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Journal of Function Spaces and Applications
Volume 2012, Article ID 163213, 47 pages
http://dx.doi.org/10.1155/2012/163213
Research Article

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 ̇𝐵𝑠𝑝,𝑞(𝑑) and ̇𝐹𝑠𝑝,𝑞(𝑑) on the Euclidean space 𝑑 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 𝑎𝑥+𝑏-group 𝒢. Spaces on 𝒢 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 [1115] 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 𝒢 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 [1115]. 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 𝑎𝑥+𝑏-group 𝒢. We recover the homogeneous spaces from Section 2 as coorbits of certain spaces on 𝒢. 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 ,,,0,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 𝑑, its elements will be denoted by 𝑥,𝑦,𝑧,, and |𝑥| is used for the Euclidean norm. We will use |𝑘|1 for the 𝑑1-norm of a vector 𝑘. For a multi-index 𝛼 and 𝑥𝑑, we write𝑥𝛼=𝑥𝛼11𝑥𝛼𝑑𝑑(1.1)

and define the differential operators 𝐷𝛼 and Δ by𝐷𝛼=𝜕|𝛼|1𝜕𝑥𝛼11𝜕𝑥𝛼𝑑𝑑,Δ=𝑑𝑘=1𝜕2𝜕𝑥2𝑘.(1.2)

If 𝑋 is a (quasi-)Banach space and 𝑓𝑋, we use 𝑓𝑋 or simply 𝑓 for its (quasi-)norm. Operator norms of linear mappings 𝐴𝑋𝑌 are denoted by 𝐴𝑋𝑌 or simply by 𝐴. As usual, the letter 𝑐 denotes a constant, which may vary from line to line but is always independent of 𝑓, unless the opposite is explicitly stated. We also use the notation 𝑎𝑏 if there exists a constant 𝑐>0 (independent of the context-dependent relevant parameters) such that 𝑎𝑐𝑏. If 𝑎𝑏 and 𝑏𝑎, we will write 𝑎𝑏.

2. Function Spaces on 𝑑

2.1. Vector-Valued Lebesgue Spaces

The space 𝐿𝑝(𝑑), 0<𝑝, denotes the collection of complex-valued functions (equivalence classes) with finite (quasi-)norm𝑓𝐿𝑝𝑑=𝑑||||𝑓(𝑥)𝑝𝑑𝑥1/𝑝,(2.1) with the usual modification if 𝑝=. The Hilbert space 𝐿2(𝑑) plays a separate role, see for instance, Section 3. Having a sequence of complex-valued functions {𝑓𝑘}𝑘𝐼 on 𝑑, where 𝐼 is a countable index set, we put𝑓𝑘𝑞𝐿𝑝𝑑=𝑘𝐼𝑓𝑘𝐿𝑝𝑑𝑞1/𝑞,𝑓𝑘𝐿𝑝𝑞,𝑑=𝑘𝐼||𝑓𝑘||(𝑥)𝑞1/𝑞𝐿𝑝𝑑,(2.2) where we modify appropriately in the case 𝑞=.

2.2. Maximal Functions

For a locally integrable function 𝑓, we denote by 𝑀𝑓(𝑥) the Hardy-Littlewood maximal function defined by(𝑀𝑓)(𝑥)=sup𝑥𝑄1||𝑄||𝑄||||𝑓(𝑦)𝑑𝑦,𝑥𝑑,(2.3) where the supremum is taken over all cubes centered at 𝑥 with sides parallel to the coordinate axes. The following theorem is due to Fefferman and Stein [19].

Theorem 2.1. For 1<𝑝< and 1<𝑞, there exists a constant 𝑐>0, such that 𝑀𝑓𝑘𝐿𝑝𝑞,𝑑𝑓𝑐𝑘𝐿𝑝𝑞(2.4) holds for all sequences {𝑓𝑘}𝑘 of locally Lebesgue-integrable functions on 𝑑.

Let us recall the classical Peetre maximal operator, introduced in [20]. Given a sequence of function {Ψ𝑘}𝑘𝒮(𝑑), a tempered distribution 𝑓𝒮(𝑑) and a positive number 𝑎>0, we define the system of maximal functionsΨ𝑘𝑓𝑎(𝑥)=sup𝑦𝑑||Ψ𝑘||𝑓(𝑥+𝑦)1+2𝑘||𝑦||𝑎,𝑥𝑑,𝑘.(2.5)

Since (Ψ𝑘𝑓)(𝑦) makes sense pointwise (see the following paragraph), everything is well-defined. However, the value “’’ is also possible for (Ψ𝑘𝑓)𝑎(𝑥). This was the reason for the problematic arguments in [4] mentioned in the introduction. We will often use dilates Ψ𝑘(𝑥)=2𝑘𝑑Ψ(2𝑘𝑥) of a fixed function Ψ𝒮(𝑑), where Ψ0(𝑥) might be given by a separate function. Also continuous dilates are needed. Let the operator 𝒟𝐿𝑝𝑡, 𝑡>0, generate the 𝑝-normalized dilates of a function Ψ given by 𝒟𝐿𝑝𝑡Ψ=𝑡𝑑/𝑝Ψ(𝑡1). If 𝑝=1, we omit the super index and use additionally Ψ𝑡=𝒟𝑡Ψ=𝒟𝐿1𝑡Ψ. We define (Ψ𝑡𝑓)𝑎(𝑥) byΨ𝑡𝑓𝑎(𝑥)=sup𝑦𝑑||Ψ𝑡||𝑓(𝑥+𝑦)||𝑦||1+/𝑡𝑎,𝑥𝑑,𝑡>0.(2.6) 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, 𝒮(𝑑) is used for the locally convex space of rapidly decreasing infinitely differentiable functions on 𝑑, where the topology is generated by the family of seminorms𝜑𝑘,=sup𝑥𝑑,|𝛼|1||𝐷𝛼||𝜑(𝑥)(1+|𝑥|)𝑘,𝜑𝒮𝑑,𝑘,0.(2.7) The space 𝒮(𝑑) is called the set of all tempered distributions on 𝑑 and defined as the topological dual of 𝒮(𝑑). Indeed, a linear mapping 𝑓𝒮(𝑑)belongs to 𝒮(𝑑) if and only if there exist numbers 𝑘,0 and a constant 𝑐=𝑐𝑓 such that||||𝑓(𝜑)𝑐𝑓sup𝑥𝑑,|𝛼|1||𝐷𝛼||𝜑(𝑥)(1+|𝑥|)𝑘(2.8)for all 𝜑𝒮(𝑑). 𝒮(𝑑) is equipped with the weak-topology.

