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

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 𝐵̇^{𝑠𝑝,𝑞}(ℝ^𝑑) and 𝐹̇^{𝑠𝑝,𝑞}(ℝ^𝑑) on the Euclidean space ℝ^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 [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 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 [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 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 ℝ,ℂ,ℕ,ℕ₀,and ℤdenote 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 ℓ^𝑑1-norm of a vector 𝑘. For a multi-index 𝛼 and 𝑥∈ℝ^𝑑, we write𝑥^𝛼=𝑥^𝛼1₁⋅…⋅𝑥^𝛼𝑑_𝑑(1.1)

and define the differential operators 𝐷^𝛼 and Δ by𝐷^𝛼=𝜕^|𝛼|1𝜕𝑥^𝛼1₁⋯𝜕𝑥^𝛼𝑑_𝑑,Δ=^𝑑_𝑘=1𝜕²𝜕𝑥^2𝑘.(1.2)

If 𝑋 is a (quasi-)Banach space and 𝑓∈𝑋, we use ‖𝑓∣𝑋‖ or simply ‖𝑓‖ for its (quasi-)norm. Operator norms of linear mappings 𝐴∶𝑋→𝑌 are denoted by ‖𝐴∶𝑋→𝑌‖ 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 𝑎≲𝑏 if there exists a constant 𝑐>0 (independent of the context-dependent relevant parameters) such that 𝑎≤𝑐𝑏. If 𝑎≲𝑏 and 𝑏≲𝑎, we will write 𝑎≍𝑏.

2. Function Spaces on ℝ^𝑑

2.1. Vector-Valued Lebesgue Spaces

The space 𝐿_𝑝(ℝ^𝑑), 0<𝑝≤∞, denotes the collection of complex-valued functions (equivalence classes) with finite (quasi-)norm‖‖𝑓∣𝐿_𝑝ℝ^𝑑‖‖=_ℝ𝑑||𝑓(𝑥)||^𝑝𝑑𝑥^1/𝑝,(2.1) with the usual modification if 𝑝=∞. The Hilbert space 𝐿₂(ℝ^𝑑) plays a separate role, see for instance, Section 3. Having a sequence of complex-valued functions {𝑓_𝑘}_𝑘∈𝐼 on ℝ^𝑑, where 𝐼 is a countable index set, we put‖‖𝑓_𝑘∣ℓ_𝑞𝐿_𝑝ℝ^𝑑‖‖=_𝑘∈𝐼‖‖𝑓_𝑘∣𝐿_𝑝ℝ^𝑑‖‖^𝑞^1/𝑞‖‖𝑓_𝑘,∣𝐿_𝑝ℓ_𝑞,ℝ^𝑑‖‖=‖‖‖‖_𝑘∈𝐼||𝑓_𝑘(𝑥)||^𝑞^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(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)=sup_𝑥∈𝑄1||𝑄||_𝑄||𝑓(𝑦)||𝑑𝑦,𝑥∈ℝ^𝑑,(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 ‖‖𝑀𝑓_𝑘∣𝐿_𝑝ℓ_𝑞,ℝ^𝑑‖‖≤𝑐‖‖𝑓_𝑘∣𝐿_𝑝ℓ_𝑞‖‖(2.4) holds for all sequences {𝑓_𝑘}_𝑘∈ℤ of locally Lebesgue-integrable functions on ℝ^𝑑.

Let us recall the classical Peetre maximal operator, introduced in [20]. Given a sequence of function {Ψ_𝑘}_𝑘∈ℕ⊂𝒮(ℝ^𝑑), a tempered distribution 𝑓∈𝒮′(ℝ^𝑑) and a positive number 𝑎>0, we define the system of maximal functionsΨ^∗𝑘𝑓_𝑎(𝑥)=sup_{𝑦∈ℝ𝑑}||Ψ_𝑘∗𝑓(𝑥+𝑦)||1+2^𝑘||𝑦||^𝑎,𝑥∈ℝ^𝑑,𝑘∈ℤ.(2.5)

Since (Ψ_𝑘∗𝑓)(𝑦) makes sense pointwise (see the following paragraph), everything is well-defined. However, the value “∞’’ is also possible for (Ψ^∗𝑘𝑓)_𝑎(𝑥). This was the reason for the problematic arguments in [4] mentioned in the introduction. We will often use dilates Ψ_𝑘(𝑥)=2^𝑘𝑑Ψ(2^𝑘𝑥) of a fixed function Ψ∈𝒮(ℝ^𝑑), where Ψ₀(𝑥) might be given by a separate function. Also continuous dilates are needed. Let the operator 𝒟^𝐿𝑝_𝑡, 𝑡>0, generate the 𝑝-normalized dilates of a function Ψ given by 𝒟^𝐿𝑝_𝑡Ψ∶=𝑡^−𝑑/𝑝Ψ(𝑡⁻¹⋅). If 𝑝=1, we omit the super index and use additionally Ψ_𝑡∶=𝒟_𝑡Ψ∶=𝒟^𝐿1_𝑡Ψ. We define (Ψ^∗𝑡𝑓)_𝑎(𝑥) byΨ^∗𝑡𝑓_𝑎(𝑥)=sup_{𝑦∈ℝ𝑑}||Ψ_𝑡∗𝑓(𝑥+𝑦)||1+||𝑦||/𝑡^𝑎,𝑥∈ℝ^𝑑,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(ℝ^𝑑) is used for the locally convex space of rapidly decreasing infinitely differentiable functions on ℝ^𝑑, where the topology is generated by the family of seminorms‖𝜑‖_𝑘,ℓ=sup_{𝑥∈ℝ𝑑,|𝛼|1≤ℓ}||𝐷^𝛼𝜑(𝑥)||(1+|𝑥|)^𝑘,𝜑∈𝒮ℝ^𝑑,𝑘,ℓ∈ℕ₀.(2.7) The space 𝒮′(ℝ^𝑑) is called the set of all tempered distributions on ℝ^𝑑 and defined as the topological dual of 𝒮(ℝ^𝑑). Indeed, a linear mapping 𝑓∶𝒮(ℝ^𝑑)→ℂbelongs to 𝒮′(ℝ^𝑑) if and only if there exist numbers 𝑘,ℓ∈ℕ₀ and a constant 𝑐=𝑐_𝑓 such that||𝑓(𝜑)||≤𝑐_𝑓sup_{𝑥∈ℝ𝑑,|𝛼|1≤ℓ}||𝐷^𝛼𝜑(𝑥)||(1+|𝑥|)^𝑘(2.8)for all 𝜑∈𝒮(ℝ^𝑑). 𝒮′(ℝ^𝑑) is equipped with the weak^∗-topology.

The convolution 𝜑∗𝜓 of two integrable functions 𝜑,𝜓 is defined via the integral(𝜑∗𝜓)(𝑥)=_ℝ𝑑𝜑(𝑥−𝑦)𝜓(𝑦)𝑑𝑦.(2.9) If 𝜑,𝜓∈𝒮(ℝ^𝑑), then (2.9) still belongs to 𝒮(ℝ^𝑑). The convolution can be generalized to 𝒮(ℝ^𝑑)×𝒮^(ℝ^𝑑) via (𝜑∗𝑓)(𝑥)=𝑓(𝜑(𝑥−⋅)), makes sense pointwise, and is a 𝐶^∞-function in ℝ^𝑑 of at most polynomial growth.

As usual, the Fourier transform defined on both 𝒮(ℝ^𝑑) and 𝒮′(ℝ^𝑑) is given by (ℱ𝑓)(𝜑)∶=𝑓(ℱ𝜑), where 𝑓∈𝒮′(ℝ^𝑑),𝜑∈𝒮(ℝ^𝑑), andℱ𝜑(𝜉)∶=(2𝜋)^−𝑑/2_ℝ𝑑𝑒^{−𝑖𝑥⋅𝜉}𝜑(𝑥)𝑑𝑥.(2.10) The mapping ℱ is a bijection (in both cases) and its inverse is given by ℱ⁻¹𝜑=ℱ𝜑(−⋅).

In order to deal with homogeneous spaces, we need to define the subset 𝒮₀(ℝ^𝑑)⊂𝒮(ℝ^𝑑). Following [1, Chapter 5], we put𝒮₀ℝ^𝑑=𝜑∈𝒮ℝ^𝑑∶𝐷^𝛼(ℱ𝜑)(0)=0foreverymulti-index𝛼∈ℕ^𝑑0.(2.11) The set 𝒮^′₀(ℝ^𝑑) denotes the topological dual of 𝒮₀(ℝ^𝑑). If 𝑓∈𝒮′(ℝ^𝑑), the restriction of 𝑓 to 𝒮₀(ℝ^𝑑) clearly belongs to 𝒮^0(ℝ^𝑑). Furthermore, if 𝑃(𝑥) is an arbitrary polynomial in ℝ^𝑑, we have (𝑓+𝑃(⋅))(𝜑)=𝑓(𝜑) for every 𝜑∈𝒮₀(ℝ^𝑑). Conversely, if 𝑓∈𝒮^0(ℝ^𝑑), then 𝑓 can be extended from 𝒮₀(ℝ^𝑑) to 𝒮(ℝ^𝑑), that is, to an element of 𝒮′(ℝ^𝑑). However, this fact is not trivial and makes use of the Hahn-Banach theorem in locally convex topological vector spaces. We may identify 𝒮^0(ℝ^𝑑) with the factor space 𝒮′(ℝ^𝑑)/𝒫(ℝ^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(ℝ^𝑑) be the collection of all systems {𝜑_𝑗(𝑥)}_𝑗∈ℕ0⊂𝒮(ℝ^𝑑) with the following properties:(i)𝜑_𝑗(𝑥)=𝜑(2^−𝑗𝑥),𝑗∈ℕ, (ii)supp𝜑₀⊂{𝑥∈ℝ^𝑑∶|𝑥|≤2},supp𝜑⊂{𝑥∈ℝ^𝑑∶1/2≤|𝑥|≤2},(iii)∑^∞𝑗=0𝜑_𝑗(𝑥)=1 for every 𝑥∈ℝ^𝑑.
(b) Moreover, the system Φ̇(ℝ^𝑑) denotes the collection of all systems {𝜑_𝑗(𝑥)}_𝑗∈ℤ⊂𝒮(ℝ^𝑑) with the following properties: (i)𝜑_𝑗(𝑥)=𝜑(2^−𝑗𝑥),𝑗∈ℤ, (ii)supp𝜑={𝑥∈ℝ^𝑑∶1/2≤|𝑥|≤2},(iii)∑^{∞𝑗=−∞}𝜑_𝑗=1 for every 𝑥∈ℝ^𝑑⧵{0}.

Remark 2.3. If we take 𝜑₀∈𝒮(ℝ^𝑑) satisfying 𝜑₀(𝑥)=10::||𝑥𝑥||≤>12(2.12) and define 𝜑(𝑥)=𝜑₀(𝑥)−𝜑₀(2𝑥), then the system {𝜑_𝑗(𝑥)}_𝑗∈ℕ0 belongs to Φ(ℝ^𝑑) and the system {𝜑_𝑗(𝑥)}_𝑗∈ℤ with 𝜑₀∶=𝜑 belongs to Φ̇(ℝ^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 {𝜑_𝑗(𝑥)}^∞𝑗=0∈Φ(ℝ^𝑑) and Φ_𝑗=ℱ⁻¹𝜑_𝑗, 𝑗∈ℕ₀. Let further −∞<𝑠<∞ and 0<𝑞≤∞. (i) If 0<𝑝≤∞, then𝐵^{𝑠𝑝,𝑞}ℝ^𝑑=⎧⎪⎨⎪⎩𝑓∈𝒮^′ℝ^𝑑∶‖‖𝑓∣𝐵^{𝑠𝑝,𝑞}ℝ^𝑑‖‖=^∞_𝑗=02^𝑗𝑠𝑞‖‖Φ_𝑗∗𝑓∣𝐿_𝑝ℝ^𝑑‖‖^𝑞^1/𝑞⎫⎪⎬⎪⎭<∞.(2.13)(ii)If 0<𝑝<∞, then𝐹^{𝑠𝑝,𝑞}ℝ^𝑑=⎧⎪⎨⎪⎩𝑓=∈𝒮‖‖‖‖′ℝ^𝑑_∞∶‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖_𝑗=02^𝑗𝑠𝑞||Φ_𝑗∗𝑓(𝑥)||^𝑞^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖<∞⎫⎪⎬⎪⎭.(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 {𝜑_𝑗(𝑥)}_𝑗∈ℤ∈Φ̇(ℝ^𝑑) and Φ_𝑗=ℱ⁻¹𝜑_𝑗. Let further −∞<𝑠<∞ and 0<𝑞≤∞. (i) If 0<𝑝≤∞, then𝐵̇^{𝑠𝑝,𝑞}ℝ^𝑑=⎧⎪⎨⎪⎩𝑓∈𝒮^0ℝ^𝑑∶‖‖𝑓∣𝐵̇^{𝑠𝑝,𝑞}ℝ^𝑑‖‖=^∞_𝑗=_−∞2^𝑗𝑠𝑞‖‖Φ_𝑗∗𝑓∣𝐿_𝑝ℝ^𝑑‖‖^𝑞^1/𝑞<∞⎫⎪⎬⎪⎭.(2.15) (ii) If 0<𝑝<∞, then𝐹̇^{𝑠𝑝,𝑞}ℝ^𝑑=⎧⎪⎨⎪⎩=𝑓∈‖‖‖‖𝒮^0_∞ℝ^𝑑∶‖‖𝑓∣𝐹̇^{𝑠𝑝,𝑞}ℝ^𝑑‖‖_𝑗=_−∞2^𝑗𝑠𝑞||Φ_𝑗∗𝑓(𝑥)||^𝑞^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖<∞⎫⎪⎬⎪⎭.(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,Φ∈𝒮(ℝ^𝑑) satisfying||(ℱ||ℱΦ)Φ(𝑥₀)||(>𝑥)||0>0onon𝜀{|𝑥|<2𝜀},2<|𝑥|<2𝜀,(2.17) for some 𝜀>0, and𝐷^𝛼(ℱΦ)(0)=0∀||𝛼||≤𝑅.(2.18) We will call the functions Φ₀ and Φ kernels for local means. Recall that Φ_𝑘=2^𝑘𝑑Φ(2^𝑘⋅), 𝑘∈ℕ, and Ψ_𝑡=𝒟_𝑡Ψ.

Theorem 2.6. Let 𝑠∈ℝ, 0<𝑝<∞, 0<𝑞≤∞, 𝑎>𝑑/min{𝑝,𝑞} and 𝑅+1>𝑠. Let further Φ₀,Φ∈𝒮(ℝ^𝑑) be given by (2.17) and (2.18). Then the space 𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑) can be characterized by 𝐹^{𝑠𝑝,𝑞}ℝ^𝑑=𝑓∈𝒮′ℝ^𝑑∶‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖_𝑖<∞,𝑖=1,…,5,(2.19) where ‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}‖‖₁=‖‖Φ₀∗𝑓∣𝐿_𝑝ℝ^𝑑‖‖+‖‖‖‖¹⁰𝑡^−𝑠𝑞||Φ_𝑡∗𝑓(𝑥)||^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐹_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(2.20)‖‖₂=+‖‖Φ‖‖‖‖₀∗₁₀𝑓∣𝐿_𝑝ℝ^𝑑‖‖𝑡^−𝑠𝑞sup_𝑧∈ℝ^𝑑||Φ_𝑡∗𝑓(𝑥+𝑧)||(1+|𝑧|/𝑡)^𝑎^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐹_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(2.21)‖‖₃=+‖‖Φ‖‖‖‖₀∗₁₀𝑓∣𝐿_𝑝ℝ^𝑑‖‖𝑡^−𝑠𝑞_|𝑧|<𝑡||Φ_𝑡∗𝑓(𝑥+𝑧)||^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞‖‖𝑓∣𝐹_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(2.22)‖‖₄=‖‖‖‖^∞_𝑘=02^𝑠𝑘𝑞sup_𝑧∈ℝ^𝑑||Φ_𝑘∗𝑓(𝑥+𝑧)||(1+2^𝑘|𝑧|)^𝑎^𝑞^1/𝑞‖‖𝑓∣𝐹_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(2.23)‖‖₅=‖‖‖‖^∞_𝑘=02^𝑠𝑘𝑞||Φ_𝑘∗𝑓(𝑥)||^𝑞^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(2.24) with the usual modification in case 𝑞=∞. Furthermore, all quantities ‖𝑓∣𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑)‖_𝑖, 𝑖=1,…,5, are equivalent (quasi-)norms in 𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑).

