Discrete Calderón s Identity, Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

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1 J Geom Anal (010) 0: DOI /s Discrete Calderón s Identity, Atomic Decomosition and Boundedness Criterion of Oerators on Multiarameter Hardy Saces Y. Han G. Lu K. Zhao Received: 7 Aril 009 / Published online: 3 March 010 Mathematica Josehina, Inc. 010 Abstract In this aer we establish a discrete Calderón s identity which converges in both L (R n+m ) (1 << ) and Hardy sace H (R n R m ) (0 < 1). Based on this identity, we derive a new atomic decomosition into (, )-atoms (1 << ) on H (R n R m ) for 0 < 1. As an alication, we rove that an oerator T, which is bounded on L (R n+m ) for some 1 <<, is bounded from H (R n R m ) to L (R n+m ) if and only if T is bounded uniformly on all (, )-roduct atoms in L (R n+m ). The similar result from H (R n R m ) to H (R n R m ) is also obtained. Keywords Boundedness Calderón-Zygmund oerator Calderón s identity Multiarameter Hardy saces Atomic decomosition Boundedness criterion of oerators Mathematics Subject Classification (000) 4B30 4B0 Communicated by Richard Rochberg. G. Lu is artly suorted by US NSF grants DMS and DMS and NSFC of China grant No K. Zhao is artly suorted by NNSF-China No Y. Han Deartment of Mathematics, Auburn University, Auburn, Alabama , USA hanyong@auburn.edu G. Lu ( ) Deartment of Mathematics, Wayne State University, Detroit, MI 480, USA gzlu@math.wayne.edu K. Zhao College of Mathematics, Qingdao University, Qingdao 66071, Shandong Province, China zkzc@yahoo.com.cn

2 Boundedness of Oerators on Multiarameter Hardy Saces Introduction The atoms and molecules were introduced (see Coifman [6], Latter [19], Coifman-Weiss [7], etc.) and were used in harmonic and wavelet analysis (see Meyer [1], Meyer-Coifman [] and Stein [4], etc.) to form the basic building blocks of various function saces. Since then, the study of the oerators acting on a sace of functions or distributions can become very simle when the elements of the sace admit atomic decomositions. Indeed, many roblems in analysis have natural formulations as uestions of continuity of linear oerators defined on saces of functions or distributions. Such uestions can often be answered by rather straightforward techniues if they can first be reduced to the study of the oerator on an aroriate class of simle functions which generate the entire sace in some aroriate sense. This fundamental rincile was alied by many eole to roblems where atomic decomositions exist. However, recently Bownik in [1] gave an examle of a linear functional defined on a dense subsace of Hardy sace H 1 (R n ), which mas all (1, )-atoms into bounded scalars, but it cannot be extended to a bounded functional on the whole sace H 1 (R n ). The construction in [1] is based on the fact due to Meyer [1], which states that uasi-norms corresonding to finite and infinite atomic decomositions in H,0< 1, are not euivalent. We note that H 1 (R n ) can be generated by (1, )-atoms. This examle shows that one has to be careful when the above fundamental rincile is used. It is then natural to ask under what extra conditions the above fundamental rincile can be used. Nevertheless, Meda et al. in [0] have recently roved that if T is a linear oerator defined on the subsace H 1, fin (Rn ) of finite linear combinations of (1,)atoms in R n with the roerty that su{ Ta Y : a is a (1,)-atom} <, 1 <<, then T admits a (uniue) continuous extension to a bounded linear oerator from H 1 (R n ) to Y (Banach sace) (see also Han and Zhao [13] for related results). Furthermore, Ricci and Verdera have shown that if 0 <<1 and T is a linear oerator defined on the subsace H, fin (R n ) of finite linear combinations of (, ) atoms in R n with the roerty that su{ Ta Y : a is a (, )-atom} <, then T admits a (uniue) continuous extension to a bounded linear oerator from H (R n ) to Y (Banach sace). Moreover, the structure of the sace F of finite linear combinations of (, ) atoms, the comletion of F and the dual saces were thoroughly studied in [3]. The main urose of this aer is to establish a discrete Calderón s identity in multiarameter setting and aly this identity to rovide a new atomic decomosition of (, )-atoms of multiarameter Hardy saces H (R n R m ). As an alication of this new atomic decomosition, we will rove the uniform boundedness criterion of linear oerators in multiarameter setting by considering its action on all (, )- atoms for 0 < 1 and 1 <<, which is similar to the one-arameter setting. Atomic decomosition in multiarameter Hardy saces is much more comlicated. It is well-known that there is a basic obstacle to the ure roduct Hardy and