The convolution 𝜑𝜓 of two integrable functions 𝜑,𝜓 is defined via the integral(𝜑𝜓)(𝑥)=𝑑𝜑(𝑥𝑦)𝜓(𝑦)𝑑𝑦.(2.9) If 𝜑,𝜓𝒮(𝑑), then (2.9) still belongs to 𝒮(𝑑). The convolution can be generalized to 𝒮(𝑑)×𝒮(𝑑) via (𝜑𝑓)(𝑥)=𝑓(𝜑(𝑥)), makes sense pointwise, and is a 𝐶-function in 𝑑 of at most polynomial growth.

As usual, the Fourier transform defined on both 𝒮(𝑑) and 𝒮(𝑑) is given by (𝑓)(𝜑)=𝑓(𝜑), where 𝑓𝒮(𝑑),𝜑𝒮(𝑑), and𝜑(𝜉)=(2𝜋)𝑑/2𝑑𝑒𝑖𝑥𝜉𝜑(𝑥)𝑑𝑥.(2.10) The mapping is a bijection (in both cases) and its inverse is given by 1𝜑=𝜑().

In order to deal with homogeneous spaces, we need to define the subset 𝒮0(𝑑)𝒮(𝑑). Following [1, Chapter 5], we put𝒮0𝑑=𝜑𝒮𝑑𝐷𝛼(𝜑)(0)=0foreverymulti-index𝛼𝑑0.(2.11) The set 𝒮0(𝑑) denotes the topological dual of 𝒮0(𝑑). If 𝑓𝒮(𝑑), the restriction of 𝑓 to 𝒮0(𝑑) clearly belongs to 𝒮0(𝑑). Furthermore, if 𝑃(𝑥) is an arbitrary polynomial in 𝑑, we have (𝑓+𝑃())(𝜑)=𝑓(𝜑) for every 𝜑𝒮0(𝑑). Conversely, if 𝑓𝒮0(𝑑), then 𝑓 can be extended from 𝒮0(𝑑) to 𝒮(𝑑), that is, to an element of 𝒮(𝑑). However, this fact is not trivial and makes use of the Hahn-Banach theorem in locally convex topological vector spaces. We may identify 𝒮0(𝑑) with the factor space 𝒮(𝑑)/𝒫(𝑑), 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 Φ(𝑑) be the collection of all systems {𝜑𝑗(𝑥)}𝑗0𝒮(𝑑) with the following properties:(i)𝜑𝑗(𝑥)=𝜑(2𝑗𝑥),𝑗, (ii)supp𝜑0{𝑥𝑑|𝑥|2},supp𝜑{𝑥𝑑1/2|𝑥|2},(iii)𝑗=0𝜑𝑗(𝑥)=1 for every 𝑥𝑑.
(b) Moreover, the system ̇Φ(𝑑) denotes the collection of all systems {𝜑𝑗(𝑥)}𝑗𝒮(𝑑) with the following properties: (i)𝜑𝑗(𝑥)=𝜑(2𝑗𝑥),𝑗, (ii)supp𝜑={𝑥𝑑1/2|𝑥|2},(iii)𝑗=𝜑𝑗=1 for every 𝑥𝑑{0}.

Remark 2.3. If we take 𝜑0𝒮(𝑑) satisfying 𝜑0(𝑥)=1:|𝑥|10:|𝑥|>2(2.12) and define 𝜑(𝑥)=𝜑0(𝑥)𝜑0(2𝑥), then the system {𝜑𝑗(𝑥)}𝑗0 belongs to Φ(𝑑) and the system {𝜑𝑗(𝑥)}𝑗 with 𝜑0=𝜑 belongs to ̇Φ(𝑑).

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 {𝜑𝑗(𝑥)}𝑗=0Φ(𝑑) and Φ𝑗=1𝜑𝑗, 𝑗0. Let further <𝑠< and 0<𝑞. (i) If 0<𝑝, then𝐵𝑠𝑝,𝑞𝑑=𝑓𝒮𝑑𝑓𝐵𝑠𝑝,𝑞𝑑=𝑗=02𝑗𝑠𝑞Φ𝑗𝑓𝐿𝑝𝑑𝑞1/𝑞<.(2.13)(ii)If 0<𝑝<, then𝐹𝑠𝑝,𝑞𝑑=𝑓𝒮𝑑𝑓𝐹𝑠𝑝,𝑞𝑑=𝑗=02𝑗𝑠𝑞||Φ𝑗||𝑓(𝑥)𝑞1/𝑞𝐿𝑝𝑑.<(2.14) In case 𝑞=, 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 {𝜑𝑗(𝑥)}𝑗̇Φ(𝑑) and Φ𝑗=1𝜑𝑗. Let further <𝑠< and 0<𝑞. (i) If 0<𝑝, theṅ𝐵𝑠𝑝,𝑞𝑑=𝑓𝒮0𝑑̇𝐵𝑓𝑠𝑝,𝑞𝑑=𝑗=2𝑗𝑠𝑞Φ𝑗𝑓𝐿𝑝𝑑𝑞1/𝑞.<(2.15) (ii) If 0<𝑝<, theṅ𝐹𝑠𝑝,𝑞𝑑=𝑓𝒮0𝑑̇𝐹𝑓𝑠𝑝,𝑞𝑑=𝑗=2𝑗𝑠𝑞||Φ𝑗||𝑓(𝑥)𝑞1/𝑞𝐿𝑝𝑑.<(2.16) In case 𝑞=, we replace the sum by a supremum in both cases.