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

Theorem 2.7. Let 𝑠∈ℝ, 0<𝑝,𝑞≤∞, 𝑎>𝑑/𝑝, and 𝑅+1>𝑠. Let further Φ₀,Φ∈𝒮(ℝ^𝑑) be given by (2.17) and (2.18). Then the space 𝐵^{𝑠𝑝,𝑞}(ℝ^𝑑) can be characterized by 𝐵^{𝑠𝑝,𝑞}ℝ^𝑑=𝑓∈𝒮′ℝ^𝑑∶‖‖𝑓∣𝐵^{𝑠𝑝,𝑞}ℝ^𝑑‖‖_𝑖<∞,𝑖=1,…,4,(2.25) where ‖‖𝑓∣𝐵^{𝑠𝑝,𝑞}‖‖₁=‖‖Φ₀∗𝑓∣𝐿_𝑝ℝ^𝑑‖‖+¹⁰𝑡^−𝑠𝑞‖‖Φ_𝑡∗𝑓(𝑥)∣𝐿_𝑝ℝ^𝑑‖‖^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐵_{𝑠𝑝,𝑞},‖‖₂=‖‖Φ₀∗𝑓∣𝐿_𝑝ℝ^𝑑‖‖+¹⁰𝑡^−𝑠𝑞‖‖‖‖sup_{𝑧∈ℝ𝑑}||Φ_𝑡∗𝑓(𝑥+𝑧)||(1+|𝑧|/𝑡)^𝑎∣𝐿_𝑝ℝ^𝑑‖‖‖‖^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐵_{𝑠𝑝,𝑞},‖‖₃=^∞_𝑘=02^𝑠𝑘𝑞‖‖‖‖sup_{𝑧∈ℝ𝑑}||Φ_𝑘∗𝑓(𝑥+𝑧)||1+2^𝑘|𝑧|^𝑎∣𝐿_𝑝ℝ^𝑑‖‖‖‖^𝑞^1/𝑞‖‖𝑓∣𝐵_{𝑠𝑝,𝑞},‖‖₄=^∞_𝑘=02^𝑠𝑘𝑞‖‖Φ_𝑘∗𝑓(𝑥)∣𝐿_𝑝ℝ^𝑑‖‖^𝑞^1/𝑞,(2.26) with the usual modification if 𝑞=∞. Furthermore, all quantities ‖𝑓|𝐵^{𝑠𝑝,𝑞}(ℝ^𝑑)‖_𝑖, 𝑖=1,…,4, are equivalent quasinorms in 𝐵^{𝑠𝑝,𝑞}(ℝ^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∈ℝ, 0<𝑝<∞, 0<𝑞≤∞, 𝑎>𝑑/min{𝑝,𝑞}, and 𝑅+1>𝑠. Let further Φ∈𝒮(ℝ^𝑑) be given by (2.17) and (2.18). Then the space 𝐹̇^{𝑠𝑝,𝑞}(ℝ^𝑑) can be characterized by 𝐹̇^{𝑠𝑝,𝑞}ℝ^𝑑=𝑓∈𝒮^0ℝ^𝑑∶‖‖𝑓∣𝐹̇^{𝑠𝑝,𝑞}ℝ^𝑑‖‖_𝑖<∞,𝑖=1,…,5,(2.27) where ‖‖𝑓∣𝐹̇^{𝑠𝑝,𝑞}‖‖₁=‖‖‖‖^∞0𝑡^−𝑠𝑞||Φ_𝑡∗𝑓(𝑥)||^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐹̇_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,‖‖₂=‖‖‖‖^∞0𝑡^−𝑠𝑞sup_𝑧∈ℝ^𝑑||Φ_𝑡∗𝑓(𝑥+𝑧)||(1+|𝑧|/𝑡)^𝑎^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐹̇_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,‖‖₃=‖‖‖‖^∞0𝑡^−𝑠𝑞_|𝑧|<𝑡||Φ_𝑡∗𝑓(𝑥+𝑧)||^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞‖‖𝑓∣𝐹̇_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,‖‖₄=‖‖‖‖^∞_𝑘_=−∞2^𝑠𝑘𝑞sup_𝑧∈ℝ^𝑑||Φ_𝑘∗𝑓(𝑥+𝑧)||(1+2^𝑘|𝑧|)^𝑎^𝑞^1/𝑞‖‖𝑓∣𝐹̇_{𝑠𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,‖‖₅=‖‖‖‖^∞_𝑘_=−∞2^𝑠𝑘𝑞||Φ_𝑘∗𝑓(𝑥)||^𝑞^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖(2.28) with the usual modification if 𝑞=∞. Furthermore, all quantities ‖𝑓|𝐹̇^{𝑠𝑝,𝑞}(ℝ^𝑑)‖_𝑖, 𝑖=1,…,5, are equivalent quasinorms in 𝐹̇^{𝑠𝑝,𝑞}(ℝ^𝑑).

For the Besov spaces, we obtain the following characterizations.

Theorem 2.9. Let 𝑠∈ℝ,0<𝑝,𝑞≤∞, 𝑎>𝑑/𝑝, and 𝑅+1>𝑠. Let further Φ∈𝒮(ℝ^𝑑) be given by (2.17) and (2.18). Then the space 𝐵̇^{𝑠𝑝,𝑞}(ℝ^𝑑) can be characterized by 𝐵̇^{𝑠𝑝,𝑞}ℝ^𝑑=𝑓∈𝒮^0ℝ^𝑑∶‖‖𝑓∣𝐵̇^{𝑠𝑝,𝑞}ℝ^𝑑‖‖_𝑖<∞,𝑖=1,…,4,(2.29) where ‖‖𝑓∣𝐵̇^{𝑠𝑝,𝑞}‖‖₁=^∞0𝑡^−𝑠𝑞‖‖Φ_𝑡∗𝑓(𝑥)∣𝐿_𝑝ℝ^𝑑‖‖^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐵̇_{𝑠𝑝,𝑞},‖‖₂=^∞0𝑡^−𝑠𝑞‖‖‖‖sup_{𝑧∈ℝ𝑑}||Φ_𝑡∗𝑓(𝑥+𝑧)||(1+|𝑧|/𝑡)^𝑎∣𝐿_𝑝ℝ^𝑑‖‖‖‖^𝑞𝑑𝑡𝑡^1/𝑞‖‖𝑓∣𝐵̇_{𝑠𝑝,𝑞},‖‖₃=^∞_𝑘_=−∞2^𝑠𝑘𝑞‖‖‖‖sup_{𝑧∈ℝ𝑑}||Φ_𝑘∗𝑓(𝑥+𝑧)||1+2^𝑘|𝑧|^𝑎∣𝐿_𝑝ℝ^𝑑‖‖‖‖^𝑞^1/𝑞‖‖𝑓∣𝐵̇_{𝑠𝑝,𝑞},‖‖₄=^∞_𝑘_=−∞2^𝑠𝑘𝑞‖‖Φ_𝑘∗𝑓(𝑥)∣𝐿_𝑝ℝ^𝑑‖‖^𝑞^1/𝑞,(2.30) with the usual modification if 𝑞=∞. Furthermore, all quantities ‖𝑓∣𝐵̇^{𝑠𝑝,𝑞}(ℝ^𝑑)‖_𝑖, 𝑖=1,…,4, are equivalent quasinorms in 𝐵̇^{𝑠𝑝,𝑞}(ℝ^𝑑).

Remark 2.10. Observe, that the (quasi-)norms ‖⋅∣𝐹̇^{𝑠𝑝,𝑞}(ℝ^𝑑)‖₃ and ‖⋅∣𝐹^{𝑠𝑝,𝑞}(ℝ^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,𝑘⁰∈𝒮(ℝ^𝑑) such that ℱ𝑘₀(0),ℱ𝑘⁰(0)≠0(2.31) and define Φ₀=𝑘₀,Φ=Δ^𝑁𝑘⁰,(2.32) where 𝑁∈ℕsuch that 2𝑁>𝑠. Then (2.20), (2.21), (2.22), (2.23), and (2.24) characterize 𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑).