3 67 Y. Han et al. BMO sace theory associated with multiarameter roduct dilations. Indeed it was conjectured that the roduct atomic Hardy sace on R R could be defined by rectangle atoms. Here a rectangle atom is a function a(x,y) suorted on a rectangle R = I J having the roerty that a R 1/, a(x,y)dx = a(x,y)dy = 0 I J for every (x, y) R. Then Hrect 1 (R R) is the sace of functions k λ ka k with each a k a rectangle atom and k λ k <. However, this conjecture was disroved by Carleson by constructing a counter-examle of a measure satisfying the roduct form of the Carleson measure, that is, the measure μ satisfies dμ C I J S(I) S(J) for all intervals I,J in R and S(I) is the Carleson region associated with I.Carleson [] showed that the measure he constructed is not bounded on the roduct Hardy sace H 1 (R R). This led to the role of cubes in the classical atomic decomosition of H (R n ) being relaced by arbitrary oen sets of finite measures in the roduct H (R n R m ) and the Hardy sace H (R n R m ) theory was finally develoed by Chang and Fefferman [3 5]. Chang and Fefferman [4] roved the following Theorem f H (R + R + ) if and only if f(x,y) = k λ ka k (x, y) where k λ k < and a k (x, y) are (, )-atoms, that is, each a k (x, y) is suorted in an oen set with finite measure satisfying the following roerties: a k 1/ 1/ ; each a k (x, y) can be further decomosed by a k (x, y) = R a R (x, y) where R = I J are dyadic rectangles in R, and a R (x, y) satisfy a R (x, y)x α dx = a R (x, y)y β dy = 0 I for 0 α, β N =[/ 4/3], and a R is a C α,α N + 1, function satisfying α x α a R(x, y) d R I α, α y α a R(x, y) d R J α with R m( ) J R d R 1 /, where m( ) is the set of all rectangles in which are maximal in both directions of x and y.

4 Boundedness of Oerators on Multiarameter Hardy Saces 673 It is worthwhile to oint out that the key tool, using the maximal characterization of the one-arameter H (R n ), to get the atomic decomosition is a refined Calderón-Zygmund decomosition; see [6, 7, 19]. This maximal characterization of the one-arameter H (R n ) is the main tool used in the works of [0, 3]. However, it seems to be difficult to carry this method out to the multiarameter roduct saces. Using the classical version of continuous Calderón s identity on the roduct sace, A. Chang and R. Fefferman established the above atomic decomosition for H (R n R m ), by use of (, )-atoms. We would like to oint out that this classical version of Calderón s identity does not seem to lead to an atomic decomosition by (, )-atoms for 1 <<,. Thus, deriving an atomic decomosition in the roduct saces using (, )-atoms for becomes interesting. One of the main uroses of this aer is to establish all (, ), 1 <<, atomic decomosition for multiarameter Hardy sace H (R n R m ) for 0 < 1 with the convergence in both L (R n+m ) and H (R n R m ) norms (see Theorem 1. below). It should be ointed out that the series in (, )-atomic decomosition in [3] converges only in the sense of distributions. As we ointed out earlier, this classical continuous version of Calderón s identity is not alicable to rovide an atomic decomosition by (, )-atoms for 1 <<,. The crucial idea to rove (, ), 0 < 1 <<, atomic decomosition is to establish a new discrete Calderón s identity on the roduct sace, which converges both in L (R n+m ) norm for 1 << and the H (R n R m ) norm (see Theorem 1.1 below). The idea of using discrete Calderón s identity in roduct saces was develoed earlier in [10, 11], but in a uite different way. To be more recise, to develo multiarameter Hardy sace theory associated with flag singular integrals and the Zygmund dilation on R 3, the first two authors introduced the multiarameter test function saces associated with flag singular integrals and the Zygmund dilation, resectively, and rovided the discrete Calderón s identities on these test function saces. Using these discrete identities, they roved the Min-Max comarison ineualities and finally the Hardy saces H F (Rn R m ) and H Z (R3 ) were well defined; see [10, 11] for more details. Moreover, boundedness of singular integral oerators from H to H and from H to L were roved in [10, 11] without using Journé s covering lemma. In articular, we develoed a general method to derive H L boundedness from H H boundedness of linear oerators for the roduct saces using discrete Calderón s identity and discrete Littlewood-Paley theory. The crucial idea there is to rove the following ineuality f L C f H for f L H, 1 <<, 0 < 1. (1.1) Using this ineuality, the H to L boundedness follows immediately from the boundedness on H saces. This general hilosohy in the roduct saces of homogeneous tye has also been established in [17]. The duality theory for multiarameter Hardy saces has been established in [16] using the idea of discretization. In this resent aer, using a different aroach from those in [10, 11], namely Coifman s decomosition of the identity oerator and the Calderón-Zygmund oerator theory on the multiarameter roduct sace, we first derive the following discrete Calderón-tye identity:

5 674 Y. Han et al. Theorem 1.1 Suose 0 < 1. Let ψ (1) (x 1 ), ψ () (x ) be Schwartz functions with suort on the unit ball satisfying the conditions: for a fixed large integer M deending on, ψ (1) ( j ξ 1 ) = 1 j for all ξ 1 R n \{0}, and R n ψ (1) (x 1 )x1 α dx 1 = 0 for all 0 α M; and ψ () ( k ξ ) = 1 k for all ξ R m \{0}, and R m ψ () (x )x β dx = 0 for all 0 β M. Let ψ j,k (x 1,x ) = jn+km ψ (1) ( j x 1 )ψ () ( k x ). Then for each function f H (R n R m ) L (R n+m ),1<<, there exists a function g H (R n R m ) L (R n+m ) with c 1 f g c f and c 1 f H g H c f H, where the constants c 1 and c are indeendent of f, such that f(x 1,x ) = j I J ψ j,k (x 1 x I,x x J )ψ j,k g(x I,y J ), (1.1) k I J where I R n and J R m are dyadic cubes, l(i) = j N,l(J) = k N for some large integer N deending on ψ and,,m; x I, y J are any fixed oints in I and J, resectively; the series in (1.1) converges in both the norms of L (R n+m ) and H (R n R m ). As a conseuence, this discrete Calderón-tye identity gives the following atomic decomosition in terms of (, )-atoms for the multiarameter roduct Hardy sace. Theorem 1. Suose 0 < 1. Let f L (R n+m ) H (R n R m ), 1 <<. Then there exists a seuence of (, )-roduct atoms {a j } of H (R n R n ) and a seuence of scalars {λ j } with ( j λ j ) 1 C f H (R n R m ), where the constant C is indeendent of f, such that f = j λ j a j, where the series converges to f in both the H (R n R m ) and L (R n+m ) norms. Here, a function a is said to be a (, )-roduct atom of H (R n R m ), 0 < 1 <<, if a satisfies (1) Su a, where is an oen set of R n+m with finite measure. () a L (R n+m ) 1 1. Moreover, a can be further decomosed into a rectangle (, )-atom a R associated to the rectangle R = I J which is suorted in 3R and such that (3a) For <,a= R m( ) a R, and ( R m( ) a R L (R n+m ) )1/ 1 1.

6 Boundedness of Oerators on Multiarameter Hardy Saces 675 (3b) For 1 <<,a = R m 1 ( ) a R + R m ( ) a R, and for any δ>0, there exists a constant C,δ>0, where C,δ deends only on and δ, such that R m 1 ( ) γ δ a R L (R n+m ) + R m ( ) 1 a R L (R n+m ) γ δ 1/ C,δ 1 1. Here and in the seuel, m( ) is the set of all maximal rectangles contained in in both directions x 1 and x and ={(x 1,x ) R n R m : M s (χ )(x 1,x )> }, where M s is the strong maximal function. m 1 ( ) is the set of all rectangles contained in and maximal in the x 1 direction and m ( ) is defined similarly. γ 1 is defined by γ 1 (R) = l I, where R = I J m ( ) and l is the maximal dyadic cube such that l J. γ (R) is similarly defined. (4) (Cancellation conditions): for 0 α, β N = n[ 1 1]+m[ 1 1], 3I 3J a R (x, y)x α dx = 0 for a.e. y 3J, a R (x, y)y β dy = 0 for a.e. x 3I. We would like to oint out that the condition given in (3b) for (, )-atoms when 1 << is different from the original one as given in (3a) for = by[9]. However, Fefferman s boundedness criterion of an oerator by considering its action on rectangle atoms [9] still works for (, )-atoms for 1 << under the condition of (3b). We refer the reader to the Aendix for the outline of the roof. We remark here that it is easy to see that L (R n+m ) H (R n R m ) is dense in H (R n R m ) for 0 < 1 <<. Finally, we are able to rove a uniform boundedness criterion for an oerator on the roduct Hardy saces by considering its action on (, )-atoms.thisisthefollowing Theorem 1.3 Suose 0 < 1. Let T be a linear oerator which is bounded on L (R n+m ) for some 1 <<. Then (1) T is bounded from H (R n R m ) to L (R n+m ) if and only if Ta L (R n+m ) C for all (, )-roduct atoms; () T is bounded on H (R n R m ) if and only if Ta H (R n R m ) C for all (, )- roduct atoms, where the constant C is indeendent of a. The organization of the aer is as follows. In Sect. we rove a new discrete Calderón-tye identity on the roduct sace which converges both in L (R n+m ) and H (R n R m ) norms (Theorem 1.1). This is the key ingredient of this aer. Next, we rovide an atomic decomosition in terms of (, )-atoms for the multiarameter roduct Hardy sace (Theorem 1.). Section 3 gives the roof of Theorem 1.3.

