Abstract

We introduce the generalized Carleson measure spaces CM that extend BMO. Using Frazier and Jawerth's -transform and sequence spaces, we show that, for and , the duals of homogeneous Triebel-Lizorkin spaces for and are CM and CM (for any ), respectively. As applications, we give the necessary and sufficient conditions for the boundedness of wavelet multipliers and paraproduct operators acting on homogeneous Triebel-Lizorkin spaces.

1. Introduction

In 1972, Fefferman and Stein [1] proved that the dual of is the space. In 1990, Frazier and Jawerth [2, Theorem  5.13] generalized the above duality to homogeneous Triebel-Lizorkin spaces . More precisely, they showed that the dual of is for and , where is the conjugate index of . Throughout the paper, is interpreted as whenever , and for . Note that and . For , , and , it is known (cf. [24]) that the dual of is . Here, we will give another characterization for the duals of in terms of the generalized Carleson measure spaces for , , and .

We say that a cube is dyadic if , for some and . Denote by the side length of and by the “left lower corner’’ of when . We use and to express the supremum and summation taken over all dyadic cubes , respectively. Also, denote the summation taken over all dyadic cubes contained in by . For any dyadic cubes and , either and are nonoverlapping or one contains the other. For any function defined on , , and dyadic cube , set It is clear that , where denotes the paring in the usual sense for in a Fréchet space and in the dual of .

Choose a fixed function in Schwartz class , the collection of rapidly decreasing functions on , satisfying For and , we say that belongs to the homogeneous Triebel-Lizorkin space if , the tempered distributions modulo polynomials, satisfies When and , the above -norm is modified to be the supremum norm as usual, and is defined to be , which is

We now introduce a new space as follows.

Definition 1.1. Let satisfy (1.2). For and , the generalized Carleson measure spaces is the collection of all satisfying , where and denotes the characteristic function of .

Remark 1.2. By definition, we immediately have for , and it is easy to check for and . Note that the zero element in means the class of polynomials. Also note that with equivalent norms for and . It follows from Proposition 3.3 that for and . In particular, , and hence the spaces generalize .

Remark 1.3. For a dyadic cube , denote by ; that is, is the integer so that . In [5, 6], Yang and Yuan introduced the so-called “unified and generalized" Triebel-Lizorkin-type spaces with four parameters by for , , , and . Note that in [5] the space was defined for , , and . It follows from [6, Theorem  3.1] that

It is clear that for , and hence “looks like" a special case of . In fact, it was proved in [7, 8] that the space is the “same" as the space .

The definition of is independent of the choice of satisfying (1.2). To show that, we need the following Plancherel-Pôlya inequalities.

Theorem 1.4 (Plancherel-Pôlya inequality for ). Let satisfy (1.2). For and , if satisfies then

Theorem 1.5 (Plancherel-Pôlya inequality for ). Let satisfy (1.2). For , if satisfies then

Remark 1.6. Let satisfy (1.2). Denote by the collection of all satisfying defined in Definition 1.1 with respect to . Then, by Theorem 1.4, Similarly, by interchanging the roles of and . Hence, the definition of is independent of the choice of and, for short, denoted by . Also, Theorem 1.5 shows that is independent of the choice of satisfying (1.2) in the same argument.

Remark 1.7. The classical Plancherel-Pôlya inequality [9] concludes that if is an appropriate set of points in , for example, lattice points, where the length of the mesh is sufficiently small, then for all with a modification if .

Using the Calderón reproducing formula (either continuous or discrete version), several authors obtain the variant Plancherel-Pôlya inequalities [1013]. These inequalities give characterizations of the Besov spaces and the Triebel-Lizorkin spaces. Moreover, using these inequalities, one can show that the Littlewood-Paley -function and Lusin area -function are equivalent in -norm.

Define a linear map from into the family of complex sequences by Let denote the family of satisfying for all . For , define a linear functional by

We now state our first main result as follows.

Theorem 1.8 (duality for ). Suppose that , , and .
(a)For , the dual of is in the following sense.(i)For , the linear functional given by (1.15), defined initially on , extends to a continuous linear functional on with .(ii)Conversely, every continuous linear functional on satisfies for some with .(b)For , the dual of is in the following sense. (i)For , the linear functional given by (1.15), defined initially on , extends to a continuous linear functional on with .(ii)Conversely, every continuous linear functional on satisfies for some with .

Remark 1.9. For and , it follows immediately from [2, 3] (Verbitsky [4] corrected a gap of the proof) and definition that (any ). Theorem 1.8 (b) shows a different approach to the duality and includes the case of .
For , we have . For , , and hence . That is, each coincides with for and .

