Hardy spaces associated with different homogeneities and boundedness of composition operators

Transcription

1 Rev. Mat. Iberoam., 1 33 c Euroean Mathematical Society Hardy saces associated with different homogeneities boundedness of osition oerators Yongsheng Han, Chincheng Lin, Guozhen Lu, Zhuoing Ruan Eric.T.Sawyer Abstract. It is well nown that stard Calderón-Zygmund singular integral oerators with the isotroic non-isotroic homogeneities are bounded on the classical H (R m ) non-isotroic H h (Rm ), resectively. In this aer, we develo a new Hardy sace theory rove that the osition of two Calderón-Zygmund singular integral oerators with different homogeneities is bounded on this new Hardy sace. It is interesting that such a Hardy sace has surrisingly a multiarameter structure associated with the underlying mixed homogeneities arising from two singular integral oerators under consideration. The Calderón-Zygmund deosition an interolation theorem hold on such new Hardy saces. 1. Introduction statement of results The urose of this aer is to develo a new Hardy sace theory rove that the osition of two Calderón-Zygmund singular integrals associated with different homogeneities, resectively, is bounded on these new Hardy saces. Indeed, the osition of oerators was considered by Calderón Zygmund when introducing the first generation of Calderón-Zygmund convolution oerators. Calderón Zygmund discovered that to ose two convolution oerators, T 1 T, it is enough to emloy the roduct of the corresonding multiliers m 1 (ξ) m (ξ). However, the symbol m 3 (ξ) =m 1 (ξ)m (ξ) does not necessarily have zero integral on the unit shere, so they considered the algebra of oerators ci + T, where c is a constant, I is the identity oerator T is the oerator introduced by them. In 1965, Calderón considered again the roblem of the symbolic calculus of the second generation of Calderón-Zygmund singular integral oerators with the minimal regularity with resect to x on ernels L 1 (x, y) L (x, y), corresonding to Mathematics Subject Classification (010): Primary 4B30; Secondary 4B0. Keywords: Hardy saces, Calderón-Zygmund oerators, discrete Calderón s identity, Almost orthogonality estimates, discrete Littlewood-Paley-Stein square functions. Corresonding author.

2 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer oerators T 1 T. This roblem reduced to the study of the mutator which was the first non-convolution oerator raised in harmonic analysis. In the resent aer, we consider the osition of two oerators associated with different homogeneities. To be more recise, let e(ξ) be a function on R m homogeneous of degree 0 in the isotroic sense smooth away from the origin. Similarly, suose that h(ξ) is a function on R m homogeneous of degree 0inthe non-isotroic sense related to the heat equation, also smooth away from the origin. Then it is well-nown that the Fourier multiliers T 1 defined by T 1 (f)(ξ) = e(ξ) f(ξ) T given by T (f)(ξ) =h(ξ) f(ξ) are both bounded on L for 1 <<, satisfy various other regularity roerties such as being of wea-tye (1, 1). It was well nown that T 1 T are bounded on the classical isotroic non-isotroic Hardy saces, resectively. Rivieré in [4] ased the question: Is the osition T 1 T still of wea-tye (1,1)? Phong Stein in [] answered this question gave a necessary sufficient condition for which T 1 T is of weatye (1,1). The oerators Phong Stein studied are in fact ositions with different ind of homogeneities which arise naturally in the -Neumann roblem. This motivates the resent wor in this aer. In order to describe more recisely questions results studied in this aer, we begin with considering all functions oerators defined on R m. We write R m = R m 1 R with x =(x,x m )wherex R m 1 x m R. We consider two inds of homogeneities δ :(x,x m ) (δx, δx m ), δ > 0 δ :(x,x m ) (δx, δ x m ), δ > 0. The first are the classical isotroic dilations occurring in the classical Calderón- Zygmund singular integrals, while the second are non-isotroic related to the heat equations (also Heisenberg grous). For x =(x,x m ) R m 1 R we denote x e =( x + x m ) 1 x h = ( x + x m ) 1. We also use notations j =min{j, } j = max{j, }. The singular integrals considered in this aer are defined by Definition 1.1. A locally integrable function K 1 on R m /{0} is said to be a Calderón- Zygmund ernel associated with the isotroic homogeneity if (1.1) α x α K 1(x) A x m α e for all α 0, (1.) r 1 < x e <r K 1 (x) dx =0 for all 0 <r 1 <r <. We say that an oerator T 1 is a Calderón-Zygmund singular integral oerator associated with the isotroic homogeneity if T 1 (f)(x) =.v.(k 1 f)(x), where K 1 satisfies conditions in (1.1) (1.).

3 Hardy saces associated with different homogeneities 3 Definition 1.. Suose K L 1 loc (Rm \{0}). K is said to be a Calderón- Zygmund ernel associated with the non-isotroic homogeneity if (1.3) α β (x ) α (x m ) β K (x,x m ) B x m 1 α β h for all α 0, β 0, (1.4) r 1< x h <r K (x) dx =0 for all 0 <r 1 <r <. We say that an oerator T is a Calderón-Zygmund singular integral oerator associated with the non-isotroic homogeneity if T (f)(x) =.v.(k f)(x), where K satisfies the conditions in (1.3) (1.4). It is well-nown that any Calderón-Zygmund singular integral oerator associated with the isotroic homogeneity is bounded on L (R m ) for 1 << is also bounded on the classical Hardy sace H (R m )with0< 1. Here the classical Hardy sace H (R m )isintroducedbyfefferman Stein in [FS]. This sace is associated with the isotroic homogeneity. To see this, let ψ (1) S(R m ) with (1.5) su ψ (1) {(ξ, ξ m ) R m 1 R : 1 ξ e }, (1.6) ψ (1) ( j ξ, j ξ m ) = 1 for all (ξ, ξ m ) R m 1 R/{(0, 0)}. j Z The Littlewood-Paley-Stein square function of f S (R m )thenisdefinedby { g(f)(x) = ψ (1) j f(x) } 1, j Z where ψ (1) j (x,x m )= jm ψ (1) ( j x, j x m ). Note that the isotroic homogeneity is involved in g(f). The classical Hardy sace H (R m ) then can be characterized by H (R m )={f S /P(R m ):g(f) L (R m )}, where S /P denotes the sace of distributions modulo olynomials. If f H (R m ), the H norm of f is defined by f H = g(f) L. As we mentioned above, a Calderón-Zygmund singular integral oerator associated with the non-isotroic homogeneity is bounded on L, 1 <<. It is not bounded on the classical Hardy sace but bounded on the non-isotroic Hardy sace. The non-isotroic Hardy sace can also be characterized by the non-isotroic Littlewood-Paley-Stein square function. To be more recise, let ψ () S(R m )with (1.7) su ψ () {(ξ, ξ m ) R m 1 R : 1 ξ h },