Corollary 2.12. Let 𝑝,𝑞,𝑠 as in Theorem 2.6. Let further Φ₀∈𝒮(ℝ^𝑑) be a radial function such that ℱΦ₀ is non-increasing and atisfying ℱΦ₀(0)≠0,𝐷^𝛼ℱΦ₀(0)=0(2.33) for 1≤|𝛼|₁≤𝑅, where 𝑅+1>𝑠. Define Φ(𝑥)=Φ₀(𝑥)−12^𝑑Φ₀𝑥2.(2.34) Then (2.23), and (2.24) characterize 𝐹^{𝑠𝑝,𝑞}(ℝ^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 𝐹^{𝑠𝑝,𝑞}(ℝ^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)=𝑒^−𝑡2(2.35) and, if 𝑎>0 is given, the tempered distribution 𝑓(𝑡)=|𝑡|^𝑛 with 𝑎<𝑛∈ℕ. Then (Ψ^∗𝑘𝑓)_𝑎(𝑥) is infinite in every point 𝑥∈ℝ. 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 0<𝑝,𝑞≤∞ and, 𝛿>0. Let {𝑔_𝑘}_𝑘∈ℕ0 be a sequence of nonnegative measurable functions on ℝ^𝑑 and put 𝐺_ℓ(𝑥)=_𝑘∈ℤ2^{−|𝑘−ℓ|𝛿}𝑔_𝑘(𝑥),𝑥∈ℝ^𝑑,ℓ∈ℤ.(2.36) Then there is some constant 𝐶=𝐶(𝑝,𝑞,𝛿), such that ‖‖‖‖𝐺_ℓ𝐺_ℓ∣ℓ_𝑞𝐿_𝑝ℝ^𝑑‖‖≤𝐶‖‖𝑔_𝑘_𝑘∣ℓ_𝑞𝐿_𝑝ℝ^𝑑_ℓ‖‖,_ℓ∣𝐿_𝑝ℓ_𝑞,ℝ^𝑑‖‖≤𝐶‖‖𝑔_𝑘_𝑘∣𝐿_𝑝ℓ_𝑞,ℝ^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, 𝐹^{𝑠𝑝,𝑞}(ℝ^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: ‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}‖‖₂≲‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}‖‖₁≲‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}‖‖₂(2.38) for every 𝑓∈𝒮^(ℝ^𝑑).Substep 1.1. Put 𝜑₀=ℱΦ₀ and 𝜑_ℓ=(ℱΦ)(2^−ℓ⋅) if ℓ≥1. Because of (2.17), it is possible to find functions 𝜓₀,𝜓∈𝒮(ℝ^𝑑) with supp𝜓₀⊂{𝜉∈ℝ^𝑑∶|𝜉|≤2𝜀}, supp𝜓⊂{𝜉∈ℝ^𝑑∶𝜀/2≤|𝜉|≤2𝜀} and 𝜓_ℓ(𝑥)=𝜓(2^−ℓ𝑥) such that _ℓ_∈ℕ0𝜑_ℓ(𝜉)⋅𝜓_ℓ(𝜉)=1.(2.39) We need a bit more. Fix a 1≤𝑡≤2. Clearly, we have also _ℓ_∈ℕ0𝜑_ℓ(𝑡𝜉)⋅𝜓_ℓ(𝑡𝜉)=1(2.40) for all 𝜉∈ℝ^𝑑. With Ψ₀=ℱ⁻¹𝜓₀ and Ψ=ℱ⁻¹𝜓, we obtain then 𝑔=_𝑚∈ℕ0Ψ_𝑚_𝑡∗Φ_𝑚_𝑡∗𝑔.(2.41) The 2^ℓ dilation gives then 𝑔_ℓ=_𝑚∈ℕ0Ψ_𝑚_𝑡2^−ℓ∗Φ_𝑚_𝑡2^−ℓ∗𝑔_ℓ(2.42) for every 𝑔∈𝒮′(ℝ^𝑑), where 𝑔_ℓ(𝜂)=𝑔(𝜂_−ℓ), 𝜂∈𝒮(ℝ^𝑑). Obviously, we can rewrite (2.42) to obtain 𝑔=_𝑚∈ℕ0Ψ_𝑚_𝑡2^−ℓ∗Φ_𝑚_𝑡2^−ℓ∗𝑔(2.43) for all 𝑔∈𝒮′(ℝ^𝑑). Let us now choose 𝑔=(Φ_ℓ)_𝑡∗𝑓. This gives the final version of the convolution identity Φ_ℓ_𝑡∗𝑓=_𝑚∈ℕ0Φ_ℓ_𝑡∗Ψ_𝑚_𝑡2^−ℓ∗Φ_𝑚_𝑡2^−ℓ∗𝑓.(2.44) For 𝑚,ℓ∈ℕ₀ we define Λ_𝑚,ℓ(𝑥)=⎧⎪⎨⎪⎩2^ℓ𝑑ΦΦ₀2^ℓ𝑥:𝑚=0,_ℓ(𝑥):𝑚>0,𝑥∈ℝ^𝑑.(2.45) Clearly, we have Φ_ℓ_𝑡∗Φ_𝑚_𝑡2^−ℓ=Λ_𝑚,ℓ_𝑡∗Φ_𝑚+ℓ_𝑡.(2.46) Plugging this into (2.44), we end up with the pointwise representation Φ_ℓ_𝑡∗𝑓(𝑦)==_𝑚∈ℕ0Ψ_𝑚₂^−ℓ_𝑡∗Λ_𝑚,ℓ_𝑡∗Φ_𝑚+ℓ_𝑡∗𝑓(𝑦)_𝑚∈ℕ0=Ψ_𝑚₂^−ℓ_𝑡∗Λ_𝑚,ℓ_𝑡∗Φ_𝑚+ℓ_𝑡∗𝑓(𝑦)_𝑚∈ℕ0_ℝ𝑑Ψ_𝑚₂^−ℓ_𝑡∗Λ_𝑚,ℓ_𝑡(𝑦−𝑧)⋅Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)𝑑𝑧(2.47) for all 𝑦∈ℝ^𝑑.Substep 1.2. Let us prove the following important inequality first. For every 𝑟>0 and every 𝑁∈ℕ₀, we have ||Φ_ℓ_𝑡∗𝑓(𝑥)||^𝑟≤𝑐_𝑘∈ℕ02^{−𝑘𝑁𝑟}2^{(𝑘+ℓ)𝑑}_ℝ𝑑||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)||^𝑟1+2^ℓ||𝑥−𝑦||^𝑁𝑟𝑑𝑦,(2.48) where 𝑐 is independent of 𝑓∈𝒮^(ℝ^𝑑), 𝑥∈ℝ^𝑑, and ℓ∈ℕ₀.
The representation (2.47) will be the starting point to prove (2.48). Namely, we have for 𝑦∈ℝ^𝑑||Φ_ℓ_𝑡∗𝑓(𝑦)||≤_𝑚∈ℕ0_ℝ𝑑≤||Ψ_𝑚_𝑡2^−ℓ∗Λ_𝑚,ℓ_𝑡(𝑦−𝑧)||⋅||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||𝑑𝑧_𝑚∈ℕ0𝑆_{𝑚,ℓ,𝑡}_ℝ𝑑||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||1+2^ℓ||𝑦−𝑧||^𝑁𝑑𝑧,(2.49) where 𝑆_{𝑚,ℓ,𝑡}=sup_{𝑥∈ℝ𝑑}||Ψ_𝑚₂^−ℓ_𝑡∗Λ_𝑚,ℓ_𝑡(𝑥)||⋅1+2^ℓ|𝑥|^𝑁.(2.50) Elementary properties of the convolution yield (compare with (2.84)) 𝑆_{𝑚,ℓ,𝑡}=2^ℓ𝑑𝑡^𝑑sup_{𝑥∈ℝ𝑑}||||Ψ_𝑚∗Λ_𝑚,ℓ₂^ℓ𝑥2^ℓ𝑡||||⋅1+2^ℓ|𝑥|^𝑁=2^ℓ𝑑𝑡^𝑑sup_{𝑥∈ℝ𝑑}||𝜓_𝑚∗𝜂_𝑚,ℓ(𝑥)||⋅(1+|𝑡𝑥|)^𝑁,(2.51) where 𝜂_𝑚,ℓ(𝑥)=ΦΦ(𝑥):ℓ≥0,𝑚>0,₀(𝑥):otherwise.(2.52) Lemma A.3 yields 𝑆_{𝑚,ℓ,𝑡}≤𝑐_𝑁2^ℓ𝑑2^−𝑚𝑁,(2.53) which we put into (2.49) to obtain ||Φ_ℓ_𝑡∗𝑓(𝑦)||≤𝐶_𝑁𝑚_∈ℕ02^−𝑚𝑁_ℝ𝑑2^{(𝑚+ℓ)𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||1+2^ℓ||𝑦−𝑧||^𝑁𝑑𝑧.(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 ℓ by 𝑘+ℓ in (2.54) and multiply on both sides with 2^−𝑘𝑁. Then we can estimate 2^−𝑘𝑁||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)||≤𝐶_𝑁𝑚_∈ℕ02^−𝑘𝑁2^−𝑚𝑁_ℝ𝑑2^{(𝑚+𝑘+ℓ)𝑑}||Φ_{𝑚+𝑘+ℓ}_𝑡∗𝑓(𝑧)||1+2^𝑘+ℓ||𝑦−𝑧||^𝑁≤𝐶_𝑁𝑑𝑧_𝑚_∈ℕ02^{−(𝑚+𝑘)𝑁}_ℝ𝑑2^{(𝑚+𝑘+ℓ)𝑑}||Φ_{𝑚+𝑘+ℓ}_𝑡∗𝑓(𝑧)||1+2^ℓ||𝑦−𝑧||^𝑁=𝐶_𝑁𝑑𝑧_𝑚∈_𝑘+ℕ02^−𝑚𝑁_ℝ𝑑2^{(𝑚+ℓ)𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||1+2^ℓ||𝑦−𝑧||^𝑁≤𝐶_𝑁𝑑𝑧,(2.55)_𝑚_∈ℕ02^−𝑚𝑁_ℝ𝑑2^{(𝑚+ℓ)𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||1+2^ℓ||𝑦−𝑧||^𝑁𝑑𝑧.(2.56) Next, we apply the elementary inequalities 1+2^ℓ||𝑦−𝑧||||⋅Φ1+_𝑚+ℓ2^ℓ||𝑥−𝑦||≥1+2^ℓ|𝑥−𝑧|,_𝑡∗𝑓(𝑧)||≤||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||^𝑟1+2^ℓ|𝑥−𝑧|^{𝑁(1−𝑟)}×sup_{𝑦∈ℝ𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑦)||^1−𝑟1+2^ℓ||𝑥−𝑦||^{𝑁(1−𝑟)},(2.57) where 0<𝑟≤1. We define the maximal function 𝑀_ℓ,𝑁(𝑥,𝑡)=sup_𝑘∈ℕ0sup_{𝑦∈ℝ𝑑}2^−𝑘𝑁||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)||1+2^ℓ||𝑥−𝑦||^𝑁,𝑥∈ℝ^𝑑,(2.58) and estimate 𝑀_ℓ,𝑁(𝑥,𝑡)≤𝐶_𝑁𝑚_∈ℕ02^−𝑚𝑁_ℝ𝑑2^{(𝑚+ℓ)𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||1+2^ℓ|𝑥−𝑧|^𝑁≤𝐶_𝑁𝑑𝑧(2.59)_𝑚_∈ℕ02^{−𝑚𝑁𝑟}2^−𝑚𝑁sup_{𝑦∈ℝ𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑦)||1+2^ℓ||𝑥−𝑦||^𝑁^1−𝑟×_ℝ𝑑2^{(𝑚+ℓ)𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||^𝑟1+2^ℓ|𝑥−𝑧|^𝑁𝑟𝑑𝑧.(2.60) Observe that we can estimate the term (⋯)^1−𝑟 in the right-hand side of (2.60) by 𝑀_ℓ,𝑁(𝑥,𝑡)^1−𝑟. Hence, if 𝑀_ℓ,𝑁(𝑥,𝑡)<∞ we obtain from (2.60) 𝑀_ℓ,𝑁(𝑥,𝑡)^𝑟≤𝐶_𝑁𝑚_∈ℕ02^{−𝑚𝑁𝑟}_ℝ𝑑2^{(𝑚+ℓ)𝑑}||Φ_𝑚+ℓ_𝑡∗𝑓(𝑧)||^𝑟1+2^ℓ|𝑥−𝑧|^𝑁𝑟𝑑𝑧,(2.61) where 𝐶_𝑁 is independent of 𝑥, 𝑓, ℓ, and 𝑡∈[1,2]. We claim that there exists 𝑁^𝑓∈ℕ₀ such that 𝑀_ℓ,𝑁(𝑥,𝑡)<∞ for all 𝑁≥𝑁^𝑓. Indeed, we use that 𝑓∈𝒮^(ℝ^𝑑), that is, there is an 𝑀∈ℕ₀ and 𝑐_𝑓>0 such that ||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)||≤𝑐_𝑓sup_{|𝛼|1≤𝑀}sup_{𝑧∈ℝ𝑑}||𝐷^𝛼Φ_𝑘+ℓ(𝑧)||⋅1+||𝑦−𝑧||^𝑀.(2.62) Assuming 𝑁>𝑀, we estimate as follows: ||Φ_ℓ_𝑡∗𝑓(𝑥)||≤≤𝑀𝑐sup_{ℓ,𝑁𝑘∈ℕ0}(𝑥,𝑡)sup_{𝑦∈ℝ𝑑}2^−𝑘𝑁||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)||1+||𝑥−𝑦||²^𝑁/2≤𝑐sup_𝑘∈ℕ0sup_{𝑦∈ℝ𝑑}2^−𝑘𝑁2^{(𝑘+ℓ)(𝑀+𝑑)}sup_{𝑧∈ℝ𝑑}sup_{|𝛼|1≤𝑀}||𝐷^𝛼𝛾_𝑘+ℓ(𝑧)||⋅1+||𝑦−𝑧||^𝑀1+||𝑥−𝑦||^𝑁≤𝑐2^{ℓ(𝑀+𝑑)}sup_𝑘∈ℕ0sup_{𝑧∈ℝ𝑑}sup_{|𝛼|1≤𝑀}||𝐷^𝛼𝛾_𝑘+ℓ(𝑧)||(1+|𝑥−𝑧|)^𝑁,(2.63) where we again used the inequality (compare with (2.57)) 1+||𝑦−𝑧||≤1+||𝑥−𝑦||(1+|𝑥−𝑧|)(2.64) and have set 𝛾_ℓ(𝑡)=Φ⁰Φ(𝑡()𝑡)∶∶ℓℓ>=00.,(2.65) Hence 𝛾_𝑘+ℓ gives us only two different functions from 𝒮(ℝ^𝑑). This implies the boundedness of 𝑀_ℓ,𝑁(𝑥,𝑡) for 𝑥∈ℝ^𝑑 if 𝑁>𝑀=𝑁^𝑓. Therefore, (2.61) together with (2.63) yield (2.48) with 𝑐=𝐶_𝑁, independent of 𝑥, 𝑓, and ℓ, for all 𝑁≥𝑁^𝑓. 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 𝑁∈ℕ₀ but with 𝑐=𝑐(𝑓)=𝐶_𝑁^𝑓  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 𝑁∈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_ℓ,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_ℓ,𝑁(𝑥,𝑡)<∞, we end up with (2.61) for all 𝑁 and 𝐶_𝑁 independent of 𝑓. Finally, from (2.61) we obtain (2.48) and are done in case 0<𝑟≤1.
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 𝑎≤𝑁 and ℓ∈ℕΦ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)^𝑟≤𝑐_𝑘∈ℕ02^{−𝑘𝑁𝑟}2^{(𝑘+ℓ)𝑑}_ℝ𝑑||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)||^𝑟1+2^ℓ||𝑥−𝑦||^𝑎𝑟𝑑𝑦.(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/𝑟𝑑𝑡𝑡^𝑟/𝑞,(2.67) on both sides and use Minkowski’s inequality for integrals, which yields ²¹|||Φ_{∗2−ℓ𝑡}𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^𝑟/𝑞≤𝑐_𝑘∈ℕ02^{−𝑘𝑁𝑟}2^{(𝑘+ℓ)𝑑}_ℝ𝑑∫²¹|Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)|^𝑞(𝑑𝑡/𝑡)^𝑟/𝑞1+2^ℓ||𝑥−𝑦||^𝑎𝑟𝑑𝑦.(2.68) If 𝑎𝑟>𝑑,then we have 𝑔_ℓ(𝑦)=2^𝑑ℓ1+2^ℓ||𝑦||^𝑎𝑟∈𝐿₁ℝ^𝑑,(2.69) and we observe ²¹|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^𝑟/𝑞≤𝑐_𝑘∈ℕ02^{−𝑘𝑁𝑟}2^𝑘𝑑2^ℓ𝑠𝑟𝑔^ℓ∗²¹|2^ℓ𝑠Φ_𝑘+ℓ_𝑡∗𝑓(⋅)|^𝑞𝑑𝑡𝑡^𝑟/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|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^𝑟/𝑞≤𝑐_𝑘∈ℕ02^ℓ𝑟𝑠2^{𝑘(−𝑁𝑟+𝑑)}𝑀²¹||Φ_𝑘+ℓ_𝑡∗𝑓(⋅)||^𝑞𝑑𝑡𝑡^𝑟/𝑞(𝑥).(2.71) An index shift on the right-hand side gives ²¹|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^𝑟/𝑞=𝑐_{𝑘∈ℓ+ℕ0}2^ℓ𝑟𝑠2^{(𝑘−ℓ)(−𝑁𝑟+𝑑)}𝑀²¹||Φ_𝑘_𝑡∗𝑓(⋅)||^𝑞𝑑𝑡=𝑐_{𝑘∈ℓ+ℕ0}𝑡^𝑟/𝑞(𝑥)2^{(ℓ−𝑘)(𝑁𝑟−𝑑+𝑟𝑠)}2^𝑘𝑟𝑠𝑀²¹||Φ_𝑘_𝑡∗𝑓(⋅)||^𝑞𝑑𝑡𝑡^𝑟/𝑞(𝑥).(2.72) Choose now 1/𝑎<𝑟<min{𝑝,𝑞}, 𝑁>max{0,−𝑠}+𝑎 and put 𝛿=𝑁+𝑠−𝑑𝑟>0.(2.73) We obtain for ℓ∈ℕ²¹|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^𝑟/𝑞≤𝑐_𝑘∈ℕ2^{−𝛿𝑟|ℓ−𝑘|}2^𝑘𝑟𝑠𝑀²¹||Φ_𝑘_𝑡∗𝑓(⋅)||^𝑞𝑑𝑡𝑡^𝑟/𝑞(𝑥).(2.74) Now we apply Lemma 2.13 in 𝐿_𝑝/𝑟(ℓ_𝑞/𝑟,ℝ^𝑑), which yields ‖‖‖‖²¹|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^𝑟/𝑞≤𝑐‖‖‖‖𝑀₂₁∣𝐿_𝑝/𝑟ℓ_𝑞/𝑟‖‖‖‖||2^𝑘𝑠Φ_𝑘_𝑡∗𝑓(⋅)||^𝑞𝑑𝑡𝑡^𝑟/𝑞∣𝐿^𝑝/𝑟ℓ_𝑞/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|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝ℓ_𝑞‖‖‖‖^𝑟=‖‖‖‖𝑀²¹|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡≲‖‖‖‖₂₁𝑡^𝑟/𝑞∣𝐿^𝑝/𝑟ℓ_𝑞/𝑟‖‖‖‖|2^𝑘𝑠Φ_𝑘_𝑡∗𝑓(⋅)|^𝑞𝑑𝑡𝑡^𝑟/𝑞=‖‖‖‖₂₁∣ℓ_𝑞/𝑟ℓ_𝑞/𝑟‖‖‖‖||2^𝑘𝑠Φ_𝑘_𝑡∗𝑓(⋅)||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝ℓ_𝑞‖‖‖‖^𝑟.(2.76) Hence, we obtain ‖‖‖‖¹⁰||𝜆^−𝑠𝑞Φ^∗𝜆𝑓_𝑎(𝑥)||^𝑞𝑑𝜆𝜆^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖≍‖‖‖‖^∞_ℓ=1²¹|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝=‖‖‖‖₂₁‖‖‖‖|||2_ℓ𝑠Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^1/𝑞≲‖‖‖‖₁₀∣𝐿_𝑝ℓ_𝑞‖‖‖‖||𝜆^−𝑠𝑞Φ_𝜆∗𝑓(𝑥)||^𝑞𝑑𝜆𝜆^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖.(2.77) This proves ‖𝑓∣𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑)‖₂≲‖𝑓∣𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑)‖₁. Since the reverse inequality is trivial, this finishes Step 1.Step 2. Let Ψ₀,Ψ∈𝒮′(ℝ^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 ‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖^Ψ4≲‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖^Φ2(2.78) for all 𝑓∈𝒮′(ℝ^𝑑).We decompose 𝑓 similar as in Step 1. Exploiting the property (2.17) for the system Φ, we find 𝒮(ℝ^𝑑)-functions 𝜆₀,𝜆∈𝒮(ℝ^𝑑)such that supp𝜆₀⊂{𝜉∈ℝ^𝑑∶|𝜉|≤2𝜀} and supp𝜆⊂{𝜉∈ℝ^𝑑∶𝜀/2≤|𝜉|≤2𝜀} and _ℓ_∈ℕ0𝜆_ℓ(𝑡𝑥)⋅𝜑_ℓ(𝑡𝑥)=1(2.79) for 𝑥∈ℝ^𝑑 and 𝑡∈[1,2] fix. Putting Λ₀=ℱ𝜆₀ and Λ=ℱΛ, we obtain the decomposition 𝑔=_ℓ∈ℕ0Λ_ℓ_𝑡∗Φ_ℓ_𝑡∗𝑔(2.80) for every 𝑔∈𝒮′(ℝ^𝑑). We put 𝑔=Ψ_ℓ∗𝑓 and see Ψ_ℓ∗𝑓=_𝑘∈ℕ0Ψ_ℓ∗Λ_𝑘_𝑡∗Φ_𝑘_𝑡∗𝑓.(2.81) Now, we estimate as follows: ||Ψ_ℓ≤∗Λ_𝑘_𝑡∗Φ_𝑘_𝑡∗𝑓(𝑦)_ℝ𝑑||≤Φ||Ψ_ℓ∗Λ_𝑘_𝑡(𝑧)||⋅||Φ_𝑘_𝑡∗𝑓(𝑦−𝑧)||𝑑𝑧_{∗2−𝑘𝑡}≤Φ𝑓_𝑎(𝑦)_ℝ𝑑||Φ_ℓ∗Λ_𝑘_𝑡(𝑧)||⋅1+2^𝑘|𝑧|^𝑎𝑑𝑧_{∗2−𝑘𝑡}𝑓_𝑎(𝑦)𝐽_ℓ,𝑘,(2.82) where 𝐽_ℓ,𝑘=_ℝ𝑑||Ψ_ℓ∗Λ_𝑘_𝑡(𝑧)||1+2^𝑘|𝑧|^𝑎𝑑𝑧.(2.83) We first observe that for 𝑥∈ℝ^𝑑 and functions 𝜇,𝜂∈𝒮(ℝ^𝑑), the following identity holds true for 𝑢,𝑣>0𝜇_𝑢∗𝜂_𝑣(𝑧)=1𝑢^𝑑𝜇∗𝜂_𝑣/𝑢𝑧𝑢=1𝑣^𝑑𝜇_𝑢/𝑣∗𝜂𝑣𝑧.(2.84) This yields in case ℓ≥𝑘 (with a minor change if 𝑘=0 or ℓ=0) 𝐽_ℓ,𝑘=_ℝ𝑑||Ψ_ℓ−𝑘_1/𝑡∗Λ(𝑧)||(1+|𝑡𝑧|)^𝑎≲sup_{𝑧∈ℝ𝑑}||Ψ_ℓ−𝑘_1/𝑡∗Λ(𝑧)||(1+|𝑧|)^𝑎+𝑑+1≲2^{(𝑘−ℓ)(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_ℓ,𝑘=≲2_ℝ𝑑||Ψ_{(𝑘−ℓ)𝑎}∗Λ_𝑘−ℓ(𝑧)||1+||2^𝑘−ℓ𝑧||^𝑎𝑑𝑥sup_{𝑧∈ℝ𝑑}||Ψ∗Λ_𝑘−ℓ_𝑡(𝑧)||(1+|𝑧|)^𝑎+𝑑+1≲2^{(ℓ−𝑘)(𝐿+1−𝑎)},(2.86) where 𝐿 can be chosen arbitrary large since Λ satisfies (𝑀_𝐿) for every 𝐿∈ℕaccording to its construction. Let us further use the estimate Ψ^∗𝑘𝑓_𝑎(𝑦)≤Ψ^∗𝑘𝑓_𝑎(𝑥)1+2^𝑘||𝑥−𝑦||^𝑎≲Ψ^∗𝑘𝑓_𝑎(𝑥)1+2^ℓ||𝑥−𝑦||^𝑎max1,2^{(𝑘−ℓ)𝑎}.(2.87)
Consequently, sup_{𝑦∈ℝ𝑑}2^𝑠ℓ||Ψ_ℓ∗Λ_𝑘_𝑡∗Φ_𝑘_𝑡∗𝑓(𝑦)||1+2^ℓ||𝑥−𝑦||^𝑎≲2^𝑘𝑠Φ^∗2−𝑘_𝑡𝑓_𝑎(𝑥)2^{(ℓ−𝑘)𝑠}max1,2^{(𝑘−ℓ)𝑎}𝐽_ℓ,𝑘≤2^𝑘𝑠Φ^∗2−𝑘_𝑡𝑓_𝑎(𝑥)2^{(ℓ−𝑘)(𝐿+1−𝑎+𝑠)}2∶𝑘>𝑙,_{(𝑘−ℓ)(𝑅+1−𝑠)}∶ℓ≥𝑘.(2.88) Plugging this into (2.81) and choosing 𝐿≥𝑎+|𝑠| and 𝛿=min{1,𝑅+1−𝑠}, we obtain the inequality 2^ℓ𝑠Ψ^∗ℓ𝑓_𝑎(𝑥)≲_𝑘∈ℕ02^{−|𝑘−ℓ|𝛿}2^𝑘𝑠Φ^∗2−𝑘_𝑡𝑓_𝑎(𝑥)(2.89) for all 𝑥∈ℝ^𝑑 and all 𝑡∈[1,2]. Suppose first that 𝑞≥1. Then we take on both sides (∫²¹|⋅|^𝑞𝑑𝑡/𝑡)^1/𝑞, which gives 2^ℓ𝑠Ψ^∗ℓ𝑓_𝑎(𝑥)≲⎛⎜⎜⎝_𝑘∈ℕ02^{−|𝑘−ℓ|𝛿}2^𝑘𝑠²¹|||Φ_{∗2−𝑘𝑡}𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡⎞⎟⎟⎠^1/𝑞.(2.90) Applying Lemma 2.13 yields ‖‖2^ℓ𝑠Ψ^∗ℓ𝑓_𝑎(𝑥)𝐿_𝑝ℓ_𝑞,ℝ^𝑑‖‖≲‖‖‖‖^∞_𝑘=12^𝑘𝑠𝑞²¹|||Φ_{∗2−𝑘𝑡}𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(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 2^ℓ𝑠Ψ^∗ℓ𝑓_𝑎(𝑥)^𝑞≲_𝑘∈ℕ02^{−|𝑘−ℓ|𝛿𝑞}2^𝑘𝑠𝑞²¹|||Φ_{∗2−𝑘𝑡}𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡.(2.92) Notice that the right-hand side is nothing more than a convolution (𝛾∗𝛽)_ℓ of the sequences 𝛾_𝑘=2^{−|𝑘|𝛿𝑞},𝛽_𝑘=2^𝑘𝑠𝑞²¹|||Φ_{∗2−𝑘𝑡}𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡.(2.93) Now we apply the ℓ₁-norm to both sides and get for all 𝑥∈ℝ^𝑑‖‖2^ℓ𝑠Ψ^∗ℓ𝑓_𝑎(𝑥)∣ℓ_𝑞‖‖^𝑞≤≲‖‖_∞𝛾∣ℓ₁‖‖⋅‖‖𝛽∣ℓ₁‖‖_𝑘=12^𝑘𝑠𝑞²¹|||Φ_{∗2−𝑘𝑡}𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡.(2.94) We take both sides to the power (⋯)^1/𝑞 and apply the 𝐿_𝑝(ℝ^𝑑)-norm. This gives (2.78).Substep 2.2. With similar arguments and obvious modifications of Substep 2.1, we obtain for all 𝑓∈𝒮^(ℝ^𝑑)‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖^Ψ2≲‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖^Φ4.(2.95)Step 3. Choosing 𝑡=1 in Step 1 and omitting the integration over 𝑡, we see immediately ‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖₅≲‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖₄≲‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖₅.(2.96)Step 4. What remains is to show that (2.22) is equivalent to the rest.Substep 4.1. Let us prove ‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖₂≲‖‖𝑓∣𝐹^{𝑠𝑝,𝑞}ℝ^𝑑‖‖₃.(2.97) We return to (2.66) in Substep 1.3. If |𝑧|<2^{−(ℓ+𝑘)}𝑡, formula (2.66) implies by shift in the integral the following: Φ^∗2−ℓ_𝑡𝑓_𝑎(𝑥)^𝑟≤𝐶_𝑁_𝑘∈ℕ02^{−𝑘𝑁𝑟}2^{(𝑘+ℓ)𝑑}_ℝ𝑑||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦+𝑧)||^𝑟1+2^ℓ||𝑥−𝑦||^𝑎𝑟𝑑𝑦.(2.98) Indeed, we have 1+2^ℓ||𝑥−𝑦||≤≲11++22^ℓ_ℓ||𝑥−(𝑦+𝑧)||+|𝑧|||𝑥−(𝑦+𝑧)||+2^−𝑘≲1+2^ℓ||𝑥−(𝑦+𝑧)||,(2.99) where the last estimate follows from the fact that 𝑘∈ℕ₀ in the sum. Instead of the integral (∫²¹|⋅|^𝑞/𝑟𝑑𝑡/𝑡)^𝑟/𝑞 (see Step 3), we take now on both sides of (2.98) the norm ²¹_|𝑧|<𝑡|⋅|^𝑞/𝑟𝑑𝑧𝑑𝑡𝑡^𝑑+1^𝑟/𝑞.(2.100) The integration over 𝑧 does not influence the left-hand side. Instead of (2.68), we obtain ²¹|||Φ_{∗2−ℓ𝑡}𝑓_𝑎(𝑥)|||^𝑞𝑑𝑡𝑡^𝑟/𝑞≤𝑐_𝑘∈ℕ02^{−𝑘𝑁𝑠}2^{(𝑘+ℓ)𝑑}_ℝ𝑑∫²¹∫_|𝑧|<𝑡||Φ_𝑘+ℓ_𝑡∗𝑓(𝑦)||^𝑞𝑑𝑧𝑑𝑡/𝑡^𝑑+1^𝑟/𝑞1+2^ℓ||𝑥−𝑦||^𝑎𝑟𝑑𝑦.(2.101) We continue with analogous arguments as after (2.68) and end up with (2.97).Substep 4.2. We prove ‖𝑓∣𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑)‖₃≲‖𝑓∣𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑)‖₂. Indeed, it is easy to see, that we have for all 𝑡>01𝑡^𝑑_|𝑧|<𝑡||Φ_𝑡∗𝑓(𝑥+𝑧)||𝑑𝑧≲sup_|𝑧|<𝑡||Φ_𝑡∗𝑓(𝑥+𝑧)||(1+1/𝑡|𝑧|)^𝑎≲Φ^∗𝑡𝑓_𝑎(𝑥),(2.102) and we are done. The proof is complete.