Remark 1.10. In Remark 1.2 we are aware that generalize by the viewpoint of spaces directly. Choosing and in Theorem 1.8, we immediately have for . In particular, . Once again, we obtain that generalize by the viewpoint of duality. It was also proved in [14] that the dual of the multiparameter product Hardy space is the generalized multiparameter Carleson measure space (cf. [14] for more details).

Remark 1.11. For , in order to make each index works, we defined to be in our earlier version and in [7]. In such a situation, for , the dual of would be . In this paper, however, we follow the referee’s suggestion and adopt a more “natural’’ definition of in Definition 1.1, that is, the limit of as . The sequence space given in Definition 2.1 has a similar story as well.

As applications, we first recall the Haar multipliers introduced in [15, 16]. Given a sequence , where the ’s are dyadic intervals in , a Haar multiplier on is a linear operator of the form where are the Haar functions corresponding to .

Using Meyer’s wavelets, we may generalize the above Haar multiplier to and obtain a necessary and sufficient condition for the boundedness on Triebel-Lizorkin spaces. Let for be Meyer’s wavelets (cf. [17], [18, pages 71–109]). Then, , where and are dyadic cubes in , is a frame for for and ; that is, for . For , define a wavelet multiplier on by for such that the above summation is well defined.

Theorem 1.12. Suppose that , and . Then,
(a)for , is bounded from into if and only if ;(b)for and , is bounded from into if and only if , where is given in Definition 2.1.

We consider another application. Let and in satisfy (1.2) and (3.1). Choose a function supported on and . For and , define the paraproduct operator by Thus, the adjoint operator is Then, and since and . Also, if , then both and are singular integral operators satisfying the weak boundedness property. Moreover, is a Calderón-Zygmund operator (i.e., is bounded on ) if and only if by David-Journé’s theorem [19] (also see [12, Theorems 5.4 and 5.8]). The authors showed a more general type of paraproduct operators in [12, page 688], which were derived from the discrete Calderón reproducing formula.

Theorem 1.13. Suppose that , and . (i)For , is bounded from into if and only if .(ii)If with and , then is bounded from into .

Remark 1.14. When , , and , Theorem 1.13 says that is bounded from into if and only if for , and is bounded from into for provided . In 1995, Youssfi [20] showed that, for , , , and , is bounded from into if and only if . The special case of Theorem 1.13(i), , generalizes Youssfi’s result to . More precisely, for , , , and , is bounded from to if and only if .

The paper is organized as follows. In Section 2, we introduce the discrete version of the generalized Carleson measure spaces and show that the duals of sequence Triebel-Lizorkin spaces for and are and (for any ), respectively. In Section 3, we prove the duals of homogeneous Triebel-Lizorkin spaces for and to be the generalized Carleson measure spaces and (for any ), respectively. In Section 4, we prove the Plancherel-Pôlya inequalities that give us the independence of the choice of for the definition of the generalized Carleson measure spaces. In the last section, we show the boundedness of wavelet multipliers and paraproduct operators. Throughout, we use to denote a universal constant that does not depend on the main variables but may differ from line to line. Also, and always mean the dyadic cubes in , and, for , we denote by the cube concentric with whose each edge is times as long.

2. Sequence Spaces

In this section, we introduce sequence spaces and then characterize the duals of by means of . Let us recall the definition of these sequence spaces defined in [2]. For and , the space consists all such sequences satisfying As before, the previous -norm is modified to the supremum norm for and . For , we adopt the norm Note that is equivalent to the Carleson norm of the measure where is the point mass at . See [2] for the details.

To study the duals of , we introduce a discrete version of the generalized Carleson measure spaces .

Definition 2.1. For and , the space is the collection of all sequences satisfying , where
It is obvious that and for . Using embedding theorem, Frazier and Jawerth [2, equation (5.14) and Theorem 5.9] obtained that, for and , the dual of is when , and the dual of is . Note that for and . Here we give the dual relationship between sequence spaces and .

Theorem 2.2 (duality for ). Suppose that , , and .
(a)For , the dual of is in the following sense.(i)For , the linear functional on given by is continuous with for .(ii)Conversely, every continuous linear functional on satisfies for some with .(b)For , the dual of is in the following sense.(i)For , the linear functional on given by is continuous with for .(ii)Conversely, every continuous linear functional on satisfies for some with .