4 4 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer (1.8) ψ () ( ξ, ξ m ) = 1 for all (ξ, ξ m ) R m 1 R \{(0, 0)}. Z We then define g h (f), the non-isotroic Littlewood-Paley-Stein square function of f S (R m ), by { g h (f)(x) = ψ () f(x) } 1, Z where ψ () (x,x m )= (m+1) ψ( x, x m ). Note again that the non-isotroic homogeneity is involved in g h (f). The non-isotroic Hardy sace H h (Rm )then can be characterized by H h (Rm )={f S /P(R m ):g h (f) L (R m )} if f H h (Rm ), the H h norm of f is defined by f H = g h(f) h L. If T 1 T are Calderón-Zygmund singular integrals with isotroic nonisotroic homogeneities, resectively, then the osition T 1 T is always bounded on L, 1 <<, however, in general, bounded neither on the classical Hardy sace H (R m ) nor on the non-isotroic Hardy sace H h (Rm ). Our goal of this aer is to develo a new Hardy sace theory associated with different homogeneities such that the osition T 1 T is bounded on this new Hardy sace. A new idea to achieve this is to establish the Littlewood-Paley-Stein theory associated with different homogeneities. More recisely, suose that ψ (1) ψ () are functions satisfying conditions in (1.5) - (1.6) (1.7) - (1.8), resectively. Let ψ j, (x) =ψ (1) j ψ () (x). Define a new Littlewood-Paley-Stein square function by { G (f)(x) = j, Z ψ j, f(x) } 1. We remar that a significant feature is that the multiarameter structure is involved in the above Littlewood-Paley-Stein square function. As in the classical case, it is not difficult to chec that for 1 <<, (1.9) G (f) L f L. The estimates above suggest us to define the H norm of f in terms of the L norm of G (f) when0< 1. However, this continuous version of the Littlewood-Paley-Stein square function G (f) is convenient to deal with the case for 1 << but not for the case when 0 < 1. See further remar about this below. The crucial idea is to relace the continuous version G (f) bythe discrete version Gψ d (f) as follows. To define the discrete version Gψ d (f), the ey tool is discrete Calderón s identity. To be more recise, we first recall classical continuous Calderón s identity on L (R m ). Let ψ (1) be a function satisfying the conditions of (1.5) (1.6). By taing the Fourier transform, we have the following classical continuous Calderón s identity: f(x) = ψ (1) j ψ (1) j f(x), j Z

5 Hardy saces associated with different homogeneities 5 where the series converges in L (R m ) in S 0 (R m ):={f S(R m ): f(x)x α dx = R m 0 for any α 0}. Note that the Fourier transforms of both ψ (1) j ψ (1) j f are actly suorted. Using a similar idea as in the Shannon samling theorem, one can deose ψ (1) j ψ (1) j f(x) by ψ (1) j (x j l)(ψ (1) j f)( j l). l Z m Then classical discrete Calderón s identity is given by (1.10) f(x) = ψ (1) j (x j l)(ψ (1) j f)( j l), j Z l Z m where the series converges in L (R m ), S 0 (R m ) S 0(R m ). See [9] [10] for more details. Now by considering ψ j, = ψ (1) j ψ () taing the Fourier transform, we obtain the following continuous Calderón s identity: (1.11) f(x) = ψ j, ψ j, f(x), j, Z where the series converges in L (R m ), S 0 (R m ) S 0(R m ). Furthermore, we will rove the following discrete Calderón s identity. Theorem 1.3. Suose that ψ (1) ψ () are functions satisfying conditions in (1.5) - (1.6) (1.7) - (1.8), resectively. Let ψ j, (x) =ψ (1) j ψ () (x). Then f(x,x m )= (m 1)(j ) (j ) (ψ j, f)( (j ) l, (j ) l m ) j, Z (l,l m) Z m 1 Z where the series converges in L (R m ), S 0 (R m ) S 0(R m ). ψ j, (x (j ) l,x m (j ) l m ) (1.1), This discrete Calderón s identity leads to the following discrete Littlewood- Paley-Stein square function. Definition 1.4. For f S 0(R m ), Gψ d (f), the discrete Littlewood-Paley-Stein square function of f, is defined by { Gψ(f)(x d,x m )= (ψ j, f) ( (j ) l, (j ) l m ) j, Z (l,l m) Z m 1 Z χ I (x )χ J (x m ) where I are dyadic cubes in R m 1 J are dyadic intervals in R with the side length l(i) = (j ) l(j) = (j ), the left lower corners of I the left end oints of J are (j ) l (j ) l m, resectively. } 1,

6 6 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer Now we formally define the Hardy saces associated with two different homogeneities by the following Definition 1.5. Let 0 < 1. H (R m )={f S 0(R m ):G d ψ (f) L (R m )}. If f H (R m ),thenormoff is defined by f H (R m ) = G d ψ (f) L (R m ). Note that, as mentioned above for the Littlewood-Paley-Stein square function, the multiarameter structures are involved again in the discrete Calderón s identity the Hardy saces H (R m ). To see that these Hardy saces are well defined, we need to show that H (R m ) is indeendent of the choice of the functions ψ (1) ψ (). This will directly follow from the following theorem Theorem 1.6. If ϕ satisfies the same conditions as ψ, then for 0 < 1 f S 0(R m ), G d ψ(f) L (R m ) G d ϕ(f) L (R m ). We would lie to oint out that one can define the Hardy sace H (R m )in terms of G (f), the Littlewood-Paley-Stein square function. Then one has to show the following su-inf rincile for all 0 << : { su u I,v J j, Z,I,J { j, Z,I,J ψ j, f(u, v) χ I (x )χ J (x m ) inf φ j, f(u, v) χ I (x )χ J (x m ) u I,v J } 1 L where I are dyadic cubes in R m 1 J are dyadic intervals in R with the side length l(i) = (j ) l(j) = (j ), ψ j, (x) =ψ (1) j ψ () (x), φ j,(x) = φ (1) j φ () (x) ψ(1), φ (1) ψ (), φ () are functions satisfying conditions in (1.5) - (1.6) (1.7) - (1.8), resectively. This will actually imly the equivalence of the L norms of the two square functions G (f) Gψ d (f) allow us to use the discrete Littlewood-Paley-Stein square function to define the Hardy sace. In the case of the multiarameter structure associated with the flag singular integrals, it was done in [15] (see Theorem 1.9 there). However, such a roof in our case is more licated than using the discrete Littlewood-Paley-Stein square function directly as we are doing here. This is why, instead of using G (f), we decide to choose Gψ d (f) to define the Hardy sace H (R m ). Indeed, by alying a similar argument as in [9], one can also show that for all 0 <<, G (f) G d ψ(f). We omit the details of the roof refer reader to [9] for further details. We now state the main results of this aer. Theorem 1.7. Let T 1 T be Calderón-Zygmund singular integral oerators with the isotroic non-isotroic homogeneity, resectively. Then for 0 < 1, the osition oerator T = T 1 T is bounded on H (R m ). } 1 L

7 Hardy saces associated with different homogeneities 7 It is well nown that the atomic deosition of the classical Hardy saces is the main tool to study the H L boundedness for classical Calderón-Zygmund oerators. See [4], [6], [1] [13]. However, to get an atomic deosition for the Hardy sace H (R m ) with miltiarameter structures, as in the classical case, one needs first to establish Journé s covering lemma in this setting. See [1], [], [3], [19], [7], [8] [1] for more details. Our aroach is quite different from this scheme. Indeed, we will rove the following theorem Theorem 1.8. Let 0 < 1. If f L (R m ) H (R m ), then there is a constant C = C() such that f L (R m ) C f H (R m ), where the constant C is indeendent of f. We remar that the roof of the above theorem does not use atomic deosition hence Journé s covering lemma is not required. As a consequence, we obtain Theorem 1.9. Let 0 < 1. Suose that T is a osition of T 1 T as given in Theorem 1.7. Then T extends to a bounded oerator from H (R m ) to L (R m ). Next we rovide the Calderón-Zygmund deosition rove an interolation theorem on H (R m ). We note that H (R m )=L (R m ) for 1 <<. Theorem (Calderón-Zygmund deosition for H) Let 0 < 1, << 1 < let α > 0 be given f H. Then we may write f = g + b where g H 1 b H such that g 1 H 1 Cα 1 f H b H Cα f, where C is an absolute constant. H Theorem (Interolation theorem on H) Let 0 < < 1 < T be a linear oerator which is bounded from H to L bounded from H 1 to L 1, then T is bounded from H to L for all << 1. Similarly, if T is bounded on H H, 1 then T is bounded on H for all << 1. Before we end this section, several remars must be in order. First, as mentioned before, the continuous version of the Littlewood-Paley-Stein square function G (f) is convenient to deal with the case for 1 << but not for the case when 0 < 1. However, we can still use this continuous version G (f) to define the Hardy saces H (R m ) for 0 < 1. More recisely, suose that ψ (1) S 0 satisfies 0 ψ () S 0 satisfies ψ (1) (tξ,tξ m ) dt t = 1 for all (ξ, ξ m ) R m 1 R/{(0, 0)}

