Some Function Spaces via Orthonormal Bases on Spaces of Homogeneous Type

Abstract

Let be a space of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure μ satisfies only the doubling property. Adapting the recently developed randomized dyadic structures of X and applying orthonormal bases of constructed recently by Auscher and Hytönen, we develop the Besov and Triebel-Lizorkin spaces on such a general setting. In this paper, we establish the wavelet characterizations and provide the dualities for these spaces. The results in this paper extend earlier related results with additional assumptions on the quasi-metric d and the measure μ to the full generality of the theory of these function spaces.

1. Introduction

The aim of this paper is to introduce the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss without any additional assumption on the quasi-metric and the doubling measure. The main tool used in this paper is the remarkable orthonormal basis constructed recently by Auscher and Hytönen [1]. It is well known that function spaces play an important role in both classical and modern analysis, ordinary and partial differential equations, and approximation theory, and so forth. Since the seventies of last century, classical theory of the Besov spaces [2] as well as the Triebel-Lizorkin spaces [3, 4] has been developed rapidly from Euclidean spaces to spaces of homogeneous type in the sense of Coifman and Weiss with some additional assumptions on the quasi-metric and the doubling measure. See, for example, [5, 6] and references therein.

Let us recall briefly spaces of homogeneous type introduced by Coifman and Weiss [7]. A function is called a quasi-metric on a set if satisfies the following: (1) for all ; (2) if and only if ; and (3) the quasi-triangle inequality holds: there is a constant such that for all , and . In addition, the quasi-metric ball with center and radius is defined by . We say that a nonzero measure satisfies the doubling condition if there is a constant such that, for all and all , We point out that the doubling condition (2) implies that there exist positive constants (the upper dimension of ) and such that, for all and ,

A space is said to be the space of homogeneous type in the sense of Coifman and Weiss [7] if is a quasi-metric on and satisfies the doubling condition. Such spaces have many applications in the theory of singular integrals and function spaces [8]. Unfortunately, for some applications, additional assumptions were imposed on these general spaces due to the facts that the original quasi-metric may have no (Hölder) regularity and quasi-metric balls, even Borel sets, may not be open.

A recent work on the Besov spaces and the Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss was developed by Han et al. [6]. More precisely, in [6], in order to apply Coifman’s construction for the approximation to the identity, they need that the quasi-metric satisfies the Hölder regularity and the doubling measure is required to satisfy the reversed doubling condition; that is, there are constants and such that for all , , and . This assumption ensures that which is the key to develop the theory of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. The main tool used in [6] is the so-called frame developed by Han [9] (see also [10]) because the Fourier transform is missing in general spaces of homogeneous type and wavelets were not available at that time. However, things seem to be changed after the quite recent work of Auscher and Hytönen [1], where remarkable orthonormal bases (wavelets) were constructed by using spline functions with the original quasi-metric and the doubling measure. These orthonormal bases open the door for developing function spaces on these general settings. See the very recent work in [11] for the theory of product , , and duality on such general spaces. Motivated by the work in [11], in this paper, we develop the theory of the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss without any additional assumption on the quasi-metric and the doubling measure. Therefore, results in this paper extend earlier related results with additional assumptions on the quasi-metric and the measure to the full generality of the theory of the Besov and Triebel-Lizorkin spaces.

This paper is organized as follows. In Section 2 we give some notations and preliminaries, including a useful almost orthogonality estimate (see Lemma 7). Wavelet characterizations and dualities of the inhomogeneous Besov and Triebel-Lizorkin spaces are given in Section 3. We introduce homogeneous Besov and Triebel-Lizorkin spaces and discuss relationships between homogeneous and inhomogeneous Besov and Triebel-Lizorkin spaces in Section 4.

2. Preliminaries and Notations

Throughout this paper, denotes a space of homogeneous type with quasi-triangle constant and . By we denote the measure of , the ball centered at with radius , and by we denote the measure of , the ball centered at with radius . In addition, we use the notation to mean that there is a constant such that and the notation to mean that . The implicit constants, , are meant to be independent of other relevant quantities.

