley-Stein square function and Hardy spaces, boundedness of flag singular integrals on Hardy spaces H p, from Hp to Lp : Proofs of Theorems.0 and. 5. Duality of Hardy spaces H p and the boundedness of flag singular integrals on BMO space: Proofs of Theorems.4,.6 and.8 6. Calderón-Zgymund decomposition and interpolation on flag Hardy spaces H p : Proofs of Theorems.9 and.0 References. ntroduction and statement of results The multi-parameter structures play a significant role in ourier analysis. On the one hand, the classical Calderón-Zygmund theory can be regarded as centering around the Hardy-Littlewood maximal operator and certain singular integrals which commute with the usual dilations on R n, given by δ x = δx,..., δx n ) for δ > 0. On the other hand, if we consider the multi-parameter dilations on R n, given by δ x = δ x,..., δ n x n ), where δ = δ,..., δ n ) R n + = R + ) n, then these n-parameter dilations are naturally associated with the strong maximal function [MZ]), given by.) M s f)x) = sup fy) dy, x R R R where the supremum is taen 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. or Calderón-Zygmund theory in this setting, one considers operators of the form T f = K f, where K is homogeneous, that is, δ...δ n Kδ x) = Kx), or, more generally, Kx) satisfies the certain differential inequalities and cancellation conditions such that δ...δ n Kδ x) also satisfy the same bounds. This type of operators has been the subject of extensive investigations in the literature, see for instances the fundamental wors of Gundy-Stein [GS]), R. efferman and Stein [S], R. efferman []), Chang and R. efferman [C], [C], [C3]), ourne [], []), Pipher [P], etc. t is well-nown that there is a basic obstacle to the pure product Hardy space theory and pure product BMO space. ndeed, the role of cubes in the classical atomic decomposition of H p R n ) was replaced by arbitrary open sets of finite measures in the product H p R n ). Suggested by a counterexample constructed by L. Carleson [Car], the very deep product BMOR n ) and Hardy space H p R n ) theory was developed by Chang and R. efferman [Ch],[C3]). Because of the complicated nature of atoms in product space, a certain geometric lemma, namely

3 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 3 ourne s covering lemma[], [] and [P]), played an important role in the study of the boundedness of product singular integrals on H p R n ) and BMOR n ). While great progress has been made in the case of pure product structure for both L p and H p theory, multi-parameter analysis has only been developed in recent years for the L p 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 Calderón reproducing formula associated with the given multiparameter structure, and thus prove a Plancherel-Pôlya type inequality in this setting. This discrete scheme of Littlewood-Paley-Stein analysis is particularly useful in dealing with the Hardy spaces H p for 0 < p. 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. n the wor of Muller-Ricci-Stein [MRS], by considering an implicit multi-parameter structure on Heisenberg-type) groups, the Marciniewicz multipliers on the Heisenberg groups yield a new class of flag singular integrals. To be more precise, let ml, it ) be the Marciniewicz multiplier operator, where L is the sub- Laplacian, T is the central element of the Lie algebra on the Heisenberg group H n, and m satisfies the Marciniewicz conditions. t was proved in [MRS] that the ernel of ml, it ) satisfies the standard one-parameter Calderón-Zygmund type estimates associated with automorphic dilations in the region where t < z, and the multi-parameter product ernel in the region where t z on the space C n R. The proof of the L p, < p <, boundedness of ml, it ) given in [MRS] requires lifting the operator to a larger group, H n R. This lifts K, the ernel of ml, it ) on H n, to a product ernel K on H n R. The lifted ernel K is constructed so that it projects to K by taen in the sense of distributions. Kz, t) = Kz, t u, u)du The operator T corresponding to product ernel K can be dealt with in terms of tensor products of operators, and one can obtain their L p, < p <, boundedness by the nown pure product theory. inally, the L p, < p <, boundedness of operator with ernel K follows from transference method of Coifman and Weiss [CW]), using the projection π : H n R H n by πz, t), u) = z, t + u). Another example of implicit multi-parameter structure is the flag singular integrals on R n studied by Nagel-Ricci-Stein [NRS]. The simplest form of flag singular integral ernel Kx, y) on R n is defined through a projection of a product ernel Kx, y, z) defined on R n+m given by.) Kx, y) = Kx, y z, z)dz. A more general definition of flag singular ernel 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 ernel is a special case of product singular ernel. As a consequence, the L p, < p <, boundedness of flag singular integral follows directly from the product theory on R n. We note the regularity satisfied by flag singular

4 4 Y. HAN AND G. LU ernels is better than that of the product singular ernels. More precisely, the singularity of the standard pure product ernel on R n, is sets {x, 0)} {0, y)} while the singularity of Kx, y), the flag singular ernel on R n defined by.), is a flag set given by {0, 0)} {0, y)}. or example, K x, y) = xy is a product ernel on R and K x, y) = xx+iy) is a flag ernel on R. The wor 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 questions arise. rom now on, we will use the subscript to express function spaces or functions associated with the multi-parameter flag structure without further explanation. Question : What is the analogous estimate when p =? Namely, do we have a satisfactory flag Hardy space H Rn ) theory associated with the flag singular integral operators? More generally, can we develop the flag Hardy space H p Rn ) theory for all 0 < p such that the flag singular integral operators are bounded on such spaces? Question : Do we have a boundedness result on a certain type of BMO R n ) space 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: s there a Calderón-Zygmund decomposition in terms of functions in flag Hardy spaces H p Rn )? urthermore, is there a satisfactory theory of interpolation on such spaces? Question 5: What is the difference and relationship between the Hardy space H p R n ) in the pure product setting and H p Rn ) in flag multiparameter setting? The original goal of our wor is to address these questions. As in the L p theory for p > considered in [MRS], one is naturally tempted to establish the Hardy space theory under the implicit multi-parameter structure associated with the flag singular ernel 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 because the transference method is not nown to be valid when p. This suggests that a different approach in dealing with the Hardy H p Rn ) space associated with this implicit multi-parameter structure is necessary. This motivated our wor in this paper. n fact, we will develop a unified approach to study multiparameter Hardy space theory. Our approach will be carried out in the order of the following steps. ) We first establish the theory of Littlewood-Paley-Stein square function g associated with the implicit multi-parameter structure and the L p estimates of g < p < ). We then develop a discrete Calderón 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 L p estimates can be used to provide a new proof of Nagel-Ricci-Stein s L p < p < ) boundedenss of flag singular integral operators. ) We next develop the theory of Hardy spaces H p associated to the multi-parameter flag structures and the boundedness of flag singular integrals on these spaces; We then establish the boundedness of flag singular integrals from H p to Lp. We refer to the reader the wor of product multi-parameter Hardy space theory by Chang-R. efferman [C-3], R. efferman [-3], ourne [-] and Pipher [P].