8 8 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer 0 ψ () (sξ,s ξ m ) ds s = 1 for all (ξ, ξ m ) R m 1 R/{(0, 0)}. Set ψ t,s = ψ (1) t ψ s (), where ψ (1) t (x,x m )=t m ψ (1) ( x t, xm t ) ψ() s (x,x m )= s m 1 ψ () ( x s, x m s ). Then one can argue that the H(R m ) norm of f defined in Definition 1.5 is equivalent to { 0 0 ψ t,s f(x,x m ) dt t ds s } 1 L. The same ideas in this aer can be carried out to the roof of the above equivalent norms of such defined two Hardy saces. Secondly, in this aer, we restrict our attention to the above two very secific dilations. However, all results in this aer can be carried out to the osition with more singular integral oerators associated with more general non-isotroic homogeneities. To see this, let T i (f)(x) =.v.k i f(x), 1 i n, be singular integral oerators associated with non-isotroic dilations given by δ i :(x 1,x,,,x m ) (δ λ i,1 i x 1, δ λ i, i x,,, δ λ i,m i x m ) for δ i > 0, λ i,l > 0, 1 i n 1 l m. For x R m λ we denote x i = x 1 i,1 + λ x i, + λ + xm i,m. Let ψ (i) S(R m )with su ψ (i) {(ξ 1, ξ, ξ m ) R m : 1 ξ i }, ψ (i) ( jiλi,1 ξ 1, jiλi, ξ,, jiλi,m ξ m ) = 1 for all (ξ 1, ξ,, ξ m ) R m /{0}. j i Z Set ψ j1,j,,j n (x) =ψ (1) j 1 ψ () j ψ (n) j n (x), where ψ (i) j i (x) = ji(λi,1+λi,+ +λi,m) ψ (i) ( jiλi,1 x 1, jiλi, x,, jiλi,m x m ). Define a Littlewood-Paley-Stein square function by { G (f)(x) = j 1,j,,j n Z ψ j1,j,,j n f(x) } 1. Alying the same line as in this aer, one can develo the Hardy sace theory associated with these more general non-isotroic dilations. The details of the roofs

9 Hardy saces associated with different homogeneities 9 seem to be rather lengthy to be written out. Therefore, we shall not discuss these in more details in this aer. Thirdly, the regularity conditions on ernels can be weaened if one considers only the H (R m ) boundedness for the certain range of. Finally, we would lie to remar that the method of discrete Littlewood-Paley- Stein analysis in the multiarameter settings used in this aer has been used in a number of other cases earlier. This method allows us to avoid using the Journé covering lemma to rove the boundedness of multiarameter singular integrals from the Hardy saces. It first aeared in [15] where the theory of the multiarameter Hardy saces associated with the flag singular integrals was develoed in [16] where the discrete Littlewood-Paley-Stein theory was established in the multiarameter structure associated with the Zygmund dilation (see also the exository article [17]). A recent develoment for the imlicit multiarameter Hardy sace the Marciniewicz multilier theory on the Heisenberg grou has been successfully carried out in [18]. We also refer to [5], [0], [3], [14] for this discrete Littlewood-Paley-Stein analysis in other settings such as weighted multiarameter Hardy saces in Euclidean saces or multiarameter theory in homogeneous saces. Section 1 deals with Theorem 1.3. The roof of Theorem 1.6 is given in section 3. The method of the roof will be alied to the roof of Theorem 1.8 Theorem To show Theorem 1.7, we rovide a discrete Calderón-tye identity, Theorem 4.1 which has its own interest. These will be given in Section 4. Theorem 1.8 Theorem 1.9 are roved in Section 5. In the last section, we rove the Calderón-Zygmund deosition interolation theorems.. Proof of Theorem 1.3 As mentioned in the revious section, by taing the Fourier transform, we obtain the following continuous Calderón s identity: (.1) f(x) = ψ j, ψ j, f(x), j, Z where the convergence of series in L (R m ), S 0 (R m ) S 0(R m ) follows from the results in the classical case. See [9] [10] for more details. To get a discrete version of Calderon s identity, we need to deose ψ j, ψ j, f in (.1). Similar to a method as in [10], set g = ψ j, f h = ψ j,.the Fourier transforms of g h are given by ĝ(ξ, ξ m )= ψ (1) ( j ξ, j ξ m ) ψ () ( ξ, ξ m ) f(ξ, ξ m ) ĥ(ξ, ξ m )= ψ (1) ( j ξ, j ξ m ) ψ () ( ξ, ξ m ). Note that the Fourier transforms of g h are both actly suorted. More recisely, su ĝ, su ĥ {(ξ, ξ m ) R m 1 R : ξ j π, ξ m j π}.

10 10 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer Thus, we first ex ĝ in a Fourier series on the rectangle R j, = {ξ R m 1, ξ m R : ξ j π, ξ m j π}: ĝ(ξ, ξ m )= (l,l m) Z m 1 Z (m 1)(j ) (j ) (π) m R j, ĝ(η, η m )e i( (j ) l η + (j ) l mη m) dη dη m e i( (j ) l ξ + (j ) l m ξ m ) then relace R j, by R m since ĝ is suorted in R j,. Finally, we obtain ĝ(ξ, ξ m )= (l,l m ) Z m 1 Z (m 1)(j ) (j ) g( (j ) l, (j ) l m ) Multilying ĥ(ξ, ξ m ) from both sides yields e i( (j ) l ξ + (j ) l m ξ m ). ĝ(ξ, ξ m )ĥ(ξ, ξ m )= (l,l m ) Z m 1 Z (m 1)(j ) (j ) g( (j ) l, (j ) l m ) ĥ(ξ, ξ m ) e i( (j ) l ξ + (j ) l mξ m). Note that ĥ(ξ, ξ m ) e i( (j ) l ξ + (j ) l mξ m) = ĥ( (j ) l, (j ) l m )(ξ, ξ m ). Therefore, alying the identity g h =(ĝ ĥ) imlies that (g h)(x,x m )= (m 1)(j ) (j ) g( (j ) l, (j ) l m ) (l,l m) Z m 1 Z h(x (j ) l,x m (j ) l m ). (.) Substituting g by ψ j, f h by ψ j, into Calderón s identity in (.1) gives the discrete Calderón s identity in (1.1) the convergence of the series in the L (R m ). It remains to rove that the series in (1.1) converges in S 0 (R m ). To do this, it suffices to show that (m 1)(j ) (j ) (ψ j, f)( (j ) l, (j ) l m ) j N 1 or N (l,l m ) Z m 1 Z ψ j, (x (j ) l,x m (j ) l m ) (.3) tend to zero in S 0 (R m ) as N 1 N tend to infinity. For the sae of convenience, we denote x I = (j ) l x J = (j ) l m. Let I be dyadic cubes in R m 1 J be dyadic intervals in R with side-length l(i) = (j ) l(j) = (j ), the left lower corners of I the left end oints of J are x I x J, resectively. Then the above limit will follow from the following estimates: for any fixed j, any given integer M>0, α 0, there exists a constant C = C(M,α) > 0 which is indeendent of j such that I J (ψ j, f)(x I,x J )(D α ψ j, )(x x I,x m x J ) C j (1 + x + x m ) M. (.4) I J