We first recall the orthonormal basis of constructed by Auscher and Hytönen [1]. Let be spline functions constructed in [1, Section 3] and let be the closed linear span in of . By [1, Theorem 5.1] we observe that Further on, by writing as the orthogonal (in ) complement of in , Auscher and Hytönen pointed out the following result.

Theorem 1 (Theorems 6.1 and 7.1, [1]). Let . There exists an orthonormal basis of   satisfying the following exponential decay: and the Hölder regularity of order : whenever for some . Here is a fixed parameter small enough, say , and , are two positive constants independent of , , , and . Clearly, is an orthonormal basis of .
Also, there exists an orthonormal basis of satisfying the following exponential decay: the Hölder regularity of order : whenever for some , and the cancellation property: where is given above and , are two positive constants independent of , , , and . Clearly, is an orthonormal basis of . In what follows, we also refer to the functions as wavelets.

In order to introduce the (inhomogeneous) Besov and Triebel-Lizorkin spaces, we need the following concepts of test functions and distributions. Refer to [12] for details and also see [11].

Definition 2. For fixed , , , where is given in Theorem 1, and . A function is said to be a test function of type centered at with width if satisfies the following decay and Hölder regularity properties:(i) for all ,(ii) for all , such that .

If is a test function of type , we write . The norm of on is defined by

We denote . It is easy to check that with equivalent norms for any fixed and . Furthermore, it is also easy to see that is a Banach space with respect to the norm on . For given , let be the completion of the space in with , . Obviously, . In addition, we say that if and there is a sequence such that as . For given , we define . Obviously, is a Banach space with respect to the norm . As usual, we denote by the dual space of consisting of all continuous linear functionals on .

Han et al. [11] showed that the wavelets constructed by Auscher and Hytönen [1] are test functions defined above.

Lemma 3 (see [11], Theorem 3.3). Let be given in Theorem 1 with the Hölder regularity of . Then for each .

By analogous arguments, we can obtain the next result.

Proposition 4. Let be given in Theorem 1 with Hölder regularity of . Then for each .

Let and be orthogonal projections onto and , respectively. We next give some estimates on the kernels of operators and .

Lemma 5. The kernel is symmetric in and , and, for any , satisfies that(i)for all , , (ii)for , (iii)for and ,

Proof. Note that, for any , and with , from the doubling property on the measure it follows that Also, note that the kernel is symmetric in and due to [1, Lemma 10.1]. Thus, it suffices to prove (13) for the cases that and . Again, by [1, Lemma 10.1] we have for all , where the constant is independent of . If , then and hence On the other hand, if , then (17) implies that By letting , we obtain (13), immediately.
The proof of (14) is based on the estimates given in [1, Lemma 10.1]. Indeed, since implies that and that , it follows from (17) and (21) that Applying yields (14), immediately. The proof of (15) is similar to that of (13) and (14), and we omit it here.

Lemma 6 (see [11], Lemma 3.6). The kernel is symmetric in and and, for any , satisfies that(i)for all , , (ii)for , (iii)for and

Let be another orthonormal basis of given by the algorithm in Theorem 1, and let be the projection onto with respect to the basis . Finally, we give the almost orthogonality estimate of the kernel of operator to end this section. The proof is similar to that in [6, Lemma 3.1], and we omit it here.

Lemma 7. Let , and let be the order of the Hölder regularity given in (8). There exists a constant , dependent only on , such that, for all , (ii) for all , whenever . The same estimate holds with and interchanged.

3. Inhomogeneous Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type

In this section, we introduce the inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and give wavelet characterizations of corresponding spaces.

We would like to point out that the wavelet expansion (Theorem 9) is the key to develop the theory of the Besov and Triebel-Lizorkin spaces. As defined previously, by and we denote the orthogonal projections onto and , respectively. These two projections provide us with regular Littlewood-Paley decompositions for spaces of homogeneous type.

Theorem 8 (Theorem 10.2, [1]). Let . Then which gives homogeneous Littlewood-Paley decomposition, while provides inhomogeneous Littlewood-Paley decomposition.

Furthermore, following the proof of Theorem  3.4 in [11], we can show that such wavelet expansions also hold in the space of distributions.

