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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 153849, 21 pages
Pointwise Multipliers of Triebel-Lizorkin Spaces on Carnot-Carathéodory Spaces
1School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2School of Geographical Science, Guangzhou University, Guangzhou 510006, China
Received 19 April 2012; Accepted 7 June 2012
Academic Editor: Yongsheng S. Han
Copyright © 2012 Yanchang Han and Fang Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a Carnot-Carathéodory space, namely, is a smooth manifold, is a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. is a nonnegative Borel regular measure on satisfying that there exists constant such that for all and diam , . Using the discrete Calderón reproducing formula and the Plancherel-Pôlya characterization of the inhomogeneous Triebel-Lizorkin spaces developed in Han et al., in press and Han et al., 2008, pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces are obtained.
The multiplier theory of function spaces has been studied for a long time, and a lot of results have been obtained. As we know, the multiplier theory is one of the important parts in the studies of the Gleason problem, function space properties, and general operator theory. The pointwise multipliers on are studied as a part of the researches of function spaces in several monographs, [1–8]. Pointwise multipliers have been found many important applications in partial differential equations.
However, it was not clear how to generalize the pointwise multipliers on to spaces of homogeneous type introduced by Coifman and Weiss (see ) because the Fourier transform is no longer available. The main purpose of this paper is to establish pointwise multipliers on inhomogeneous Triebel-Lizorkin spaces in the setting of Carnot-Carathéodory spaces. To be more precisely, we first recall some necessary definitions. In this paper, we always assume that is a metric space with a regular Borel measure such that all balls defined by have finite and positive measures. In what follows, set diam, and for any and , set .
Definition 1.1 (see ). Let be a metric space with a Borel regular measure such that all balls defined by have finite and positive measures. The triple is called a space of homogeneous type if there exists a constant such that for all and ,
Remark 1.2. We point out that the doubling condition (1.1) implies that there exist positive constants and such that for all and ,
where is independent of and . Denote by the homogeneous “dimension” of as in .
A space of homogeneous type is called an RD space, if there exist constants , such that for all and , that is, some “reverse” doubling condition holds.
Clearly, any Ahlfors -regular metric measure space (which means that there exists some such that for and ) is a -space (see ), also is an RD space and space of homogeneous type in the sense of Coifman in . In other words, satisfies the doubling condition which is weaker than Ahlfors -regular metric measure space and RD-spaces.
Another typical such a space is Carnot-Carathéodory space. One example with unbounded total measure studied in  is that arises as the boundary of an unbounded model polynomial domain in . Let , where is a real, subharmonic, non-harmonic polynomial of degree . Then can be identified with . The basic Levi vector field is then , and we write . The real vector fields and their commutators of order span the tangent space to at each point. See [10, 12] for more details and references therein.
We will also suppose that for all . For any and , set and . It follows from (1.1) that . The following notion of approximations of the identity on RD spaces was first introduced in . Let .
We begin with recalling the definition of an approximation to the identity, which plays the same role as the heat kernel does in Nagel-Stein's theory .
Definition 1.3 (see [10, 12]). A sequence of operators is said to be an approximation to the identity (in short, ATI) if there exists constant such that for all and all , and , , the kernel of satisfies the following conditions:
for and ,
The space of test functions plays a key role in this paper; see .
Definition 1.4. Fix two exponents and . A function defined on is said to be a test function of type centered at with width if there exists a nonnegative constant such that satisfies the following conditions:
If is a test function of type centered at with width , we write , and the norm of in is defined by
We denote by the class of all . It is easy to see that with the equivalent norms for all and . Furthermore, it is also easy to check that is a Banach space with respect to the norm in .
In what follows, for given , we let be the completion of the space in when . Obviously . Moreover, if and only if when and there exists such that as . If , we then 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 functionals from to with the property that there exists a constant such that 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 .
The following constructions, which provide an analogue of the grid of Euclidean dyadic cubes on spaces of homogeneous type, were given by Christ in .
Lemma 1.5. Let be a space of homogeneous type. Then there exist a collection of open subsets, where is some (possible finite) index set, and constants and such that(i) for each fixed and if ;(ii)for any with , either or ;(iii)for each and each there is a unique such that ;(iv);(v)each contains some ball , where .
In fact, we can think of as being a dyadic cube with a diameter roughly and centered at . In what follows, we always suppose . See  for how to remove this restriction. Also, in the following, for , we will denote by , the set of all cubes , where is a fixed large positive integer.
Now, we can introduce the inhomogeneous Triebel-Lizorkin spaces via the approximation in Definition 1.3. Note that the Triebel-Lizorkin spaces have been already investigated for decades in the study of partial differential equations, interpolation theory, and approximation theory.
Definition 1.6. Suppose that is an ATI and let , and for . Let be a fixed large positive integer, be as above. Suppose that .
The inhomogeneous Triebel-Lizorkin space for and is the collection of , for some and satisfying such that where are averages of over .
