Relationship between Hardy Spaces Associated with Different Homogeneities and One-Parameter Hardy Spaces

The first are the classical isotropic dilations occurring in the classical Calderón-Zygmund singular integrals, while the second are nonisotropic and related to the heat equations (also the Heisenberg groups). Let e(ξ) and h(ξ) be functions onR homogeneous of degree 0 in the isotropic sense and in the nonisotropic sense, respectively, and both smooth away from the origin. Then, it is well known that the Fourier multipliers T1 defined by T1(f)(ξ) = e(ξ)f(ξ) and T2 given by T2(f)(ξ) = h(ξ)f(ξ) are both bounded on L p


Introduction
For  = (  ,   ) ∈ R −1 × R and  > 0, we consider two kinds of homogeneities on R  : ∘  (  ,   ) = (  ,   ) , ∘ ℎ (  ,   ) = (  ,  2   ) . (1) The first are the classical isotropic dilations occurring in the classical Calderón-Zygmund singular integrals, while the second are nonisotropic and related to the heat equations (also the Heisenberg groups).Let () and ℎ() be functions on R  homogeneous of degree 0 in the isotropic sense and in the nonisotropic sense, respectively, and both smooth away from the origin.Then, it is well known that the Fourier multipliers  1 defined by T1 ()() = () f() and  2 given by T2 ()() = ℎ() f() are both bounded on   for 1 <  < ∞, of weak-type (1,1), and bounded on the classical isotropic Hardy spaces   iso and, nonisotropic Hardy spaces   non , respectively.Riviere in [1] asked the question is the composition  1 ∘  2 still of weak-type (1, 1)?Phong and Stein in [2] answered this question and gave a necessary and sufficient condition for which  1 ∘  2 is of weak-type (1,1).The operators Phong and Stein studied are in fact a composition of operators with different kinds of homogeneities which arise naturally in the -Neumann problem.Recently, Han et al. [3] developed a theory of the Hardy spaces   com , 0 <  < ∞, associated with the different homogeneities and proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on   com (R  ).Weighted function spaces associated with different homogeneities and boundedness of composition of operators on them were recently investigated in [4][5][6].
The Hardy spaces   com introduced in [3] have surprising multiparameter structures which reflect the mixed homogeneities arising from the two operators under consideration.A natural question arises: Is there any relationship between   com and the two classical one-parameter Hardy spaces   iso and   non ?The main purpose of this paper is to answer this question.We shall prove that   com are continuously embedded into the intersection of   iso and   non .As an application, we show that any operator boundedness from either   iso or   non into   must be bounded from   com into   .Our methods are to use the partially discrete Calderón-type formula and the Littlewood-Paley theory in this context, which are appropriately developed.
Before stating the results more precisely, we first recall some notions and notations.
The isotropic discrete square function   iso () is defined by where  (1)   () = 2 −  (1) (2 − ∘  ), Q  iso denotes the set of all dyadic cubes with sidelength 2  , and   denotes the left-lower corner of .The isotropic Hardy spaces   iso (R  ), 0 <  < ∞, are defined by The nonisotropic discrete square function   non () is defined by where  (2)   () = 2 −(+1)  (2) (2 , and   is the left-lower corner of .The nonisotropic Hardy spaces   non , 0 <  < ∞, are defined by For ,  ∈ Z, let  , =  (1)    *  (2)   .The discrete square function   com associated with different homogeneities is given by where R , denote the set of dyadic rectangles is the left-lower corner of .The Hardy spaces   com , 0 <  < ∞, associated with different homogeneities are defined by The main result of this paper is as follows.
Theorem 1.Let 0 <  < ∞.One has More precisely, there is a constant  depending on  and  such that, for all In [3], it was proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on   com (see [3,Theorem 1.9]).This result can be improved by the following.
Lemma 3.For  ∈  2 (R  ), there is a sufficiently large integer  such that where the two series converge in  2 (R  ) and   denotes any fixed point of .
The Calderón-type identities in Lemma 3 lead to the following square functions: where  is the sufficiently large integer in Lemma 3. The purpose of this section is to prove the following.
To verify (19), we need the following lemma.
We first assume that  ∈ where we have used ∑   ∈Z 2 It follows that Finally, taking the   norm on both sides and applying the Fefferman-Stein vector-valued inequality yield (19).Since  2 ∩   com is dense in   com (see [3]), a limiting argument concludes the proof of Theorem 4.

Proofs of Theorem 1 and Corollary 2
We first give the following.
Denote that where  is the sufficiently large integer in Lemma 3.
Applying the discrete Calderón-type reproducing formula in Lemma 3, where   :=  (2)   *  and the series converges in the where in the second inequality, we have used Minkowski's inequality, for ℓ 2 and in the third inequality used the inequality Thus, to finish the proof of Theorem 1, it suffices to verify claim (29).Since  (1) is supported in unit ball of R  , for (, ) ∈ B  , { (1)   (⋅ −   )} are supported in .