11 Hardy saces associated with different homogeneities 11 To show (.4), we aly the classical almost orthogonality argument. To be more recise, for any given ositive integers L 1 L, there exists a constant C = C(L 1,L ) > 0 such that (.5) ψ (1) j ψ (1) j j L 1 (j j )m j (x,x m ) C (1 + (j j ) x + (j j ) x m ) L (.6) ψ () ψ () (x L 1 ( )(m+1),x m ) C (1 + ( ) x + ( ). x m ) L Alying (.6) with ψ () 0 = f,l 1 = L +M + m + 1 L = M, where L M are any fixed ositive integers, we obtain (ψ () f)(x (L+M+m+1) ( 0)(m+1),x m ) C (1+ ( 0) x + ( 0) x m ) M C L 1 (1+ x + x m ) M, where the last inequality is obvious if 0, when 0, Note that ψ () (m+1) (1 + x + x m ) M (M+m+1) 1 (1 + x + x m ) M. f S 0 (R m ). Similarly, we have that (.7) (ψ (1) j (ψ () f))(u,u m ) C L j L 1 (1 + u + u m ) M. From the size conditions of the functions ψ (1) ψ (), we have that for any fixed large M, D α ψ j, (u,u m ) = D α (ψ (1) j ψ () )(u,u m ) C j α + α j(m 1) (m+1) (1 + j u v + j u m v m ) M (1 + v + v m ) M dv dv m C j α + α (j )(m 1) (j ) (1 + j u + j u m ) M C j (M+m+ α ) (M++ α ) 1 (1 + u + u m ) M. (.8) Estimates in (.7) (.8) yield I J (D α ψ j, )(x x I,x m x J )(ψ j, f)(x I,x J ) I J C (L M α ) j (L M m α ) 1 I J (1 + x I + x J ) M (1 + x x I + x m x J ) M I J = C (L M α ) j (L M m α ) dy dy m I J I J (1 + x I + x J ) M (1 + x x I + x m x J ) M. (.9)

12 1 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer Note that if y I y m J, thenl(i)+ x x I l(i)+ x y, l(i)+ x I l(i)+ y, l(j)+ x m x J l(j)+ x m y m, l(j)+ x J l(j)+ y m. The simle calculation gives 1 (1 + x x I + x m x J ) M j M 3M (l(i)+l(j)+ x x I + x m x J ) M Similarly, j 4M 6M (1 + x y + x m y m ) M. 1 (1 + x I + x J ) M j 4M 6M (1 + y + y m ) M. This imlies that the last term in (.9) is dominated by C (L 0M α ) j (L 0M m α ) 1 (1 + x + x m ) M. Choosing L = 0M + α + m + 3, we derive the estimates in (.4) hence the series in (.3) converges to zero as N 1 N tend to infinity. Therefore, the series in (1.1) converges in S 0 (R m ). By the duality argument, we obtain the series in (1.1) converges in S 0(R m ). The roof of Theorem 1.3 is concluded. 3. Proof of Theorem 1.6 In this section, we first derive almost orthogonality estimates in Lemma 3.1 discrete version of maximal estimate in Lemma 3.. Lemmas together with Theorem 1.3 yield Theorem 1.6. Lemma 3.1. (Almost orthogonality estimates) Suose that ψ j, ϕ j, satisfy the same conditions in (1.5)-(1.8). Then for any given integers L M, there exists a constant C = C(L, M) > 0 such that ψ j, ϕ j, (x,x m ) C j j L L (j j )(m 1) (1 + j j x ) (M+m 1) Proof: We first write (ψ j, ϕ j, )(x,x m )= R m 1 R j j ( ) (1 + j j ( ) x m ). (M+1) (ψ (1) j ϕ (1) j )(x y,x m y m )(ψ () By the almost orthogonal estimates as in (.4) (.5), we have ϕ() )(y,y m )dy dy m. (3.1) ψ (1) j ϕ (1) (j j )m j j L j (u,u m ) C (1 + (j j ) u ) (M+m 1) (1 + (j j ) u m ). (M+1)

13 Hardy saces associated with different homogeneities 13 (3.) ψ () ϕ () (y ( )(m+1) L,y m ) C (1 + ( ) y ) (M+m 1) (1 + ( ) y m ). (M+1) The estimates in (3.1) (3.) imly that (3.3) (ψ j, ϕ j, )(x,x m ) C j j L L AB, where A = (j j ) ( ) R (1 + (j j ) y m ) (M+1) (j j )(m 1) C (1 + j j x ) (M+m 1) (1 + ( ) x m y m ) (M+1) dy m B = Rm 1 (j j )(m 1) )(m 1) ( (1 + (j j ) y ) (M+m 1) (1 + ( ) x y ) j j ( ) C (1 + j j ( ) x m ). (M+1) (M+m 1) dy This imlies the conclusion of Lemma 3.1. Now we rove the following estimate of the discrete version of the maximal function. Lemma 3.. Let I,I be dyadic cubes in R m 1 J, J be dyadic intervals in R with the side lengths l(i) = (j ), l(i ) = (j ) l(j) = (j ), l(j )= (j ), the left lower corners of I,I the left end oints of J, J are (j ) l, (j ) l, (j ) l m (j ) l m, resectively. Then for any u,v m 1 I, u m,v m J, any M+m 1 < δ 1, (l,l m ) Zm 1 Z C 1 { M s [( (m 1)(j j ) j j (m 1)(j ) (j ) (1 + j j u (j ) l ) (M+m 1) (ϕ j, ) l, f)( (j (j ) l m) (1 + j j u m (j ) l m ) (M+1) δ/ ]} 1/δ (ϕ j, ) l, f)( (j (j ) l m) χ I χ J ) (v,v m ) (l,l m ) Zm 1 Z where C 1 = C (m 1)( 1 δ 1)(j j ) + ( 1 δ 1)(j j ) +, here (a b) + = max{a b, 0}, M s is the strong maximal function. Before roving Lemma 3., we would lie to oint out that this lemma is the ey tool to show Theorem The discrete version lays a crucial

14 14 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer role for this maximal function estimate. And this is why we choose the discrete Littlewood-Paley-Stein square function use it to define the Hardy sace. Proof of Lemma 3.: For the sae of convenience, we denote by x I = (j ) l,x I = (j ) l the left lower corners of I,I by x J = (j ) l m,x J = (j ) l m the left end of oints of J, J, resectively. Set A 0 = { I : u x I } { 1, B (j j ) 0 = J : u m x J } 1, (j j ) for r 1 s 1, A r = { I : r 1 < u x I { r}, B (j j ) s = J : s 1 < u m x J s}. (j j ) For any fixed r, s 0, denote E = {(w,w m ) R m 1 R : w u r (j j ) + (j ), w m u m r (j j ) + (j ) }. Then A r B s E for any (v,v m ) I J, (v,v m ) E. Obviously, E C (m 1)[r (j j )] [s (j j )]

15 Hardy saces associated with different homogeneities 15 Thus for m 1 M+m 1 < δ 1, (l,l m ) Zm 1 Z (m 1)(j j ) j j (m 1)(j ) (j ) (1 + j j u (j ) l ) (M+m 1) (ϕ j, ) l, f)( (j (j ) l m) (1 + j j u m (j ) l m ) (M+1) C r(m+m 1) s(m+1) (m 1)(j j ) j j (m 1)(j ) (j ) r,s 0 ( ) 1/δ (ϕ j, f)(x I,x J ) δ I J A r B s = C r(m+m 1) s(m+1) (m 1)(j j ) j j I J E 1/δ r,s 0 { } 1/δ 1 I 1 J 1 (ϕ j, E f)(x I,x J ) δ χ I χ J dx E I J A r B s C r(m+m 1) s(m+1) (m 1)(j j ) j j I 1 1 δ J 1 1 δ E 1/δ r,s 0 { ( M s C 1 { M s ( = C 1 { M s [( = C 1 { M s [( I J A r B s (ϕ j, f)(x I,x J ) δ χ I χ J I J (ϕ j, f)(x I,x J ) δ χ I χ J I J (ϕ j, f)(x I,x J ) χ I χ J (l,l m ) Zm 1 Z ) } 1/δ (v,v m ) ) } 1/δ (v,v m ) ) δ/ ] } 1/δ (v,v m ) (ϕ j, f)( (j ) l, (j ) l m) χ I χ J ) δ/ ] (v,v m )} 1/δ. Now we return to Proof of Theorem 1.6: Let f S 0(R m ). We denote x I = (j ) l,x J = (j ) l m,x I = (j ) l x J = (j ) l m. Discrete Calderón s identity on S /P(R m m 1 ) the almost orthogonality estimates yield that for M+m 1 < δ <