5 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 5 3) We then establish the duality theory of the flag Hardy space H p and introduce the dual space CMO p, in particular, the duality of H and the space BMO. We then establish the boundedness of flag singular integrals on BMO. t is worthwhile to point out that in the classical one-parameter or pure product case, BMOR n ) or BMOR n ) is related to the Carleson measure. The space CMO p for all 0 < p, as the dual space of Hp introduced in this paper, is then defined by a generalized Carleson measure. 4) We further establish a Calderón-Zygmund decomposition lemma for any H p Rn ) function 0 < p < ) in terms of functions in H p Rn ) and H p Rn ) with 0 < p < p < p <. Then an interpolation theorem is established between H p Rn ) and H p Rn ) for any 0 < p < p < it is noted that H p Rn ) = L p R n+m ) for < p < ). n the present paper, we will use the above approach to study the Hardy space theory associated 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. nspired by the idea of lifting method of proving the L p R n ) boundedness given in [MRS], we will use a lifting method to construct a test function defined on R n, given by the non-standard convolution on the second variable only:.3) ψx, y) = ψ ) ψ ) x, y) = ψ ) x, y z)ψ ) z)dz, where ψ ) SR n+m ), ψ ) S ), and satisfy for all ξ, ξ ) R n \{0, 0)}, and for all η \{0}, and the moment conditions ψ ) j ξ, j ξ ) = j ψ ) η) = R n+m x α y β ψ ) x, y)dxdy = z γ ψ ) z)dz = 0 for all multi-indices α, β, and γ. We remar here that it is this subtle convolution which provides a rich theory for the implicit multi-parameter analysis. or f L p, < p <, g f), the Littlewood-Paley-Stein square function of f, is defined by.4) g f)x, y) = ψ j, fx, y) j where functions ψ j, x, y) = ψ ) j ψ ) x, y),

6 6 Y. HAN AND G. LU ψ ) j x, y) = n+m)j ψ ) j x, j y) and ψ ) z) = m ψ ) z). We remar here that the terminology implicit multi-parameter structure is clear from the fact that the dilation ψ j, x, y) is not induced from ψx, y) explicitly. By taing the ourier transform, it is easy to see the following continuous version of the Calderón reproducing formula holds on L R n+m ),.5) fx, y) = ψ j, ψ j, fx, y). j Note that if one considers the summation on the right hand side of.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 Calderón reproducing formula on L allows us to obtain the L p, < p <, estimates of g. Theorem.:. Let < p <. Then there exist constants C and C depending on p such that for C f p g f) p C f p. n order to state our results for flag singular integrals, we need to recall some definitions given in [NRS]. ollowing closely from [NRS], we begin with the definitions of a class of distributions on an Euclidean space R N. A normalized bump function on a space R N is a C function supported on the unit ball with C norm bounded by. As pointed out in [NRS], the definitions given below are independent of the choices of, and thus we will simply refer to normalized bump function without specifying. or the sae of simplicity of presentations, we will restrict our considerations to the case R N = R n+m. We will rephrase Definition.. in [NRS] of product ernel in this case as follows: Definition.:. A product ernel on R n+m is a distribution K on R n+m+m which coincides with a C function away from the coordinate subspaces 0, 0, z) and x, y, 0), where 0, 0) R n+m and x, y) R n+m, and satisfies ) Differential nequalities) or any multi-indices α = α,, α n ), β = β,, β m ) and γ m = γ,, γ m ) α x β y γ z Kx, y, z) C α,β,γ x + y ) n m α β z m γ for all x, y, z) R n with x + y = 0 and z = 0. ) Cancellation Condition) x α y β Kx, y, z)φ δz)dz C α,β x + y ) n m α β for all multi-indices α, β and every normalized bump function φ on and every δ > 0; z γ Kx, y, z)φ δx, δy)dxdy C γ z m γ

7 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 7 for every multi-index γ and every normalized bump function φ on R n+m and every δ > 0; and Kx, y, z)φ 3 δ x, δ y, δ z)dxdydz C R n+m+m for every normalized bump function φ 3 on R n+m+m and every δ > 0 and δ > 0. Definition.3:. A flag ernel on R n is a distribution on R n+m which coincides with a C function away from the coordinate subspace {0, y)} R n+m, where 0 R n and y and satisfies ) Differential nequalities) or any multi-indices α = α,, α n ), β = β,, β m ) for all x, y) R n with x = 0. ) Cancellation Condition) α x β y Kx, y) C α,β x n α x + y ) m β x α Kx, y)φ δy)dy C α x n α for every multi-index α and every normalized bump function φ on and every δ > 0; y β Kx, y)φ δx)dx C γ y m β R n for every multi-index β and every normalized bump function φ on R n and every δ > 0; and Kx, y)φ 3 δ x, δ y)dxdy C R n+m for every normalized bump function φ 3 on R n+m and every δ > 0 and δ > 0. By a result in [MRS], we may assume first that a flag ernel K lies in L R n+m ). Thus, there exists a product ernel K on R n+m such that Kx, y) = K x, y z, z)dz. Conversely, if a product ernel K lies in L R n+m ), then Kx, y) defined as above is a flag ernel on R n. As pointed out in [MRS], we may always assume that Kx, y), a flag ernel, is integrable on R n by using a smooth truncation argument. As a consequence of Theorem., we give a new proof of the L p, < p <, boundedness of flag singular integrals due to Nagel, Ricci and Stein in [NRS]. More precisely, let T f)x, y) = K fx, y) be a flag singular integral on R n. Then K is a projection of a product ernel K on R n+m.

