Journal of Function Spaces and Applications

Volume 2012, Article ID 163213, 47 pages

http://dx.doi.org/10.1155/2012/163213

## Continuous Characterizations of Besov-Lizorkin-Triebel Spaces and New Interpretations as Coorbits

Hausdorff Center for Mathematics, Institute for Numerical Simulation, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Received 7 April 2010; Accepted 20 July 2010

Academic Editor: Hans Triebel

Copyright © 2012 Tino Ullrich. 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.

#### Abstract

We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces (H. Triebel 1983, 1992, and 2006) in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions (tent spaces) and spaces involving the Peetre maximal function to apply the classical coorbit space theory according to Feichtinger and Gröchenig (H. G Feichtinger and K. Gröchenig 1988, 1989, and 1991). This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.

#### 1. Introduction

This paper deals with Besov-Lizorkin-Triebel spaces and on the Euclidean space and their interpretation as coorbits. For this purpose we prove a number of characterizations for homogeneous and inhomogeneous spaces for the full range of parameters. Classically introduced in Triebel’s monograph [1, Section 2.3.1] by means of a dyadic decomposition of unity, we use more general building blocks and provide in addition continuous characterizations in terms of Lusin and maximal functions. Equivalent (quasi-) normings of this kind were first given by Triebel in [2]. His proofs use in an essential way the fact that the function under consideration belongs to the respective space. Therefore, the obtained equivalent (quasi-)norms could not yet be considered as a definition or characterization of the space. Later on, Triebel was able to solve this problem partly in his monograph [3, Sections 2.4.2, 2.5.1] by restricting to the Banach space case. Afterwards, Rychkov [4] completed the picture by simplifying a method due to Bui et al. [5, 6]. However, [4] contains some problematic arguments. One aim of the present paper is to provide a complete and self-contained reference for general characterizations of discrete and continuous type by avoiding these arguments. We use a variant of a method from Rychkov’s subsequent papers [7, 8], which is originally due to Strömberg and Torchinsky developed in their monograph [9, Chapter 5].

In a different language, the results can be interpreted in terms of the continuous wavelet transform (see Appendix A.1) belonging to a function space on the -group . Spaces on considered here are mixed norm spaces like tent spaces [10] and Peetre-type spaces. The latter are indeed new and received their name from the fact that quantities related to the classical Peetre maximal function are involved. This leads to the main intention of the paper. We use the established characterizations for the homogeneous spaces in order to embed them in the abstract framework of coorbit space theory originally due to Feichtinger and Gröchenig [11–15] in the 1980s. This connection was already observed by them in [11, 14, 15]. They worked with Triebel’s equivalent continuous normings from [2] and the results on tent spaces, which were introduced more or less at the same time by Coifman et al. [10] to interpret Lizorkin-Triebel spaces as coorbits. On the one hand the present paper gives a late justification, and on the other hand, we observe that Peetre-type spaces on are a much better choice for this issue. Their two-sided translation invariance is immediate and much more transparent as we will show in Section 4.1. Furthermore, generalizations in different directions are now possible. In a forthcoming paper, we will show how to apply a generalized coorbit space theory due to Fornasier and Rauhut [16] in order to recover generalized inhomogeneous spaces based on the characterizations given here. Moreover, the extension of the results to quasi-Banach spaces using a theory developed by Rauhut in [17, 18] is possible.

Once we have interpreted classical homogeneous Besov-Lizorkin-Triebel spaces as certain coorbits, we are able to benefit from the achievements of the abstract theory in [11–15]. The main feature is a powerful discretization machinery which leads in an abstract universal way to atomic decompositions. We are now able to apply this method, which results in atomic decompositions and wavelet bases for homogeneous spaces. More precisely, sufficient conditions in terms of vanishing moments, decay, and smoothness properties of the respective wavelet function are given. Compact support of the used atoms does not play any role here. In particular, we specify the order of a suitable orthonormal spline wavelet system depending on the parameters of the respective space.

The paper is organized as follows. After giving some preliminaries, we start in Section 2 with the definition of classical Besov-Lizorkin-Triebel spaces and their characterization via continuous local means. In Section 3, we give a brief introduction to abstract coorbit space theory, which is applied in Section 4 on the -group . We recover the homogeneous spaces from Section 2 as coorbits of certain spaces on . Finally, several discretization results in terms of atomic decompositions and wavelet isomorphisms are established. The underlying decay result of the continuous wavelet transform and some basic facts about orthonormal wavelet bases are shifted to the appendix.