16 16 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer 1 any v I,v m J, (ψ j, f)(x I,x J ) = (m 1)(j ) (j ) (ψ j, ϕ j, )(x I x I,x J x J )(ϕ j, f)(x I,x J ) j, (l,l m ) C j j L L (m 1)(j j ) j j (m 1)(j ) (j ) (1 + j j x j, I x I ) (M+m 1) (l,l m ) (ϕ j, f)(x I,x J ) (1 + j j x J x J ) (M+1) C { [( j j L L C 1 M s j, (l,l m ) (ϕ j, f)(x I,x J ) χ I χ J where the last inequality follows from Lemma 3.. Squaring both sides, then multilying χ I, χ J, summing over all j, Z (l, l m ) Z m 1 Z, finally alying Hölder s inequality we obtain that for any x I,x m J, max{ m+1 L+m+1, m 1 M+m 1 } < δ < 1, ) δ/ ] } 1/δ (v,v m ) Gψ(f)(x d,x m ) C { } j j L L (m 1)( 1 δ 1)(j j ) + ( 1 δ 1)(j j ) + j, j, { j j L L (m 1)( 1 δ 1)(j j ) + ( 1 δ 1)(j j ) + j, [( {M s (ϕ j, ) l, f)( (j (j ) ) δ ] } } l m) /δ χ I χ J (x,x m ) (l,l m ) Zm 1 Z { [( C {M s j, (l,l m ) Zm 1 Z (ϕ j, f)( (j ) l, (j ) l m) χ I χ J ) δ ] } } /δ (x,x m ), where in the last inequality we use the facts that (j j ) + j j +, (j j ) + j j + if choose L>(m + 1)( 1 δ 1) then j, j j L L (m 1)( 1 δ 1)(j j ) + ( 1 δ 1)(j j ) + C j j L L (m 1)( 1 δ 1)(j j ) + ( 1 δ 1)(j j ) + C. j, Alying Fefferman-Stein s vector-valued strong maximal inequality on L /δ (l /δ ) yields G d ψ(f) L (R m ) C G d ϕ(f) L (R m ).

17 Hardy saces associated with different homogeneities 17 The conclusion of Theorem 1.6 follows. As a consequence of Theorem 1.6, L (R m ) H(R m )isdenseinh(r m ). Indeed we have the following result Corollary 3.3. S 0 (R m ) is dense in H (R m ). Proof: Let f H (R m ). For any fixed N>0, denote E = {(j,, l, l m ): j N, N, l N, l m N}, f N (x,x m ):= (j,,l,l m ) E (m 1)(j ) (j ) (ψ j, f)( (j ) l, (j ) l m ) ψ j, (x (j ) l,x m (j ) l m ) where ψ j, is the same as in Theorem 1.3. Since ψ j, S 0 (R m ), we obviously have f N S 0 (R m ). Reeating the roof of Theorem 1.6, we can conclude that f N H (R m ) C f H (R ). To see that m f N tends to f in H, by the discrete Calderón s identity in S 0(R m ) in Theorem 1.3, (f f N )(x,x m )= (m 1)(j ) (j ) (ψ j, f)( (j ) l, (j ) l m ) (j,,l,l m) E c where the series converges in S 0(R m ). Therefore, ψ j, (x (j ) l,x m (j ) l m ), G ψ (f f N ): = = { j, (l,l m ) { j, (l,l m ) ψ j, (f f N )( (j ) l, (j ) l m) χ I χ J ψ j, (ψ j, f)( (j ) l, (j ) l m ) (j,,l,l m ) E c (m 1)(j ) (j ) } 1/ ψ j, ψ j,( (j ) l (j ) l, (j ) l m (j ) l m ) χ I χ J } 1/ Reeating the roof of Theorem 1.6, G ϕ (f f N ) L (R m ) C ψ j, f( (j ) l, (j ) l m ) χ I χ J } 1 (j,,l,l m ) E c L (R m ), where the last term tends to 0 as N tends to infinity. This imlies that f N tends to f in the H(R m ) norm as N tend to infinity.

18 18 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer 4. Proof of Theorem 1.7 To show Theorem 1.7, we need a discrete Calderón-tye identity on L (R m ) H (R m ), which has its own interests. To do this, let φ (1) S(R m )withsu φ (1) B(0, 1), (4.1) φ (1) ( j ξ) = 1 for all ξ R m \{0}, j Z (4.) R m φ (1) (x)x α dx = 0 for all α 10M, where M is a fixed large ositive integer deending on. We also let φ () S(R m ) with su φ () B(0, 1), (4.3) φ () ( ξ, ξ m ) = 1 for all (ξ, ξ m ) R m 1 R \{(0, 0)}, Z (4.4) R m φ () (x)x β dx = 0 for all β 10M. Set φ j, = φ (1) j φ (), where φ(1) j (x) = jm φ (1) ( j x) φ () (x,x m )= (m+1) φ () ( x, x m ). The discrete Calderón-tye identity is given by the following Theorem 4.1 Let φ (1) φ () satisfy conditions from (4.1) to (4.4). Then for any f L (R m ) H(R m ), there exists h L (R m ) H(R m ) such that for asufficiently large N N, f(x,x m )= I J φ j, (x (j ) N l,x m (j ) N l m ) j, Z (l,l m ) Z m 1 Z (φ j, h)( (j ) N l, (j ) N l m ), (4.5) where the series converges in L,Iare dyadic cubes in R m 1 J are dyadic intervals in R with side-length l(i) = (j ) N l(j) = (j ) N, the left lower corners of I the left end oints of J are (j ) N l (j ) N l m, resectively. Moreover, (4.6) f L (R m ) h L (R m ), (4.7) f H (R m ) h H (R m ). We oint out that the main difference between the discrete Calderón-tye identity above the discrete Calderón s identity given in Theorem 1.3 is that for

19 Hardy saces associated with different homogeneities 19 any fixed j, Z, φ j, (x,x m ) in (4.5) have act suorts but ψ j, (x,x m )in (1.17) don t. Being of act suort allows to use the orthogonality argument in the roof of Theorem 1.7. Proof of Theorem 4.1: By taing the Fourier transform, we have that for any f L (R m ), f(x,x m )= φ j, φ j, f(x,x m ). j, Alying Coifman s deosition of the identity oerator, we obtain f(x,x m ) = I J φ j, (x (j ) N l,x m (j ) N l m ) j, where (l,l m) (φ j, f)( (j ) N l, (j ) N l m )+R N (f)(x,x m ) := T N (f)(x,x m )+R N (f)(x,x m ), R N (f)(x,x m ) = [φ j, (x y,x m y m )(φ j, f)(y,y m ) j, I J = j, + j, (l,l m ) φ j, (x (j ) N l,x m (j ) N l m )(φ j, f)( (j ) N l, (j ) N l m )]dy dy m [φ j, (x y,x m y m ) φ j, (x (j ) N l,x m (j ) N l m )] (l,l m ) (l,l m ) I J (φ j, f)(y,y m )dy dy m φ j, (x (j ) N l,x m (j ) N l m ) I J := R 1 N (f)(x,x m )+R N (x,x m ), [φ j, f(y,y m ) φ j, f( (j ) N l, (j ) N l m )]dy dy m here I are dyadic cubes in R m 1 J are dyadic intervals in R with side-length l(i) = (j ) N l(j) = (j ) N the left lower corners of I the left end oints of J are (j ) N l (j ) N l m, resectively. We claim that for i =1,, (4.8) R i N (f) L (R m ) C N f L (R m ), (4.9) R i N (f) H (R m ) C N f H (R m ), where C is the constant indeendent of f N. Assume the claim for the moment, then, by choosing sufficiently large N, T 1 N = (R N ) n is bounded in both L H, which imlies that n=0 T 1 N (f) L (R m ) f L (R m )