8 8 Y. HAN AND G. LU Theorem.4:. Suppose that T is a flag singular integral defined on R n with the flag ernel Kx, y) = K x, y z, z)dz, where the product ernel K satisfies the conditions of Definition. above. Then T is bounded on L p for < p <. Moreover, there exists a constant C depending on p such that for f L p, < p <, T f) p C f p. n order to use the Littlewood-Paley-Stein square function g to define the Hardy space, one needs to extend the Littlewood-Paley-Stein square function to be defined on a suitable distribution space. or this purpose, we first introduce the product test function space on R n+m. Definition.5:. A Schwartz test function fx, y, z) defined on R n is said to be a product test function on R n+m if.6) fx, y, z)x α y β dxdy = fx, y, z)z γ dz = 0 for all multi-indices α, β, γ of nonnegative integers. f f is a product test function on R n+m we denote f S R n+m ) and the norm of f is defined by the norm of Schwartz test function. We now define the test function space S on R n associated with the flag structure. Definition.6:. A function fx, y) defined on R n is said to be a test function in S R n ) if there exists a function f S R n+m ) such that.7) fx, y) = f x, y z, z)dz. f f S R n ), then the norm of f is defined by f S R n ) = inf{ f S R n+m ) : for all representations of f in.7)}. We denote by S ) the dual space of S. We would lie to point out that the implicit multi-parameter structure is involved in S. Since the functions ψ j, constructed above belong to S R n ), so the Littlewood-Paley-Stein square function g can be defined for all distributions in S ). ormally, we can define the flag Hardy space as follows. Definition.7:. Let 0 < p. H p Rn ) = {f S ) : g f) L p R n )}. f f H p Rn ), the norm of f is defined by.8) f H p = g f) p. A natural question arises whether this definition is independent of the choice of functions ψ j,. Moreover, to study the H p -boundedness of flag singular integrals and establish the duality result of H p, this formal definition is not sufficiently good. We need to discretize the norm of Hp. n order to obtain such a discrete H p norm we will prove the Plancherel-Pôlya-type inequalities. The main tool to provide such inequalities is the Calderón reproducing formula.5). To be more specific, we will prove that the formula.5) still holds on test function space S R n ) and its dual space S ) see Theorem 3.6 below). urthermore, using an approximation procedure and the almost orthogonality argument, we prove the following discrete Calderón reproducing formula.

9 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 9 Theorem.8:. Suppose that ψ j, are the same as in.4). Then.9) fx, y) = j ψ j, x, y, x, y )ψ j, fx, y ) where ψ j, x, y, x, y ) S R n ), R n,, are dyadic cubes with side-length l) = j N and l) = N + j N for a fixed large integer N, x, y are any fixed points in,, respectively, and the series in.9) converges in the norm of S R n ) and in the dual space S ). The discrete Calderón reproducing formula.9) provides the following Plancherel-Pôlya-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.9:. Suppose ψ ), φ ) SR n+m ), ψ ), φ ) S ) and ψx, y) = ψ ) x, y z)ψ ) z)dz, φx, y) = φ ) x, y z)ψ ) z)dz, and ψ j, φ j satisfy the conditions in.4). Then for f S ) and 0 < p <, { j sup u,v ψ j, fu, v) χ x)χ y)} p.0) { j inf φ j, fu, v) χ x)χ y)} p u,v where ψ j, x, y) and φ j, x, y) are defined as in.4), R n,, are dyadic cubes with side-length l) = j N and l) = N + j N for a fixed large integer N, χ and χ are indicator functions of and, respectively. The Plancherel-Pôlya-type inequalities in Theorem.9 give the discrete Littlewood-Paley- Stein square function.) g d f)x, y) = ψ j, fx, y ) χ x)χ y) j where,, x, and y are the same as in Theorem.8 and Theorem.9. rom this it is easy to see that the Hardy space H p in.8) is well defined and the Hp norm of f is equivalent to the L p norm of g d. By use of the Plancherel-Pôlya-type inequalities, we will prove the boundedness of flag singular integrals on H p.

10 0 Y. HAN AND G. LU Theorem.0:. Suppose that T is a flag singular integral with the ernel Kx, y) satisfying the same conditions as in Theorem.4. Then T is bounded on H p, for 0 < p. Namely, for all 0 < p there exists a constant C p such that T f) H p C p f H p. To obtain the H p Lp boundedness of flag singular integrals, we prove the following general result: Theorem.. Let 0 < p. f T is a linear operator which is bounded on L R n+m ) and H p Rn ), then T can be extended to a bounded operator from H p Rn ) to L p R n+m ). rom 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. n particular, for flag singular integral we can deduce from this general result the following Corollary.:. Let T be a flag singular integral as in Theorem.4. Then T is bounded from H p Rn ) to L p R n+m ) for 0 < p. To study the duality of H p, we introduce the space CMOp. Definition.3:. Let ψ j, be the same as in.4). We say that f CMO p if f S ) and it has the finite norm f CMO p defined by.) sup Ω Ω p j, Ω,: Ω ψ j, fx, y) χ x)χ y)dxdy for all open sets Ω in R n with finite measures, and R n,, are dyadic cubes with side-length l) = j and l) = + j respectively. Note that the Carleson measure condition is used and the implicit multi-parameter structure is involved in CMO p space. When p =, as usual, we denote by BMO the space CMO. To see the space CMO p is well defined, one needs to show the definition of CMOp is independent of the choice of the functions ψ j,. This can be proved, again as in the Hardy space H p, by the following Plancherel-Pôlya-type inequality. Theorem.4:. Suppose ψ, φ satisfy the same conditions as in Theorem.9. Then for f S ), sup sup ψ Ω Ω p j, fu, v) u,v.3) sup Ω Ω p j j Ω Ω inf φ j, fu, v) u,v

