Research Article | Open Access
Pointwise Multipliers on Spaces of Homogeneous Type in the Sense of Coifman and Weiss
By applying the remarkable orthonormal basis constructed recently by Ausher and Hytönen on spaces of homogeneous type in the sense of Coifman and Weiss, pointwise multipliers of inhomogeneous Besov and Triebel-Lizorkin spaces are obtained. We make no additional assumptions on the quasi-metric or the doubling measure. Hence, the results of this paper extend earlier related results to a more general setting.
The main purpose of this paper is to provide pointwise multipliers of inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss. By a pointwise multiplier from a function space into another function space , we meant that a function defines a bounded linear mapping from into by pointwise multiplication. Pointwise multipliers arise in many different areas of mathematical analysis and have many applications; for example, coefficients of differential operators and symbols of more general pseudodifferential operators may be considered as pointwise multipliers. For the theory of pointwise multipliers acting on several function spaces such as Sobolev, Besov, and Triebel-Lizorkin spaces on we refer to . See also [2–5] for more details.
It was well known that the Fourier transform is a crucial tool to study pointwise multipliers on . However, it was not clear how to generalize pointwise multipliers on to spaces of homogeneous type introduced by Coifman and Weiss  because the Fourier transform is no longer available on spaces of homogenous type. To be more precise, let us first recall briefly these spaces. A quasi-metric on a set is a function : satisfying that (i) if and only if ; (ii) for all ; (iii) there exists a constant such that for all and , Any quasi-metric defines a topology, for which the balls for all and all form a basis. We say that is a space of homogeneous type in sense of Coifman and Weiss if is a quasi-metric and is a nonnegative Borel regular measure on satisfying the doubling condition; that is, for all , , then and where is assumed to be defined on a -algebra which contains all Borel sets and all balls and the constant is independent of and . Spaces of homogenous type in the sense of Coifman and Weiss have many applications in analysis. For example, Coifman and Weiss introduced atomic Hardy space for in [6, 7] that proved that if is a Calderón-Zygmund singular integral operator and is bounded on , then extends a bounded operator from to for suitable . However, note that the quasi-metric, in contrast to a metric, may not be Hölder regular and quasi-metric balls may not be open. For this reason, in many applications, the additional assumptions on the quasi-metric and the measure are required. For instance, in order to provide the maximal function characterization of the Hardy spaces on spaces of homogenous type, Macías and Segovia in  showed that the quasi-metric can be replaced by another quasi-metric such that the topologies induced on by and coincide, and, moreover, has the following Hölder regularity: there exist positive constant and such that for all and all Furthermore, if balls are defined by , that is, , then Macías and Segovia provided the maximal function characterization of the Hardy spaces for , on spaces of homogeneous type with the regularity condition (4) on and property (5) on the measure .
A fundamental result for these spaces is the theorem of David-Journé-Semmes , where and satisfy (4) and (5), respectively. The crucial tool in the proof of the theorem is the existence of a suitable approximation to the identity. The construction of such an approximation to the identity is due to Coifman. We would like to point out that for Coifman’s construction the additional assumptions (4) on and (5) on are crucial. Later, based on the conditions in (4) and (5), the Calderón reproducing formula, test function spaces and distributions, the Littlewood-Paley theory, and function spaces on were developed in [10–12].
In , Nagel and Stein developed the product () theory in the setting of the Carnot-Carathéodory spaces formed by vector fields satisfying Hörmander’s finite rank condition. The particular Carnot-Carathéodory spaces studied in  are spaces of homogeneous type with a smooth quasi-metric and a measure satisfying the conditions for and for .
Recently, pointwise multiplier theorems of Besov and Triebel-lizorkin spaces were obtained by the first author on spaces of homogeneous type with the additional assumptions (1.3) and (1.4) in  and with the conditions (1.3) and (1.5) in [15, 16].
A natural question arises: whether pointwise multipliers still hold on spaces of homogeneous type in the sense of Coifman and Weiss with only the original quasi-metric and a doubling measure?
Very recently, Auscher and Hytönen constructed an orthonormal basis with Hölder regularity and exponential decay on spaces of homogeneous type . This result is remarkable since there are no additional assumptions other than those defining spaces of homogeneous type in the sense of Coifman and Weiss. Motivated by Auscher and Hytönen’s orthonormal basis on spaces of homogeneous type, the purpose of the current paper is to answer the above question. More precisely, in this paper, we will provide pointwise multipliers on spaces of homogeneous type in the sense of Coifman and Weiss with the original quasi-metric and doubling measure .
The main tool used in this paper is the orthonormal basis constructed by Auscher and Hytönen . We now briefly recall the orthonormal basis constructed in  and inhomogeneous Besov and Triebel-Lizorkin spaces obtained in  on spaces of homogeneous type in the sense of Coifman and Weiss.
The orthonormal basis of constructed by Auscher and Hytönen  is given by the following.
Theorem 1 (see  Theorem 7.1). Let be a space of homogeneous type in the sense of Coifman and Weiss. There exists an orthonormal basis , , , of , having exponential decay Hölder-regularity for some and for , and the cancellation property Moreover, in the sense of .
Here is a fixed small parameter, say , and and are constants independent of , , , and ; see  for more details. In what follows, we also refer to the functions as wavelets.
To develop function spaces such as the Hardy, Besov and Triebel-Lizorkin spaces, the key point is to introduce test function and distributions spaces. For this purpose, the following definitions were introduced in [18, 19].