20 0 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer T 1 N (f) H (R m ) f H (R m ). Moreover, for any f L (R m ) H(R m ), set h = T 1 N (f), then f(x,x m ) = T N (T 1 N (f))(x,x m ) = I J φ j, (x (j ) N l,x m (j ) N l m ) j, Z (l,l m ) Z m 1 Z (φ j, h)( (j ) N l, (j ) N l m ), where the series converges in L. Now we show the claim. Since the roofs for R 1 N R N are similar, we only give the roof for R 1 N. Roughly seaing, the roof is similar to Theorem 1.6. To see this, let f L (R m ) H(R m ). Alying discrete Calderón s identity in L (R m ) in Theorem 1.3 yields ψ j, R1 N (f)(x,x m ) = j, Z (l,l m) Z m 1 Z I J ψ j, [φ j,( y, y m ) φ j, ( (j ) N l, (j ) N l m )](x,x m )(φ j, f)(y,y m )dy dy m = ψ j, [φ j,( y, y m ) j, Z (l,l m) Z m 1 Z I J φ j, ( (j ) N l, (j ) N l m )](x,x m ) { φ j, I J ψ j, ( ) l, (j (j ) l m) j, Z (l,l m ) Zm 1 Z } (ψ j, ) l, f)( (j (j ) l m) (y,y m )dy dy m, (4.10) where I are dyadic cubes in R m 1 J are dyadic intervals in R with the side length l(i )= (j ) J are dyadic intervals in R with the side length l(j )= (j ), the left lower corners of I the left end oints of J are (j ) l (j ) l m,resectively. Set φ j, = φ j, (z y,z m y m ) φ j, (z (j ) N l,z m (j ) N l m ). Then by the almost orthogonality arguments in Lemma 3.1, we obtain ψ j, φ j, (x,x m ) N 10M j j 10M (j )(m 1) j (1 + j x m y m ) (M+1) (1 + j x y ) (M+m 1)

21 Hardy saces associated with different homogeneities 1 similarly, for y I,y m J, φ j, ψ j, (y (j ) l,y m (j ) l m) 10M j j 10M (j )(m 1) (1 + j y (j ) l ) (M+m 1) j (1 + j y m (j ) l m) ). (M+1) Substituting these estimates into the last term in (4.10) yields ψ j, R1 N (f)(x,x m ) I J (ψ j, ) l, f)( (j (j ) l m) j, Z (l,l m ) Zm 1 Z j, Z (l,l m ) Z m 1 Z I J N j j 3M 3M (j )(m 1) j (1 + j x y ) (M+m 1) (1 + j x m y m ) j j 3M 3M (M+1) (j )(m 1) j (1 + j y (j ) l ) (M+m 1) (1 + j y m (j ) l m) ) (M+1) dy dy m N j j, (l,l m ) j 3M 3M I J (j j ) ( ) (j j )(m 1) (1 + j j x (j ) l ) (M+m 1) (1 + (j j ) ( ) x m (j ) l m ) (ψ (M+1) j, ) f)( (j l, (j ) l m). By the L boundedness of the discrete Littlewood-Paley-Stein square function (f), we have G d ψ R 1 N (f) L Gψ(R d 1 N f)(x,x m ) L N { N f L. j, Z (l,l m ) Zm 1 Z Reeating the same roof as in Theorem 1.6 imlies (ψ j, f)( (j ) l, (j ) l m) χ I χ J} 1 L R 1 N (f) H Gψ(R d 1 N f)(x,x m ) L N { (ψ j, ) l, f)( (j (j ) l m) χ I χ J} 1 L N f H. j, Z (l,l m ) Zm 1 Z The claim is concluded hence Theorem 4.1 follows. Reeating the same roof of Theorem 1.6, we have

22 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer Corollary 4.1. Let 0 < 1. Suose φ j, satisfies the same conditions as in Theorem 4.1 with a large M deending on. Then for a large N as in Theorem 4.1 f L H, f H ( j, Z (l,l m) Z m 1 Z (φ j, f)( (j ) N l, (j ) N l m ) χ I χ J ) 1/ L We now rove Theorem 1.7. Proof of Theorem 1.7: We may assume that K i is the ernel of the convolution oerator T i, i =1,, K is the ernel of the osition oerator T = T 1 T. Then T (f) =K f K = K 1 K. For f L H, 0 < 1, by the L boundedness of T alying discrete Calderon s identity of f on L H in Theorem 4.1, we conclude T (f) H C { (φ j, K f)( (j ) N l, (j ) N l m ) χ I χ J } 1 L j, (l,l m ) = C { ) (j ) (φ j, ) N l, h)( (j (j ) N l m) j, j, (l,l m ) (l,l m ) (m 1)(j (K φ j, φ j, )( (j ) N l (j ) N l, (j ) N l m (j ) N l m) χ I χ J } 1 L, where φ j,, φ j,,h N are the same as in Theorem 4.1. We claim that for any given M>0, (4.11) K 1 φ (1) (4.1) K φ () m (x,x m ) C (1 + x ) M+m 1 (1 + x m ) M+1, (m+1) (x,x m ) C (1 + x ) M+m+1 (1 + x m ) M+1. We only show (4.1) here since the roof of (4.11) is similar. following two cases: Case 1. x h : In this case, x x m 4, which imly that We consider the 1+ x x m 1. By the fact su φ () {x : x h } the cancellation condition in (4.4),

23 Hardy saces associated with different homogeneities 3 K φ () (x) is bounded by K φ () (x) = lim ε 0 K (x y)φ () ε x y h 10 (y)dy = lim K (x y) [φ () ε 0 (y) φ() (x)] dy ε x y h 3 C (m+1) ( x y ) (m 1)+1 dy x m y m + dy m x y 3 (m+1) x m y m 9 C (m+1) C (1 + x ) M+m 1 (1 + x m ) M+1. Case. x h > : In this case, x > or x m > 4, which imly that 1+ x x or 1 + x m x m. By the cancellation condition of φ () with order 4M in (4.4) the size condition of K in (1.3), K φ () y (x) = 1 [K (x y) h α! Dα1 x Dα x m K (x,x m )y α ] φ () (y)dy y h ( y h ) 4M+1 C ( x h ) α = α 1 + α 4M φ() m+1+4m+1 (m+1) (y) dy C (1 + x ) M+m 1 (1 + x m ) M+1. Thus the claim follows. By the classical orthogonality argument, for any fixed L amd M, (4.13) φ (1) j φ (1) j j L m(j j ) j (x,x m ) C (1 + (j j ) x ) (M+m 1) (1 + (j j ) x m ), (M+1) (4.14) φ () φ () (x L ( )(m+1),x m ) C (1 + ( ) x ) (M+m 1) (1 + ( ) x m ). (M+1) Estimates from (4.11) to (4.14) yield that K φ j, φ j, (x,x m ) = [K 1 φ (1) j φ (1) j ] [K ψ () ψ () ](x,x m ) j j L L (j j )(m 1) j j C. (4.15) (1 + j j x ) (M+m 1) (1 + j j x m ) (M+1) Using the estimates in (4.15) alying the same roof as in Theorem 1.6 yield that for f L H 0 < δ < 1, { } 1 T (f) H C {M s [( (φ j, h)( (j ) N l, (j ) N l m) χ I χ J ) δ ]} δ L j, (l,l m ) C h H C f H.

