Some Classes of Function Spaces, Their Properties, and ApplicationsView this Special Issue
A Note on Weighted Besov-Type and Triebel-Lizorkin-Type Spaces
Let , and ω be in the class of Muckenhoupt. We introduce the weighted Besov-type spaces and weighted Triebel-Lizorkin-type spaces for and then establish the φ-transform characterizations of these new spaces in the sense of Frazier and Jawerth.
Function spaces have been a central topic in modern analysis and are now of increasing applications in areas such as harmonic analysis and partial differential equations.
Since Besov spaces and Triebel-Lizorkin spaces were introduced in [1–3], these spaces became the focus of many scholars. Bui studied weighted Besov and Triebel-Lizorkin spaces in [4, 5]. In 2010, Yuan et al. gave a unified treatment of Morrey, Campanato, Besov, Lizorkin, and Triebel spaces in . A series of research results on these topics can be found in [6–15]. They develop a theory of spaces of Besov-Triebel-Lizorkin type built on Morrey spaces. Based on their work, this note will generalize these spaces to the weighted cases.
Let be the space of all Schwartz functions on . Let and be functions on satisfying where . Throughout the paper, for all and , we put .
As in , we set Let be the topological dual of , namely, the set of all continuous linear functionals on . Endowed with the weak -topology, then is complete; see .
Let . If , then there exist independent positive constants and , , such that for each cube and each subcube .
Before stating our theorems on the weighted Triebel-Lizorkin-type spaces, we first give the definition of these spaces.
Definition 1. Let , , and let be a Schwartz function satisfying (1) through (3).(i) Let . The weighted Besov-type space is defined to be the set of all such that , where with suitable modifications made when or .(ii) Let . The weighted Triebel-Lizorkin-type space is defined to be the set of all such that , where with suitable modifications made when , where is the side length of dyadic cube , , and the supremum is taken over all dyadic cubes .
Obviously, letting , the above spaces are weighted Besov space and weighted Triebel-Lizorkin space in [5, 18]. If , then . For simplicity, in what follows, we use to denote either or . If means , then the case is excluded. Below are the main results of the paper.
Let and satisfy (1) through (4). Recall that the -transform is defined to be the map taking each to the sequence , where for all dyadic cubes ; the inverse -transform is defined to be the map taking a sequence to ; see, for example, [19, 20]. The following theorem is about the -transform characterizations of the spaces .
Theorem 2. Let , , , and and satisfy (1) through (4). Then the operators and are bounded. Furthermore, is the identity on .
Next is the result of an -almost diagonal operator (defined in Section 3) on .
Theorem 3. Let , , , , and . Then all -almost diagonal operators are bounded on .
Remark 4. Let , then and . Then all -almost diagonal operators are bounded on , which is just the results established by Yuan et al.; see [6, Theorem 3.1] and [10, Theorem 4.1].
This paper is organized as follows. In Section 2, we establish the -transform characterizations of the spaces . And the boundedness of almost diagonal operators on is considered in Section 3.
At the end of this section, we make some conventions on notation. Throughout the paper, denotes unspecified positive constants, possibly different at each occurrence; the symbol means that there exists a positive constant such that , and means . For any , we set for all . For and , denotes the dyadic cube and . We denote by the lower left-corner of . Throughout the paper, when dyadic cube appears as an index, such as and , it is understood that runs over all dyadic cubes in . For each cube , we denote its side length by and its center by , and for , we denote by the cube concentric with having the side length . Let be a set of . Denote by its characteristic function and its interior. Also, set and .
2. -Transform Characterizations of
In this section, we establish the -transform characterizations of the spaces . To this end, we introduce their corresponding sequence spaces as follows.
Definition 5. Let , , and .(i) Let . The sequence space is defined to be the set of all sequences such that , where and .(ii)Let . The sequence space is defined to be the set of all sequences such that , where
Obviously, we have
In a similar manner to consider , we use to denote either or . If means , then the case is excluded.
To prove Theorem 2, we need some technical lemmas. The next lemma is a special case of [21, Lemma 2.11].