2.5. Inhomogeneous Spaces

Essential for the sequel are functions Φ0,Φ𝒮(𝑑) satisfying||Φ0||||||𝜀(𝑥)>0on{|𝑥|<2𝜀},(Φ)(𝑥)>0on2,<|𝑥|<2𝜀(2.17) for some 𝜀>0, and𝐷𝛼||(Φ)(0)=0𝛼||𝑅.(2.18) We will call the functions Φ0 and Φ kernels for local means. Recall that Φ𝑘=2𝑘𝑑Φ(2𝑘), 𝑘, and Ψ𝑡=𝒟𝑡Ψ.

Theorem 2.6. Let 𝑠, 0<𝑝<, 0<𝑞, 𝑎>𝑑/min{𝑝,𝑞} and 𝑅+1>𝑠. Let further Φ0,Φ𝒮(𝑑) be given by (2.17) and (2.18). Then the space 𝐹𝑠𝑝,𝑞(𝑑) can be characterized by 𝐹𝑠𝑝,𝑞𝑑=𝑓𝒮𝑑𝑓𝐹𝑠𝑝,𝑞𝑑𝑖<,𝑖=1,,5,(2.19) where 𝑓𝐹𝑠𝑝,𝑞1=Φ0𝑓𝐿𝑝𝑑+10𝑡𝑠𝑞||Φ𝑡||𝑓(𝑥)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑑,(2.20)𝑓𝐹𝑠𝑝,𝑞2=Φ0𝑓𝐿𝑝𝑑+10𝑡𝑠𝑞sup𝑧𝑑||Φ𝑡||𝑓(𝑥+𝑧)(1+|𝑧|/𝑡)𝑎𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑑,(2.21)𝑓𝐹𝑠𝑝,𝑞3=Φ0𝑓𝐿𝑝𝑑+10𝑡𝑠𝑞|𝑧|<𝑡||Φ𝑡||𝑓(𝑥+𝑧)𝑞𝑑𝑡𝑡𝑑+11/𝑞𝐿𝑝𝑑,(2.22)𝑓𝐹𝑠𝑝,𝑞4=𝑘=02𝑠𝑘𝑞sup𝑧𝑑||Φ𝑘||𝑓(𝑥+𝑧)(1+2𝑘|𝑧|)𝑎𝑞1/𝑞𝐿𝑝𝑑,(2.23)𝑓𝐹𝑠𝑝,𝑞5=𝑘=02𝑠𝑘𝑞||Φ𝑘||𝑓(𝑥)𝑞1/𝑞𝐿𝑝𝑑,(2.24) with the usual modification in case 𝑞=. Furthermore, all quantities 𝑓𝐹𝑠𝑝,𝑞(𝑑)𝑖, 𝑖=1,,5, are equivalent (quasi-)norms in 𝐹𝑠𝑝,𝑞(𝑑).

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

Theorem 2.7. Let 𝑠, 0<𝑝,𝑞, 𝑎>𝑑/𝑝, and 𝑅+1>𝑠. Let further Φ0,Φ𝒮(𝑑) be given by (2.17) and (2.18). Then the space 𝐵𝑠𝑝,𝑞(𝑑) can be characterized by 𝐵𝑠𝑝,𝑞𝑑=𝑓𝒮𝑑𝑓𝐵𝑠𝑝,𝑞𝑑𝑖<,𝑖=1,,4,(2.25) where 𝑓𝐵𝑠𝑝,𝑞1=Φ0𝑓𝐿𝑝𝑑+10𝑡𝑠𝑞Φ𝑡𝑓(𝑥)𝐿𝑝𝑑𝑞𝑑𝑡𝑡1/𝑞,𝑓𝐵𝑠𝑝,𝑞2=Φ0𝑓𝐿𝑝𝑑+10𝑡𝑠𝑞sup𝑧𝑑||Φ𝑡||𝑓(𝑥+𝑧)(1+|𝑧|/𝑡)𝑎𝐿𝑝𝑑𝑞𝑑𝑡𝑡1/𝑞,𝑓𝐵𝑠𝑝,𝑞3=𝑘=02𝑠𝑘𝑞sup𝑧𝑑||Φ𝑘||𝑓(𝑥+𝑧)1+2𝑘|𝑧|𝑎𝐿𝑝𝑑𝑞1/𝑞,𝑓𝐵𝑠𝑝,𝑞4=𝑘=02𝑠𝑘𝑞Φ𝑘𝑓(𝑥)𝐿𝑝𝑑𝑞1/𝑞,(2.26) with the usual modification if 𝑞=. Furthermore, all quantities 𝑓|𝐵𝑠𝑝,𝑞(𝑑)𝑖, 𝑖=1,,4, are equivalent quasinorms in 𝐵𝑠𝑝,𝑞(𝑑).

2.6. Homogeneous Spaces

The homogeneous spaces can be characterized in a similar way. Here we do not have a separate function Φ0 anymore. We put Φ0=Φ.

Theorem 2.8. Let 𝑠, 0<𝑝<, 0<𝑞, 𝑎>𝑑/min{𝑝,𝑞}, and 𝑅+1>𝑠. Let further Φ𝒮(𝑑) be given by (2.17) and (2.18). Then the space ̇𝐹𝑠𝑝,𝑞(𝑑) can be characterized by ̇𝐹𝑠𝑝,𝑞𝑑=𝑓𝒮0𝑑̇𝐹𝑓𝑠𝑝,𝑞𝑑𝑖<,𝑖=1,,5,(2.27) where ̇𝐹𝑓𝑠𝑝,𝑞1=0𝑡𝑠𝑞||Φ𝑡||𝑓(𝑥)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑑,̇𝐹𝑓𝑠𝑝,𝑞2=0𝑡𝑠𝑞sup𝑧𝑑||Φ𝑡||𝑓(𝑥+𝑧)(1+|𝑧|/𝑡)𝑎𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑑,̇𝐹𝑓𝑠𝑝,𝑞3=0𝑡𝑠𝑞|𝑧|<𝑡||Φ𝑡||𝑓(𝑥+𝑧)𝑞𝑑𝑡𝑡𝑑+11/𝑞𝐿𝑝𝑑,̇𝐹𝑓𝑠𝑝,𝑞4=𝑘=2𝑠𝑘𝑞sup𝑧𝑑||Φ𝑘||𝑓(𝑥+𝑧)(1+2𝑘|𝑧|)𝑎𝑞1/𝑞𝐿𝑝𝑑,̇𝐹𝑓𝑠𝑝,𝑞5=𝑘=2𝑠𝑘𝑞||Φ𝑘||𝑓(𝑥)𝑞1/𝑞𝐿𝑝𝑑(2.28) with the usual modification if 𝑞=. Furthermore, all quantities ̇𝐹𝑓|𝑠𝑝,𝑞(𝑑)𝑖, 𝑖=1,,5, are equivalent quasinorms in ̇𝐹𝑠𝑝,𝑞(𝑑).