11 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS where R n,, are dyadic cubes with side-length l) = j N and l) = N + j N for a fixed large integer N respectively, and Ω are all open sets in R n with finite measures. To show that space CMO p is the dual space of Hp, we also need to introduce the sequence spaces. Definition.5:. Let s p be the collection of all sequences s = {s } such that s s p = s χ x)χ y) <, j,, where the sum runs over all dyadic cubes R n, with side-length l) = j N and l) = N + j N for a fixed large integer N, and χ, and χ are indicator functions of and respectively. Let c p be the collection of all sequences s = {s } such that.4) s c p = sup s Ω Ω p j,,: Ω L p <, where Ω are all open sets in R n with finite measures and the sum runs over all dyadic cubes R n,, with side-length l) = j N and l) = N + j N for a fixed large integer N. We would lie to point out again that certain dyadic rectangles used in s p and c p reflect the implicit multi-parameter structure. Moreover, the Carleson measure condition is used in the definition of c p. Next, we obtain the following duality theorem. Theorem.6:. Let 0 < p. Then we have s p ) = c p. More precisely, the map which maps s = {s } to < s, t > s t defines a continuous linear functional on s p with operator norm t s p ) t c p, and moreover, every l sp ) is of this form for some t c p. When p =, this theorem in the one-parameter setting on R n was proved in []. The proof given in [] depends on estimates of certain distribution functions, which seems to be difficult to apply to the multi-parameter case. or all 0 < p we give a simple and more constructive proof of Theorem.6, which uses the stopping time argument for sequence spaces. Theorem.6 together with the discrete Calderón reproducing formula and the Plancherel-Pôlyatype inequalities yields the duality of H p. Theorem.7:. Let 0 < p. Then H p ) = CMO p. More precisely, if g CMOp, the map l g given by l g f) =< f, g >, defined initially for f S, extends to a continuous linear functional on H p with l g g CMO p. Conversely, for every l Hp ) there exists some g CMO p so that l = l g. n particular, H ) = BMO. As a consequence of the duality of H and the H -boundedness of flag singular integrals, we obtain the BMO -boundedness of flag singular integrals. urthermore, we will see that L BMO and, hence, the L BMO boundedness of flag singular integrals is also obtained. These provide the endpoint results of those in [MRS] and [NRS]. These can be summarized as follows:

12 Y. HAN AND G. LU Theorem.8:. Suppose that T is a flag singular integral as in Theorem.4. Then T is bounded on BMO. Moreover, there exists a constant C such that T f) BMO C f BMO. Next we prove the Calderón-Zygmund decomposition and interpolation theorems on the flag Hardy spaces. We note that H p Rn ) = L p R n+m ) for < p <. Theorem.9. Calderón-Zygmund decomposition for flag Hardy spaces) Let 0 < p, p < p < p < and let α > 0 be given and f H p Rn ). Then we may write f = g + b where g H p Rn ) with p < p < and b H p Rn ) with 0 < p < p such that g p H p Cα p p f p H p and b p H p Cα p p f p, where C is an absolute constant. H p Theorem.0. nterpolation theorem on flag Hardy spaces) Let 0 < p < p < and T be a linear operator which is bounded from H p to Lp and bounded from H p to Lp, then T is bounded from H p to Lp for all p < p < p. Similarly, if T is bounded on H p and Hp, then T is bounded on H p for all p < p < p. We point out that the Calderón-Zygmund decomposition in pure product domains was established for all L p functions < p < ) into H and L functions by using atomic decomposition on H space see for more precise statement in Section 6). We end the introduction of this paper with the following remars. irst of all, our approach in this paper will enable us to revisit the pure product multi-parameter theory using the corresponding discrete Littlewood-Paley-Stein theory. This will provide alternative proofs of some of the nown results of Chang-R. efferman, R. efferman, ourne, Pipher and establish some new results in the pure product setting. We will clarify all these in the future. Second, as we can see from the definition of flag ernels, the regularity satisfied by flag singular ernels is better than that of the product singular ernels. t is thus natural to conjecture that the Hardy space associated with flag singular integrals should be larger than the classical pure product Hardy space. This is indeed the case. n fact, if we define the flag ernel on R n by Kx, y) = Kx z, z, y)dz, R n where Kx, z, y) is a pure product ernel on R n R n+m, and let H p be the flag Hardy space associated with this structure, thus we have shown in a forthcoming paper that H p R n ) = H p Rn ) H p Rn ). Results in [MRS] and [NRS] together with those in this paper demonstrate that the implicit multi-parameter structure, the geometric property of sets of singularities and regularities of singular ernels and multipliers are closely related. Third, the authors have carried out in [HL] the discrete Littlewood-Paley-Stein analysis and Hardy space theory in the multi-parameter structure induced by the Zygmund dilation and proved the endpoint estimates such as boundedness of singular integral operators considered by Ricci-Stein [RS] on H p Z 0 < p ) and BMO Z, the Hardy and BMO spaces associated with the Zygmund dilation, e.g., on R 3, given by δ x, y, z) = δ x, δ y, δ δ z), δ, δ > 0, where the L p < p < ) boundedness has been established see [RS] and [P]). This paper is organized as follows. n Section, we establish the L p estimates for the multiparameter Littlewood-Paley-Stein g function for < p < and prove Theorems. and.4.

13 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 3 n Section 3 we first introduce the test function spaces associated with the multi-parameter flag structure and show the Calderón reproducing formula in.5) still holds on test function space S R n ) and its dual space S ), and then prove the almost orthogonality estimates and establish the discrete Calderón reproducing formula on the test function spaces, i.e., Theorem.8. Some crucial strong maximal function estimates are given e.g. Lemma 3.7) and together with the discrete Calderón reproducing formula we derive the Plancherel-Pôlya-type inequalities,i.e., Theorem.9. Section 4 deals with numerous properties of Hardy space H p and a general result of bounding the L p norm of the function by its H p norm see Theorem 4.3), and then prove the H p boundedness of flag singular integrals for all 0 < p <, i.e., Theorem.0. The boundedness from H p to Lp for all 0 < p for the flag singular integral operators, i.e., Theorem., is thus a consequence of Theorem.0 and Theorem 4.3. The duality of the Hardy space H p is then established in Section 5. The boundedness of flag singular integral operators on BMO space is also proved in Section 5. Thus the proofs of Theorem.4,.6,.7 and.8 will be all given in the Section 5. n Section 6, we prove a Calderón-Zygmund decomposition in flag multi-parameter setting and then derive an interpolation theorem. Acnowledgement. The authors wish to express their sincere thans to Professor E. M. Stein for his encouragement over the past ten years to carry out the program of developing the Hardy space theory in the implicit multi-parameter structure and his suggestions during the course of this wor. We also lie to than Professor. Pipher for her interest in this wor and her encouragement to us.. L p estimates for Littlewood-Paley-Stein square function: Proofs of Theorems. and.4 The main purpose of this section is to show that the L p p > ) norm of f is equivalent to the L p norm of g f), and thus use this to provide a new proof of the L p boundedness of flag singular integral operators given in [MRS]. Our proof here is quite different from those in [MRS] in the sense that we do not need to apply the lifting procedure used in [MRS] directly. We first prove the L p estimate of the Littlewood-Paley-Stein square function g. Proof of Theorem.: The proof is similar to that in the pure product case given in [S] and follows from iteration and standard vector-valued Littlewood-Paley-Stein inequalities. To see this, define : R n+m H = l by x, y) = {ψ ) j fx, y)} with the norm H = { j ψ ) j fx, y) }. When x is fixed, set g )x, y) = { ψ ) x, )y) H}. t is then easy to see that g )x, y) = g f)x, y). f x is fixed, by the vector-valued Littlewood- Paley-Stein inequality, g ) p x, y)dy C p H dy. However, p H = { j ψ ) j fx, y) } p, so integrating with respect to x together with the