7 676 Y. Han et al. Discrete Calderón s Identity on Product Saces As we ointed out in the introduction, the discrete Calderón s identity in roduct saces lays a crucial role in establishing the (, )-atoms in roduct saces for all 1 <<. The main urose of this section is to rove this identity which is of indeendent interest. We first recall some basic definitions. Let ψ (1) (x 1 ), ψ () (x ) be Schwartz functions satisfying j ψ (1) ( j ξ 1 ) = 1 for all ξ 1 R n \{0}, and R n ψ (1) (x 1 )x α 1 dx 1 = 0 for all α 0; and k ψ () ( k ξ ) = 1 for all ξ R m \{0}, and R m ψ () (x )x β dx = 0 for all β 0. Definition.1 Suose that f(x 1,x ) S (R n+m ), the sace of temered distributions. Let ψ (1) (x 1 ), ψ () (x ) be functions as above. The Littlewood-Paley function of f,g(f ), is defined by { } 1 g(f )(x 1,x ) = ψ jk f(x 1,x ), where ψ jk (x 1,x ) = jn+km ψ (1) ( j x 1 )ψ () ( k x ). j,k Similar to [4], the above Littlewood-Paley function can be used to characterize the roduct Hardy saces as follows. Definition. The roduct Hardy sace f H (R n R m ) is defined by H (R n R m ) ={f S (R n+m ) : g(f ) L (R n+m )} with f H (R n R m ) = g(f ) L (R n+m ). We now recall a variant of the boundedness result of oerators on H (R n R m ) roved in [14, 15], which is sufficient for our urose here. Theorem.1 For given 0 < 1, suose that T is a bounded oerator on L (R n+m ) and it is associated with the kernel K(x 1,x,y 1,y ) given by Tf(x 1,x ) = K(x 1,x,y 1,y )f (y 1,y )dy 1 dy, (.1) where the kernel K(x 1,x,y 1,y ) is defined on R n R m R n R m and there exist constants C>0 and a large integer M> n+m such that for all 0 α, β, θ, γ M, α x 1 β x θ y 1 γ y K(x 1,x,y 1,y ) C x 1 y 1 n+ α + θ, (.) x y m+ β + γ

8 Boundedness of Oerators on Multiarameter Hardy Saces 677 and K(x 1,x,y 1,y )x α 1 dx 1 = 0, for all x R m,y 1 R n,y R m, K(x 1,x,y 1,y )x β dx = 0, for all x 1 R n,y 1 R n,y R m, K(x 1,x,y 1,y )y θ 1 dy 1 = 0, for all x 1 R n,x R m,y R m, and K(x 1,x,y 1,y )y γ dy = 0, for all x 1 R n,x R m,y 1 R n. Then T is bounded on H (R n R m ). We remark that the necessary and sufficient conditions of the roduct H boundedness of oerators of Journé s class nonconvolution tye whose kernels satisfy weaker smoothness conditions are given in [15] for the two-arameter setting and in [14] for an arbitrary number of arameters. We also mention that the roof of Theorem 1.1 follows from the same idea in [11] of the discrete Calderón s identity in the case of flag singular integrals (but simler in our ure roduct case here). But we will rovide a somewhat different roof here by invoking Theorem.1. Proof of Theorem 1.1 Based on the conditions on ψ j,k, where M is chosen by M> + 100, the classical Calderón s identity on L (R n+m ) is given by n+m f(x 1,x ) = j Using Coifman s decomosition of the identity, we write f(x 1,x ) = j = j = j k ψ j,k ψ j,k f(x 1,x ). (.3) k R n R m ψ j,k (x 1 u, x v)ψ j,k f(u,v)dudv k I J I J ψ j,k (x 1 u, x v)ψ j,k f(u,v)dudv I J ψ j,k (x 1 x I,x x J )ψ j,k f(x I,x J ) k I J + R N (f )(x 1,x ), (.4) where I R n and J R m are dyadic cubes, l(i) = j N,l(J)= k N for some large integer N which will be chosen later; x I, y J are any fixed oints in I and J,