For the Besov spaces, we obtain the following characterizations.

Theorem 2.9. Let 𝑠,0<𝑝,𝑞, 𝑎>𝑑/𝑝, and 𝑅+1>𝑠. Let further Φ𝒮(𝑑) be given by (2.17) and (2.18). Then the space ̇𝐵𝑠𝑝,𝑞(𝑑) can be characterized by ̇𝐵𝑠𝑝,𝑞𝑑=𝑓𝒮0𝑑̇𝐵𝑓𝑠𝑝,𝑞𝑑𝑖<,𝑖=1,,4,(2.29) where ̇𝐵𝑓𝑠𝑝,𝑞1=0𝑡𝑠𝑞Φ𝑡𝑓(𝑥)𝐿𝑝𝑑𝑞𝑑𝑡𝑡1/𝑞,̇𝐵𝑓𝑠𝑝,𝑞2=0𝑡𝑠𝑞sup𝑧𝑑||Φ𝑡||𝑓(𝑥+𝑧)(1+|𝑧|/𝑡)𝑎𝐿𝑝𝑑𝑞𝑑𝑡𝑡1/𝑞,̇𝐵𝑓𝑠𝑝,𝑞3=𝑘=2𝑠𝑘𝑞sup𝑧𝑑||Φ𝑘||𝑓(𝑥+𝑧)1+2𝑘|𝑧|𝑎𝐿𝑝𝑑𝑞1/𝑞,̇𝐵𝑓𝑠𝑝,𝑞4=𝑘=2𝑠𝑘𝑞Φ𝑘𝑓(𝑥)𝐿𝑝𝑑𝑞1/𝑞,(2.30) with the usual modification if 𝑞=. Furthermore, all quantities ̇𝐵𝑓𝑠𝑝,𝑞(𝑑)𝑖, 𝑖=1,,4, are equivalent quasinorms in ̇𝐵𝑠𝑝,𝑞(𝑑).

Remark 2.10. Observe, that the (quasi-)norms ̇𝐹𝑠𝑝,𝑞(𝑑)3 and 𝐹𝑠𝑝,𝑞(𝑑)3 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 Φ0 and Φ, we refer mainly to Triebel [3, Section 3.3].

The most prominent nontrivial examples (besides the one given in Remark 2.3) of functions Φ0 and Φ satisfying (2.17) and (2.18) are the classical local means. The name comes from the compact support of Φ0,Φ, which is admitted in the following statement.

Corollary 2.11. Let 𝑝,𝑞,𝑠 as in Theorem 2.6. Let further 𝑘0,𝑘0𝒮(𝑑) such that 𝑘0(0),𝑘0(0)0(2.31) and define Φ0=𝑘0,Φ=Δ𝑁𝑘0,(2.32) where 𝑁such that 2𝑁>𝑠. Then (2.20), (2.21), (2.22), (2.23), and (2.24) characterize 𝐹𝑠𝑝,𝑞(𝑑).

Corollary 2.12. Let 𝑝,𝑞,𝑠 as in Theorem 2.6. Let further Φ0𝒮(𝑑) be a radial function such that Φ0 is non-increasing and atisfying Φ0(0)0,𝐷𝛼Φ0(0)=0(2.33) for 1|𝛼|1𝑅, where 𝑅+1>𝑠. Define Φ(𝑥)=Φ01(𝑥)2𝑑Φ0𝑥2.(2.34) Then (2.23), and (2.24) characterize 𝐹𝑠𝑝,𝑞(𝑑).