Proof of Theorem 2.8. The proof of Theorem 2.8 works almost analogously to the previous one. It is even a bit simpler, since we do not have to deal with a separate function Φ₀, which causes some technical difficulties. However, there are still some technical obstacles which have to be discussed. Although we are in the homogeneous world, we use the same decomposition as used in (2.43), even with the inhomogeneity Φ₀. In the definition of Λ_𝑚,ℓ(𝑥) in (2.45), we have to add Φ(𝑥) if ℓ=0 and 𝑚>0. The consequence is (2.47) for every ℓ∈ℤ, where Φ₀=Φ. Hence, the inhomogeneity is buried in Λ_𝑚,ℓ. This yields (2.66) for all ℓ∈ℤ, where 𝑘 still runs trough ℕ₀. We need this for the argument in Substep 4.1. In contrast to the previous decomposition, we use (2.80) now for ℓ∈ℤ, where Φ₀=Φ, Λ₀=Λ, and 𝑘∈ℤ. This works since we assume 𝑔∈𝒮^0(ℝ^𝑑). The rest is clear.

Proof of Corollaries 2.11 and 2.12. Corollary 2.11 is more or less clear. We know that Δ^𝑁 gives (∑^𝑑𝑘=1|𝜉_𝑘|²)^𝑁 as factor on the Fourier side. This gives (2.18) immediately, and together with (2.31) we have (2.17) for 𝜀>0 small enough.
In the case of Corollary 2.12, the situation is a bit delicate. Clearly, Condition (2.18) holds true. But the problem here is, that (2.17) may be violated for all 𝜀>0. However, we argue as follows. In Step 2 in the proof above we have seen, that we do not need (2.17) for the system (Ψ₀,Ψ). Hence, we can estimate (2.20), (2.21), (2.22), (2.23), and (2.24) from above by a different characterization of 𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑). For the remaining estimates we apply Theorem 2.6 with the system (Φ₀,Φ), where Φ=Φ₀(𝑥)−12^𝑘𝑑Φ₀2𝑥^𝑘(2.103) and 𝑘∈ℕis chosen in such a way that (2.17) is satisfied. What remains is a simple consequence of the fact that Φ=Φ+Φ₋₁+⋯+Φ_{−(𝑘−1)}(2.104) and the triangle inequality. This type of argument is due to Triebel [3, Section 3.3.3].

3. Classical Coorbit Space Theory

In [11–13, 15], a general theory of Banach spaces related to integrable group representations was developed. The ingredients are a locally compact group 𝒢 with identity 𝑒, a Hilbert space ℋ, and an irreducible, unitary, and continuous representation 𝜋 on ℋ, which is at least integrable. Then one can associate a Banach space Co𝑌 to any solid, translation-invariant Banach space 𝑌 of functions on the group 𝒢. The main achievement of this abstract theory is a powerful discretization machinery for Co𝑌, that is, a universal approach to atomic decompositions and Banach frames. It allows to transfer certain questions concerning Banach space or interpolation theory from the function space to the associated sequence space level, see [12, 13, 26]. In connection with smoothness spaces of Besov-Lizorkin-Triebel type, the philosophy of this approach is to measure smoothness of a function in decay properties of the continuous wavelet transform 𝑊_𝑔𝑓, see the appendix for details. Indeed, homogeneous Besov and Lizorkin-Triebel-type spaces turn out to be coorbits of properly chosen spaces 𝑌 on the 𝑎𝑥+𝑏-group 𝐺.

The are many more examples according to this abstract theory. One main class of examples refers to the Heisenberg group ℍ and the short-time Fourier transform and leads to the well-known modulation spaces as coorbits of weighted 𝐿_𝑝(ℍ) spaces, see [11, Section 7.1] and also [27].

3.1. Function Spaces on 𝒢

Integration on 𝒢 will always be with respect to the left Haar measure 𝑑𝜇(𝑥). The Haar module on 𝒢 is denoted by Δ. We define further 𝐿_𝑥𝐹(𝑦)=𝐹(𝑥⁻¹𝑦) and 𝑅_𝑥𝐹(𝑦)=𝐹(𝑦𝑥), 𝑥,𝑦∈𝒢, the left and right translation operators. A Banach function space 𝑌 on the group 𝒢 is supposed to have the following properties:(i)𝑌 is continuously embedded in 𝐿^1loc(𝒢), (ii)𝑌 is invariant under left and right translation 𝐿_𝑦 and 𝑅_𝑦, which represent continuous operators on 𝑌,(iii) 𝑌 is solid, that is, 𝐻∈𝑌 and |𝐹(𝑥)|≤|𝐻(𝑥)| a.e. imply 𝐹∈𝑌 and ‖𝐹∣𝑌‖≤‖𝐻|𝑌‖.

The continuous weight 𝑤 is called submultiplicative if 𝑤(𝑥𝑦)≤𝑤(𝑥)𝑤(𝑦) for all 𝑥,𝑦∈𝒢. Further, another weight 𝑚 is called 𝑤-moderate if 𝑚(𝑥𝑦𝑧)≤𝑤(𝑥)𝑚(𝑦)𝑤(𝑧), 𝑥,𝑦,𝑧∈𝒢. The space 𝐿^𝑤𝑝(𝒢) of functions 𝐹 on the group 𝒢 is defined via the norm‖‖𝐹∣𝐿^𝑤𝑝(𝒢)‖‖=_𝒢||𝐹(𝑥)𝑤(𝑥)||^𝑝𝑑𝜇(𝑥)^1/𝑝,(3.1) where 1≤𝑝≤∞ (modification if 𝑝=∞). If 𝑤≡1, then we simply write 𝐿_𝑝(𝒢). These spaces provide left and right translation invariance, which is easy to show. Later, in Section 4.1, we are going to introduce certain mixed norm spaces where the translation invariance is not longer automatic.