Theorem 9. Let , . Then wavelet expansion (29) holds in with and and hence holds in with and as well.

Proof. Since and (29) holds in the sense of , (29) holds for such test function in the (a.e.) pointwise sense (for some subsequence with respect to the convergence).
By analogous arguments given in [11, Theorem 3.4], we can show the convergence of and in by using the size and smooth conditions of kernels and , respectively. Therefore, in the sense of . The second equality of (29) holds for the same reason.

We now introduce the inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type.

Definition 10. Let , and . The inhomogeneous Besov space is defined by where
For and , the inhomogeneous Triebel-Lizorkin space is defined by where

Remark 11. Observe that as well as

The next result shows that the Besov and Triebel-Lizorkin norms given in Definition 10 are both independent of choice of the orthonormal basis of and hence, they are independent of the choice of operators and with . Thus, the Besov and Triebel-Lizorkin spaces given above are both well defined. To see this, let be another orthonormal basis of given in Theorem 1, and let and be projections onto and , respectively, with respect to the basis .

Proposition 12. Let , and let . For all , , then and for all and

Proof. Clearly, the equivalence of and is a direct consequence of the size condition (13) and the wavelet expansion (29). We next show that Indeed, by the wavelet expansion (29) and almost orthogonal estimate (26), we have where denotes the Hardy-Littlewood maximal operator. Thus, due to the fact that . This proves (39), and hence, we obtain (37) symmetrically.
We now prove (38). Clearly, by symmetry, it suffices to show that To this end, from (40) it follows that due to the fact that and for . Thus, applying the Fefferman-Stein vector-valued maximal inequality, we obtain (42), immediately. The proof is complete.

Let be an orthonormal basis of   and let be an orthonormal basis of as shown in Theorem 1. Applying the wavelet expansion, given in Theorem 9, we now give the wavelet characterizations of the inhomogeneous Besov and Triebel-Lizorkin spaces.

Theorem 13. Let and let , be that given in Theorem 1 with . For , , then and for , , then

Proof. Observe that, by analogous arguments in the proof of (26), we have In order to prove (45), by Remark 11, it suffices to show that By the wavelet expansion (44), we have
so that By analogous arguments, we also have and hence, This implies that
On the other hand, by using the wavelet expansion and the size condition, we have as well as and hence, by using the Hölder inequality, we have further that This completes the proof of (48).
Analogously, we can also obtain (46) by using the Fefferman-Stein vector-valued maximal inequality.

As an application of wavelet characterizations, we now turn to discuss the dual spaces of the inhomogeneous Besov and Triebel-Lizorkin spaces. For convenience, we rewrite (44) as where , , , and for . The following concepts of spaces of sequences were first introduced by Frazier and Jawerth [13].

Definition 14. Let , and let and , . The space of sequences is defined by with and, for , the space is defined by with

Remark 15. It is easy to see that and due to (57), (45), and (46).

The next result shows that is dense in as well as .

Proposition 16. Let , , and . Then is dense in both for , and for and .

Proof. Let . Since , to show the conclusion, it suffices to show that is dense in and . Let . For any fixed , write By the estimates (26) and (27), we can show that and . Further on, we can also show that This is a direct consequence of the wavelet expansion (29) and estimate (26). Indeed, where is defined as the same as for . By wavelet expansion (29), we then have This implies that in as , and hence, is dense in . By analogous arguments, we can also show that is dense in .

We are ready to give the dualities of the Besov and Triebel-Lizorkin spaces.

Theorem 17. Let , and , and let . Then . More precisely, given , then defines a linear functional on with such that and this linear functional can be extended to with norm at most .
Conversely, if is a linear functional on , then there exists a unique such that defines a linear functional on with , and is the extension of with .

Remark 18. Theorem 17 also holds with and replaced by and , respectively.