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 𝐹𝑠𝑝,𝑞(𝑑). 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 𝑑=1 the functionsΨ0(𝑡)=Ψ1(𝑡)=𝑒𝑡2(2.35) and, if 𝑎>0 is given, the tempered distribution 𝑓(𝑡)=|𝑡|𝑛 with 𝑎<𝑛. Then (Ψ𝑘𝑓)𝑎(𝑥) is infinite in every point 𝑥. The mentioned incorrect argument was inherited to some subsequent papers dealing with similar topics, for instance [2224]. 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 0<𝑝,𝑞 and, 𝛿>0. Let {𝑔𝑘}𝑘0 be a sequence of nonnegative measurable functions on 𝑑 and put 𝐺(𝑥)=𝑘2|𝑘|𝛿𝑔𝑘(𝑥),𝑥𝑑,.(2.36) Then there is some constant 𝐶=𝐶(𝑝,𝑞,𝛿), such that 𝐺𝑞𝐿𝑝𝑑𝑔𝐶𝑘𝑘𝑞𝐿𝑝𝑑,𝐺𝐿𝑝𝑞,𝑑𝑔𝐶𝑘𝑘𝐿𝑝𝑞,𝑑(2.37) 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 (Φ0,Φ) to a system (Ψ0,Ψ). 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 (Φ0,Φ) satisfying (2.17) and (2.18), namely, 𝐹𝑠𝑝,𝑞(𝑑). Indeed, Definition 2.4 can be seen as a special case of (2.24).Step 1. We are going to prove the following inequalities: 𝑓𝐹𝑠𝑝,𝑞2𝑓𝐹𝑠𝑝,𝑞1𝑓𝐹𝑠𝑝,𝑞2(2.38) for every 𝑓𝒮(𝑑).Substep 1.1. Put 𝜑0=Φ0 and 𝜑=(Φ)(2) if 1. Because of (2.17), it is possible to find functions 𝜓0,𝜓𝒮(𝑑) with supp𝜓0{𝜉𝑑|𝜉|2𝜀}, supp𝜓{𝜉𝑑𝜀/2|𝜉|2𝜀} and 𝜓(𝑥)=𝜓(2𝑥) such that 0𝜑(𝜉)𝜓(𝜉)=1.(2.39) We need a bit more. Fix a 1𝑡2. Clearly, we have also 0𝜑(𝑡𝜉)𝜓(𝑡𝜉)=1(2.40) for all 𝜉𝑑. With Ψ0=1𝜓0 and Ψ=1𝜓, we obtain then 𝑔=𝑚0Ψ𝑚𝑡Φ𝑚𝑡𝑔.(2.41) The 2 dilation gives then 𝑔=𝑚0Ψ𝑚𝑡2Φ𝑚𝑡2𝑔(2.42) for every 𝑔𝒮(𝑑), where 𝑔(𝜂)=𝑔(𝜂), 𝜂𝒮(𝑑). Obviously, we can rewrite (2.42) to obtain 𝑔=𝑚0Ψ𝑚𝑡2Φ𝑚𝑡2𝑔(2.43) for all 𝑔𝒮(𝑑). Let us now choose 𝑔=(Φ)𝑡𝑓. This gives the final version of the convolution identity Φ𝑡𝑓=𝑚0Φ𝑡Ψ𝑚𝑡2Φ𝑚𝑡2𝑓.(2.44) For 𝑚,0 we define Λ𝑚,2(𝑥)=𝑑Φ02𝑥Φ:𝑚=0,(𝑥):𝑚>0,𝑥𝑑.(2.45) Clearly, we have Φ𝑡Φ𝑚𝑡2=Λ𝑚,𝑡Φ𝑚+𝑡.(2.46) Plugging this into (2.44), we end up with the pointwise representation Φ𝑡(𝑓𝑦)=𝑚0Ψ𝑚2𝑡Λ𝑚,𝑡Φ𝑚+𝑡(=𝑓𝑦)𝑚0Ψ𝑚2𝑡Λ𝑚,𝑡Φ𝑚+𝑡=𝑓(𝑦)𝑚0𝑑Ψ𝑚2𝑡Λ𝑚,𝑡Φ(𝑦𝑧)𝑚+𝑡𝑓(𝑧)𝑑𝑧(2.47) for all 𝑦𝑑.Substep 1.2. Let us prove the following important inequality first. For every 𝑟>0 and every 𝑁0, we have ||Φ𝑡||𝑓(𝑥)𝑟𝑐𝑘02𝑘𝑁𝑟2(𝑘+)𝑑𝑑||Φ𝑘+𝑡||𝑓(𝑦)𝑟1+2||||𝑥𝑦𝑁𝑟𝑑𝑦,(2.48) where 𝑐 is independent of 𝑓𝒮(𝑑), 𝑥𝑑, and 0.
The representation (2.47) will be the starting point to prove (2.48). Namely, we have for 𝑦𝑑||Φ𝑡||𝑓(𝑦)𝑚0𝑑||Ψ𝑚𝑡2Λ𝑚,𝑡||||Φ(𝑦𝑧)𝑚+𝑡||𝑓(𝑧)𝑑𝑧𝑚0𝑆𝑚,,𝑡𝑑||Φ𝑚+𝑡||𝑓(𝑧)1+2||||𝑦𝑧𝑁𝑑𝑧,(2.49) where 𝑆𝑚,,𝑡=sup𝑥𝑑||Ψ𝑚2𝑡Λ𝑚,𝑡||(𝑥)1+2|𝑥|𝑁.(2.50) Elementary properties of the convolution yield (compare with (2.84)) 𝑆𝑚,,𝑡=2𝑑𝑡𝑑sup𝑥𝑑||||Ψ𝑚Λ𝑚,2𝑥2𝑡||||1+2|𝑥|𝑁=2𝑑𝑡𝑑sup𝑥𝑑||𝜓𝑚𝜂𝑚,||(𝑥)(1+|𝑡𝑥|)𝑁,(2.51) where 𝜂𝑚,Φ(𝑥)=Φ(𝑥):0,𝑚>0,0(𝑥):otherwise.(2.52) Lemma A.3 yields 𝑆𝑚,,𝑡𝑐𝑁2𝑑2𝑚𝑁,(2.53) which we put into (2.49) to obtain ||Φ𝑡||𝑓(𝑦)𝐶𝑁𝑚02𝑚𝑁𝑑2(𝑚+)𝑑||Φ𝑚+𝑡||𝑓(𝑧)1+2||||𝑦𝑧𝑁𝑑𝑧.