9 678 Y. Han et al. resectively; and R N (f )(x 1,x ) = j k I J I J [ψ j,k (x 1 u, x v)ψ j,k f(u,v) ψ j,k (x 1 x I,x x J )ψ j,k f(x I,x J )] dudv. Now, we rove that R N is bounded on L (R n+m ) and H (R n R m ). More recisely, we show and R N (f ) L (R n+m ) C N f L (R n+m ) (.5) R N (f ) H (R n R m ) C N f H (R n R m ). (.6) To do this, we rewrite R N (f )(x 1,x ) for R N (f )(x 1,x ) = {[ψ j,k (x 1 u, x v) ψ j,k (x 1 x I,x x J )]ψ j,k f(u,v) j,k I,J I J + ψ j,k (x 1 x I,x x J )[ψ j,k f(u,v) ψ j,k f(x I,x J )]} dudv = R N (x 1,x,y 1,y )f (y 1,y )dy 1 dy, R n R m where R N (x 1,x,y 1,y ), the kernel of R N (f ), is given by R N (x 1,x,y 1,y ) = [ψ j,k (x 1 u, x v) ψ j,k (x 1 x I,x x J )] j,k I,J I J ψ j,k (u y 1,v y )dudv + [ψ j,k (u y 1,v y ) ψ j,k (x I y 1,x J y )] j,k I,J I J ψ j,k (x 1 x I,x x J )dudv = R N,1 (x 1,x,y 1,y ) + R N, (x 1,x,y 1,y ). We now verify that the kernel R N (x 1,x,y 1,y ) satisfies the condition (.) with constant C N and then rove (.5). For R N,1 (x 1,x,y 1,y ), since ψ is a Schwartz function, thus α x 1 β x ψ j,k (x 1 u, x v) α x 1 β x ψ j,k (x 1 x I,x x J ) α x 1 β x ψ j,k (x 1 u, x v) α x 1 β x ψ j,k (x 1 x I,x v) + α x 1 β x ψ j,k (x 1 x I,x v) α x 1 β x ψ j,k (x 1 x I,x x J )

10 Boundedness of Oerators on Multiarameter Hardy Saces 679 [( u x I C j + x 1 u [ + C ) j ( j + x 1 x I ) n+m+1 j α j(m+1) k(m+1) ( j + x 1 u ) n+m+1 ( k + x v ) m+m+1 ( ) v x J k + x v C N j α j k β k ( j + x 1 u ) n+m+1 ( k + x v ) m+m+1, ] k β k(m+1) ( k + x v ) m+m+1 where we use the facts that u, x I I,v,x J J and l(i) = j N,l(J)= k N. Noting that 0 α, β M, then α x 1 β x θ y 1 γ y R N,1 (x 1,x,y 1,y ) Rn Rm j α j(m+1) ] C N j,k ( j + x 1 y 1 ) n+m+1 k β k(m+1) ( k + x y ) m+m+1 θ y 1 γ y ψ j,k (u y 1,v y ) dudv C N 1 x 1 y 1 n+ α + θ 1 x y m+ β + γ. This imlies that R N,1 (x 1,x,y 1,y ) satisfies (.) with constant C N.Thesame results hold for R N, (x 1,x,y 1,y ), and hence (.) holds for R N (x 1,x,y 1,y ) with the constant C N where C deends only on ψ and M. To show (.5), we use Cotlar-Stein s lemma on L (R n+m ). To be more recise, let R N,1 (f ) = j,k R j,k(f ) where R j,k (f )(x 1,x ) = [ψ j,k (x 1 u, x v) ψ j,k (x 1 x I,x x J )]ψ j,k f(u,v)dudv. I,J I J By the roof of the estimates for the kernel of R N given above, it is easy to see that R j,k (f ) L (R n+m ) C N f L (R n+m ) for all j,k. Using an almost orthogonality argument, see [1, ], yields R j,k R j,k (x 1,x,y 1,y ) C N j j k k (j j ) ( (j j ) + x 1 y 1 ) n+1 (k k ) ( (k k ) + x y ) m+1, where a b is the minimum of a and b, and R j,k R j,k (x 1,x,y 1,y ) is the kernel of oerator R j,k R j,k. This imlies R j,k R j,k (f ) L (R n+m ) C N j j k k f L (R n+m ).

11 680 Y. Han et al. Similarly, we obtain R j,k R j,k (f ) L (R n+m ) C N j j k k f L (R n+m ). Thus the roof of the estimate in (.5) follows from Cotlar-Stein s lemma. We remark that the estimate in (.5) can also be obtained by use of Journé s T 1 theorem on roduct sace L (R n+m ). Indeed, by the regularity and cancellation conditions of R N, one only needs to check the weak boundedness conditions with the bound by C N ;see[8] for a similar roof for one-arameter case. We leave the details to the interested reader. Alying Theorem.1 yields (.6). If we choose N big enough so that C N < 1, since the identity oerator I = T N + R N, then TN 1 is also bounded on H (R n R m ) and L (R n+m ), where T N f(x 1,x ) = I J ψ j,k (x 1 x I,x x J )ψ j,k f(x I,x J ). j k I J By interolation, R N is bounded on L (R n+m ) with the bound C N for all 1 < <. Therefore, TN 1 is bounded on L (R n+m ). Set g(x 1,x ) = TN 1 (f )(x 1,x ), then g C f, 1 <<, and moreover, f(x 1,x ) = I J ψ j,k (x 1 x I,x x J )ψ j,k g(x I,x J ), (.7) j k I J where the above series converges in both L (R n+m ) and H (R n R m ) norms. To comlete the roof of Theorem 1.1, we only need to show that the series (.7) converges also in L (R n+m ) for any 1 <<. Note first that by the Littlewood- Paley estimates on L (R n+m ), 1 <<, it is easy to see that I J ψ j,k (x 1 x I,x x J )ψ j,k f(x I,x J ) j k I J C f L (R n+m ). L (R n+m ) Thus it suffices to show that the series (.7) converges in L (R n+m ) for each function f L (R n+m ) L (R n+m ) since the subsace L (R n+m ) L (R n+m ) is dense in L (R n+m ). Indeed, the convergence of the series (.7) inl (R n+m ) imlies the convergence for almost every (x 1,x ) R n+m. For f L (R n+m ) L (R n+m ) set B l ={R = I J : l(i) = j N,l(J)= k N,I B 1 (0,l), J B (0,l), j, k l}, where B 1 (0,l) and B (0,l) are balls centered at 0 with the radius l in R n and R m, resectively. Write ψ R = ψ j,k, then it suffices to show that for each function f L (R n+m ) L (R n+m ) and any ositive integer L, R ψ R (x 1 x I,x x J )ψ R (g)(x I,x J ) l>l L (R n+m )