24 4 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer Since L H is dense in H, we conclude the roof of Theorem Proofs of Theorems In this section using Theorem 4.1, we rove Theorem 1.8. Theorem 1.9 then follows directly from Theorem 1.8. Proof of Theorem 1.8: For any f L (R m ) H (R m ), set Ω i = {(x,x m ) R m 1 R : G d φ(f)(x,x m ) > i }, where G d φ(f) = { (φ j, h)( (j ) N l, (j ) N l m ) χ I χ J } 1, j, Z (l,l m ) Z m 1 Z here φ j, h are given by Theorem 4.1. Denote B i = {(j,, I, J) : (I J) Ω i > 1 I J, (I J) Ω i+1 1 I J }, where I are dyadic cubes in R m 1 J are dyadic intervals in R with the side lengths l(i) = (j ) N l(j) = (j ) N, the left lower corners of I the left end oints of J are (j ) N l (j ) N l m,resectively. By Theorem 4.1, we write f(x,x m )= I J φ j, (x (j ) N l,x m (j ) N l m ) i (j,,i,j) B i (φ j, h)( (j ) N l, (j ) N l m ), where the series converges in the L norm. We claim that I J φ j, ( (j ) N l, (j ) N l m ) (j,,i,j) B i (φ j, h)( (j ) N l, (j ) N l m ) C i Ω i, which together with the fact 0 < 1 yields L f L C i i Ω i C G d φ(f) L C h H C f H. Now we show the claim. Note that functions φ (1) φ () are suorted in unit balls. Hence if (j,, I, J) B i,thenφ j, are suorted in Ω i = {(x,x m ):M s (χ Ωi )(x,x m ) > m }.

25 Hardy saces associated with different homogeneities 5 For the sae of convenience, we denote x I = (j ) N l x J = (j ) N l m. Since Ω i C Ω i, by Hölder s inequality we obtain (j,,i,j) B i I J φ j, ( x I, x J )(φ j, h)(x I,x J ) L Ω i 1 (j,,i,j) B i I J φ j, ( x I, x J )(φ j, h)(x I,x J ). By the duality argument, we estimate the L norm of (j,,i,j) B i I J φ j, ( x I, x J )(φ j, h)(x I,x J ) as follows: For all g L with g 1, While, < I J φ j, ( x I, x J )(φ j, h)(x I,x J ),g > (j,,i,j) B i 1 C I J (φ j, h)(x I,x J ) I J (φ j, g)(x I,x J ) (j,,i,j) B i (j,,i,j) B i = (j,,i,j) B i I J (φ j, g)(x I,x J ) R m 1 R R m 1 R g L. 1 (φ j, g)(x I,x J )) χ I (x )χ J (x m ) dx dx m (j,,i,j) B i G d φ(g)(x,x m ) dx dx m 1. In addition, C i Ω i [ G φ(f)(x, d y)] dx dx m Ω i \Ω i+1 (φ j, h)(x I,x J ) (I J) Ω i \Ω i+1 (j,,i,j) B i 1 I J (φ j, h)(x I,x J ), (j,,i,j) B i where in the last inequality we use the fact that (I J) Ω i \Ω i+1 > 1 I J when (j,, I, J) B i. This letes the roof of Theorem 1.8.

26 6 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer Proof of Theorem 1.9: Suose f H L. By Theorem 1.7, T is bounded on H, which together with the fact that T is also bounded on L yields that T (f) H L, so alying first Theorem 1.8 then Theorem 1.7 we obtain T (f) L C T (f) H C f H for any f L H. Since H L is dense in H, the osition oerator T extends to a bounded oerator from H to L. 6. Proofs of Theorems We now rove the Calderón-Zygmund deosition the interolation theorem on H (R m ). Proof of Theorem 1.10: We first assume f L H. Let α > 0 Ω l = {x R m : Gd φ (f)(x) > α l },where G φ d (f) is defined in the the roof of Theorem 1.8. Let R 0 = {I J : (I J) Ω 0 < 1 } I J for l 1 R l = {I J : (I J) Ω l 1 1 I J, (I J) Ω l < 1 } I J, where I are dyadic cubes in R m 1 J are dyadic intervals in R with the side lengths l(i) = (j ) N l(j) = (j ) N, the left lower corners of I the left end oints of J are (j ) N l (j ) N l m,resectively. By the discrete Calderón-tye identity in Theorem 4.1, f(x,x m ) = I J φ j, (x x I,x m y J )φ j, h(x I,y J ) j, I,J = I J φ j, (x x I,x m y J )φ j, h(x I,y J ) j, l 1 I J R l + I J φ j, (x x I,x m y J )φ j, h(x I,y J ) j, I J R 0 = b(x,x m )+g(x,x m ), where x I = (j ) N l y J = (j ) N l m. When 1 > 1, using duality argument as in the roof of Theorem 1.8, it is easy to show 1 g 1 C φ j, h(x I,y J ) χ I χ J 1. I J R 0 j,

27 Hardy saces associated with different homogeneities 7 Next, we estimate g when 0 < H Clearly, the duality argument will not wor here. Nevertheless, we can estimate the H 1 norm directly by using discrete Calderón s identity in Theorem 1.3. To this end, we note that g H 1 (ψ j g)(x I,y J ) χ I (x)χ J (y) I,J j, 1 L 1, where I are dyadic cubes in R m 1 J are dyadic intervals in R with the side lengths l(i )= (j ) l(j )= (j ), the left lower corners of I the left end oints of J are (j ) l (j ) l m,resectively. Since (ψ j, g)(x I,y J )= I J (ψ j φ j,)(x I x I,y J y J )φ j, h(x I,y J ) j, I J R 0 Reeating the same roof of Theorem 1.6, we have 1 (ψ j g)(x I,y J ) χ I (x)χ J (y) j, I,J 1 C φ j, h(x I,y J ) χ I χ J 1. I J R 0 j, This shows that for all 0 < 1 < Claim 1: G d (f)(x,x m) α g H 1 C φ j, h(x I,y J ) χ I χ J I J R 0 j, ( G d (f)) 1 (x,x m )dx dx m C φ j, h(x I,y J ) χ I χ J I J R 0 j, This claim imlies g 1 H 1 C ( G d (f)) 1 (x,x m )dx dx m G d (f)(x,x m) α Cα 1 ( G d (f)) (x,x m )dx dx m G d (f)(x,x m ) α Cα 1 f H.

28 8 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer To show Claim 1, we choose 0 <q< 1 note that = ( G d (f)) 1 (x,x m )dx dx m G d (f)(x,x m) α φ j, h(x I,y J ) χ I (x )χ J (x m ) G d (f)(x,x m ) α C Ω c 0 = C C C R m 1 R R m 1 R j, I,J j, R R 0 1 dx dx m 1 φ j, h(x I,y J ) χ I χ J dx dx m j, R=I J R 0 1 φ j, h(x I,y J ) χ R Ω c 0 (x,x m ) dx dx m j, R R 0 q ( ( Ms φj, h(x I,y J ) q ) χ R Ω c 0 (x,x m ) ) q R m 1 R φ j, h(x I,y J ) χ R (x,x m ) R R 0 j, 1 dx dx m 1q dx dx m where in the last inequality we have used the fact that Ω c 0 R 1 R for R = I J R 0, thus χ I (x )χ J (x m ) 1 q Ms (χ R Ω c 0 ) 1 q (x,x m ) in the second to the last inequality we have used the vector-valued Fefferman- Stein inequality for strong maximal functions ( ) 1 ( r ) 1 r (M s (f )) r C f r =1 =1 with the exonents r =/q > 1 = 1 /q > 1. Thus the claim follows. We now recall Ω l = {(x,x m ) R m 1 R : M s (χ Ωl ) > 1 }. Claim : For any 0 < 1 l 1, j, I J φ j, (x x I,x m y J )φ j, h(x I,y J ) H C( l α) Ω l 1. I J R l