##### 1.1. Notation

Let us first introduce some basic notation. The symbols and denote the real numbers, complex numbers, natural numbers, natural numbers including 0, and the integers. The dimension of the underlying Euclidean space for function spaces is denoted by , its elements will be denoted by , and is used for the Euclidean norm. We will use for the -norm of a vector . For a multi-index and , we write

and define the differential operators and by

If is a (quasi-)Banach space and , we use or simply for its (quasi-)norm. Operator norms of linear mappings are denoted by or simply by . As usual, the letter denotes a constant, which may vary from line to line but is always independent of , unless the opposite is explicitly stated. We also use the notation if there exists a constant (independent of the context-dependent relevant parameters) such that . If and , we will write .

#### 2. Function Spaces on

##### 2.1. Vector-Valued Lebesgue Spaces

The space , , denotes the collection of complex-valued functions (equivalence classes) with finite (quasi-)norm with the usual modification if . The Hilbert space plays a separate role, see for instance, Section 3. Having a sequence of complex-valued functions on , where is a countable index set, we put where we modify appropriately in the case .

##### 2.2. Maximal Functions

For a locally integrable function , we denote by the Hardy-Littlewood maximal function defined by where the supremum is taken over all cubes centered at with sides parallel to the coordinate axes. The following theorem is due to Fefferman and Stein [19].

Theorem 2.1. *For and , there exists a constant , such that
**
holds for all sequences of locally Lebesgue-integrable functions on .*

Let us recall the classical Peetre maximal operator, introduced in [20]. Given a sequence of function , a tempered distribution and a positive number , we define the system of maximal functions

Since makes sense pointwise (see the following paragraph), everything is well-defined. However, the value “’’ is also possible for . This was the reason for the problematic arguments in [4] mentioned in the introduction. We will often use dilates of a fixed function , where might be given by a separate function. Also continuous dilates are needed. Let the operator , , generate the -normalized dilates of a function given by . If , we omit the super index and use additionally . We define by We will refer to this construction later on. It turned out that this maximal function construction can be used to interpret classical smoothness spaces as coorbits of certain function spaces on the group.

##### 2.3. Tempered Distributions and Fourier Transform

As usual, is used for the locally convex space of rapidly decreasing infinitely differentiable functions on , where the topology is generated by the family of seminorms The space is called the set of all tempered distributions on and defined as the topological dual of . Indeed, a linear mapping belongs to if and only if there exist numbers and a constant such thatfor all . is equipped with the -topology.

The convolution of two integrable functions is defined via the integral If , then (2.9) still belongs to . The convolution can be generalized to via , makes sense pointwise, and is a -function in of at most polynomial growth.

As usual, the Fourier transform defined on both and ) is given by , where , and The mapping is a bijection (in both cases) and its inverse is given by .

In order to deal with homogeneous spaces, we need to define the subset . Following [1, Chapter 5], we put The set denotes the topological dual of . If , the restriction of to clearly belongs to . Furthermore, if is an arbitrary polynomial in , we have for every . Conversely, if , then can be extended from to , that is, to an element of . However, this fact is not trivial and makes use of the Hahn-Banach theorem in locally convex topological vector spaces. We may identify with the factor space , since two different extensions differ by a polynomial.

##### 2.4. Besov-Lizorkin-Triebel Spaces

Let us first introduce the concept of a dyadic decomposition of unity, see also [1, Section 2.3.1].

*Definition 2.2. *(a) Let be the collection of all systems with the following properties:(i),
(ii)(iii) for every .

(b) Moreover, the system denotes the collection of all systems with the following properties: (i),
(ii)(iii) for every .

*Remark 2.3. *If we take satisfying
and define , then the system belongs to and the system with belongs to .

Now we are ready for the definition of the Besov and Lizorkin-Triebel spaces. See for instance [1, Section 2.3.1] for details and further properties.

*Definition 2.4. *Let and *, **. *Let further and *. *(i) If , then(ii)If , then
In case , we replace the sum by a supremum in both cases.

The homogeneous counterparts are defined as follows. For details, further properties and how to deal with ocurring technicalities we refer to [1, Chapter 5].

*Definition 2.5. *Let and *. *Let further and *. * (i) If , then (ii) If , then
In case , we replace the sum by a supremum in both cases.