12 Boundedness of Oerators on Multiarameter Hardy Saces 681 tends to zero as L goes to infinity. Let h L (R n+m ) L (R n+m ) ( = 1). By duality argument, Cauchy s ineuality and Hölder s ineuality, then we have R ψ R (x 1 x I,x x J )ψ R g(x I,x J ) l>l = su R ψ R (x 1 x I,x x J )ψ R g(x I,x J ), h h 1 l>l = su R ψ R h(x I,x J )ψ R g(x I,x J ) h 1 l>l su ψ R h(x I,x J ) ψ R g(x I,x J ) χ R (x 1,x )dx 1 dx h 1 l>l su h 1 { l>l ψ R h(x I,x J ) χ R (x 1,x ) { } 1 ψ R g(x I,x J ) χ R (x 1,x ) dx1 dx l>l { su ψ R h(x I,x J ) χ R (x 1,x ) h 1 l>l { ψ R g(x I,x J ) χ R (x 1,x ) l>l { C su h ψ R g(x I,x J ) χ R (x 1,x ) h 1 l>l } 1. By the L (R n+m ) norm ineuality of the discrete Littlewood-Paley function, that is, for 1 <<, { 1 I J ψ j,k f(x I,x J ) χ R (x 1,x )} j k I J C f, the last term tends to zero as L goes to infinity and this comletes the roof of Theorem 1.1. Before ending this section, we remark that by use of discrete Calderón-tye identity given in Theorem 1.1, one can rovide the following Min-Max tye ineuality:

13 68 Y. Han et al. for 0 < 1 << and each f L (R n+m ) H (R n R m ), [ 1 ψ j,k f(u,v) χ I (x 1 )χ J (x )] j,k I,J su (u,v) I J [ 1 inf ψ j,k f(u,v) χ I (x 1 )χ J (x )], (.8) (u,v) I J j,k I,J where ψ,i,j and are the same as in Theorem 1.1; see[10, 11] for the similar but more difficult roofs in the flag and Zygmund dilation settings. The Min-Max tye ineuality in (.8) easily imlies that for 0 < 1 << and each f L (R n+m ) H (R n R m ), { ψ j,k f(x 1,x ) j,k { 1 ψ j,k f(x I,x J ) χ I (x 1 )χ J (x )}. (.9) j,k I,J This rovides the discrete Littlewood-Paley characterization of H (R n R m ), that is, for each f L (R n+m ) H (R n R m ), 0 < 1 <<, { 1 ψ j,k f(x I,x J ) χ I (x 1 )χ J (x )} f H (R n+m ). (.10) j,k I,J L (R n+m ) 3 Atomic Decomosition and Boundedness Criterion of Oerators in Product Hardy Saces By the discrete Calderón s identity in (1.1), we can first establish the atomic decomosition into (, )-atoms for 0 < 1 <<, namely Theorem 1.. Proof of Theorem 1. Let f L (R n+m ) H (R n R m ), 0 < 1 <<, and ψ R = ψ j,k be functions mentioned in the roof of Theorem 1.1. Set, by Theorem 1.1, { l = (x 1,x ) : [ j,k I,J ] 1 ψ j,k g(x I,x J ) χ I (x 1 )χ J (x ) > l} and B l = {R = I J : R l > 1 R and R l+1 1 } R, where χ I and χ J are the characteristic functions of I and J, resectively, and I and J are dyadic rectangles in R n and R m, resectively. Then, by reroducing the discrete