14 4 Y. HAN AND G. LU standard Littlewood-Paley-Stein inequality yields g f) p x, y)dydx C ψ ) j fx, y) } p dydx C f p p, R n which shows that g f) p C f p. R n { The proof of the estimate f p C g f) p is routine and it follows from the Calderon reproducing formula.5) on L R n+m ), for all f L L p, g L L p and p + p =, and the inequality g f) p C f p, which was just proved. This completes the proof of Theorem.. Q.E.D. Remar.: Let ψ ) SR n+m ) be supported in the unit ball in R n+m and ψ ) S ) be supported in the unit ball of and satisfy for all ξ, ξ ) R n \{0, 0)}, and 0 0 j ψ ) tξ, tξ ) 4 dt t = ψ ) sη) 4 ds s = for all η \{0}. We define ψ x, y, z) = ψ ) x, y z)ψ ) z). Set ψ ) t x, y) = t n m ψ ) x t, y t ) and ψ s ) z) = s m ψ z s ) and ψ t,s x, y) = t x, y z)ψ ) z)dz. ψ ) Repeating the same proof as that of Theorem., we can get for < p <.) { and.) f p { 0 0 ψ t,s fx, y) dt t 0 0 s ds s } p C f p, ψ t,s ψ t,s fx, y) dt t ds s } p. The L p boundedness of flag singular integrals is then a consequence of Theorem. and Remar.. We give a detailed proof of this below. Proof of Theorem.4: We may first assume that K is integrable function and shall prove the L p, < p <, boundedness of T is independent of the L norm of K. The conclusion for general K then follows by the argument used in [MRS]. or all f L p, < p <, by.).3) T f) p C { 0 0 ψ t,s ψ t,s K fx, y) dt t ds s } p.

15 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 5 Now we claim the following estimate: for f L p, < p <,.4) ψ t,s K fx, y) CM s f)x, y), where C is a constant which is independent of the L norm of K and M s f) is the maximal function of f defined in the first section. Assuming.4) for the moment, we obtain from.3).5) T f p C { 0 0 M s ψ t,s f)x, y)) dt t ds s } p C f p, where the last inequality follows from the efferman-stein vector-valued maximal function and Remar.. We now prove the claim.4). Note that ψ t,s Kx, y) = ψ t,s K x, y z, z)dz, where ψ t,sx, y, z) is given in Remar. and Kx, y) = K x, y z, z)dz, where K x, y, z) is a product ernel satisfying the conditions of definition.. in [NRS] or Definition. in our paper). The estimate in.4) will follow by integrating with respect to z variable from the following estimate:.6) ψ t,s K t s x, y, z) C t + x + y ) n+m+ s + z ) m+, where the constant C is independent of the L norm of K. The estimate.6) follows from that in the pure product setting R n+m given by R. efferman and Stein [S]. Q.E.D. 3. Test function spaces, almost orthogonality estimates and discrete Calderón reproducing formula: Proofs of Theorems.8 and.9 n this section, we develop the discrete Calderón reproducing formula and the Plancherel- Pôlya-type inequalities on test function spaces. These are crucial tools in establishing the theory of Hardy spaces associated with the flag type multi-parameter dilation structure. The ey ideas to provide the discrete Calderón reproducing formula and the Plancherel-Pôlya-type inequalities are the continuous version of the Calderón reproducing formula on test function spaces and the almost orthogonality estimates. To be more precise, we say that a function ax, y) for x, y) R n R n belongs to the class S R n R n ) if ax, y) is smooth and satisfies the differential inequalities 3.) α x β y ax, y) A N,α,β + x y ) N and the cancellation conditions 3.) ax, y)x α dx = ax, y)y β dy = 0 for all positive integers N and multi-indices α, β of nonnegative integers. The following almost orthogonality estimate is the simplest one and its proof can be adapted to the more complicated orthogonal estimates in subsequent steps.

16 6 Y. HAN AND G. LU Lemma 3.. f ψ and φ are in the class S R n R n ), then for any given positive integers L and M, there exists a constant C = CL, M) depending only on L, M and the constants A N,α,β in 3.) such that for all t, s > 0 3.3) ψ t x, z)φ s z, y)dz C t s s t s) L t )M t s + x y ), n+l) R n where ψ t x, z) = t n ψ x t, z t ) and φ sz, y) = s n φ z s ), and t s = mint, s), t s = maxt, s). Proof: We only consider the case M = L = and t s. Then ψ t x, z)φ s z, y)dz R m = [ψ t x, z) ψ t x, y)] φ s z, y)dz R m = + = + z y t+ x y ) z y t+ x y ) We use the smoothness condition for ψ t and size condition for φ s to estimate term. To estimate term, we use the size condition for both ψ t and φ s. or the case M > and L >, we only need to use the Taylor expansion of ψ t x, ) at y and use the moment condition of ψ t. We shall not give the details. Q.E.D. Similarly, if ψ x, y, z, u, v, w) for x, y, z), u, v, w) R n is a smooth function and satisfies the differential inequalities α x β y γ z α u β v γ w ψ x, y, z, u, v, w) 3.4) A N,M,α,α,β,β,γ,γ + x u + y v ) N + z w ) M and the cancellation conditions ψ x, y, z, u, v, w)x α y β dxdy = ψ x, y, z, u, v, w)z γ dz 3.5) = ψ x, y, z, u, v, w)u α v β dudv = ψ x, y, z, u, v, w)w γ dw = 0, and for fixed x 0 R n, y 0, φ x, y, z, x 0, y 0 ) S R n+m ) and satisfies α x β y γ z φ x, y, z, x 0, y 0 ) 3.6) B N,M,α,β,γ, + x x 0 + y y 0 ) N + z ) M, for all positive integers N, M and multi-indices α, α, β, β, γ, γ of nonnegative integers. Then we have the following almost orthogonality estimate:

17 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 7 Lemma 3.. or any given positive integers L, L and K, K, there exists a constant C = CL, L, K, K ) depending only on L, L, K, K and the constants in 3.4) and 3.6) such that for all positive t, s, t, s we have 3.7) R n+m+m ψt,sx, y, z, u, v, w)φ t,s u, v, w, x 0, y 0 )dudvdw C t t t t )L s s s s )L t t ) K t t + x x 0 + y y 0 ) n+m+k ) where ψ t,sx, y, z, u, v, w) = t n m s m ψ x t, y t, z s, u t, v t, w s ) and s s ) K s s + z ) m+k ), φ t,sx, y, z, x 0, y 0 ) = t n m s m φ x t, y t, z s, x 0 t, y 0 t, ). The proofs of the almost orthogonality estimate in 3.7) is similar to that in 3.3). We will only provide a brief proof. Proof of Lemma 3.: We only consider the case L = L = K = K =, t t and s s. Thus ψt,sx, y, z, u, v, w)φ,s u, v, w, x t 0, y 0 )dudvdw = R n A Bdudvdw where and R n A = ψ t,sx, y, z, u, v, w) ψ t,sx, y, z, x 0, y 0, w) B = φ t,s u, v, w, x 0, y 0 ) φ t,s u, v, z, x 0, y 0 ). n the above, we have used the cancelation properties R n φ t,s u, v, w, x 0, y 0 )u α v β dudv = 0, for all multi-indices α, β and γ. Next, = R n A B dudvdw u x 0 + v y 0 t+ x x 0 + y y 0 ), w z s + z ) u x 0 + v y 0 t+ x x 0 + y y 0 ), w z s + z ) u x 0 + v y 0 t+ x x 0 + y y 0 ), w z s + z ) u x 0 + v y 0 t+ x x 0 + y y 0 ), w z s + z ) = V ψ t,sx, y, z, u, v, w)w γ = 0 A B dudvdw A B dudvdw A B dudvdw A B dudvdw

18 8 Y. HAN AND G. LU or term, we use the smoothness conditions on ψ t,s and φ t,s ; or term, we use the smoothness conditions on ψ t,s and size conditions on φ t,s ; or term, we use the size conditions on ψ t,s and smoothness conditions on φ t,s ; or term V, use the size conditions on both ψ t,s and φ t,s. We shall not provide the details here. Q.E.D. The crucial feature, however, is that the almost orthogonality estimate still holds for functions in S R n ). To see this, we first derive the relationship of convolutions on R n and R n+m. We will use this relationship frequently in this paper. Lemma 3.3. Let ψ, φ S R n ), and ψ, φ S R n+m ) such that ψx, y) = ψ x, y z, z)dz, φx, y) = φ x, y z, z)dz. Then ψ φ)x, y) = ψ φ ) x, y z, z)dz. Lemma 3.3 can be proved very easily. Using this lemma and the almost orthogonality estimates on R n+m, we can get the following Lemma 3.4. or any given positive integers L, L and K, K, there exists a constant C = CL, L, K, K ) depending only on L, L, K, K such that if t t s s, then ψ t,s φ t,s x, y) and if t t s s, then C t t t t )L s s s s )L ψ t,s φ t,s x, y) C t t t t )L s s s s )L t t ) K t t + x ) n+k ) t t ) K t t + x ) n+k ) s s ) K s s + y ) m+k ), t t ) K t t + y ) m+k ). Proof of Lemma 3.4: We first remar that we will prove this lemma with K, K, L, L replaced by K, K, L, L. Thus, we are given any fixed K, K, L, L. Note that ψ t,s φ t,s x, y) = ψ t,s φ t,s x, y z, z)dz, where ψ, φ S R n+m ), and ψ t,s φ t,s x, y, z) = ψ t,sx u, y v, z w)φ t,s u, v, w)dudvdw, R n

19 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 9 Then by the estimate in 3.7), for any given positive integers L, L and K, K, there exists a constant C = CL, L, K, K ) depending only on L, L, K, K such that 3.8) ψ t,s φ t,s x, y) C t t t t )L s s s s )L t t ) K s s ) K t t + x + y z ) n+m+k ) s s + z ) dz. m+k ) Case : f t t s s and y s s, write = t is easy to see that t t ) K s s ) K t t + x + y z ) n+m+k ) s s + z ) dz m+k ) = + y z y z y, or z y + t t ) K C t t + x + y ) n+m+k ) t t ) K t t ) K C t t + x ) n+k ) y K t t ) K C t t + x ) n+k ) s s ) m+k s s + y ) m+k where we have taen K = K + K and used the fact that t t s s and y s s. Next, we estimate s s ) K s s + y ) m+k ) s s ) K C s s + y ) m+k ) s s ) K C s s + y ) m+k ) t t ) K t t + x + y z ) n+m+k ) dz t t ) K t t + x ) n+k ) t t ) K t t + x ) n+k ) where we have used K K and K = K + K > K. Case : f t t s s and y s s, then t t ) K s s ) K t t + x + y z ) n+m+k ) s s + z ) dz m+k ) s s ) m s s ) K C s s + y ) m+k t t ) K t t + x + y z ) n+m+k ) dz t t ) K t t + x ) n+k ).