3.2. Sequence Spaces

Definition 3.1. Let 𝑋={𝑥_𝑖}_𝑖∈𝐼 be some discrete set of points in 𝒢 and 𝑉 a relatively compact neighborhood of 𝑒∈𝒢. (i) 𝑋 is called 𝑉-dense if 𝒢=⋃_𝑖∈𝐼𝑥_𝑖𝑉.(ii) 𝑋 is called relatively separated if for all compact sets 𝐾⊂𝒢,there exists a constant 𝐶_𝐾 such that sup_𝑗∈𝐼♯𝑖∈𝐼∶𝑥_𝑖𝐾∩𝑥_𝑗𝐾≠∅≤𝐶_𝐾.(3.2) (iii) 𝑋 is called 𝑉-well-spread (or simply well-spread) if it is both relatively separated and 𝑉-dense for some 𝑉.

Definition 3.2. For a family 𝑋={𝑥_𝑖}_𝑖∈𝐼 which is V-well-spread with respect to a relatively compact neighborhood 𝑉 of 𝑒∈𝒢 we define the sequence space 𝑌^𝑏 and 𝑌^♯ associated to 𝑌 as 𝑌𝑌^𝑏=𝜆_𝑖_𝑖∈𝐼∶‖‖𝜆_𝑖_𝑖∈𝐼∣𝑌^𝑏‖‖=‖‖‖‖_𝑖∈𝐼||𝜆_𝑖||||𝑥_𝑖𝑉||⁻¹𝜒_𝑥𝑖𝑉∣𝑌‖‖‖‖<∞_♯,=𝜆_𝑖_𝑖∈𝐼∶‖‖𝜆_𝑖_𝑖∈𝐼∣𝑌^♯‖‖=‖‖‖‖_𝑖∈𝐼||𝜆_𝑖||𝜒_𝑥𝑖𝑉∣𝑌‖‖‖‖<∞.(3.3)

Remark 3.3. For a well-spread family 𝑋, the spaces 𝑌^𝑏 and 𝑌^♯ do not depend on the choice of 𝑉, that is, different sets 𝑉 define equivalent norms on 𝑌^𝑏 and 𝑌^♯, respectively. For more facts on these sequence spaces, confer [12].

3.3. Coorbit Spaces

The starting point is a Hilbert space ℋ and an integrable, irreducible, unitary and continuous representation 𝜋 of 𝒢 on ℋ. Then the general voice transform 𝑉_𝑔𝑓 is a function on the group 𝒢 given by𝑉_𝑔𝑓(𝑥)=⟨𝜋(𝑥)𝑔,𝑓⟩,(3.4) where the brackets denote the inner product in ℋ.

The continuous wavelet transform 𝑊_𝑔𝑓 (appendix) is a voice transform for the situation ℋ=𝐿₂(ℝ^𝑑) and the 𝑎𝑥+𝑏-group.

Definition 3.4. For a weight 𝑤(⋅)≥1 on 𝒢, we define the space 𝐴_𝑤⊂ℋ of admissible vectors by 𝐴_𝑤=𝑔∈ℋ∶𝑉_𝑔𝑔∈𝐿^𝑤1(𝒢).(3.5) If 𝐴_𝑤≠{0} and 𝑔∈𝐴_𝑤, we define further ℋ^1𝑤ℝ^𝑑=𝑓∈ℋ∶‖‖𝑓∣ℋ^1𝑤‖‖=‖‖𝑉_𝑔𝑓∣𝐿^𝑤1(𝒢)‖‖<∞.(3.6) Finally, we denote by (ℋ^1𝑤)^∼ the canonical antidual of ℋ^1𝑤, that is, the space of conjugate linear functionals on ℋ^1𝑤.

We see immediately that 𝐴_𝑤⊂ℋ^1𝑤⊂ℋ. The voice transform (3.4) can now be extended to ℋ_𝑤×(ℋ^1𝑤)^∼ by the usual dual pairing. The space ℋ^1𝑤 can be considered as the space of test functions, whereas the space (ℋ^1𝑤)^∼ serves as reservoir, that is, distributions.

Let now 𝑌 be a space on 𝒢 such that (i)–(iii) in Section 3.1 hold true. We define further𝑤_𝑌(𝑥)=max‖‖𝐿_𝑥‖‖,‖‖𝐿_𝑥−1‖‖,‖‖𝑅_𝑥‖‖,Δ𝑥⁻¹‖‖𝑅_𝑥⁻¹‖‖,𝑥∈𝒢,(3.7) where the operator norms are considered from 𝑌 to 𝑌.

Definition 3.5. Let 𝑌 be a space on 𝒢 satisfying (i)–(iii) and 𝑤_𝑌(𝑥) be given by (3.7). Let further 𝑔∈𝐴_𝑤. We define the space Co𝑌, called coorbit space of Y, through C𝑜𝑌=𝑓∈ℋ^1𝑤^∼∶𝑉_𝑔𝑓∈𝑌with‖𝑓∣Co𝑌‖=‖‖𝑉_𝑔𝑓∣𝑌‖‖.(3.8)
The following result is of crucial importance. See [14] for details.

Lemma 3.6 (Correspondence principle). Let 𝐺=𝑉_𝑔𝑔∈𝐿^𝑤1(𝒢) for a fixed analyzing vector 𝑔∈𝐴_𝑤 with proper normalization. Then a function 𝐹∈𝑌 is of the form 𝑉_𝑔𝑓 for some 𝑓∈C𝑜𝑌 if and only if 𝐹(𝑥)=(𝐹∗𝐺)(𝑥)∶=_𝒢𝐹(𝑦)𝐿_𝑦𝐺(𝑥)𝑑𝜇(𝑦),𝑥∈𝒢.(3.9) In other words, Co𝑌 is isometrically isomorphic to the closed subspace 𝑌∗𝐺 of 𝑌.

The following basic properties are proved for instance, in [28, Theorem 4.5.13].

Theorem 3.7. (i) The space C𝑜𝑌 is independent of the analyzing vector 𝑔∈𝐴_𝑤.
(ii) The definition of the space Co𝑌 is independent of the reservoir in the following sense. Assume that 𝑆⊂ℋ^1𝑤 is a nontrivial locally convex vector space, which is invariant under 𝜋. Assume further that there exists a nonzero vector 𝑔∈𝑆∩𝐴_𝑤 for which the reproducing formula 𝑉_𝑔𝑓=𝑉_𝑔𝑔∗𝑉_𝑔𝑓(3.10) holds for all 𝑓∈𝑆^∼. Then we have Co𝑌=𝑓∈ℋ^1𝑤^∼∶𝑉_𝑔𝑓∈𝑌=𝑓∈𝑆^∼∶𝑉_𝑔𝑓∈𝑌.(3.11)

3.4. Discretizations

This section collects briefly the basic facts concerning atomic (frame) decompositions in coorbit spaces. We are interested in atoms of type {𝜋(𝑥_𝑖)𝑔}_𝑖∈𝐼, where {𝑥_𝑖}_𝑖∈𝐼⊂𝒢 represents a discrete subset, whereas 𝑔 denotes a fixed admissible analyzing vector.

Definition 3.8. A family {𝑔_𝑖}_𝑖∈𝐼 in a Banach space 𝐵 is called an atomic decomposition for B if there exists a family of bounded linear functionals {𝜆_𝑖}_𝑖∈𝐼⊂𝐵^′ (not necessary unique) and a Banach sequence space 𝐵^♯=𝐵^♯(𝐼) such that (a){𝜆_𝑖(𝑓)}_𝑖∈𝐼∈𝐵^♯ for all 𝑓∈𝐵 and there exists a constant 𝐶₁>0 with‖‖𝜆_𝑖(𝑓)_𝑖∈𝐼∣𝐵^♯‖‖≤𝐶₁‖𝑓∣𝐵‖.(3.12)(b)For all 𝑓∈𝐵, we have 𝑓=_𝑖∈𝐼𝜆_𝑖(𝑓)𝑔_𝑖(3.13) in some suitable topology. (c) If {𝜆_𝑖}_𝑖∈𝐼∈𝐵^♯, then ∑_𝑖∈𝐼𝜆_𝑖𝑔_𝑖∈𝐵 and there exists a constant 𝐶₂>0 such that‖‖‖‖_𝑖∈𝐼𝜆_𝑖𝑔_𝑖∣𝐵‖‖‖‖≤𝐶₂‖‖𝜆_𝑖_𝑖∈𝐼∣𝐵^♯‖‖.(3.14)

Definition 3.9. A family {ℎ_𝑖}_𝑖∈𝐼⊂𝐵^ is called a Banach frame for 𝐵 if there exists a Banach sequence space 𝐵^𝑏=𝐵^𝑏(𝐼) and a linear bounded reconstruction operator Θ∶𝐵^𝑏→𝐵 such that(a){ℎ_𝑖(𝑓)}_𝑖∈𝐼∈𝐵^𝑏 for all 𝑓∈𝐵 and there exist constants 𝐶₁,𝐶₂ such that𝐶₁‖𝑓∣𝐵‖≤‖‖ℎ_𝑖(𝑓)_𝑖∈𝐼∣𝐵^𝑏‖‖≤𝐶₂‖𝑓∣𝐵‖,(3.15)(b)Θ({ℎ_𝑖(𝑓)}_𝑖∈𝐼)=𝑓.

Remark 3.10. This setting differs slightly from the understanding of Triebel in [3, 29].

The following abstract result for the atomic decomposition in Co𝑌 is due to Feichtinger and Gröchenig (see [12, Theorem 6.1]).

Theorem 3.11. Let 𝑌 be a function space on the group 𝒢 which satisfies the hypotheses (i)–(iii). Let further 𝑤_𝑌(𝑥)∶=max‖‖𝐿_𝑥‖‖,‖‖𝐿_𝑥−1‖‖,‖‖𝑅_𝑥‖‖,Δ𝑥⁻¹‖‖𝑅_𝑥⁻¹‖‖,(3.16) where the operator norms are taken from 𝑌 to𝑌, and 𝑔∈𝐴_𝑤 such that _𝒢sup_{𝑦∈𝑥𝑉}||⟨𝜋(𝑦)𝑔,𝑔⟩||𝑤(𝑥,𝑡)𝑑𝜇(𝑥)<∞.(3.17) Then there exists a neighborhood 𝑈 of 𝑒∈𝒢 and constants 𝐶₀,𝐶₁>1 such that for every 𝑈-well-spread discrete set 𝑋={𝑥_𝑖}_𝑖∈𝐼⊂𝒢, the following is true.(i) (Analysis) Every 𝑓∈Co𝑌 has a representation 𝑓=_𝑖∈𝐼𝜆_𝑖𝜋𝑥_𝑖𝑔(3.18) with coefficients {𝜆_𝑖}_𝑖∈𝐼 depending linearly on 𝑓 and satisfying the estimate ‖‖𝜆_𝑖_𝑖∈𝐼∣𝑌^♯‖‖≤𝐶₀‖𝑓∣𝑌‖.(3.19)(ii) (Synthesis) Conversely, for any sequence {𝜆_𝑖}_𝑖∈𝐼∈𝑌^♯, the element 𝑓=∑_𝑖∈𝐼𝜆_𝑖𝜋(𝑥_𝑖)𝑔 is in Co𝑌 and one has ‖𝑓∣Co𝑌‖≤𝐶₁‖‖𝜆_𝑖_𝑖∈𝐼∣𝑌^♯‖‖.(3.20) In both cases, convergence takes place in the norm of Co𝑌 if the finite sequences are norm dense in 𝑌^♯, and in the weak^∗-sense of (ℋ^1𝑤)^∼ otherwise.

Remark 3.12. According to Definition 3.8, the family {𝜋(𝑥_𝑖)𝑔}_𝑖∈𝐼 represents an atomic decomposition for Co𝑌.

Theorem 3.13. Under the same conditions as in Theorem 3.11, the system {𝜋(𝑥_𝑖)𝑔}_𝑖∈𝐼 represents a Banach frame for Co𝑌, that is, ‖𝑓∣Co𝑌‖≍‖‖𝜋𝑥_𝑖𝑔,𝑓∣𝑌^𝑏‖‖.(3.21)

The following powerful result goes back to Gröchenig [14] and was generalized by Rauhut [17].

Theorem 3.14. Suppose that the functions 𝑔_𝑟,𝛾_𝑟, 𝑟=1,…,𝑛, satisfy (3.17). Let 𝑋={𝑥_𝑖}_𝑖∈𝐼 be a well-spread set such that 𝑓=^𝑛_𝑟=1_𝑖∈𝐼𝜋𝑥_𝑖𝛾_𝑟,𝑓𝜋𝑥_𝑖𝑔_𝑟(3.22) for all 𝑓∈𝐻. Then expansion (3.22) extends to all 𝑓∈Co𝑌. Moreover, 𝑓∈(ℋ^1𝑤)^∼ belongs to Co𝑌 if and only if {⟨𝜋(𝑥_𝑖)𝛾_𝑟,𝑓⟩}_𝑖∈𝐼 belongs to 𝑌^♯ for each 𝑟=1,…,𝑛. The convergence is considered in Co𝑌 if the finite sequences are dense in 𝑌^♯. In general, we have weak^∗-convergence.

Proof. The proof of this result relies on the fact that there exists an atomic decomposition {𝜋(𝑦_𝑖)𝑔}_𝑖∈𝐼 by Theorem 3.11 with a certain 𝑔 satisfying (3.17) and a corresponding sequence of points 𝑍={𝑦_𝑖}_𝑖∈𝐼. This has to be combined with Theorem 3.13 and Theorem 3.11/(ii) and we are done. See [14] for the details.

4. Coorbit Spaces on the 𝑎𝑥+𝑏-Group

Let 𝒢=ℝ^𝑑⋊ℝ^∗+ the 𝑑-dimensional 𝑎𝑥+𝑏-group. Its multiplication is given by(𝑥,𝑡)(𝑦,𝑠)=(𝑥+𝑡𝑦,𝑠𝑡).(4.1)

The left Haar measure 𝜇 on 𝒢 is given by 𝑑𝑡/𝑡^𝑑+1, the Haar module is Δ(𝑥,𝑡)=𝑡^−𝑑. Giving a function 𝐹 on 𝒢, the left and right translation 𝐿_𝑦=𝐿_(𝑦,𝑟) and 𝑅_𝑦=𝑅_(𝑦,𝑟) are given by𝐿_(𝑦,𝑟)𝐹(𝑥,𝑡)=𝐹(𝑦,𝑟)⁻¹(𝑥,𝑡)=𝐹𝑥−𝑦𝑅_(𝑦,𝑟)𝑟,𝑟𝑡,𝐹(𝑥,𝑡)=𝐹((𝑥,𝑡)(𝑦,𝑟))=𝐹(𝑥+𝑡𝑦,𝑟𝑡).(4.2)

4.1. Peetre-Type Spaces on 𝒢

The present paragraph is devoted to the definition of certain mixed norm spaces on the group. Such spaces were considered in various papers, see [10, 11, 14, 15]. Especially tent spaces have some nice applications in harmonic analysis. In particular, they were used by many authors in order to apply coorbit space theory to Lizorkin-Triebel spaces.