14 Boundedness of Oerators on Multiarameter Hardy Saces 683 Calderón formula in Theorem 1.1, wehave f(x 1,x ) = j = l I J ψ j,k (x 1 x I,x x J )ψ j,k g(x I,x J ) k I J R ψ R (x 1 x I,x x J )ψ R g(x I,x J ), where ψ R = ψ j,k and x I I and x J J. To see the decomosition above gives an atomic decomosition in terms of (, )- atoms, rewrite f(x 1,x ) = l λ l a l (x 1,x ), (3.1) where a l = 1 λ l and when <, and when 1 <<, R ψ R (x 1 x I,x x J )ψ R g(x I,x J ), { λ l = C ψ R g(x I,x J ) χ R (x 1,x ) l 1 1 { λ l = C ψ R g(x I,x J ) χ R (x 1,x ) l 1 1. In the above exressions we have set l ={(x 1,x ) : M s (χ l )(x 1,x )> }. Since R B l = 4R l, thus 4R l, this imlies that a l is suorted in an oen set l, and hence a l satisfies (1) of Theorem 1.. To see that a l satisfies () of Theorem 1., as in the roof of Theorem 1.1, let h L (R n+m ) L (R n+m ) ( = 1). By Cauchy s and Hölder s ineualities and the discrete Littlewood-Paley suare function estimates on L, 1 <<, then we have R ψ R (x 1 x I,x x J )ψ R g(x I,x J ) = su R ψ R (x 1 x I,x x J )ψ R g(x I,x J ), h h 1 su ψ R h(x I,x J ) ψ R g(x I,x J ) χ R (x 1,x )dx 1 dx h 1

15 684 Y. Han et al. { su ψ R h(x I,x J ) χ R (x 1,x ) h 1 { ψ R g(x I,x J ) χ R (x 1,x ) C su h 1 which yields that when <, { h ψ R g(x I,x J ) χ R (x 1,x ) ( { a l = C ψ R g(x I,x J ) χ R (x 1,x ) l 1 1 R ψ R (x 1 x I,x x J )ψ R g(x I,x J ) l 1 1. Note that a l is suorted in l. Thus if 1 <<, the similar estimate and the definition of λ l yield, ) 1 ( { ) a l = C ψ R g(x I,x J ) 1 χ R (x 1,x ) l 1 1 R ψ R (x 1 x I,x x J )ψ R g(x I,x J ) ( { ) C ψ R g(x I,x J ) 1 χ R (x 1,x ) l 1 1 l 1 1 R ψ R (x 1 x I,x x J )ψ R g(x I,x J ) l 1 1, which imlies that a l satisfies the size condition () of (, )-atoms in Theorem 1.. To verify that a l satisfies the conditions (3) and (4) of Theorem 1., note that a l = Q m( l ) a l,q, where a l,q = 1 λ l,r Q R ψ R (x 1 x I,x x J )ψ R g(x I,x J ).

16 Boundedness of Oerators on Multiarameter Hardy Saces 685 It suffices to verify that each a l,q is a rectangle (, )-atom satisfying (3) and (4) in Theorem 1.. The suort and cancellation conditions follow directly from the conditions on ψ and thus (4) follows easily. Therefore, it remains to show (3). To see that when <,a l satisfies the estimate of (3a), by the same roof for the estimate of a l, we have { a l,q C,R Q ψ R g(x I,x J ) χ R (x 1,x )} 1 and hence, the fact that < and the definition of λ l yield a l,q a l Q m( l ) which, by the estimate () for a l, imlies that a l satisfies (3a) of Theorem 1.. When 1 <<, we have γ δ (Q) a l,q Q m 1 ( l ) C λ l C λ l C λ l γ δ Q m 1 ( l ) { Q m 1 ( l ) {,R Q Q m 1 ( l ) { (Q) γ δ (Q) Q 1,R Q ψ R g(x I,x J ) χ R (x 1,x ) ψ R g(x I,x J ) χ R (x 1,x )dx 1 dx } γ δ (Q) Q } 1 { ψ R g(x I,x J ) χ R (x 1,x )dx 1 dx C,δ l (1 ) l = C δ l 1, where the last ineuality follows from Journé s Lemma [18]. The other summation in (3b) can be roved by the same manner. This shows that a l satisfies (3b) of Theorem 1.. Note that by the maximal theorem l C l. Since if (x 1,x ) R B l then M s (χ R l \ l+1 )(x 1,x )> 1,wehave χ R (x 1,x ) M s (χ R l \ l+1 )(x 1,x ) } = χ R (x 1,x ) 4M s (χ R l \ l+1 )(x 1,x ).