20 0 Y. HAN AND G. LU Case 3: We now consider the case t t s s and y t t. Then t t ) K s s ) K t t + x + y z ) n+m+k ) s s + z ) dz m+k ) t t ) K C t t + x ) n+m+k ) t t ) K t t + x ) n+k ) by noticing that K = K + K. t t ) K t t + x + y ) m+k ). Case 4: f we assume t t s s and y t t, then we divide the integral into two parts and as in the case t t s s. Thus we have where we have taen K = K + K. To estimate, we have t t ) K t t + x + y ) n+m+k ) t t ) K C t t + x ) n+m+k ) t t ) K C t t + x ) n+k ) t t ) K C t t + x ) n+k ) t t ) K t t + y ) m+k ) s s ) K s s + y ) m+k ) t t ) K t t + y ) m+k ) Q.E.D. Roughly speaing, ψ t,s φ t,s x, y) satisfies the one-parameter almost orthogonality when t t s s and the product multi-parameter almost orthogonality when t t s s. More precisely, we have the following Corollary 3.5. Given any positive integers L, L, K and K, there exists a constant C = CL, L, K, K ) > 0 such that i) f t s we obtain the one-parameter almost orthogonality 3.8) ψ t,s φ t,s x, y) C t t t t )L s s s s )L t K t + x ) n+k ) and, if t s the product multi-parameter almost orthogonality is given by 3.9) ψ t,s φ t,s x, y) C t t t t )L s s s s )L ii) Similarly, if t s, t K t + x ) n+k ) t K t + y ) m+k ) s K s + y ) m+k ). 3.0) ψ t,s φ t,s x, y) C t t t t )L s s s t K t K s )L t + x ) n+k ) t + y ), m+k )

21 and if t s, DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 3.) ψ t,s φ t,s x, y) C t t t t )L s s s t K s K s )L t + x ) n+k ) s + y ). m+k ) Corollary 3.5 is actually what we will use frequently in the subsequent parts of the paper. The proof of Corollary 3.5 is a case by case study and can be checed with patience. We shall omit the details of the proof here. All these estimates will be used to prove the following continuous version of the Calderón reproducing formula on test function space S R n ) and its dual space S ). Theorem 3.6. Suppose that ψ j, are the same as in.4). Then 3.) fx, y) = ψ j, ψ j, fx, y), j where the series converges in the norm of S and in dual space S ). Proof: Suppose f S and fx, y) = f x, y z, z)dz, where f S R n+m ). Then, by the classical Calderón reproducing formula as mentioned in the first section, for all f L, 3.3) f x, y, z) = ψ j, ψ j, f x, y, z), j where ψ j, x, y, z) = ψ) j x, y)ψ ) z). We claim that the above series in 3.3) converges in S R n+m ). This claim yields fx, y) ψ j, ψ j, fx, y) S = N j N M M [f x, y z, z) f x, y, z) N j N M M N j N M M ψ j, ψ j, f x, y z, z)]dz S ψ j, ψ j, f x, y, z) S where the last term above goes to zero as N and M tend to infinity by the above claim. To show the claim, it suffices to prove that all the following three summations ψ j, ψ j, f, ψ j, ψ j, f, ψ j, ψ j, f j >N M j N >M j >N >M tend to zero in S R n+m ) as N and M tend to infinity. Since all proofs are similar for each of the summations, we only prove the assertion for the first summation which we denote by f N,M. Note that f N,M x, y, z) = ψ j, ψ j, x u, y v, z w)f u, v, w)dudvdw j >N M R n

22 Y. HAN AND G. LU where ψ ψ x u, y v, z w) satisfies the conditions 3.4) and 3.5), and f u, v, w) satisfies the conditions 3.6) with x 0 = y 0 = 0. The almost orthogonality estimate in 3.7) with t = j, s =, t = s = and x 0 = y 0 = 0, implies ψ j, ψ j, x u, y v, z w)f u, v, w)dudvdw R n C j L L j ) K ) K j + x + y ) n+m+k ) + z ). m+k ) This, by taing L > K and L > K, gives us lim N,M sup + x + y ) n+m+k + z ) m+k f N,M x, y, z) = 0. x R n,y,z Since α x β y γ z f N,M )x, y, z) = α x β y γ z f ) N,M x, y, z) and applying the above estimate to α x β y γ z f which also satisfies the conditions in 3.6) with x 0 = y 0 = 0, we obtain lim N,M sup + x + y ) n+m+k + z ) m+k x α y β z γ f ) N,M x, y, z) = 0, x R n,y,z which shows the claim. The convergence in dual space follows from the duality argument. The proof of Theorem 3.6 is complete. Q.E.D. Using Theorem 3.6, we prove the discrete Calderón reproducing formula. Proof of Theorem.8: We first discretize 3.) as follows. or f S, by 3.) and using an idea similar to that of decomposition of the identity operator due to Coifman, we can rewrite fx, y) = ψ j, x u, y w) ψ j, f) u, w)dudw j,, 3.4) = j,, ψ j, x u, y w)dudw ψ j, f) x, y ) + Rf)x, y). We shall show that R is bounded on S with the small norm when and are dyadic cubes in R n and with side length j N and N + j N for a large given integer N, and x, y are any fixed points in,, respectively. To do this, assuming fx, y) = f x, y z, z)dz, where f S R n+m ). When j, we write ψ j, f) u, w) ψ j, f) x, y ) = ψ j f ) u, v, w v)dv ψ j f ) x, v, y v)dv R [ m ] = ψ j u u, v v, w v w ) ψ j x u, v v, y v w ) f u, v, w )du dv dw dv

23 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 3 where the last integral above is over R n. When > j, we write ψ j, f) u, w) ψ j, f) x, y ) = ψ j f ) u, w v, v)dv ψ j f ) x, y v, v)dv R [ m ] = ψ j u u, w v v, v w ) ψ j x u, y v v, v w ) f u, v, w )du dv dw dv We now set Rf)x, y) = ψ j, x u, y w) [ψ j, f) u, w) ψ j, f) x, y )] dudw j,, = R x, y z, z, u, v, w )f u, v, w )du dv dw dz = R f )x, y z, z)dz, where R x, y, z, u, v, w ) is the ernel of R and R x, y z, z, u, v, w ) = ψ ) j x u, y z w)ψ ) z) j, [ ] ψ ) j u u, v v )ψ ) w v w ) ψ ) j x u, v v )ψ ) y w v) dudwdv + ψ ) j x u, y z w)ψ ) z) j >j [ ] ψ ) j u u, w v v ) ψ ) j x u, y v v ) ψ ) v w )dudwdv. Using the change of variables from z to z + v w in the term of j, and z to z v in the term of > j, we can rewrite R x, y, z, u, v, w ) = ψ ) j x u, y v)ψ ) z + v w) j, [ ] ψ ) j u u, v v )ψ ) w w v) ψ ) j x u, v v )ψ ) y w v) dudwdv + ψ ) j x u, y + v w)ψ ) z v) j >j [ ] ψ ) j u u, w v v ) ψ ) j x u, y v v ) ψ ) v w )dudwdv = A j + B j. j >j