Proof of Theorem 17. (i)Suppose that and . By Remark 15, we observe that is an element of , and hence, . Also, is an element of , and hence, . Write now By the Hölder inequality and Minkowski inequality, we have and hence, is a linear functional defined on . Clearly, has bounded extension to due to the density of in . This proves that .(ii)Suppose that . We need to prove that there exists a such that Note that and define a linear functional on by From (71) it follows that By the Hahn-Banach theorem, can be extended to as a continuous linear functional. In addition, by arguments analogous to that in ([13, Remark 5.11]), we can show that , and hence, there exists a unique sequence such that and Thus, for each , we have Let . By the orthogonality of , we have and . Note that is dense in . The linear functional can be extended to . Thus, for all .

The next result gives elementary embedding properties of Besov and Triebel-Lizorkin spaces which can be proved by an approach analogous to that of Proposition  2 in [14, Section 2.3.2] and we omit the details here.

Proposition 19. (i)Let , and . Then (ii)Let and . Then (iii)Let , , and . Then

4. Homogeneous Besov and Triebel-Lizorkin Spaces

In this section, we develop the theory of the homogeneous Besov and Triebel-Lizorkin spaces and discuss relationships between homogeneous and inhomogeneous versions.

In order to introduce the homogeneous Besov and Triebel-Lizorkin spaces, we recall again the spaces of test functions and distributions on spaces of homogeneous type given in [11]. Let be given in Definition 2. By we denote the collection of all test functions in by . For , let be the completion of the space in the norm of . For , we define .

The distribution space is defined to be the set of all linear functionals from to with the property that there exists such that, for all ,

The homogeneous Besov and Triebel-Lizorkin spaces are defined as follows.

Definition 20. Let and , . For , , the homogeneous Besov space is defined by with and for , the homogeneous Triebel-Lizorkin space is defined by with

The authors in [11] showed that the wavelet expansion (28) holds in as well as with . The next result shows that Definition 20 is independent of choice of for .

Proposition 21. Let and . For , then and for and , then

We next give some relationships between homogeneous and inhomogeneous spaces.

Proposition 22. Let , and . For , then and for , , then

We only give the proof of Proposition 22. Others can be proved by arguments analogous to that of inhomogeneous versions.

Proof. We first show that . Let . By the wavelet expansion (29) and the Minkowski inequality, we have and Applying the wavelet expansion (29) and the size condition of (26), we also have and Thus, and .
Conversely, we show that . Indeed, let . By the size condition of , we have and hence, . This implies that .
By analogous arguments, we can also obtain the second desired equality for the inhomogeneous Triebel-Lizorkin spaces.

We now introduce the homogeneous Besov and Triebel-Lizorkin spaces by wavelet coefficients.

Definition 23. Let , , and . The homogeneous Besov space is defined by where
For , the homogeneous Triebel-Lizorkin space is defined by where
We are ready to give the wavelet characterizations of the homogeneous Besov and Triebel-Lizorkin spaces.

Theorem 24. Let and . Then

Now we recall the homogenous spaces of sequences and ; refer to [13] for details. Let . For and , the space of sequence is defined by with and, for , the space is defined by with

Remark 25. Suppose that be the same as in Theorem 1 for , using Theorem 24 and [11, Corollary 3.5] then, we have and .

Proposition 26. Let . Then is dense in both for and and for and .

Remark 27. As the proof of Proposition 16, we can also obtain that (28) converges in both the norm of for , and the norm of for , .

The homogeneous version of Theorem 17 is given as follows.

Theorem 28. Let , , and , and let . Then . More precisely, given , then defines a linear functional on with such that and this linear functional can be extended to with norm at most .
Conversely, if is a linear functional on , then there exists a unique such that defines a linear functional on with , and is the extension of with .

Remark 29. Theorem 28 also holds with and replaced by and , respectively.

Finally, we give the following embedding properties in homogeneous Besov and Triebel-Lizorkin spaces to end this section.

Proposition 30. (i)Let , , and . Then (ii)Let , and . Then

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Chuang Chen was supported by the NNSF (Grant no. 11201319) of China and the Science Foundation for Youth Scholars of Sichuan University (Grant no. 2012SCU11048), and Fanghui Liao was supported by NNSF (Grant no. 11171345) of China, the Doctoral Fund of Ministry of Education of China (Grant no. 20120023110003), and China Scholarship Council.