Here we use a different approach and define a new scale of function spaces on the group 𝒢. We call them Peetre-type spaces since the Peetre maximal function is involved in the definition. It turned out that they are very easy to handle in connection with translation invariance. Compared with the tent space approach, they are the more natural choice for considering Lizorkin-Triebel spaces as coorbit spaces. Additionally they seem to be the suitable choice for inhomogeneous spaces and more general situations like weighted spaces, which will be studied in a subsequent contribution to the subject.

Definition 4.1. Let 𝑠∈ℝ,0<𝑝,𝑞≤∞, and 𝑎>0. We define the spaces 𝐿̇^{𝑠𝑝,𝑞}(𝒢), 𝑇̇^{𝑠𝑝,𝑞}(𝒢), and 𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢) on the group 𝒢 via the finiteness of the following (quasi-)norms: ‖‖𝐹∣𝐿̇^{𝑠𝑝,𝑞}(𝒢)‖‖=^∞0𝑡^−𝑠𝑞‖‖𝐹(⋅,𝑡)∣𝐿_𝑝ℝ^𝑑‖‖^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞‖‖𝐹∣𝑇̇_{𝑠𝑝,𝑞},(𝒢)‖‖=‖‖‖‖^∞0𝑡^−𝑠𝑞_𝐵(0,𝑡)||𝐹(𝑥+𝑧,𝑡)||^𝑞𝑑𝑧𝑑𝑡𝑡^𝑑+1^1/𝑞‖‖𝐹∣𝑃̇_{𝑠,𝑎𝑝,𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(𝒢)‖‖=‖‖‖‖^∞0𝑡^−𝑠𝑞sup_𝑦∈ℝ^𝑑||𝐹(𝑥+𝑦,𝑡)||(1+||𝑦||/𝑡)^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖,(4.3) using the usual modification in case 𝑞=∞.
Proposition 4.2. The spaces 𝐿̇^{𝑠𝑝,𝑞}(𝒢),𝑇̇^{𝑠𝑝,𝑞}(𝒢) and 𝑃̇^{𝑠𝑝,𝑞}(𝒢), are left and right translation invariant. Precisely, we have ‖‖‖‖𝑅𝐿_{(𝑧,𝑟)(𝑧,𝑟)}∶𝐿̇^{𝑠𝑝,𝑞}(𝒢)⟶𝐿̇^{𝑠𝑝,𝑞}(𝒢)‖‖=𝑟^{𝑑(1/𝑝−1/𝑞)−𝑠},‖‖𝐿_(𝑧,𝑟)∶𝐿̇^{𝑠𝑝,𝑞}(𝒢)⟶𝐿̇^{𝑠𝑝,𝑞}(𝒢)‖‖=𝑟^{𝑠+𝑑/𝑞},‖‖𝑅_(𝑧,𝑟)∶𝑇̇^{𝑠𝑝,𝑞}(𝒢)⟶𝑇̇^{𝑠𝑝,𝑞}(𝒢)‖‖=𝑟^{𝑑/𝑞−𝑠},∶𝑇̇^{𝑠𝑝,𝑞}(𝒢)⟶𝑇̇^{𝑠𝑝,𝑞}(𝒢)‖‖≤𝐶𝑟^{𝑑/𝑞+𝑠}max1,𝑟^−𝑏(1+|𝑧|)^𝑏,(4.4) where 𝑏>0 is a constant depending on 𝑑, 𝑝, and 𝑞, and ‖‖‖‖𝑅𝐿_{(𝑧,𝑟)(𝑧,𝑟)}∶𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)⟶𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖=𝑟^{𝑑(1/𝑝−1/𝑞)−𝑠},∶𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)⟶𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖≤𝑟^{𝑠+𝑑/𝑞}max{1,𝑟^−𝑎}(1+|𝑧|)^𝑎.(4.5)Proof. Step 1. The left and right translation invariance of 𝐿̇^{𝑠𝑝,𝑞}(𝒢) and 𝑇̇^{𝑠𝑝,𝑞}(𝒢) was shown in [28, Lemma. 4.7.10].Step 2. Let us consider 𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢). Clearly, we have for 𝐹∈𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖𝐿_(𝑧,𝑟)𝐹∣𝑃̇^{𝑠𝑝,𝑞}(𝒢)‖‖=‖‖‖‖^∞0𝑡^−𝑠𝑞sup_𝑦∈ℝ^𝑑||𝐹((𝑥+𝑦−𝑧)/𝑟,𝑡/𝑟)||(1+||𝑦||/𝑡)^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞=𝑟_𝑑/𝑝∣𝐿_𝑝ℝ^𝑑‖‖‖‖‖‖‖‖_∞0𝑡^−𝑠𝑞sup_𝑦∈ℝ^𝑑||𝐹(𝑥+𝑦,𝑡/𝑟)||(1+𝑟||𝑦||/𝑡)^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞=𝑟_{𝑑(1/𝑝−1/𝑞)−𝑠}∣𝐿_𝑝ℝ^𝑑‖‖‖‖‖‖‖‖_∞0𝑡^−𝑠𝑞sup_𝑦∈ℝ^𝑑||𝐹(𝑥+𝑦,𝑡)||(1+||𝑦||/𝑡)^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞∣𝐿_𝑝(ℝ^𝑑₎‖‖‖‖.(4.6) Hence, we obtain ‖‖𝐿_(𝑧,𝑟)∶𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)⟶𝑃̇^{𝑠𝑝,𝑞}(𝒢)‖‖=𝑟^{𝑑(1/𝑝−1/𝑞)−𝑠}.(4.7) The right translation invariance is obtained by ‖‖𝑅_(𝑧,𝑟)𝐹∣𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖=‖‖‖‖^∞0𝑡^−𝑠𝑞sup_𝑦∈ℝ^𝑑||𝐹(𝑥+𝑡𝑧+𝑦,𝑡𝑟)||1+||𝑦||/𝑡^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞=‖‖‖‖_∞0∣𝐿_𝑝ℝ^𝑑‖‖‖‖𝑡^−𝑠𝑞sup_𝑦∈ℝ^𝑑||𝐹(𝑥+𝑦,𝑡𝑟)||(1+|𝑦−𝑡𝑧|/𝑡)^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞=𝑟_{𝑠+𝑑/𝑞}∣𝐿_𝑝ℝ^𝑑‖‖‖‖‖‖‖‖_∞0𝑡^−𝑠𝑞sup_𝑦∈ℝ^𝑑||𝐹(𝑥+𝑦,𝑡)||1+||𝑦−𝑡𝑧||𝑟/𝑡^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖.(4.8) Observe that sup_{𝑦∈ℝ𝑑}||𝐹(𝑥+𝑦,𝑡)||1+||𝑦−𝑡𝑧||𝑟/𝑡^𝑎=sup_{𝑦∈ℝ𝑑}||𝐹1(+𝑥+||𝑦||𝑦/𝑡,𝑡^𝑎)||⋅1+||𝑦||/𝑡^𝑎1+||𝑦||/𝑡_𝑎1+||𝑦−𝑡𝑧||𝑟/𝑡^𝑎,1+||𝑦−𝑡𝑧||𝑟/𝑡^𝑎≤1+||𝑦−𝑡𝑧||/𝑡+|𝑧|^𝑎1+||𝑦−𝑡𝑧||𝑟/𝑡^𝑎=(1+|𝑦−𝑡𝑧|/𝑡)^𝑎)(1+|𝑧|)^𝑎1+||𝑦−𝑡𝑧||𝑟/𝑡^𝑎.(4.9) This yields sup_{𝑦∈ℝ𝑑}||𝐹(𝑥+𝑦,𝑡)||1+||𝑦−𝑡𝑧||𝑟/𝑡^𝑎≤max{1,𝑟^−𝑎}(1+|𝑧|)^𝑎sup_{𝑦∈ℝ𝑑}||𝐹(𝑥+𝑦,𝑡)||1+||𝑦||/𝑡^𝑎(4.10) and consequently ‖‖𝑅_(𝑧,𝑟)∶𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)⟶𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖≤𝑟^{𝑠+𝑑/𝑞}max{1,𝑟^−𝑎}(1+|𝑧|)^𝑎.(4.11)

Remark 4.3. Note that we did neither use the translation invariance of the Lebesgue measure nor any change of variable in order to prove the right translation invariance of 𝑃̇^{𝑠𝑝,𝑞}(𝒢). This gives room for generalizations, that is, replacing the space 𝐿_𝑝(ℝ^𝑑) by some weighted Lebesgue space 𝐿_𝑝(ℝ^𝑑,𝜔) in the definition.

4.2. New Old Coorbit Spaces

We start with ℋ=𝐿₂(ℝ^𝑑) and the representation𝜋(𝑥,𝑡)=𝑇_𝑥𝒟^𝐿2_𝑡,(4.12) which is unitary continuous on ℋ. This representation is not irreducible. However, if we restrict to radial functions 𝑔∈𝐿₂(ℝ^𝑑), then span{𝜋(𝑥,𝑡)𝑔∶(𝑥,𝑡)∈𝒢} is dense in 𝐿₂(ℝ^𝑑). Another possibility to overcome this obstacle is to extend the group by 𝑆𝑂(𝑑), which is somehow equivalent. See [11, 12] for details.

The voice transform in this special situation is represented by the so-called continuous wavelet transform 𝑊_𝑔𝑓, see Appendix A.1 in the appendix. Recall the abstract definition of the space ℋ^1𝑤 and 𝒜_𝑤 from Definition 3.4. The following lemma is proved for instance in [28, Lemma 4.7.11] and is also a consequence of our Lemma A.3 on the decay of the continuous wavelet transform. It states under which conditions on the weight 𝑤 the space ℋ^1𝑤 is nontrivial.

Lemma 4.4. If the weight function 𝑤(𝑥,𝑡)≥1 satisfies the condition 𝑤(𝑥,𝑡)≤(1+|𝑥|)^𝑟𝑡^𝑠+𝑡^−𝑠′(4.13) for some 𝑟,𝑠,𝑠′≥0, then 𝒮₀ℝ^𝑑↪ℋ^1𝑤.(4.14)

This is a kind of minimal condition which we need in order to define coorbit spaces in a reasonable way. Instead of (𝐻^1𝑤)^∼, one may use 𝒮^0(ℝ^𝑑) as reservoir and a radial 𝑔∈𝒮₀(ℝ^𝑑) as analyzing vector. Considering (3.7), we have to restrict to such function spaces 𝑌 on 𝒢 satisfying (i),(ii),(iii) in Section 3.1 where additionally (recall the definition of 𝑤_𝑌 in (3.16))

(iv)𝑤_𝑌(𝑥,𝑡)≲(1+|𝑥|)^𝑟𝑡^𝑠+𝑡^−𝑠′(4.15)

for some 𝑟,𝑠,𝑠′≥0. The following theorem shows how the spaces of Besov-Lizorkin-Triebel type from Section 2 can be recovered as coorbit spaces.

Theorem 4.5. (i) For 1≤𝑝,𝑞≤∞, and 𝑠∈ℝ,we have 𝐵̇^{𝑠𝑝,𝑞}(ℝ^𝑑)=Co𝐿̇^{𝑠+𝑑/2−𝑑/𝑞𝑝,𝑞}(𝒢),
(ii) for 1≤𝑝<∞, 1≤𝑞≤∞ and 𝑠∈ℝwe have 𝐹̇^{𝑠𝑝,𝑞}ℝ^𝑑=Co𝑇̇^{𝑠+𝑑/2𝑝,𝑞}(𝒢)(4.16)
(iii) and if additionally 𝑎>𝑑/min{𝑝,𝑞}, we obtain 𝐹̇^{𝑠𝑝,𝑞}ℝ^𝑑=Co𝑃̇^{𝑠+𝑑/2−𝑑/𝑞,𝑎𝑝,𝑞}(𝒢).(4.17)

Proof. Theorem 4.5 is a direct consequence of Proposition 4.2, Theorems 2.8, 2.9, and the abstract result in Theorem 3.7.

Remark 4.6. (a) The assertions (i) and (ii) are not new. They appear for instance in [11, 14, 15]. They rely on the characterizations given by Triebel in [2] and [3, Sections 2.4, 2.5], see in particular, [3, Section 2.4.5] for the variant in terms of tent spaces, which were invented in [10]. From the deep result in [10, Proposition 4], it follows that 𝑇̇^{𝑠𝑝,𝑞}(𝒢) are translation invariant Banach function spaces on 𝒢, which makes them feasible for coorbit space theory.
(b) Assertion (iii) is indeed new and makes the rather complicated tent spaces 𝑇̇^{𝑠𝑝,𝑞}(𝒢) obsolete for this issue. We showed that 𝑌=𝑃^{𝑠,𝑎𝑝,𝑞}(𝒢) is a much better choice since the right translation invariance is immediate and gives more transparent estimates for its norm. Once we are interested in reasonable conditions for atomic decompositions, this is getting important, see Section 4.5.

4.3. Sequence Spaces

In the following, we consider a compact neighborhood of the identity element in 𝒢 given by 𝒰=[0,𝛼]^𝑑×[𝛽⁻¹,1], where 𝛼>0 and 1<𝛽. Furthermore, we consider the discrete set of points𝑥_𝑗,𝑘=𝛼𝑘𝛽^−𝑗,𝛽^−𝑗∶𝑗∈ℤ,𝑘∈ℤ^𝑑.(4.18) Then the family {𝑥_𝑗,𝑘𝒰}_{𝑗∈ℤ,𝑘∈ℤ}^𝑑 defines a partition of 𝒢. Indeed,𝑥_𝑗,𝑘𝒰=𝑄_𝑗,𝑘×𝛽^−(𝑗+1),𝛽^−𝑗,(4.19)

where𝑄_𝑗,𝑘=𝛼𝑘₁𝛽^−𝑗,𝛼𝑘₁+1𝛽^−𝑗×⋯×𝛼𝑘_𝑑𝛽^−𝑗,𝛼𝑘_𝑑+1𝛽^−𝑗.(4.20)

Note that in this case the spaces 𝑌^♯ and 𝑌^𝑏 coincide. We will further use the notation𝜒_𝑗,𝑘(𝑥)=1:𝑥∈𝑄^𝑗,𝑘0:otherwise.(4.21)

Definition 4.7. Let 𝑌 be a function space on 𝒢 satisfying Section 3/(i), (ii), (iii). We put 𝑌^♯(𝛼,𝛽)=𝜆_𝑗,𝑘_𝑗,𝑘∶‖‖𝜆_𝑗,𝑘_𝑗,𝑘∣𝑌^♯(𝛼,𝛽)‖‖<∞,(4.22) where ‖‖𝜆_𝑗,𝑘_𝑗,𝑘∣𝑌^♯(𝛼,𝛽)‖‖=‖‖‖‖_𝑗,𝑘||𝜆_𝑗,𝑘||𝜒_𝑗,𝑘(𝑥)𝜒_[𝛽^𝑗_,𝛽^𝑗+1_](𝑡)∣𝑌‖‖‖‖.(4.23)

Theorem 4.8. Let 1≤𝑝,𝑞≤∞, 𝑠∈ℝ,and 𝑎>𝑑/min{𝑝,𝑞}. Then ‖‖𝜆_𝑗,𝑘‖‖_𝑗,𝑘𝜆∣_𝑗,𝑘𝑃̇^{𝑠,𝑎𝑝,𝑞}^♯(𝛼,𝛽)‖‖∽‖‖‖‖_ℓ∈ℤ_{𝑘∈ℤ𝑑}𝛽^{ℓ(𝑠+𝑑/𝑞)𝑞}||𝜆_ℓ,𝑘||^𝑞𝜒_ℓ,𝑘(𝑥)^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖,_𝑗,𝑘∣𝐿̇^{𝑠𝑝,𝑞}^♯(𝛼,𝛽)‖‖∽⎛⎜⎜⎝_ℓ∈ℤ𝛽^{ℓ(𝑠+𝑑/𝑞−𝑑/𝑝)𝑞}_{𝑘∈ℤ𝑑}||𝜆_ℓ,𝑘||^𝑝^𝑞/𝑝⎞⎟⎟⎠^1/𝑞.(4.24)