24 4 Y. HAN AND G. LU We claim that R is bounded in S R n+m ). To see this, write A j = ψ ) j x u, y v)ψ ) z + v w) [ψ ) j u u, v v ) ψ ) j x u, v v )]ψ ) w w v)dudwdv + ψ ) j x u, y v)ψ ) z + v w) ψ ) j x u, v v )[ψ ) w w v) ψ ) y w v)]dudwdv =A ) j x, y, z, u, v, w ) + A ) j x, y, z, u, v, w ) t is not difficult to chec that ψ ) z + v w)ψ) w w v)dw satisfies all the conditions as ψ ) z w ) does with the comparable constants of S R n ) norm and that ψ ) j x u, y v)[ψ ) j u u, v v ) ψ ) j x u, v v )]dudv satisfies all conditions as ψ ) j x u, y v ) but with the constants of S R n ) norm replaced by C N. This follows from the smoothness condition on ψ ) j say the mean-value theorem) and the fact that u, x and l) = N j. f we write A ) j x, y, z, u, v, w ) = ψ ) j x u, y v)[ψ ) j u u, v v ) ψ ) j x u, v v )] [ ] z + v w)ψ) w w v)dw dudv ψ ) then the function A ) j x, y, z, u, v, w ) satisfies all conditions as ψ ) j x u, y v ) ψ ) z w ) does but with the S R n ) norm constant replaced by C N. By the proof of Theorem 3.6, we conclude that A ) j x, y, z, u, v, w )f u, v, w )du dv dw is a test function in S R n+m ) and its test function norm is bounded by C N. Similarly, ψ ) z + v w)[ψ) w w v) ψ ) y w v)]dw

25 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 5 satisfies all conditions as ψ ) z w ) but with the constant replaced by C N, which follows from the smoothness condition on ψ ) and the fact w, y and l) = N j + N and j. We also note that ψ ) j x u, y v)ψ ) j x u, v v )dudv satisfies the same conditions as ψ ) j x u, y v ) does with comparable S R n ) norm constant. Thus, we conclude that A ) j x, y, z, u, v, w )f u, v, w )du dv dw is a test function in S R n+m ) and its test function norm is bounded by C N. Therefore, A j x, y, z, u, v, w )f u, v, w )du dv dw is a test function in S R n+m ) and its test function norm is bounded by C N. Similarly, we can conclude that B j x, y, z, u, v, w )f u, v, w )du dv dw is also a test function in S R n+m ) and its test function norm is bounded by C N. This shows that R f )x, y, z) S R n+m ) and R f ) S R n+m ) C N f S R n+m ), which implies that Rf) S R n ) and 3.5) Rf) S R n ) C N f S R n ). By 3.4) together with the boundedness of R on S with the norm at most C N, if N is chosen large enough, then we obtain fx, y) = R i ψ j, u, v)dudv x, y) ψ j, f) x, y ). j i=0 Set R i i=0 ψ j, u, v)dudv x, y) = ψ j, x, y, x, y ). t remains to show ψ j, x, y, x, y ) S. This, however, follows easily from 3.5).

26 6 Y. HAN AND G. LU Q.E.D We next establish a relationship between test function in f S R n+m ) and test function fx, y) = f x, y z, z)dz under the actions of R, R and the implicit multi-parameter dilations. To be more precise, we define f j, x, y) = f ) j, x, y z, z)dz, where f ) j, x, y, z) = n+m)j m f j x, j y, z). Note that R f ) j, )x, y, z) = R x, y, z, u, v, w )f ) j u, v, w )du dv dw = n+m)j m R x, y, z, u, v, w )f j u, j v, w )du dv dw = R x, y, z, j u, j v, w )f u, v, w )du dv dw and R j x, j y, z, j u, j v, w ) = n+m)j m R x, y, z, u, v, w ). Thus we have This implies R f ) j, )x, y, z) = R f )) j x, y, z). Rf j, )x, y) = R f ) j, )x, y z, z)dz R m = R f )) j x, y z, z)dz = Rf)) j x, y) t is worthwhile to point out that ψ j, x, y, x, y ) = ψ j, x, y z, z, x, y )dz, where ψ j, x, y, z, x, y ) = ψ ) j, x, y, z, x, y ), ψ x, y, z, x, y ) S R n+m ) and satisfies the condition in 3.6) with x 0 = x, y 0 = y.

27 DSCRETE LTTLEWOOD-PALEY-STEN MULT-PARAMETER ANALYSS 7 Remar 3.: f we begin with discretizing 3.) by fx, y) = j ψ j, x x, y y ) ψ j, f) u, v)dudv + Rf)x, y), and repeating the similar proof, then the discrete Calderón reproducing formula can also be given by the following form fx, y) = j ψ j, x x, y y ) ψj, f)x, y ), where ψj, f)x, y ) = ψ j, R) i f)u, v)dudv. We leave the details of these proofs to the reader. i=0 Before we prove the Plancherel-Pôlya-type inequality, we first prove the following lemma. Lemma 3.7. Let,,, be dyadic cubes in R n and respectively such that l) = j N, l) = j N + N, l ) = j N and l ) = j N + N. Thus for any u and v we have and j, C j j j >j, C j j + )K j + )K j + u x ) n+k + v y ) m+k φ j, fx, y ) K K { M s φ j, fx, y ) χ χ ) r } r u, v) j j + )K j K j + u x ) n+k j + v y ) m+k φ j, fx, y ) j j K K >j { M φ j, fx, y ) χ χ ) r } r u, v) where M is the Hardy-Littlewood maximal { function on R} n+m, M s is the strong maximal function on R n n as defined in.), and max < r. Proof: We set and for l, i n+k, m m+k A 0 = { : l ) = j N, B 0 = { : l ) = j N + N, u x l ) } v y l ) A l = { : l ) = j N, l < u x l ) } l }.