Lemma 6. Let , . Then there exist positive constants and such that for all ,
For all , , , and , we see that by Minkowski’s inequality. The following conclusions is easily verified similarly to the proof of lemma 2.7 in . Here we omit the proof.
Lemma 7. Let , , , and , . Then for all , converges in ; moreover, is continuous.
For a sequence , , and a fixed , set and . We have the following estimates.
Lemma 8. Suppose and . Let and . Then for each and , we have
Proof. Assume . Let
Denote , then Summing on and taking the th roots yield the result.
Lemma 9. Let , then
Proof. For all , we have . Then it is obvious that
Let , choose , then . Using weighted Fefferman-Stein inequality, we have Therefore, . Similarly, we can verify that .
Lemma 10. Let , , , , and . Then there exists a constant such that for all ,
Proof. Notice that holds for all dyadic cubes . This observation immediately implies that , where .
To see the converse, fix a dyadic cube . Let if and otherwise, and let . Set and . Then for all dyadic cubes , we have
Applying the fact of Lemma 9 that for each sequence , and , we then have and similarly,
On the other hand, let be a dyadic cube with for some . Suppose is any dyadic cube with and for some , where . Then and . For we have When , by , we have When , by Hölder’s inequality and , we obtain Therefore, by (21), .
To complete the proof, for any , , and dyadic cube , set Recall that for any and , and that for all and , Similarly to the proof of Lemma (see [18, Remark ]), we obtain that for all and , where herein and in what follows, denotes the Hardy-Littlewood maximal function on . Let . Then . Applying Minkowski’s inequality, Fefferman-Stein’s weighted vector-valued inequality, and Hölder’s inequality, we have Therefore, by (21) again, , which completes the proof of Lemma 10.
Let satisfy (1) through (3). Since also satisfy (1) through (3), we may take in place of in the definition of . For any and with , define the sequence by setting , and for any , the sequence by setting . Choosing as in the proof of [6, Lemma 2.9], we have the following estimates.
Lemma 11. Let , , , and be sufficiently large. Then there exists a constant such that for all , .
Proof of Theorem 2. With Lemmas 7, 10, and 11, the argument for Theorem 2 follows from the method pioneered by Frazier and Jawerth (see [18, pages 50-51]); see also the proof of [10, Theorem 3.1]. We omit the details.
From Theorem 2, we immediately deduce the following conclusion.
Corollary 12. With all the notations as in Definition 1, then the spaces are independent of the choice of .
3. Almost Diagonal Operators on
As an application of Theorem 2, we study boundedness of operators in by first considering their boundedness in corresponding . In this section, we show that almost diagonal operators are bounded on for appropriate indices, which generalize the classical results on and ; see [16, 18].
Definition 13. Let , , for the space , for the space , and . An operator associated with a matrix , namely, for all sequences , , is called -almost diagonal on if the matrix satisfies where
We remark that an -almost diagonal operator is also an almost diagonal operator introduced by Frazier and Jawerth in . In [18, Section 9], Frazier and Jawerth showed that certain appropriate Calderón-Zygmund operators and certain classes of Fourier multiplier operators correspond to almost diagonal matrices, and hence, the -transform simultaneously “almost diagonalizes” these operators. Moreover, Yang and Yuan proved that all almost diagonal operators are bounded on and ; see [9, 10]. These results can be generalized into the weighted Besov- and Triebel-Lizorkin-type spaces. We turn to prove it.
Proof of Theorem 3. Let and let be an -almost diagonal operator associated with the matrix and . Without loss of generality, we may assume that . Indeed, if the conclusion holds for , let and let be the operator associated with the matrix , where for all . Then we have , which deduces the desired conclusion.
We now consider the space in the case . For all , we write with and . By Definition 13, we see that for all , and therefore
For all and , set and and and . Then we have , where denotes the cardinality of . Notice that for all . Since , ,
Let . If , , , then here we use and .
If , using Minkowski’s inequality, we have
Therefore, Thus, by ,
For , let and be the same as in the proof of Lemma 10. We see that Applying Lemma 8 with , for all , we have Hence Hölder’s inequality and the boundedness for of the Hardy-Littlewood maximal operator yield