Proof. We prove the first statement. The proof for the second one is even simpler. Let 𝐹(𝑥,𝑡)=_𝑗,𝑘||𝜆_𝑗,𝑘||𝜒_𝑗,𝑘(𝑥)⋅𝜒_[𝛽^−(𝑗+1)_,𝛽^−𝑗_](𝑡).(4.25) Discretizing the integral over 𝑡 by 𝑡∼𝛽^−ℓ, we obtain ‖𝐹∣𝑌‖=‖‖‖‖^∞0𝑡^−𝑠𝑞sup_𝑤||𝐹(𝑥+𝑤,𝑡)||(1+|𝑤|/𝑡)^𝑎^𝑞𝑑𝑡𝑡^𝑑+1^1/𝑞≍‖‖‖‖_ℓ∈ℤ∣𝐿_𝑝ℝ^𝑑‖‖‖‖𝛽^{ℓ(𝑠+𝑑/𝑞)𝑞}^𝛽−ℓ_{𝛽−(ℓ+1)}sup_𝑤||𝐹(𝑥+𝑤,𝑡)||1+𝛽^ℓ|𝑤|^𝑎^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖.(4.26) With 𝑡∈[𝛽^−(ℓ+1),𝛽^−ℓ], we observe 𝐹(𝑥,𝑡)=_𝑘||𝜆_ℓ,𝑘||𝜒_ℓ,𝑘(𝑥)(4.27) and estimate ‖𝐹∣𝑌‖≤‖‖‖‖_ℓ∈ℤ𝛽^{ℓ(𝑠+𝑑/𝑞)𝑞}sup_{𝑤∈ℝ𝑑}11+𝛽^ℓ|𝑤|^𝑎_𝑘||𝜆_ℓ,𝑘||𝜒_ℓ,𝑘(𝑥+𝑤)^𝑞^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖.(4.28) In order to include also the situation min{𝑝,𝑞}≤1, we use the following trick. Obviously, we can rewrite and estimate (4.28) with 0<𝑟<1 in the following way: ‖𝐹∣𝑌‖≤‖‖‖‖‖⎛⎜⎜⎝_ℓ∈ℤ_𝑘𝛽^{ℓ(𝑠+𝑑/𝑞)𝑟}||𝜆_ℓ,𝑘||^𝑟sup_{𝑤∈ℝ𝑑}𝜒_ℓ,𝑘(𝑥+𝑤)1+𝛽^ℓ|𝑤|^𝑎𝑟^𝑞/𝑟⎞⎟⎟⎠^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖‖.(4.29)
The next observation is the useful estimate sup_𝑤||𝜒_ℓ,𝑘(𝑥+𝑤)||1+𝛽^ℓ|𝑤|^𝑎𝑟≲11+𝛽^ℓ||𝑥−𝑘𝛽^−ℓ||^𝑎𝑟≲𝜒_ℓ,𝑘(⋅)∗𝛽^ℓ𝑑1+𝛽^ℓ|⋅|^𝑎𝑟(𝑥).(4.30) Indeed, the first estimate is obvious. Let us establish the second one 𝜒^ℓ,𝑘(⋅)∗11+𝛽^ℓ|⋅|^𝑎𝑟(𝑥)=_{|𝑦𝑖−𝑘𝑖𝛽−ℓ|≤𝛽−ℓ𝑖=1,…,𝑑}11+𝛽^ℓ||𝑥−𝑦||^𝑎𝑟≳𝑑𝑦_{||𝑦||≤𝑐𝛽−ℓ}11+𝛽^ℓ||𝑥−𝑘𝛽^−ℓ−𝑦||^𝑎𝑟≳𝑑𝑦_{||𝑦||≤𝑐𝛽−ℓ}11+𝛽^ℓ||𝑥−𝑘𝛽^−ℓ||+𝛽^ℓ||𝑦||^𝑎𝑟≳𝛽_−ℓ𝑑𝑑𝑦¹⁰𝑢^𝑑−11+𝛽^ℓ||𝑥−𝑘𝛽^−ℓ||+𝑢^𝑎𝑟≳𝛽𝑑𝑢_−ℓ𝑑1+𝛽^ℓ||𝑥−𝑘𝛽^−ℓ||^𝑎𝑟.(4.31) Note, that the functions 𝑔_ℓ(𝑥)=𝛽^ℓ𝑑1+𝛽^ℓ|⋅|^𝑎𝑟(4.32) belong to 𝐿₁(ℝ^𝑑) with uniformly bounded norm where we need 𝑎𝑟>𝑑. Putting (4.31) and (4.30) into (4.29), we obtain ‖𝐹∣𝑌‖^𝑟≤‖‖‖‖‖⎛⎜⎜⎝_ℓ∈ℤ𝑔^ℓ∗_{𝑘∈ℤ𝑑}𝛽^{ℓ(𝑠+𝑑/𝑞)𝑟}||𝜆_ℓ,𝑘||^𝑟𝜒_ℓ,𝑘(𝑥)^𝑞/𝑟⎞⎟⎟⎠^𝑟/𝑞∣𝐿_𝑝/𝑟ℝ^𝑑‖‖‖‖‖.(4.33) Now we are in a position to use the majorant property of the Hardy/Littlewood maximal operator (see Section 2.2 and [25, Chapter 2]), which states that a convolution of a function 𝑓 with an 𝐿₁(ℝ^𝑑)-function (having norm one) can be estimated from above by the Hardy/Littlewood maximal function of 𝑓. We choose 𝑟<𝑚𝑖𝑛{𝑝,𝑞} and apply Theorem 2.1 for the 𝐿_𝑝/𝑟(ℓ_𝑞/𝑟) situation. This gives ‖𝐹∣𝑌‖^𝑟≲‖‖‖‖‖⎛⎜⎜⎝_ℓ∈ℤ_{𝑘∈ℤ𝑑}𝛽^{ℓ(𝑠+𝑑/𝑞)𝑟}||𝜆_ℓ,𝑘||^𝑟𝜒_ℓ,𝑘(𝑥)^𝑞/𝑟⎞⎟⎟⎠^𝑟/𝑞=‖‖‖‖_ℓ∈ℤ∣𝐿_𝑝/𝑟ℝ^𝑑‖‖‖‖‖_𝑘_∈ℤ𝑑𝛽^{ℓ(𝑠+𝑑/𝑞)𝑞}||𝜆_ℓ,𝑘||^𝑞𝜒_ℓ,𝑘(𝑥)^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖^𝑟(4.34) and finishes the upper estimate. Both conditions, 𝑎𝑟>𝑑 and 𝑟<min{𝑝,𝑞}, are compatible if 𝑎>𝑑/min{𝑝,𝑞} is assumed at the beginning.
For the estimate from below, we go back to (4.26) and observe sup_𝑤||𝐹(𝑥+𝑤,𝑡)||1+𝛽^ℓ|𝑤|^𝑎≥||𝐹(𝑥,𝑡)||,(4.35) which results in ‖𝐹∣𝑌‖≳‖‖‖‖_ℓ∈ℤ𝛽^{ℓ(𝑠+𝑑/𝑞)𝑞}^𝛽−ℓ_{𝛽−(ℓ+1)}||𝐹(𝑥,𝑡)||^𝑞𝑑𝑡𝑡^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖.(4.36) A further use of (4.27) gives finally ‖𝐹∣𝑌‖≳‖‖‖‖_{ℓ∈ℤ𝑘}_∈ℤ𝑑𝛽^{ℓ(𝑠+𝑑/𝑞)𝑞}||𝜆_ℓ,𝑘||^𝑞𝜒_ℓ,𝑘(𝑥)^1/𝑞∣𝐿_𝑝ℝ^𝑑‖‖‖‖.(4.37) The proof is complete.

4.4. Atomic Decompositions

The following theorem is a direct consequence of the abstract results in Theorems 3.11, and 3.13.

Theorem 4.9. Let 1≤𝑝,𝑞≤∞, 𝑎>𝑑/min{𝑝,𝑞}, and 𝑠∈ℝ. Let further 𝑔∈𝒮₀(ℝ^𝑑) be a radial function. Then there exist numbers 𝛼₀>0 and 𝛽₀>1 such that for all 0<𝛼≤𝛼₀ and 1<𝛽≤𝛽₀ the family 𝑔_𝑗,𝑘_{𝑗∈ℤ,𝑘∈ℤ}^𝑑=𝑇_𝛼𝑘𝛽^𝑗𝒟^𝐿2_𝛽𝑗𝑔(4.38) has the following properties.(i){𝑔_𝑗,𝑘}_{𝑗∈ℤ,𝑘∈ℤ}^𝑑 forms a Banach frame for Co𝐿̇^{𝑠𝑝,𝑞}(𝒢) and Co𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢), that is, we have a dual frame {𝑒_𝑗,𝑘}_{𝑗∈ℤ,𝑘∈ℤ}^𝑑⊂𝒮₀(ℝ^𝑑)with 𝑓=∑_{𝑗∈ℤ,𝑘∈ℤ𝑑}⟨𝑔_𝑗,𝑘,𝑓⟩𝑒_𝑗,𝑘 and the norm equivalences ‖‖𝑓∣Co𝐿̇^{𝑠𝑝,𝑞}(𝒢)‖‖≍‖‖𝑔_𝑗,𝑘,𝑓∣𝐿̇^{𝑠𝑝,𝑞}(𝒢)^♯(𝛼,𝛽)‖‖,𝑓∈Co𝐿̇^{𝑠𝑝,𝑞}(𝒢)(4.39) as well as ‖‖𝑓∣Co𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖≍‖‖𝑔_𝑗,𝑘,𝑓∣𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)^♯(𝛼,𝛽)‖‖,𝑓∈Co𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢).(4.40)(ii){𝑔_𝑗,𝑘}_{𝑗∈ℤ,𝑘∈ℤ}^𝑑 is an atomic decomposition, that is, for 𝑓∈Co𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢), we have a (not necessary unique) decomposition ∑_{𝑗∈ℤ,𝑘∈ℤ𝑑}𝜆_𝑗,𝑘(𝑓)𝑔_𝑗,𝑘 such that ‖‖𝜆_𝑗,𝑘(𝑓)_𝑗,𝑘∣𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)^♯(𝛼,𝛽)‖‖≲‖‖𝑓∣Co𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖.(4.41) Conversely, if {𝜆_𝑗,𝑘}_{𝑗∈ℤ,𝑘∈ℤ}^𝑑∈(𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢))^♯(𝛼,𝛽), then 𝑓=∑_{𝑗∈ℤ,𝑘∈ℤ𝑑}𝜆_𝑗,𝑘𝑔_𝑗,𝑘 converges and belongs to Co𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢) and moreover, ‖‖𝑓∣Co𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)‖‖≲‖‖𝜆_𝑗,𝑘_𝑗,𝑘∣𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢)^♯(𝛼,𝛽)‖‖(4.42) (analogously for Co𝐿̇^{𝑠𝑝,𝑞}(𝒢)). Convergence is considered in the strong topology if the finite sequences are dense in (𝑃̇^{𝑠,𝑎𝑝,𝑞}(𝒢))^♯(𝛼,𝛽) and in the weak^∗-topology otherwise.

Remark 4.10. (i) Since the analyzing function or atom 𝑔 can be chosen arbitrarily, we allow more flexibility here than in the results given in Frazier/Jawerth [30] and Triebel [3, 29].
(ii) Instead of regular families of sampling points (𝛼𝛽^−𝑗𝑘,𝛽^−𝑗) rather irregular families of points in 𝒢 are allowed as long as they are distributed sufficiently dense, see Theorem 3.11.

4.5. Wavelet Frames

In the sequel, we consider wavelet bases on ℝ^𝑑 in the sense of Lemma A.5 from Appendix A.2 in the appendices. We have given an orthonormal scaling function Ψ⁰ and the associated wavelet Ψ¹ on ℝand consider the tensor products Ψ^𝑐, 𝑐∈𝐸. Our aim is to specify, that is, give sufficient conditions to Ψ⁰, Ψ¹, such that (A.19) represents an unconditional basis in 𝐵^{𝑠𝑝,𝑞}(ℝ^𝑑)and 𝐹^{𝑠𝑝,𝑞}(ℝ^𝑑). We will apply the abstract Theorem 3.14.

In order to do so we need to have (3.17) for all functions Ψ^𝑐. We will state certain smoothness, decay, and moment conditions to Ψ¹ and Ψ⁰, see Definition A.1 in the appendix, to ensure this. Let us fix the neighborhood 𝑉=[−1,1]^𝑑×(1/2,1]⊂𝒢 of 𝑒∈𝒢.

Proposition 4.11. Let 𝐿∈ℕ, 𝐾>0, and Ψ⁰ be an orthogonal scaling function with associated wavelet Ψ¹ on ℝ. The function Ψ⁰ is supposed to satisfy (𝐷) and (𝑆_𝐾) and Ψ₁ is supposed to satisfy (𝐷), (𝑆_𝐾), and (𝑀_𝐿−1). For 𝑟₁,𝑟₂∈ℝ,the weight 𝑤(𝑥,𝑡) is given by 𝑤(𝑥,𝑡)=(1+|𝑥|)^𝑣(𝑡^𝑟2+𝑡^−𝑟1),(𝑥,𝑡)∈𝒢.(4.43) If now 𝑟₁<min{𝐿,𝐾}−𝑑2,𝑟²<min{𝐿,𝐾}+𝑑2−𝑣,(4.44) then we have _ℝ𝑑^∞0sup_{(𝑦,𝑠)∈(𝑥,𝑡)𝑉}||⟨𝜋(𝑦,𝑠)Ψ^𝑐,Ψ^𝑐⟩||𝑤(𝑥,𝑡)𝑑𝑡𝑡^𝑑+1𝑑𝑥<∞.(4.45)

Proof. With Lemma A.3, we obtain for 𝑊_Ψ¹Ψ¹ the following estimates: ||𝑊_Ψ1Ψ¹(𝑠,𝑡)||≲𝑡^{min{𝐿,𝐾}+1/2}(1+𝑡)^{2min{𝐿,𝐾}+1}⋅1(1+|𝑠|/(1+𝑡))^𝑁.(4.46) And in addition ||𝑊_Ψ𝑖Ψ^𝑖(𝑠,𝑡)||≲𝑡^1/2(𝑡+1)(1+|𝑠|/1(1+𝑡))^𝑁,𝑖=1,2.(4.47) Hence, for any 𝑐∈𝐸, the tensor product structure gives (assume without restriction that 𝑐_𝑑=1) ||𝑊_Ψ𝑐Ψ^𝑐(𝑥,𝑡)||≲𝑡^{min{𝐿,𝐾}}(1+𝑡)^{2min{𝐿,𝐾}}⋅𝑡^𝑑/2(1+𝑡)^𝑑𝑑_𝑖=111+||𝑥_𝑖||/(1+𝑡)^𝑁.(4.48) The expression sup_{(𝑦,𝑠)∈(𝑥,𝑡)𝑉}|𝑊_Ψ^𝑐Ψ^𝑐(𝑦,𝑠)| can be estimated similar sup_{(𝑦,𝑠)∈(𝑥,𝑡)𝑉}||𝑊_Ψ𝑐Ψ^𝑐(𝑦,𝑠)||=≲sup𝑡_{||𝑦𝑖−𝑥𝑖||≤𝑡𝑡/2≤𝑠≤𝑡}||𝑊_Ψ𝑐Ψ^𝑐(𝑦,𝑠)_{min{𝐿,𝐾}}||(1+𝑡)^{2min{𝐿,𝐾}}𝑡^𝑑/2(1+𝑡)^𝑑𝑑_𝑖=111+||𝑥_𝑖||/(1+𝑡)^𝑁≲𝑡^{min{𝐿,𝐾}}(1+𝑡)^{2min{𝐿,𝐾}}𝑡^𝑑/2(1+𝑡)^𝑑1(1+|𝑥|/(1+𝑡))^𝑁.(4.49) Fubini’s theorem and a change of variable yields _ℝ𝑑^∞0sup_{(𝑦,𝑠)∈(𝑥,𝑡)𝑉}||⟨𝜋(𝑦,𝑠)Ψ^𝑐,Ψ^𝑐⟩||𝑤(𝑥,𝑡)𝑑𝑡𝑡^𝑑+1𝑑𝑥≲^∞0𝑡^{min{𝐿,𝐾}}(1+𝑡)^{2min{𝐿,𝐾}}𝑡^𝑑/2⋅(𝑡^𝑟2+𝑡^−𝑟1)𝑑𝑡𝑡^𝑑+1.(4.50) Finally it is easy to see that the latter is finite if the conditions in (4.44) are valid. This proves Proposition 4.11.