17 686 Y. Han et al. Thus, by the Fefferman-Stein vector valued ineuality, for all 1 <<, { ψ R g(x I,x J ) χ R (x 1,x ) { } = ψ R g(x I,x J ) χ R (x 1,x ) dx1 dx R n R m { } C ψ R g(x I,x J )M s (χ R R n R m l \ l+1 )(x 1,x ) dx1 dx { } C ψ R g(x I,x J ) χ R (x 1,x ) dx1 dx l \ l+1 C l l. Therefore, when <, we have λ l = C l l C l { ψ R g(x I,x J ) χ R (x 1,x ) l 1 l l l 1 = C l l l C l l l { C ψ R g(x I,x J ) χ R (x 1,x ) l C g H C f H, and when 1 <<, λ l = C l l C l { ψ R g(x I,x J ) χ R (x 1,x ) l 1 l l l 1 = C l l l C l l l

18 Boundedness of Oerators on Multiarameter Hardy Saces 687 { C ψ R g(x I,x J ) χ R (x 1,x ) l C g H C f H. Finally, the fact that the atomic decomosition in (3.1) converges in L (R n+m ) follows from the same roof as given in Theorem 1.1. This ends the roof of Theorem 1.. We are now ready to rovide the Proof of Theorem 1.3 We only need to rove the if arts of Theorem 1.3 because by the same roof as in [9], it is easy to check that if a is a (, )-roduct atom then a H (R n R m ) and a H (R n R m ) C where C is a constant uniformly for all (, )- roduct atoms. See the Aendix for the outline of this roof. If T(a) L (R n+m ) C uniformly on all (, )-roduct atoms in L (R n+m ), then by Theorem 1., forf H (R n R m ) L (R n+m ), Tf = l λ l Ta l since T is bounded and f = l λ la l on L (R n+m ). Thus Tf λ l Ta l C λ l C f H. l l () If T(a) H (R n R m ) C uniformly on all (, )-roduct atoms in H (R n R m ), then by Theorem 1. and notes that the H norm of f is the L norm of the Littlewood-Paley function g(f ),forf H (R n R m ) L (R n+m ), Tf H l λ l Ta l H C l λ l C f H. Since H (R n R m ) L (R n+m ) is dense in H (R n R m ), the roof of Theorem 1.3 is comlete. Acknowledgements The authors wish to sincerely thank Professor Fulvio Ricci for his encouragement to comlete this work and his suggestions. The authors are also grateful to the referee s very careful reading and many useful suggestions and comments which imrove the exosition of the aer. Aendix We give the outline of the roof that if a is a (, )-roduct atom for 1 <<, then a H (R n R m ) C, where C is a constant indeendent of a. The roof for this fact when is basically done in [9]. For simlicity, we only consider the case where n = m = 1. But from the roof given below, the roof for the general n and m can be done in the same manner with

19 688 Y. Han et al. minimal modifications. To show a H (R n R m ) C, it will be enough to rove that g(a) C, where g(a) is the Littlewood-Paley g function of a given in Definition.1 and C is a constant indeendent of a. Recall that a is an atom suorted in satisfying conditions (1), () in Theorem 1. and the following (i.e., (3b) in Theorem 1.): a = a R + a R, R m 1 ( ) R m ( ) and for any δ>0, there exists a constant C,δ>0, where C,δ deends only on and δ and a R satisfies condition (4) in Theorem 1. such that γ δ a R L (R n+m ) + 1/ γ δ 1 a R L (R n+m ) C,δ 1 1. R m 1 ( ) R m ( ) We will use the same notation and follow the similar outline as given on age 10 of [9]. Let ={(x 1,x ) : M s (χ )(x 1,x )> 1 } and = ( ). To estimate g(a) dx 1 dx, alying Hölder s ineuality and the boundedness of g on L we get { } g(a) dx 1 dx g(a) dx 1 dx 1 C a 1 C. To handle c g(a) dx 1 dx, we note the roerties of ψ jk in the definition of the g function and follow the outline given on. 11 of [9]. Let R = I J m ( ),let Iˆ be the maximal dyadic cube in R n such that I ˆ J, let ˆ I be the double of Iˆ and ( ˆ I) c be the comlement of ˆ I. Thus, we have ˆ I c R Summing over R gives γ δ 1 (R) a R R 1 R m ( ) { R m ( ) g(a R ) dx 1 dx Cγ δ 1 (R) a R R 1. } { γ δ 1 (R) a R R m ( ) } 1 γ δ 1 (R) R C, where δ = δ and δ = δ ( ). Here the last ineuality above follows from Journé s Lemma and the condition of (3b) of Theorem 1.. The similar roof works when the Littlewood-Paley suare function g is relaced by the oerator T as given in [9]. We leave the details to the interested reader. This means if we assume Fefferman s condition of the oerator on the rectangle atom, then we can show T(a) C for any (, )-atom when 1 <<. Therefore, Fefferman s criterion holds in this case as well. This demonstrates that our definition of (, )-atoms is consistent with the existing multiarameter H theory.

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