Abstract and Applied Analysis

Volume 2012, Article ID 879073, 26 pages

http://dx.doi.org/10.1155/2012/879073

## Generalized Carleson Measure Spaces and Their Applications

^{1}Department of Mathematics, National Central University, Chung-Li 320, Taiwan^{2}Department of Applied Mathematics, National Dong Hwa University, Hualien 970, Taiwan

Received 10 October 2011; Revised 20 February 2012; Accepted 12 March 2012

Academic Editor: Stevo Stevic

Copyright © 2012 Chin-Cheng Lin and Kunchuan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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. [2–4]) 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 [10–13]. 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 3.1–3.3 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