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
Then, the following discrete multiparameter local Calderón identity is proved in . For some other discrete Calderón’s identities, we refer the readers to [25–27].
Theorem 2. For , suppose that and are functions satisfying conditions (4)-(6), respectively. Then, where the series converges in , , and .
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 .
This result can be obtained if we prove the following Theorem 5. We will give a direct proof in Section 2.
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 .
It is easy to see that, for , . Moreover, due to Carleson, there is a well-known counterexample to show that this inclusion is true inclusion when [4, 34].
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 .
Theorem 12. Let . For , suppose that and satisfy conditions (4)-(6), respectively. Then, the operators and are all bounded; moreover, is the identity on .
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 .
Theorem 13. Let . For , suppose that and are functions satisfying conditions (4)-(6), respectively. Then, the 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, We only estimate since the other three terms can be considered by the same routine. Details can be seen in [26, 29].
For each integer , let . Denote , Then, can be rewritten as To estimate , we first estimate the following: Note that if , one has For , we consider the following four cases:
Case 1. , .
Case 2. , .
Case 3. , .
Case 4. , .
From the definition of , one can see that, if Case 2, then which implies Case 2 is an empty set. For the same reason, Case 3 and Case 4 are also empty. Hence, Since and , one has To estimate , we should compare the side length of with the side length of . Hence, we divide into four categories.
Category 1. , .
Category 2. , .
Category 3. , .
Category 4. , .
In Category 1, by (52), one has for some integer , since are all dyadic. Note that, for fixed , these should be bigger than some positive integer to ensure , , that is, satisfying . Moreover, for each fixed , the number of such ’s must be less than since . Therefore, For Category 2, , , from (52), one has which implies that Hence, for some integer . Note that for each fixed , the number of such ’s must be less than because . Moreover, from , one has for some integer with . For each fixed , the number of such ’s must be less than since . So Similar as Category 2, for Category 3, one can obtain the following estimate: For Category 4, , , from (52), one has which implies that . Furthermore, by , one has for some integer and for each fixed , the number of such ’s must be less than since . Hence, Then, From the definition of , it is easy to see that , which implies Hence, by choosing proper .
We complete the proof.☐
Proof of Theorem 4. Suppose that are functions satisfying Definition 3. We may assume that For , set , and . Then, that is, .
By (19), . Hence, by Theorem 13, Thus, we complete the proof.
3. Duality of Sequence Spaces
In this section, we establish the dual relationship between and .
Theorem 14. Let . Then,
Proof. Let . For any , denote and for , Then, Let , where is the strong maximal function. Then, obviously, by the boundedness of , and . Hence, On the other hand, implies . Therefore, The above two estimates yield that which implies Conversely, let ; then, there exists some , such that for every , , and To complete the proof, it suffices to show that . To do this, for any open set with finite measure, denote , and let be a measure on such that if , and when . Furthermore, let be the sequence space such that if ,
Note that On the other hand, by Hölder’s inequality. It gives that that is, for any open set with finite measure.
We thus complete the proof.☐
4. Duality of
Once the duality of sequence spaces are established, we can obtain the dual relationship between and by multiparameter -transform.
Proof of Theorem 5. It is known that is dense in (). Let , . Then, by (20), we have By Theorem 14, Theorem 12, and Theorem 13, This implies that for any, the map, given by, is a continuous linear functional on, a dense subset of, and hence can be extended to a continuous linear functional onwith.
Conversely, suppose that . Then, by Theorem 12. So there exists by Theorem 14 such that for all , and for the boundedness of . Obviously, sinceis identified by Theorem 12. If we denote , then for , one has by (21), which implies that . At last, by Theorem 13, we have Thus, we complete the proof of Theorem 5.
5. The Spaces and
Proof of Theorem 8. Given a dyadic rectangles , obviously, Set such that and .
Hence, since , where the notation means .
To prove the second inequality in Theorem 8, we first claim that Then, if we obtain one can complete the proof. Now, we prove (85). Set , and define For any , denote Then, . By locally reproducing formula (7), one has where is denoted by if .
Hence, Note that implies ; then, which follows that by the Fefferman-Stein vector-valued strong maximal inequality. Since in the domain , one has Hence, which implies (85).
At last, to prove (86), using the continuous-type norm of , by almost orthogonality estimates, one has Thus, we have completed the proof of Theorem 8.
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
This study was supported by National Natural Science Foundation of China grants (No. 11771223).