##### 2.5. Inhomogeneous Spaces

Essential for the sequel are functions satisfying for some , and We will call the functions and kernels for local means. Recall that , , and .

Theorem 2.6. *Let , , , and . Let further be given by (2.17) and (2.18). Then the space can be characterized by
**
where
**
with the usual modification in case . Furthermore, all quantities , , are equivalent (quasi-)norms in .*

For the inhomogeneous Besov spaces, we have the following characterizations.

Theorem 2.7. *Let , , , and . Let further be given by (2.17) and (2.18). Then the space can be characterized by
**
where
**
with the usual modification if . Furthermore, all quantities , , are equivalent quasinorms in .*

##### 2.6. Homogeneous Spaces

The homogeneous spaces can be characterized in a similar way. Here we do not have a separate function anymore. We put .

Theorem 2.8. *Let , , , , and . Let further be given by (2.17) and (2.18). Then the space can be characterized by
**
where
**
with the usual modification if . Furthermore, all quantities , , are equivalent quasinorms in .*

For the Besov spaces, we obtain the following characterizations.

Theorem 2.9. *Let , , and . Let further be given by (2.17) and (2.18). Then the space can be characterized by
**
where
**
with the usual modification if . Furthermore, all quantities , , are equivalent quasinorms in .*

*Remark 2.10. *Observe, that the (quasi-)norms and are characterizations via Lusin functions, see [3, Section 2.4.5] and [1, Section 2.12.1] and the references given there. We will return to it later when defining tent spaces, see Definition 4.1 and (4.3).

##### 2.7. Particular Kernels

For more details concerning particular choices for the kernels and , we refer mainly to Triebel [3, Section 3.3].

The most prominent nontrivial examples (besides the one given in Remark 2.3) of functions and satisfying (2.17) and (2.18) are the classical local means. The name comes from the compact support of , which is admitted in the following statement.

Corollary 2.11. *Let as in Theorem 2.6. Let further such that
**
and define
**
where such that . Then (2.20), (2.21), (2.22), (2.23), and (2.24) characterize .*

Corollary 2.12. *Let as in Theorem 2.6. Let further be a radial function such that is non-increasing and atisfying
**
for , where . Define
**
Then (2.23), and (2.24) characterize .*

##### 2.8. Proofs

We give the proof for Theorem 2.6 in full detail. The proof of Theorem 2.8 is more or less the same, even a bit simpler. We refer to the next paragraph for the necessary modifications. The proofs in the Besov scale are analogous, so we omit them completely. The strategy is a modification of Rychkov [4], where he proved the discrete case, that is, that (2.23) and (2.24) characterize . However, Hansen [21, Remark 3.2.4] recently observed that the arguments used for proving (23) and (23′) in [4] are somehow problematic. The finiteness of the Peetre maximal function is assumed. But this is not true in general under the stated assumptions. Consider for instance in dimension the functions and, if is given, the tempered distribution with . Then is infinite in every point . The mentioned incorrect argument was inherited to some subsequent papers dealing with similar topics, for instance [22–24]. Anyhow, the stated results hold true. An alternative strategy, in order to avoid the problematic Lemma 3 in [4], is given in Rychkov [7] as well as [8]. A variant of this method, which is originally due to Strömberg/Torchinsky [9, Chapter V], is also used in our proof below.

We start with a convolution-type inequality which will be often needed below. The following lemma is essentially [4, Lemma 2].

Lemma 2.13. *Let and, . Let be a sequence of nonnegative measurable functions on and put
**
Then there is some constant , such that
**
hold true.*

*Proof of Theorem 2.6. *The strategy of the proof is as follows. First, we prove the equivalence of the “continuous’’ characterizations (2.20) and (2.21). The next step is to build the bridge between the “continuous’’ (2.21) and the “discrete’’ characterization (2.23) and to change from the system to a system . The equivalence of (2.23) and (2.24) goes parallel to (2.20) and (2.21). This was the original proof by Rychkov in [4]. So, up to this point, we have that (2.20), (2.21), (2.23), and (2.24) generate the same space for every chosen functions satisfying (2.17) and (2.18), namely, . Indeed, Definition 2.4 can be seen as a special case of (2.24).*Step 1. *We are going to prove the following inequalities:
for every .*Substep 1.1. *Put and if . Because of (2.17), it is possible to find functions with , and such that
We need a bit more. Fix a . Clearly, we have also
for all . With and , we obtain then
The dilation gives then
for every , where , . Obviously, we can rewrite (2.42) to obtain
for all . Let us now choose . This gives the final version of the convolution identity
For we define
Clearly, we have
Plugging this into (2.44), we end up with the pointwise representation
for all .*Substep 1.2. *Let us prove the following important inequality first. For every and every , we have
where is independent of , , and .