Remark 2.3. For and , sequence spaces and (for any ) by definitions. Theorem 2.2 shows that , which gives a different but simpler proof of Frazier-Jawerth’s result for the duality of (cf. [2, Theorem 5.9]).

Proof of Theorem 2.2. For and , set and to be Then, . Also, Without loss of generality, we may assume that .
We first consider the case . Let and define a linear functional on by For , let For , let where is the Hardy-Littlewood maximal function. Then, for each dyadic cube , there exists exactly a such that . For every , let denote the maximal dyadic cube in containing . Then all of such ’s are pairwise disjoint. Thus, by Hölder’s inequality for and the inequality for , Since implies , the disjointness of ’s and Hölder’s inequality yield We claim that for and . Assume the claim for the moment. The weak boundedness of gives , and hence To prove the claim, we note that, for and , which implies
For , with a modification, we have
On the other hand, suppose that is a continuous linear functional on . For each dyadic cube , write to be the sequence defined by Let and . Then, for , Fix a dyadic cube . For , let be the sequence space consisting of , and define a counting measure on dyadic cubes by . Then, Note that Thus, and hence . For , consider defined before. Then, and Hence, . This completes the proof.

3. Proof of the Main Theorem

Let us recall the -transform identity given by Frazier and Jawerth [2]. Choose a function satisfying (1.2). Then there exists a function satisfying the same conditions as such that for . The -transform identity is given by where the identity holds in the sense of , , and -norm.

Define a linear map from into the family of complex sequences by and another linear map from the family of complex sequences into by Then, is the identity on by [2, Theorem 2.2].

Proposition 3.1. Suppose that and, , and in satisfy (1.2) and (3.1). The linear operators and defined by (3.2) and (3.3), respectively, are bounded. Furthermore, is the identity on . In particular, and can be identified with a complemented subspace of .

Figures 1 and 2 illustrate the relationship among , , , and .

One recalls the almost diagonality given by Frazier and Jawerth [2]. For and , let . One says that a matrix is -almost diagonal if there exists such that where

Lemma 3.2. For and , an -almost diagonal matrix is bounded on . Furthermore, when , an -almost diagonal matrix is bounded on .

We postpone the proof of Lemma 3.2 until the end of Section 4.

Let . For , we have and . Thus, and are bounded by Proposition 3.1. For and , let . Then, the -transform identity (3.1) shows that and . In particular, . Furthermore, for , where is -almost diagonal (cf. [2, Lemma 3.6]) and hence is bounded on by Lemma 3.2. Therefore, is bounded from to and is bounded from to .

We summarize that is also the identity on .

Proposition 3.3. For or , the linear operators and are bounded. Furthermore, is the identity on and . In particular, for and , and for .

Theorem 1.8 can be proved as a consequence of Propositions 3.13.3 and a duality result between two sequence spaces.

Proof of Theorem 1.8. First let us consider the case for . Let . Then, by Proposition 3.3, . It follows from Theorem 2.2 that is a continuous linear functional on and . Hence, for , Since is dense in , the functional can be extended to a continuous linear functional on satisfying .
Conversely, let , and set on . By Proposition 3.1, . Thus, by Theorem 2.2, there exists such that and . For , we have So, for and letting , It follows from [2, equations (2.7)-(2.8)] that and for and . This shows that for . Proposition 3.3 and Theorem 2.2 give
A similar argument gives the desired result for with a slight modification, and hence the proof is finished.

Remark 3.4. As pointed out by one of the referees, Yang and Yuan [8, Theorem 1] show that if and , then , where the definition of is given in Remark 1.3. Thus, for and , which demonstrates a different approach to the duality.

4. Proofs of the Plancherel-Pôlya Inequalities

In this section we demonstrate the Plancherel-Pôlya inequalities.

Proof of Theorem 1.4. Without loss of generality, we may assume that . By (3.1), we rewrite as Using the inequality [2, page 151, equation (B.5)] where and , we obtain Thus, for , where the last inequality is followed by Hölder’s inequality and Denote by Since can be replaced by any point in in the last inequality, Given a dyadic cube with , the above estimates yield where Then, can be further decomposed as There are dyadic cubes in with the same side length as , so Thus,
Next we decompose the set of dyadic cubes into according to the distance between each and . Namely, for each , where denotes the center of . Then, we obtain Since for each and for , the right-hand side of (4.14) is dominated by There are at most cubes in , and hence
To estimate , for and , set Then, for and Since, for , and the number of dyadic cubes contained in is at most , where the condition is used in the last equality. Combining the estimates of and , we prove Theorem 1.4.

By modifying the proof above, we may easily show Theorem 1.5. Detailed verifications are left to the reader.