(2.54) 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 𝑘+ in (2.54) and multiply on both sides with 2𝑘𝑁. Then we can estimate 2𝑘𝑁||Φ𝑘+𝑡||𝑓(𝑦)𝐶𝑁𝑚02𝑘𝑁2𝑚𝑁𝑑2(𝑚+𝑘+)𝑑||Φ𝑚+𝑘+𝑡||𝑓(𝑧)1+2𝑘+||||𝑦𝑧𝑁𝑑𝑧𝐶𝑁𝑚02(𝑚+𝑘)𝑁𝑑2(𝑚+𝑘+)𝑑||Φ𝑚+𝑘+𝑡||𝑓(𝑧)1+2||||𝑦𝑧𝑁𝑑𝑧=𝐶𝑁𝑚𝑘+02𝑚𝑁𝑑2(𝑚+)𝑑||Φ𝑚+𝑡||𝑓(𝑧)1+2||||𝑦𝑧𝑁𝑑𝑧,(2.55)𝐶𝑁𝑚02𝑚𝑁𝑑2(𝑚+)𝑑||Φ𝑚+𝑡||𝑓(𝑧)1+2||||𝑦𝑧𝑁𝑑𝑧.(2.56) Next, we apply the elementary inequalities 1+2||||𝑦𝑧1+2||||𝑥𝑦1+2,||Φ|𝑥𝑧|𝑚+𝑡||||Φ𝑓(𝑧)𝑚+𝑡||𝑓(𝑧)𝑟1+2|𝑥𝑧|𝑁(1𝑟)×sup𝑦𝑑||Φ𝑚+𝑡||𝑓(𝑦)1𝑟1+2||||𝑥𝑦𝑁(1𝑟),(2.57) where 0<𝑟1. We define the maximal function 𝑀,𝑁(𝑥,𝑡)=sup𝑘0sup𝑦𝑑2𝑘𝑁||Φ𝑘+𝑡||𝑓(𝑦)1+2||||𝑥𝑦𝑁,𝑥𝑑,(2.58) and estimate 𝑀,𝑁(𝑥,𝑡)𝐶𝑁𝑚02𝑚𝑁𝑑2(𝑚+)𝑑||Φ𝑚+𝑡||𝑓(𝑧)1+2|𝑥𝑧|𝑁𝑑𝑧(2.59)𝐶𝑁𝑚02𝑚𝑁𝑟2𝑚𝑁sup𝑦𝑑||Φ𝑚+𝑡||𝑓(𝑦)1+2||||𝑥𝑦𝑁1𝑟×𝑑2(𝑚+)𝑑||Φ𝑚+𝑡||𝑓(𝑧)𝑟1+2|𝑥𝑧|𝑁𝑟𝑑𝑧.(2.60) Observe that we can estimate the term ()1𝑟 in the right-hand side of (2.60) by 𝑀,𝑁(𝑥,𝑡)1𝑟. Hence, if 𝑀,𝑁(𝑥,𝑡)< we obtain from (2.60) 𝑀,𝑁(𝑥,𝑡)𝑟𝐶𝑁𝑚02𝑚𝑁𝑟𝑑2(𝑚+)𝑑||Φ𝑚+𝑡||𝑓(𝑧)𝑟1+2|𝑥𝑧|𝑁𝑟𝑑𝑧,(2.61) where 𝐶𝑁 is independent of 𝑥, 𝑓, , and 𝑡[1,2]. We claim that there exists 𝑁𝑓0 such that 𝑀,𝑁(𝑥,𝑡)< for all 𝑁𝑁𝑓. Indeed, we use that 𝑓𝒮(𝑑), that is, there is an 𝑀0 and 𝑐𝑓>0 such that ||Φ𝑘+𝑡||𝑓(𝑦)𝑐𝑓sup|𝛼|1𝑀sup𝑧𝑑||𝐷𝛼Φ𝑘+||||||(𝑧)1+𝑦𝑧𝑀.(2.62) Assuming 𝑁>𝑀, we estimate as follows: ||Φ𝑡||𝑓(𝑥)𝑀,𝑁(𝑥,𝑡)𝑐sup𝑘0sup𝑦𝑑2𝑘𝑁||Φ𝑘+𝑡||𝑓(𝑦)||||1+𝑥𝑦2𝑁/2𝑐sup𝑘0sup𝑦𝑑2𝑘𝑁2(𝑘+)(𝑀+𝑑)sup𝑧𝑑sup|𝛼|1𝑀||𝐷𝛼𝛾𝑘+(||||||𝑧)1+𝑦𝑧𝑀||||1+𝑥𝑦𝑁𝑐2(𝑀+𝑑)sup𝑘0sup𝑧𝑑sup|𝛼|1𝑀||𝐷𝛼𝛾𝑘+||(𝑧)(1+|𝑥𝑧|)𝑁,(2.63) where we again used the inequality (compare with (2.57)) ||||||||1+𝑦𝑧1+𝑥𝑦(1+|𝑥𝑧|)(2.64) and have set 𝛾Φ(𝑡)=0(𝑡)Φ(𝑡)=0,>0.(2.65) Hence 𝛾𝑘+ gives us only two different functions from 𝒮(𝑑). This implies the boundedness of 𝑀,𝑁(𝑥,𝑡) for 𝑥𝑑 if 𝑁>𝑀=𝑁𝑓. Therefore, (2.61) together with (2.63) yield (2.48) with 𝑐=𝐶𝑁, independent of 𝑥, 𝑓, and , for all 𝑁𝑁𝑓. But this is not yet what we want. Observe that the right-hand side of (2.48) decreases as 𝑁 increases. Therefore, we have (2.48) for all 𝑁0 but with 𝑐=𝑐(𝑓)=𝐶𝑁𝑓  depending on 𝑓. This is still not yet what we want. Now we argue as follows: starting with (2.48) where 𝑐=𝑐(𝑓) and 𝑁0 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 𝑓. 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 𝑀,𝑁(𝑥,𝑡)<.
We assume rhs (2.48) <. Otherwise, there is nothing to prove in (2.48). Returning to (2.60) and having in mind that now 𝑀,𝑁(𝑥,𝑡)<, we end up with (2.61) for all 𝑁 and 𝐶𝑁 independent of 𝑓. Finally, from (2.61) we obtain (2.48) and are done in case 0<𝑟1.