29 Hardy saces associated with different homogeneities 9 Claim imlies b H l 1( l α) Ω l 1 C l 1( l α) Ω l 1 C ( G d ) (f)(x,x m )dx dx m G d (f)(x,y)>α Cα ( G d ) (f)(x,x m )dx dx m Cα f H. G d (f)(x,y)>α To show Claim, again we have I J φ j, (x x I,x m y J )φ j, h(x I,y J ) H j, I J R l C I J (ψ j I,J φ j,)(x I x I,y J y J )φ j, h(x I,y J ) I J R l j j, C φ j, h(x I,y J ) χ I χ J I J R l j, where we can use a similar argument in the roof of Theorem 1.8 to rove the last inequality. However, as in the roof of the claim 1, choosing 0 <q< q< imlies that 1 L 1 L ( l α) Ω l 1 Gd (f) (x,x m )dx dx m = Ω l 1 \Ω l Ω l 1 \Ω l = C R m 1 R φ j, h(x I,y J ) χ I (x )χ J (x m ) j, R m 1 R I,J dx dx m φ j, h(x I,y J ) χ (I J) Ωl 1 \Ω l ) (x,x m ) j, I,J j, I,J dx dx m (M s ( φ j, h(x I,y J ) q χ (I J) Ωl 1 \Ω l ) (x,x m ) { } C R m 1 R φ j, h(x I,y J ) χ I (x )χ J (x m ) dx dx m. I J R l q )) q q dx dx m

30 30 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer In the above string of inequalities, we have used the fact that for I J R l we have (I J) Ω l 1 > 1 I J (I J) Ω l 1 I J consequently I J Ω l 1. Therefore (I J) ( Ω l 1 \Ω l ) > 1 I J for I J R l.thus χ I (x )χ J (x m ) 1 q Ms (χ (I J) Ωl 1 \Ω l ) ) 1 q (x,x m ). This gives the roof of the claim. Since L (R m ) H is dense in H. We are now ready to rove the interolation theorem on Hardy saces H for all 0 <<. Proof of Theorem 1.11: Suose that T is bounded from H to L from H 1 to L 1. For any given λ > 0 f H, by the Calderón-Zygmund deosition, f(x) =g(x)+b(x) with g 1 H 1 Cλ 1 f H b H Cλ f H. Moreover, we have roved the estimates g 1 H 1 C G d (f) 1 (x,x m )dx dx m which imlies that Thus, Tf = G d (f)(x,x m ) α b H C G d (f) (x,x m )dx dx m G d (f)(x,x m )>α α 1 {(x,x m ): Tf(x,x m ) > λ} dα { α 1 (x,x m ): Tg(x,x m ) > λ } dα { (x,x m ): Tb(x,x m ) > λ α 1 0 α 1 0 C f H G d (f)(x,x m ) α α 1 G d (f)(x,x m)>α Tf C f H } dα G d (f) 1 (x,x m )dx dx m dα for any << 1. Hence, T is bounded from H to L. G d (f) (x,x m )dx dx m dα

31 Hardy saces associated with different homogeneities 31 To rove the second assertion that T is bounded on H for << 1, for any given λ > 0 f H, by the Calderón-Zygmund deosition again {(x,x m ): g(tf)(x,x m ) > α} { (x,x m ): g(tg)(x,x m ) > α } + { (x,x m ): g(tb)(x,x m ) > α } Cα 1 Tg 1 H 1 Cα 1 g 1 H 1 + Cα Tb H + Cα b H Cα 1 G G d d (f)(x,x m (f) 1 (x,x ) α m )dx dx m +Cα G G d d (f)(x,x m (f) (x,x )>α m )dx dx m which, as above, shows that Tf H C g(tf) C f H for any 0 < << 1 <. Acnowledgement The authors wish to exress their sincere thans to the referee for his/her valuable ments suggestions. References [1] S.Y.A. Chang, R. Fefferman: Some recent develoments in Fourier analysis H -theory on roduct domains.. Bull.Amer.Math.Soc. vol 1 (1985), no. [1], [] S.Y.A. Chang, R. Fefferman: A continuous version of duality of H 1 with BMO on the bidisc. Ann. of Math. vol 11 (1980), no. [1], [3] S.Y.A. Chang, R. Fefferman: The Caldern-Zygmund deosition on roduct domains. Amer. J. Math. vol 104 (198), no. [3], [4] R.R.Coifman, G.Weiss: Extensions of Hardy saces their use in analysis. Bull.Amer.Math.Soc. vol 83 (1977), no. 4, [5] Y. Ding, Y. Han, G. Lu, X. Wu: Boundedness of singular integrals on weighted multiarameter Hardy saces H w(r n R m ). Potential Analysis ublished online, 011. [6] R. Fefferman: The atomic deosition of H 1 in roduct saces. Adv. in Math. vol 55 (1985), [7] R. Fefferman: Harmonic analysis on roduct saces. Ann. of Math. vol 16 () (1987), no. [1], [8] S. Ferguson, M. Lacey: A characterization of roduct BMO by mutators.. Acta Math. vol 189 (00), no. [], [9] M. Frazier, B. Jawerth: A discrete transform deosition of distribution. J. Func.Anal. vol 93 (1990), no. [1], [10] M. Frazier, B. Jawerth, G.Weiss: Littlewood-Paley theory the study of function saces. CBMS Regional conference series in Mathematics vol 79 (1991).

32 3 Y.Han, C.Lin, G.Lu, Z.Ruan E.Sawyer [11] C.Fefferman, E.M.Stein: H saces of several variables. Acta. Math. vol 19 (198), no. [3-4], [1] G.B. Foll, E.M.Stein: Hardy saces on homogeneous grous. Princeton Univ.Press Univ.Toyo Press, Princeton, 198. [13] J.Garcia Cuerva, J.L.Rubio de Francia: Weighted norm inequalities related toics. North Holl, Amsterdam, [14] Y. Han, J. Li, G. Lu: Duality of multiarameter Hardy saces H on saces of homogeneous tye. Ann. Sc. Norm. Suer. Pisa Cl. Sci. (5) 9 (010), no. [4], [15] Y.Han, G.Lu: Discrete Littlewood-Paley-Stein theory multi-arameter Hardy saces associated with flag singular integrals. arxiv: [16] Y.Han, G.Lu: Endoint estimates for singular integral oerators on multiarameter Hardy saces associated with the Zygmund dilations. Prerint (007). [17] Y.Han, G.Lu: Some recent wors on multiarameter Hardy sace theory discrete Littlewood-Paley analysis. Trends in artial differential equations, , Adv. Lect. Math. (ALM), 10, Int. Press, Somerville, MA, 010. [18] Y.Han, G.Lu, E. Sawyer: Imlicit multiarameter Hardy saces Marciniewicz multiliers on the Heisenberg grou. Prerint (010). [19] J.L. Journé: Calderón-Zygmund oerators on roduct sace. Rev. Mat. Iberoam. vol 1 (1985), no. [3], [0] G. Lu, Z. Ruan: Duality theory of weighted Hardy saces with arbitrary number of arameters. Forum Mathematicum, to aear. [1] J. Piher: Journé s covering lemma its extension to higher dimensions. Due Math. J. vol 53 (1986), no. [3], [] D.H.Phong, E.M.Stein: Some further classes of seudo-differential singular integral oerators arising in boundary valve roblems I, osition of oerators. Amer.J.Math. vol 104 (198), [3] Z. Ruan: Weighted Hardy saces in three arameter case. J. Math. Anal. Al. vol 367 (010), [4] S.Wainger, G.Weiss: Proceedings of Sym. in Pure Math. 35 (1979). Received?? Yongshen Han: Deartment of Mathematics Auburn University Auburn, AL 36849, USA hanyong@mail.auburn.edu Chincheng Lin: Deartment of Mathematics National Central University Taiwan, 3054, Taiwan clin@math.ncu.edu.tw Research by the authors are suorted by NCU Center for Mathematics Theoretic Physics, NSC of Taiwan under Grant #NSC M MY3The, NSF grant DMS , NSCF grant No Canadian NSERC grant, resectively.

33 Hardy saces associated with different homogeneities 33 Guozhen Lu: Deartment of Mathematics Wayne State University Detroit, MI 480, USA Zhuoing Ruan: Deartment of Mathematics & IMS Nanjing University Nanjing, 10093, P.R. China Eric T. Sawyer: Deartment of Mathematics & Statistics McMaster University Hamilton, Ontario, L8S 4K1, Canada