We now return to show Lemma 3.2.

Proof of Lemma 3.2. For , , and hence the result holds. For , , and so the matrix is bounded by [2, Theorem 3.3]. To complete the proof, it suffices to show the boundedness of -almost diagonal matrices for the case .
We may assume that since the case implies the general case. The proof is similar to the proof of Theorem 1.4. Here, we only outline the proof. First let us consider the case for . Let be an -almost diagonal matrix. Then, for , due to Hölder’s inequality. Given a dyadic cube with , where Then, can be further decomposed as The same argument showed in the proof of Theorem 1.4 for the term gives us
To estimate , for and , let Then, using the same argument as Theorem 1.4 for , we have Both estimates for and show the desired result for .
When , we modify the previous proof by replacing Hölder’s inequality with -triangle inequality to get the result.
When and , the space , and hence an -almost diagonal matrix is bounded on by Proposition 5.3.

Remark 4.1. Note that . By a duality argument and [2, Theorem 3.3 and page 81], one can show that the -almost diagonal matrix is bounded on . When and , we can prove Lemma 3.2 by duality in Theorem 2.2. Let be an -almost diagonal matrix. Also define the transpose of by . For and , let . Then, . Since is -almost diagonal, is -almost diagonal by a calculation for a different value of . Thus, by Theorem 2.2 (a) and Proposition 5.3, is bounded on .

5. Applications

We define another wavelet multiplier on by using -transform identity as follows. Let and in satisfy (1.2) and (3.1). For a sequence , where the are dyadic cubes in , define the wavelet multiplier by for such that the above summation is well defined. Thus, we have the following characterization.

Theorem 5.1. Suppose that , , and . Then,
(a)for , is bounded from into if ,(b)for and , is bounded from into if .

Proof. We show the case only, which implies the general case by (2.7). For , , and , let and . It follows from Theorem 2.2 and Proposition 3.1 that This shows that is bounded from into and . A similar argument yields the boundedness of for the case .

In order to prove Theorem 1.12, we demonstrate a similar result in sequence spaces first. For a sequence , define by

Theorem 5.2. Suppose that , , and . Then,
(a)for , is extendible to be bounded from into if and only if ,(b)for and , is extendible to be bounded from into if and only if .

Proof. We still assume that . For , , and , let and . It follows from Theorem 2.2 that
Conversely, suppose that maps from into boundedly. For , let . Define a linear functional by Then, The assumption shows that is a continuous linear functional on . Using Theorem 2.2, we have , and hence .
For , a similar argument gives the desired result of (b).

Proof of Theorem 1.12. The “if’’ part follows from Theorem 5.1. To show the “only if’’ part, define by The boundedness of says that is bounded from into . Clearly, It follows from Proposition 3.1 that is bounded from into , and hence for and for and by Theorem 5.2.

In order to study the boundedness of the paraproduct operators acting on Triebel-Lizorkin spaces, we need more results described as follows.

Proposition 5.3 ([2, pages 54 and 81]). For and , an -almost diagonal matrix is bounded on .

Lemma 5.4. Define a matrix by . Then, for and , is -almost diagonal and hence is bounded on .

Proof. For , since for all , by [2, page 150, Lemma  B.1], we have for and , where and is independent of and . For , by [2, page 152, Lemma B.2], we obtain Choosing , we obtain the result.

We now can prove Theorem 1.13.

Proof of Theorem 1.13. To simplify notations, let and . The requirement guarantees that . Now assume that and . To prove part (i), by (3.1) we rewrite as where . Proposition 3.1 and Theorem 2.2 give It is clear that and Hence, by Propositions 3.1 and 3.3, and Lemma 5.4,
Next suppose that is bounded from into . Without lost of generality, we may assume that . A computation yields Fix an integer . Choose a function satisfying on , if and for all multi-indices with . By the molecular theory [2, page 56], it follows that . For each dyadic cube , define by Then, for all dyadic cubes and by the translation invariance and the dilation properties of . By Proposition 3.1, and hence, by the boundedness of , Taking the supremum on , we show that .
To prove part (ii), assume that and . Let and . By Proposition 3.1, where is the transpose of . Since , by Lemma 5.4, is -almost diagonal and hence is bounded on . Following the same argument as the proof of part (i), we get which completes the proof.

Acknowledgments

The authors are grateful to the referees for many invaluable suggestions. Research by both authors was supported by NSC of Taiwan under Grant nos. NSC 100–2115-M-008-002-MY3 and NSC 100–2115-M-259-001, respectively.