Of course, (2.48) also holds true for 𝑟>1 with a much simpler proof. In that case, we use (2.54) with 𝑁+1 instead of 𝑁 and apply Hölder’s inequality with respect to 1/𝑟+1/𝑟=1 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 𝑎𝑁 and Φ2𝑡𝑓𝑎(𝑥)𝑟𝑐𝑘02𝑘𝑁𝑟2(𝑘+)𝑑𝑑||Φ𝑘+𝑡||𝑓(𝑦)𝑟1+2||||𝑥𝑦𝑎𝑟𝑑𝑦.(2.66) We proved that the inequality (2.66) holds for all 𝑡[1,2] where 𝑐>0 is independent of 𝑡. If we choose 𝑟<min{𝑝,𝑞}, we can apply the norm 21||𝑞/𝑟𝑑𝑡𝑡𝑟/𝑞,(2.67) on both sides and use Minkowski’s inequality for integrals, which yields 21|||Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡𝑟/𝑞𝑐𝑘02𝑘𝑁𝑟2(𝑘+)𝑑𝑑21|Φ𝑘+𝑡𝑓(𝑦)|𝑞(𝑑𝑡/𝑡)𝑟/𝑞1+2||||𝑥𝑦𝑎𝑟𝑑𝑦.(2.68) If 𝑎𝑟>𝑑,then we have 𝑔2(𝑦)=𝑑1+2||𝑦||𝑎𝑟𝐿1𝑑,(2.69) and we observe 21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡𝑟/𝑞𝑐𝑘02𝑘𝑁𝑟2𝑘𝑑2𝑠𝑟𝑔21|2𝑠Φ𝑘+𝑡𝑓()|𝑞𝑑𝑡𝑡𝑟/𝑞(𝑥).(2.70) Now we use the majorant property of the Hardy-Littlewood maximal operator (see Section 2.2 and [25, Chapter 2]) and continue estimating 21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡𝑟/𝑞𝑐𝑘02𝑟𝑠2𝑘(𝑁𝑟+𝑑)𝑀21||Φ𝑘+𝑡||𝑓()𝑞𝑑𝑡𝑡𝑟/𝑞(𝑥).(2.71) An index shift on the right-hand side gives 21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡𝑟/𝑞=𝑐𝑘+02𝑟𝑠2(𝑘)(𝑁𝑟+𝑑)𝑀21||Φ𝑘𝑡||𝑓()𝑞𝑑𝑡𝑡𝑟/𝑞(𝑥)=𝑐𝑘+02(𝑘)(𝑁𝑟𝑑+𝑟𝑠)2𝑘𝑟𝑠𝑀21||Φ𝑘𝑡||𝑓()𝑞𝑑𝑡𝑡𝑟/𝑞(𝑥).(2.72) Choose now 1/𝑎<𝑟<min{𝑝,𝑞}, 𝑁>max{0,𝑠}+𝑎 and put 𝑑𝛿=𝑁+𝑠𝑟>0.(2.73) We obtain for 21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡𝑟/𝑞𝑐𝑘2𝛿𝑟|𝑘|2𝑘𝑟𝑠𝑀21||Φ𝑘𝑡||𝑓()𝑞𝑑𝑡𝑡𝑟/𝑞(𝑥).(2.74) Now we apply Lemma 2.13 in 𝐿𝑝/𝑟(𝑞/𝑟,𝑑), which yields 21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡𝑟/𝑞𝐿𝑝/𝑟𝑞/𝑟𝑀𝑐21||2𝑘𝑠Φ𝑘𝑡||𝑓()𝑞𝑑𝑡𝑡𝑟/𝑞𝐿𝑝/𝑟𝑞/𝑟.(2.75) The Fefferman-Stein inequality (see Section 2.2 /Theorem 2.1, having in mind that 𝑝/𝑟,𝑞/𝑟>1) gives 21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑞𝑟=𝑀21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡𝑟/𝑞𝐿𝑝/𝑟𝑞/𝑟21|2𝑘𝑠Φ𝑘𝑡𝑓()|𝑞𝑑𝑡𝑡𝑟/𝑞𝑞/𝑟𝑞/𝑟=21||2𝑘𝑠Φ𝑘𝑡||𝑓()𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑞𝑟.(2.76) Hence, we obtain 10||𝜆𝑠𝑞Φ𝜆𝑓𝑎||(𝑥)𝑞𝑑𝜆𝜆1/𝑞𝐿𝑝𝑑=121|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝=21|||2𝑠Φ2𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑞10||𝜆𝑠𝑞Φ𝜆||𝑓(𝑥)𝑞𝑑𝜆𝜆1/𝑞𝐿𝑝𝑑.(2.77) This proves 𝑓𝐹𝑠𝑝,𝑞(𝑑)2𝑓𝐹𝑠𝑝,𝑞(𝑑)1. Since the reverse inequality is trivial, this finishes Step 1.Step 2. Let Ψ0,Ψ𝒮(𝑑) be functions satisfying (2.18). In fact, the condition (2.17) for Ψ0,Ψ is not necessary for what follows.Substep 2.1. We are going to prove 𝑓𝐹𝑠𝑝,𝑞𝑑Ψ4𝑓𝐹𝑠𝑝,𝑞𝑑Φ2(2.78) for all 𝑓𝒮(𝑑).We decompose 𝑓 similar as in Step 1. Exploiting the property (2.17) for the system Φ, we find 𝒮(𝑑)-functions 𝜆0,𝜆𝒮(𝑑)such that supp𝜆0{𝜉𝑑|𝜉|2𝜀} and supp𝜆{𝜉𝑑𝜀/2|𝜉|2𝜀} and 0𝜆(𝑡𝑥)𝜑(𝑡𝑥)=1(2.79) for 𝑥𝑑 and 𝑡[1,2] fix. Putting Λ0=𝜆0 and Λ=Λ, we obtain the decomposition 𝑔=0Λ𝑡Φ𝑡𝑔(2.80) for every 𝑔𝒮(𝑑). We put 𝑔=Ψ𝑓 and see Ψ𝑓=𝑘0ΨΛ𝑘𝑡Φ𝑘𝑡𝑓.