The representation (2.47) will be the starting point to prove (2.48). Namely, we have for
where
Elementary properties of the convolution yield (compare with (2.84))
where
Lemma A.3 yields
which we put into (2.49) to obtain
We prefer the strategy used by Rychkov in [7, Theorem 3.2] and [8, Lemma 2.9], which is a variant of the Strömberg/Torchinsky technique introduced in [9, Chapter V].

Let us continue by replacing by in (2.54) and multiply on both sides with . Then we can estimate
Next, we apply the elementary inequalities
where . We define the maximal function
and estimate
Observe that we can estimate the term in the right-hand side of (2.60) by . Hence, if we obtain from (2.60)
where is independent of , , , and . We claim that there exists such that for all . Indeed, we use that , that is, there is an and such that
Assuming , we estimate as follows:
where we again used the inequality (compare with (2.57))
and have set
Hence gives us only two different functions from . This implies the boundedness of for if . Therefore, (2.61) together with (2.63) yield (2.48) with , independent of , , and , for all . But this is not yet what we want. Observe that the right-hand side of (2.48) decreases as increases. Therefore, we have (2.48) *for all * but with *depending on *. This is still not yet what we want. Now we argue as follows: starting with (2.48) where and arbitrary, we apply the same arguments as used from (2.55) to (2.56), switch to the maximal function (2.58) with the help of (2.57), and finish with (2.61) instead of (2.59) but with a constant that depends on . But this does not matter now. It is Important, that a finite right-hand side of (2.61) (which is the same as rhs(2.48)) implies .

We assume rhs (2.48) . Otherwise, there is nothing to prove in (2.48). Returning to (2.60) and having in mind that now , we end up with (2.61) for all and independent of . Finally, from (2.61) we obtain (2.48) and are done in case .

Of course, (2.48) also holds true for with a much simpler proof. In that case, we use (2.54) with instead of and apply Hölder’s inequality with respect to first for integrals and then for sums.*Substep 1.3. *The inequality (2.48) implies immediately a stronger version of itself. Using (2.57) again, we obtain for and
We proved that the inequality (2.66) holds for all where is independent of . If we choose , we can apply the norm
on both sides and use Minkowski’s inequality for integrals, which yields
If then we have
and we observe
Now we use the majorant property of the Hardy-Littlewood maximal operator (see Section 2.2 and [25, Chapter 2]) and continue estimating
An index shift on the right-hand side gives
Choose now , and put
We obtain for
Now we apply Lemma 2.13 in , which yields
The Fefferman-Stein inequality (see Section 2.2 /Theorem 2.1, having in mind that ) gives
Hence, we obtain
This proves . Since the reverse inequality is trivial, this finishes Step 1.*Step 2. *Let be functions satisfying (2.18). In fact, the condition (2.17) for is not necessary for what follows.*Substep 2.1. *We are going to prove
for all We decompose similar as in Step 1. Exploiting the property (2.17) for the system , we find -functions such that and and
for and fix. Putting and , we obtain the decomposition
for every . We put and see
Now, we estimate as follows:
where
We first observe that for and functions , the following identity holds true for
This yields in case (with a minor change if or )
where we used Lemma A.3 for the last estimate.

If , we change the roles of and to obtain again with Lemma A.3
where can be chosen arbitrary large since satisfies for every according to its construction. Let us further use the estimate

Consequently,
Plugging this into (2.81) and choosing and , we obtain the inequality
for all and all . Suppose first that . Then we take on both sides , which gives
Applying Lemma 2.13 yields
which gives the desired result.

In case , we argue as follows. The quantity is not longer a norm, but a norm. This gives
Notice that the right-hand side is nothing more than a convolution of the sequences
Now we apply the -norm to both sides and get for all
We take both sides to the power and apply the -norm. This gives (2.78).*Substep 2.2. *With similar arguments and obvious modifications of Substep 2.1, we obtain for all *Step 3. *Choosing in Step 1 and omitting the integration over , we see immediately