Dual Spaces of Multiparameter Local Hardy Spaces
In this paper, we study the duality theory of the multiparameter local Hardy spaces , and we prove that , where are defined by discrete Carleson measure. Moreover, we discuss the relationship among , , and rectangle .
The classical theory of one-parameter harmonic analysis may be considered as centering around the Hardy-Littlewood maximal operator and its relationship with certain singular integral operators which commute with the usual one-parameter dilations on , given by . If this isotropic dilation is replaced by more general nonisotropic groups of dilations, then many nonisotropic variants of the classical theories can be produced, such as the strong maximal functions and multiparameter singular integral operators, corresponding to the multiparameter dilations , and . Such a multiparameter theory has been developed extensively over the past decades. We refer the reader to the work in [1–18].
Since space is well suited only to the pure Fourier analysis and not stable under multiplication by test functions which associated with PDE, in , Goldberg introduced the class of localizable Hardy spaces . Let with and
Then, , where the symbol has its usual meaning, the Schwartz class of rapidly decreasing test functions, and is the dual space of . Goldberg proved that is dense in and is preserved by pseudodifferential operators. Similar as Hardy space , local Hardy space can be characterized by Poisson maximal function, Poisson area integral integral , and local Riesz transforms . In 2001, Rychkov in  pointed out that this space has its continuous Littlewood-Paley characterization. Recently, its discrete Littlewood-Paley characterization has also been obtained in .
Motivated by those, multiparameter local Hardy spaces were discussed in [23, 24]. To be precise, let be functions defined on satisfying where denotes the set of all smooth functions with compact support on . Then, the multiparameter local Hardy spaces are introduced in .
Definition 1. Let . For , suppose that satisfies condition (2), and . Then, the multiparameter local Hardy space is defined to be the set of such that
In this paper, for , set , , , , , and . For and any , denote are dyadic cubes in with the side length , and the left lower corners of are , }, , and , .
Similar to the classical one-parameter local Hardy space theory, it is proved in  that is dense in and , if Recently, we pointed out in  that this multiparameter local Hardy space can also be characterized by a discrete Littlewood-Paley norm. More precisely, for let with
With the above discrete multiparameter local Calderón identity, multiparameter local Hardy spaces can also be defined by discrete Littlewood-Paley norm ; that is, for every , the norm is equivalent to
The multiparameter Hardy space was introduced by Gundy and Stein in the 1970s in  and was developed by Chang and Fefferman in [1, 3]. Moreover, Chang and Fefferman in [1, 28] characterized duality of the product by the product Carleson measure. The generalized Carleson measure space was first introduced in . It has been proved in  that the duality of multiparameter Hardy space is . We refer the readers to [26, 31] for the recent development of generalized Carleson measure spaces.
The main goal of this paper is to identify the duality of with a new product Carleson measure space .
Definition 3. Let . For , suppose that and are functions satisfying conditions (4)-(6), respectively. The multiparameter local is defined to be the set of such that where ranges over all open sets in with finite measures.
To see that the space is well defined, one needs to show the following theorem.
Theorem 4. Let . Suppose that both and satisfy the same conditions in Definition 3. Then, one has for all .
Theorem 5. Let . Then, Namely, if , the map , given by , defined initially for , extends to a continuous linear functional on with . Conversely, every satisfies for some with .
Certainly, the dual spaces of one-parameter local Hardy spaces can be also characterized by the above form, namely, . Details can be seen in . On the other hand, in , Goldberg showed that the duality of is , which is defined as the set of such that equipped with the norm , where is the mean of over , i.e., . For , let . Then, the duality of is for . We refer the reader to  for more details of when . It is convenient to denote . Since and are all dualities of for , should be coincident with . In , we give a direct proof to identify them with equivalent norms. Moreover, we proved that, for , is coincident with the local Lipschitz space defined by where
A natural question, does this result hold in multiparameter setting? First, we introduce the following multiparameter local Lipschitz spaces.
Definition 6. Let . Suppose that satisfy the conditions in Definition 3. The multiparameter local Lipschitz space is defined by where
Some results about Lipschitz spaces associated with mixed homogeneities can be seen in . In this paper, one of our main results is to discuss the relationship among , , and the following multiparameter local rectangle .
Definition 7. Let . For , suppose that and are functions satisfying conditions (4)-(6), respectively. The multiparameter local rectangle is defined as the set of such that where the supremum is taken over all possible dyadic rectangles .
Theorem 8. Let . Then, for every , one has
One may ask whether or is true. It would be an interesting question.
The organization of this paper is as follows. In Section 2, we introduce the multiparameter -transform and its inverse -transform . These transforms correspond between and sequence spaces. Using these -transform , we prove Theorem 4. In Section 3, we establish the duality of sequence spaces. The proof of Theorem 5 is placed in Section 4. In the last section, we prove Theorem 8.
Finally, we make some conventions. Throughout the paper, denotes a positive constant that is independent of the main parameters involved, but whose value may vary from line to line. Constants with subscript, such as , do not change in different occurrences. We denote by . If , we write .
2. Multiparameter -Transform
In order to prove the duality theorems, following the idea of Frazier and Jawerth in , we first define and study the corresponding sequence spaces. For any , setting , then by (7), it is easy to have
Definition 9. The multiparameter -transform is the map taking to the sequence , where . Define the inverse multiparameter -transform as the map taking a sequence to .
By (19), for and , one has
For a sequence , one also has the following identity:
Now, we define two discrete sequence spaces corresponding to and , respectively.
Definition 10. Let . Sequence space is defined to be the collection of all complex-valued sequences such that where .
Definition 11. For , define to be the collection of all complex-valued sequences such that where ranges over all open sets in with finite measures, and is defined as in Definition 10.
Then, we have the following generalization of the fundamental result of Theorem 2.2 in .
Proof. The boundedness of is obvious since
by Definition 10 and (8).
We now prove the boundedness of . Similar proofs can be seen in [25, 31]. For a sequence , denote Then, by almost orthogonality estimates , for any positive integers , there exists a constant such that where the notation means . Hence, for any where , which can be sufficiently small if is large enough, and is the strong maximal operator. Summing over and , one has Then, by Cauchy’s inequality, we obtain Applying Fefferman-Stein’s vector-valued strong maximal inequality on provided , we complete the proof.☐
In the next theorem, we discuss the actions of the multiparameter -transform and its inverse -transform on the space and discrete sequence space , respectively. We will obtain that operators and are bounded, and is the identity on .
Proof. We only need to prove that is bounded. Let and . For , suppose that and are functions satisfying conditions (4)-(6), respectively. We are going to prove
Given any , by classical almost orthogonality estimates, for any positive integers , there exists a constant such that
which implies that, for any positive integers ,
On the other hand, by conditions (4) and (5), one has that if or ; it follows that
Combining the above two estimates, one has
which gives that
for any positive integers .
Thus, where To obtain the inequality (29), by (35), we only need to prove Let Given any , let be the set of dyadic rectangles defined as follows: and for , and for , and for , and , Obviously, for any , one has that ; moreover, if and . It is easy to see that, for , which gives that Therefore,