Definition 2. For fixed , , , where is given in Theorem 1. 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 with , If is a test function of type centered at with width , we write . The norm of on is defined by
We denote by the class of all . 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, . Moreover, if and only if with , and there exists such that as . If , we define . Obviously, is a Banach space and we also have for the above chosen .
We denote by the dual space of consisting of all linear functional from to with the property that there exists a constant , for all ,
We denote by the natural pairing of elements and . Since with the equivalent norms for all and . Thus, for all , is well defined for all with and .
We now give definitions of inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss. Denote and . Let and .
Definition 3 (see ). Let , and . The inhomogeneous Besov space is defined by
The inhomogeneous Triebel-Lizorkin space is defined by where
We would like to point out that on , are the Bessel-potential spaces (Lebesgue spaces, Liouville spaces). If and , then are the usual Sobolev spaces. If and , then coincides with the classical Besov spaces (Lipschitz spaces .
In this paper, we will consider the following.
Definition 4. Suppose that is a given function on . Then is called a pointwise multiplier for if admits a bounded linear mapping from into itself. Similarly, is called a pointwise multiplier for if admits a bounded linear mapping from into itself.
The main results in this paper are as follows.
Theorem 5. Let , , ; then is a multiplier for with . Moreover, there exists a positive constant such that for all and all .
Theorem 6. Let , , ; then is a multiplier for with . Moreover, there exists a positive constant such that for all and all .
Here, the Hölder space is defined as the collection of such that
We remark that Theorems 5 and 6 were proved in  on based on Fourier transform. As mentioned before, the Fourier transform on spaces of homogeneous type is not available and hence the idea used in  does not work for this more general setting. A recent work on pointwise multipliers of Besov and Triebel-Lizorkin spaces on Carnot-Carathéodory spaces was developed in [14–16]. However, all results in those papers require the additional assumptions on both the quasi-metric and the measure . Therefore, results in the present paper extend all results given in [4, 14–16].
Throughout this paper, we use to denote positive constants, whose value may change from one occurrence to the next. For the measure of ball , we sometimes use the abbreviations
2. Proof of Theorem 5
Let and be orthogonal projections onto and with , respectively. The next lemma gives some estimates on kernels of operators and .
Lemma 7 (see [17, 18]). Let be the Hölder regularity, and . Suppose that and are kernels of and , respectively. Then there exists a constant such that(i)(ii) for , (iii)(iv)(v)for , (vi)Note that (ii), (iii), (v), and (vi) still hold with and interchanged.
The key tool used in this paper, as mentioned, is the following version of the wavelet expansion.
For our purpose, we need the following lemmas.
Lemma 10. If with , , then and with , , .
Proof. Suppose that with , . We claim that
We first verify (31). By the size condition of and definition of test functions, we have
To estimate (32), by the cancellation condition on , we have where and .
For , for any , we have where .
For , implies ; then where and .
To estimate , since , then where . The claim is concluded.
We now return to the proof of Lemma 10 and only prove that since the proof of is similar. By applying (31) and (32), it follows that where . Thus .
Lemma 11. Let and . For , where denotes the minimum of and .
Proof. We first consider (39). By the size conditions of and the definition of Hölder space , we have
For (40), we only consider that the case for and the proof for are similar. In fact, if , we have
We estimate by further splitting it into where and .
For , for any , we have where .
For , note that implies that ; then we have where .
For , we have Denoting , then Thus where and we obtain
We now return to verify . By the size condition of , and the definition of , we have where . Then where .
The proofs of (41) and (42) are similar to the proof of (40) and we omit the details.
We are now ready to prove Theorem 5.
Proof of Theorem 5. We first show Theorem 5 for the special case; that is, if with , and with , then
To verify (54), we write
By the wavelet expansion, Hölder’s inequality, and the estimates in (39) and (41), we obtain
where we use the fact that and in the third inequality, by (39) and (41), we use the estimates , and .
For , instead of using (40) and (42), we have where and . Thus, This completes the proof of (54).
To show Theorem 5 for , note that if , in general, could be a distribution and the multiplication of is not well defined even for . For this purpose, we make the following observation: for any with , , and with , there exists a sequence such that with , and converges for any with satisfying , . Indeed, for any with , , , set where , for . By the Proposition 4.4 of , , and . Now we prove that converges for any with satisfying , . To do this, for , , by duality in  and the estimate in (54), we have Note that and tend to zero as tend to infinity. This implies that as with , , .
Now for any with and with , , , by the above observation, exists. Therefore, we define for with satisfying , . It is easy to see that limit is independent of the choice of . By Fatou’s lemma and (54), we have which gives the proof of Theorem 5.
3. Proof of Theorem 6
We first prove the following technical version of Theorem 6.
Lemma 12. For any , when , , then where , , , and .
Proof. Applying the wavelet expansion, for any , when , , we have
The estimate of is the same as in the proof of Theorem 5. We only estimate , , and . By applying the inequality (39)–(42), the Hölder inequality and the Fefferman-stein vector-valued maximal function inequality for , in , it follows that where we use the fact that . This verifies Lemma 12.
To show Theorem 6, we also need the following technical lemma.
Lemma 13 (see ). For any , there exists a sequence for , such that in with , where , , .
The above estimate implies that exists and the limit is independent of the choice of . Therefore, for , we define where for , and is a sequence defined in Lemma 13.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The second and third authors are supported by NNSF-China (Grant no. 11171345), the Doctoral Fund of Ministry of Education of China (Grant no. 20120023110003). The second author is also supported by China Scholarship Council. The third author is also supported by NNSF-China (Grant no. 51234005).
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