The restrictions (1.12) guarantee that the definitions of the inhomogeneous Triebel-Lizorkin space for and are independent of the choices of and satisfying these conditions and in .
The classical scale of inhomogeneous Triebel-Lizorkin spaces contains many well-known function spaces. For example, if , one recovers the Hölder-Zygmund spaces , that is, The space is defined as the collection of such that
If and , then are the Bessel potential spaces (Lebesgue spaces, Liouville spaces). If and , then are the usual Sobolev spaces. If , then are the inhomogeneous Hardy spaces, which are closely related to the Hardy spaces in [15, 16] (more precisely: the homogeneous spaces coincide with the usual Hardy spaces ). We will use here the notation . Then these spaces will be denoted as the (inhomogeneous) Hardy-Sobolev spaces, which include the above Lebesgue-Sobolev spaces for .
The inhomogeneous Triebel-Lizorkin spaces have the following Plancherel-Pôlya characterizations in , which will be one of the the basic tools to prove the main results of this paper.
We now introduce the following definition of the pointwise multiplier.
Definition 1.8. Suppose that is a given function on . Then is called a pointwise multiplier for if admits a bounded linear mapping from into itself.
The main result in this paper is the following.
Theorem 1.9. Let , and and , then with is a multiplier for . In other words, yields a bounded linear mapping from into itself and there is a positive constant such that holds for all and .
We would like to point out that the study of pointwise multipliers is one of important problems in the theory of function spaces. It has attracted a lot of attention in the decades since starting with . Pointwise multipliers in general spaces , where have been studied in great detail in [4, 5, 8].
Theorem 1.9 was proved in  for pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces on based on the Fourier transform. In the present setting, however, we do not have the Fourier transform at our disposal. Since the Fourier transform on Carnot-Carathéodory spaces is not available and hence the idea used in  does not work for this more general setting. A new idea to prove Theorem 1.9 is to use the discrete Calderón reproducing formula, which was developed in . Therefore, this scheme easily extends to geometrical settings where the Fourier transform does not exist. The Fourier transform is missing but a version of pointwise multiplier is still present.
Throughout, we also denote by a positive constant independent of main parameters involved, which may vary at different occurrences. Constants with subscripts do not change through the whole paper. We use and to denote and , respectively. If , we then write . For any , set . If , set .
2. Proof of Theorem 1.9
In this section, we will prove Theorem 1.9. Since there is no the Fourier transforms on spaces of homogeneous type, the proof of Theorem 1.9 is quite different from the proof of Theorem 2.8.2 in . The key new ingredient in the proof of Theorem 1.9 is to apply the following discrete Calderón reproducing formulae established in [10, 12]. This formula can be stated as follows.
Lemma 2.1. Suppose that is an approximation to the identity as in Definition 1.3. Set for and . Then for any fixed large enough, there exists a family of functions and such that for any fixed and and all with and
where the series converges in the norm of with and , and for with and , and for with and . Moreover, and , the kernels of and , satisfy the similar estimates but with and interchanged in (2.3): for ,
when when .
To prove Theorem 1.9, we first show the following lemma.
Lemma 2.2. Let and be two approximations to the identity as in Lemma 2.1 above and for and . For any given and with , then where .
Proof. We first show the inequality (2.5) of the case above. In fact, in this case, it follows that
We next consider the case for and or and . We write
For , we can obtain where .
We may rewrite this integral as where and .
Observe further, a simple argument yields the same estimate for as follows: where we used .
For , we have
Combining the estimate for , we conclude that if , then (2.5) holds. The case is similar to above. This finishes the proof of Lemma 2.2.
Now we show the following technical version of Theorem 1.9.
Proposition 2.3. For any with and satisfying (1.12), then where , and and .
Proof. Using the Calderón reproducing formula, for any , We write
Applying the Hölder inequality for and for all and , for , it follows that where we used the fact that for any .
For , in fact for all and , and all , From the inequality (2.5), the Hölder inequality for and (2.14) for , the Fefferman-Stein vector-valued maximal function inequality in  and Lemma 1.7, it follows that where we choose satisfying .
By the inequality (2.5), similarly, where can be any number in .
We now consider the estimate of , Similar to the estimate of , using the equality (2.5), the Hölder inequality if and (2.14) if , the Fefferman-Stein vector-valued maximal function inequality in , we obtain where the first and the second inequalities follow from the estimates with , can be any number in , which verifies Proposition 2.3.
To introduce our definition of pointwise multiplication, the interesting estimate is needed.
Proof. We first prove inequality (2.22) with . In fact, since if , then
For , since , it follows that
For , the fact implies that , then That is, (2.22) with holds.
To prove (2.22) with , we write
For , since if , we obtain
For , by implies that and Thus
For , by implies that Thus
To obtain the estimate of the , we write where and .
For , we have
For , similar to , we have thus
For , obviously
Combining the estimate of and , we obtain that for Thus, (2.22) also holds. This finishes the proof of Lemma 2.4.
If and for , , , it is not clear in general what is meant by (pointwise multiplication). Our approach is the following.