## Some Classes of Function Spaces, Their Properties, and Applications

View this Special IssueResearch Article | Open Access

# A Note on Weighted Besov-Type and Triebel-Lizorkin-Type Spaces

**Academic Editor:**JĂłzef BanaĹ›

#### Abstract

Let , and *Ď‰* be in the class of Muckenhoupt. We introduce the weighted Besov-type spaces and weighted Triebel-Lizorkin-type spaces for and then establish the *Ď†*-transform characterizations of these new spaces in the sense of Frazier and Jawerth.

#### 1. Introduction

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 [1â€“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 [6]. A series of research results on these topics can be found in [6â€“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 [16], 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 [17].

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 Minkowskiâ€™s inequality. The following conclusions is easily verified similarly to the proof of lemmaâ€‰â€‰2.7 in [6]. 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
for .

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 HĂ¶lderâ€™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 Minkowskiâ€™s inequality, Fefferman-Steinâ€™s weighted vector-valued inequality, and HĂ¶lderâ€™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 [18]. In [18, Section 9], Frazier and Jawerth showed that certain appropriate CalderĂłn-Zygmund operators and certain classes of Fourier multiplier operators correspond to almost diagonal matrices, and hence, the -transform simultaneously â€śalmost diagonalizesâ€ť 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 Minkowskiâ€™s inequality, we have

Therefore,
Thus, by ,