(2.81) Now, we estimate as follows: ||ΨΛ𝑘𝑡Φ𝑘𝑡||𝑓(𝑦)𝑑||ΨΛ𝑘𝑡||||Φ(𝑧)𝑘𝑡||Φ𝑓(𝑦𝑧)𝑑𝑧2𝑘𝑡𝑓𝑎(𝑦)𝑑||ΦΛ𝑘𝑡||(𝑧)1+2𝑘|𝑧|𝑎Φ𝑑𝑧2𝑘𝑡𝑓𝑎(𝑦)𝐽,𝑘,(2.82) where 𝐽,𝑘=𝑑||ΨΛ𝑘𝑡(||𝑧)1+2𝑘|𝑧|𝑎𝑑𝑧.(2.83) We first observe that for 𝑥𝑑 and functions 𝜇,𝜂𝒮(𝑑), the following identity holds true for 𝑢,𝑣>0𝜇𝑢𝜂𝑣1(𝑧)=𝑢𝑑𝜇𝜂𝑣/𝑢𝑧𝑢=1𝑣𝑑𝜇𝑢/𝑣𝑧𝜂𝑣.(2.84) This yields in case 𝑘 (with a minor change if 𝑘=0 or =0) 𝐽,𝑘=𝑑||Ψ𝑘1/𝑡(||(Λ𝑧)1+|𝑡𝑧|)𝑎sup𝑧𝑑||Ψ𝑘1/𝑡||Λ(𝑧)(1+|𝑧|)𝑎+𝑑+12(𝑘)(𝑅+1),(2.85) where we used Lemma A.3 for the last estimate.
If 𝑘>𝑙, we change the roles of Ψ and Λ to obtain again with Lemma A.3𝐽,𝑘=𝑑||ΛΨ𝑘(||||2𝑧)1+𝑘𝑧||𝑎𝑑𝑥2(𝑘)𝑎sup𝑧𝑑||ΛΨ𝑘𝑡||(𝑧)(1+|𝑧|)𝑎+𝑑+12(𝑘)(𝐿+1𝑎),(2.86) where 𝐿 can be chosen arbitrary large since Λ satisfies (𝑀𝐿) for every 𝐿according to its construction. Let us further use the estimate Ψ𝑘𝑓𝑎Ψ(𝑦)𝑘𝑓𝑎(𝑥)1+2𝑘||||𝑥𝑦𝑎Ψ𝑘𝑓𝑎(𝑥)1+2||||𝑥𝑦𝑎max1,2(𝑘)𝑎.(2.87)
Consequently, sup𝑦𝑑2𝑠||ΨΛ𝑘𝑡Φ𝑘𝑡||𝑓(𝑦)1+2||||𝑥𝑦𝑎2𝑘𝑠Φ2𝑘𝑡𝑓𝑎(𝑥)2(𝑘)𝑠max1,2(𝑘)𝑎𝐽,𝑘2𝑘𝑠Φ2𝑘𝑡𝑓𝑎2(𝑥)(𝑘)(𝐿+1𝑎+𝑠)2𝑘>𝑙,(𝑘)(𝑅+1𝑠)𝑘.(2.88) Plugging this into (2.81) and choosing 𝐿𝑎+|𝑠| and 𝛿=min{1,𝑅+1𝑠}, we obtain the inequality 2𝑠Ψ𝑓𝑎(𝑥)𝑘02|𝑘|𝛿2𝑘𝑠Φ2𝑘𝑡𝑓𝑎(𝑥)(2.89) for all 𝑥𝑑 and all 𝑡[1,2]. Suppose first that 𝑞1. Then we take on both sides (21||𝑞𝑑𝑡/𝑡)1/𝑞, which gives 2𝑠Ψ𝑓𝑎(𝑥)𝑘02|𝑘|𝛿2𝑘𝑠21|||Φ2𝑘𝑡𝑓𝑎(|||𝑥)𝑞𝑑𝑡𝑡1/𝑞.(2.90) Applying Lemma 2.13 yields 2𝑠Ψ𝑓𝑎(𝑥)𝐿𝑝𝑞,𝑑𝑘=12𝑘𝑠𝑞21|||Φ2𝑘𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡1/𝑞𝐿𝑝𝑑,(2.91) which gives the desired result.
In case 𝑞<1, we argue as follows. The quantity (21||𝑞𝑑𝑡/𝑡)1/𝑞 is not longer a norm, but a 𝑞-norm. This gives 2𝑠Ψ𝑓𝑎(𝑥)𝑞𝑘02|𝑘|𝛿𝑞2𝑘𝑠𝑞21|||Φ2𝑘𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡.(2.92) Notice that the right-hand side is nothing more than a convolution (𝛾𝛽) of the sequences 𝛾𝑘=2|𝑘|𝛿𝑞,𝛽𝑘=2𝑘𝑠𝑞21|||Φ2𝑘𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡.(2.93) Now we apply the 1-norm to both sides and get for all 𝑥𝑑2𝑠Ψ𝑓𝑎(𝑥)𝑞𝑞𝛾1𝛽1𝑘=12𝑘𝑠𝑞21|||Φ2𝑘𝑡𝑓𝑎|||(𝑥)𝑞𝑑𝑡𝑡.(2.94) We take both sides to the power ()1/𝑞 and apply the 𝐿𝑝(𝑑)-norm. This gives (2.78).
Substep 2.2. With similar arguments and obvious modifications of Substep 2.1, we obtain for all 𝑓𝒮(𝑑)𝑓𝐹𝑠𝑝,𝑞𝑑Ψ2𝑓𝐹𝑠𝑝,𝑞𝑑Φ4.(2.95)Step 3. Choosing 𝑡=1 in Step 1 and omitting the integration over 𝑡, we see immediately 𝑓𝐹𝑠𝑝,𝑞𝑑5𝑓𝐹𝑠𝑝,𝑞𝑑4𝑓𝐹𝑠𝑝,𝑞𝑑5.(2.96)Step 4. What remains is to show that (2.22) is equivalent to the rest.Substep 4.1. Let us prove 𝑓𝐹𝑠𝑝,𝑞𝑑2𝑓𝐹𝑠𝑝,𝑞𝑑3.(2.97) We return to (2.66) in Substep 1.3. If |𝑧|<2(+𝑘)𝑡, formula (2.66) implies by shift in the integral the following: Φ2𝑡𝑓𝑎(𝑥)𝑟𝐶𝑁𝑘02𝑘𝑁𝑟2(𝑘+)𝑑𝑑||Φ𝑘+𝑡||𝑓(𝑦+𝑧)𝑟1+2||||𝑥𝑦𝑎𝑟𝑑𝑦.(2.98) Indeed, we have 1+2||||𝑥𝑦1+2||||𝑥(𝑦+𝑧)+|𝑧|1+2