Research Article | Open Access
Chin-Cheng Lin, Kunchuan Wang, "Generalized Carleson Measure Spaces and Their Applications", Abstract and Applied Analysis, vol. 2012, Article ID 879073, 26 pages, 2012. https://doi.org/10.1155/2012/879073
Generalized Carleson Measure Spaces and Their Applications
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.
In 1972, Fefferman and Stein  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  the space was defined for , , and . It follows from [6, Theorem 3.1] that
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  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  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  that the dual of the multiparameter product Hardy space is the generalized multiparameter Carleson measure space (cf.  for more details).
Remark 1.11. For , in order to make each index works, we defined to be in our earlier version and in . 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. , [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  (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  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 . 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  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 . 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.
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 .
One recalls the almost diagonality given by Frazier and Jawerth . 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 .
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