Theorem 4.12. Let 𝐿∈ℕ, 𝐾>0, and Ψ⁰ be an orthogonal scaling function with associated wavelet Ψ¹ on ℝ. The function Ψ⁰ is supposed to satisfy (𝐷) and (𝑆_𝐾), and Ψ₁ is supposed to satisfy (𝐷), (𝑆_𝐾) and (𝑀_𝐿−1). (a) If 1≤𝑝,𝑞≤∞ and −min{𝐿,𝐾}+𝑑𝑝<𝑠<min{𝐿,𝐾}−𝑑1−1𝑝,(4.51) then (A.19) is a Banach frame for 𝐵̇^{𝑠𝑝,𝑞}(ℝ^𝑑) in the sense of (3.22).(b) If 1≤𝑝,𝑞<∞ and −min{𝐿,𝐾}+2𝑑max1𝑝,1𝑞<𝑠<min{𝐿,𝐾}−𝑑max1𝑝,1𝑞,1−1𝑝,(4.52) then (A.19) is a Banach frame for 𝐹̇^{𝑠𝑝,𝑞}(ℝ^𝑑)in the sense of (3.22).

Proof. Let us prove (a). First of all, we apply Theorem 4.5/(i). Afterwards, we use Proposition 4.2 in order to estimate the weight 𝑤_𝑌(𝑦,𝑡) for 𝑌=𝐿̇^{𝑠+𝑑/2−𝑑/𝑞𝑝,𝑞}(𝒢). We obtain 𝑤_𝑌(𝑥,𝑡)=≤ma𝑡x_−𝑟1𝑡^{𝑑(1/𝑝−1/2)−𝑠},𝑡^{𝑠−𝑑(1/𝑝−1/2)},𝑡^𝑠+𝑑/2,𝑡^{−𝑠+𝑑/2}𝑡∶0<𝑡<1,_𝑟2∶𝑡≥1.(4.53) Let us distinguish the cases, 𝑠≥0 and 𝑠<0. In the first case we can put 𝑟₁=max{𝑠−𝑑(1/𝑝−1/2),−𝑠+𝑑(1/𝑝−1/2),𝑠−𝑑/2}, 𝑟₂=max{𝑠+𝑑/2,−𝑠+𝑑(1/𝑝−1/2)}, and 𝑣=0. Now we apply first Proposition 4.11. This gives the condition 0≤𝑠<min{𝐿,𝐾}−𝑑(1−1/𝑝).(4.54) In the second case we put 𝑟₁=max{𝑠−𝑑(1/𝑝−1/2),−𝑠+𝑑(1/𝑝−1/2),−𝑠−𝑑/2}, 𝑟₂=max{−𝑠+𝑑/2,−𝑠+𝑑(1/𝑝−1/2)} and 𝑣=0. With Proposition 4.11, we obtain the condition −min{𝐿,𝐾}+𝑑/𝑝<𝑠<0.(4.55) Finally (4.54), (4.55), and Theorem 3.14 yield (a).
Step 2. We prove (b). We apply Theorem 4.5/(iii) and afterwards Proposition 4.2 and obtain for 𝑌=𝑃̇^{𝑠+𝑑/2−𝑑/𝑞,𝑎𝑝,𝑞}(𝒢)𝑤_𝑌(𝑥,𝑡)=max𝑡𝑡^{𝑑(1/𝑝−1/2)−𝑠}_𝑠+𝑑/2,𝑡^{𝑠−𝑑(1/𝑝−1/2)},≤(1+|𝑥|)m_𝑎ax{1,𝑡^−𝑎}(1+|𝑥|^𝑎),𝑡^{−𝑠+𝑑/2}max{𝑡^−𝑎,𝑡^𝑎}(1+|𝑥|)^𝑎𝑡^−𝑟1𝑡:0<𝑡<1,_𝑟2:𝑡≥1.(4.56) First, we consider the case 𝑠≥0. We can put 𝑟₁=max{𝑠+𝑎−𝑑/2,𝑠+𝑑/2−𝑑/𝑝}, 𝑟₂=max{𝑠+𝑑/2,−𝑠+𝑑/2+𝑎}, and 𝑣=𝑎. Proposition 4.11 gives the condition 0≤𝑠<min{𝐿,𝐾}−max𝑎,𝑑1−1𝑝(4.57) which can be rewritten to 0≤𝑠<min{𝐿,𝐾}−𝑑max1𝑝,1𝑞,1−1𝑝(4.58) since 𝑎 can be chosen arbitrarily greater than 𝑑max{1/𝑝,1/𝑞}. This gives the upper bound in (b). Now we consider 𝑠<0. We put 𝑟₁=max{−𝑠+𝑑/𝑝−𝑑/2,𝑠+𝑑/2−𝑑/𝑝}, 𝑟₂=−𝑠+𝑑/2+𝑎. This yields −min{𝐿,𝐾}+2𝑎<𝑠<0(4.59) and can be rewritten to −min{𝐿,𝐾}+2𝑑max{1/𝑝,1/𝑞}<𝑠<0.(4.60) This yields the lower bound in (b) and we are done.

The following corollary is a direct consequence of Theorem 4.12 and the facts in Appendix A.2.

Corollary 4.13. Let 𝑚>0 and (Ψ₀,Ψ₁)=(𝜑_𝑚,𝜓_𝑚) the spline wavelet system of order 𝑚. Then (a) If 1≤𝑝,𝑞≤∞ and −𝑚+1+𝑑𝑝<𝑠<𝑚−1−𝑑1−1𝑝,
then (A.19) is a Banach frame for 𝐵̇^{𝑠𝑝,𝑞}(ℝ^𝑑) in the sense of (3.22);
(b) If 1≤𝑝,𝑞≤∞ and −𝑚+1+2𝑑max1𝑝,1𝑞<𝑠<𝑚−1−𝑑max1𝑝,1𝑞,1−1𝑝,(4.61)
then (A.19) is a Banach frame for 𝐹̇^{𝑠𝑝,𝑞}(ℝ^𝑑) in the sense of (3.22).

Remark 4.14. The (optimal) smoothness conditions in [31] are slightly weaker than (a) in case 𝑑=1. However, compared to the approach of Triebel [29, 32], we admit some more degree of freedom. The wavelet or atom does not have to be compactly supported. Additionally, in case 𝑑=1, we do not need that 𝜓∈𝐶^𝑢(ℝ) where 𝑢>𝑠. Indeed, the conditions in (a) and (b) are slightly weaker.

Remark 4.15. More examples can be obtained by using compactly supported Daubechies’ wavelets of order 𝑁 or Meyer’s wavelets. Based on the underlying abstract result in Theorem 3.14 even biorthogonal wavelet systems which provide sufficiently high smoothness and vanishing moments are suitable for this issue.

Appendix

A. Wavelets

A.1. The Continuous Wavelet Transform

The vector 𝑔 is said to be the analyzing vector for a function 𝑓∈𝐿₂(ℝ^𝑑).The continuous wavelet transform 𝑊_𝑔𝑓 is then defined through𝑊_𝑔𝑓(𝑥,𝑡)=𝑇_𝑥𝒟^𝐿2_𝑡𝑔,𝑓,𝑥∈ℝ^𝑑,𝑡>0,(A.1) where the bracket ⟨⋅,⋅⟩ denotes the inner product in 𝐿₂(ℝ^𝑑). We call 𝑔 an admissible wavelet if𝑐_𝑔∶=_ℝ𝑑||ℱ𝑔(𝜉)||²||𝜉||^𝑑𝑑𝜉<∞.(A.2)

If this is the case, then the family {𝑇_𝑥𝒟^𝐿2_𝑡𝑔}_{𝑡>0,𝑥∈ℝ}^𝑑 represents a tight continuous frame in 𝐿₂(ℝ). In particular, this means that the above family is dense in 𝐿₂(ℝ^𝑑). For a proof we refer to Theorem  1.5.1 in [28].

Let us now specify the conditions (𝑀_𝐿), (𝐷), and (𝑆_𝐾), which we intend to impose on functions Φ,Ψ∈𝐿₂(ℝ^𝑑) in order to obtain a good decay of the continuous wavelet transform |𝑊_ΨΦ(𝑥,𝑡)|.

Definition A.1. Let 𝐿+1∈ℕ₀, 𝐾>0, and fix the conditions (𝐷), (𝑀_𝐿), and (𝑆_𝐾) for a function Ψ∈𝐿₂(ℝ^𝑑). (𝐷) For every 𝑁∈ℕ,there exists a constant 𝑐_𝑁 such that ||Ψ(𝑥)||≤𝑐^𝑁(1+|𝑥|)^𝑁.(A.3) (𝑀_𝐿) We have vanishing moments 𝐷^𝛼ℱΨ(0)=0(A.4) for all |𝛼|₁≤𝐿.(𝑆_𝐾) The function1+||𝜉||^𝐾||𝐷^𝛼ℱΨ(𝜉)||(A.5)
belongs to 𝐿₁(ℝ^𝑑)for every multi-index 𝛼∈ℕ^𝑑0.

Remark A.2. If a function 𝑔∈𝐿₂(ℝ^𝑑) satisfies (𝑆_𝐾) for some𝐾>0 then by well-known properties of the Fourier transform we have 𝑔∈𝐶^⌊𝐾⌋(ℝ^𝑑).

The following lemma provides a useful decay result for the continuous wavelet transform under certain smoothness, decay, and moment conditions, see also [4, 30, 33] for similar results in a different language. It represents a continuation of [4, Lemma  1] where one deals with 𝒮(ℝ^𝑑)-functions.

Lemma A.3. Let 𝐿∈ℕ₀, 𝐾>0, and Φ,Ψ,Φ₀∈𝐿₂(ℝ^𝑑). (i) Let Φ satisfy (𝐷), (𝑀_𝐿−1), and let Φ₀ satisfy (𝐷), (𝑆_𝐾). Then for every 𝑁∈ℕthere exists a constant 𝐶_𝑁 such that the estimate ||𝑊_ΦΦ₀(𝑥,𝑡)||≤𝐶_𝑁𝑡^{min{𝐿,𝐾}+𝑑/2}(1+|𝑥|)^𝑁(A.6) holds true for 𝑥∈ℝ^𝑑and 0<𝑡<1.(ii) Let Φ,Ψ satisfy (𝐷), (𝑀_𝐿−1) and (𝑆_𝐾). For every 𝑁∈ℕthere exists a constant 𝐶_𝑁 such that the estimate ||𝑊_ΦΨ(𝑥,𝑡)||≤𝐶_𝑁𝑡^{min{𝐿,𝐾}+𝑑/2}(1+𝑡)^{2min{𝐿,𝐾}+𝑑}1+1|𝑥+|𝑡^−𝑁(A.7) holds true for 𝑥∈ℝ^𝑑and0<𝑡<∞.

Proof. Step 1. Let us prove (i). We follow the proof of Lemma  1 in [4]. This reference deals with 𝒮(ℝ^𝑑)-functions, which makes the situation much more easy. We argue as follows. Clearly, ||𝑊_ΦΦ₀(𝑥,𝑡)||=𝑡^𝑑/2||𝒟_𝑡Φ∗Φ₀(𝑥)||.(A.8) Fix 0<𝑡<1. Obviously, the convolution (𝒟_𝑡Φ)∗Φ₀ satisfies (𝐷). By well-known properties of the Fourier transform, the derivative 𝐷^𝛼ℱ((𝒟_𝑡Φ)∗Φ₀))(𝜉) exists for every multi-index 𝛼∈ℕ^𝑑0. For fixed 𝛼, we estimate by using Leibniz’ formula ||𝐷^𝛼ℱ𝒟_𝑡Φ∗Φ₀)(𝜉)||=≤||𝑐𝐷^𝛼_𝛼ℱΦ(𝑡𝜉)⋅ℱΦ₀(𝜉)||_𝛽≤𝛼𝑡^|𝛽|1|||𝐷_𝛽≤𝑐_′𝛼ℱΦ(𝑡𝜉)⋅𝐷^𝛼−𝛽ℱΦ₀(𝜉)|||𝑡^𝐿1+||𝜉||^𝐿_𝛽≤𝛼|||𝐷_𝛼−𝛽ℱΦ₀(𝜉)|||.(A.9) In the last step we used property (𝑀_𝐿−1). Assuming 𝐾≥𝐿 and exploiting (𝑆_𝐾), we obtain that the left-hand side of (A.9) belongs to 𝐿₁(ℝ^𝑑) and ‖‖𝐷^𝛼ℱ𝒟_𝑡Φ∗Φ₀)(𝜉)∣𝐿₁ℝ^𝑑‖‖≤𝑐^′_𝛼𝑡^𝐿.(A.10) We proceed as follows: max_{|𝛼|1≤𝑁+1}‖‖𝐷^𝛼ℱ𝒟_𝑡Φ∗Φ₀(𝜉)∣𝐿₁ℝ^𝑑‖‖≥≥𝑐_𝑁max_{|𝛼|1≤𝑁+1}‖‖ℱ⁻¹𝐷^𝛼ℱ𝒟_𝑡Φ∗Φ₀∣𝐿_∞ℝ^𝑑‖‖‖‖(1+|𝑥|)^𝑁𝒟_𝑡Φ∗Φ₀(𝑥)∣𝐿_∞ℝ^𝑑‖‖.(A.11) This estimate together with (A.8) and (A.10) yields (A.6).
Let us finally assume 𝐾<𝐿 and return to (A.9). Clearly, the resulting inequality keeps valid if we replace the exponent 𝐿 by 𝐿′∈ℕ₀ with 𝐿′≤𝐿. It is even possible to extend (A.9) to every 0≤𝐿^<𝐿 by the following argument. Let 𝐿^∉ℕ. We have on the one hand LHS(A.9)≤𝑐^′_𝛼𝑡^{⌊𝐿′′⌋}1+||𝜉||^{⌊𝐿′′⌋}𝐺(𝜉)(A.12) and on the other hand LHS(A.9)≤𝑐^′_𝛼𝑡^{⌊𝐿′′⌋+1}1+||𝜉||^{⌊𝐿′′⌋+1}𝐺(𝜉),(A.13) where 𝐺(𝜉)=∑_𝛽≤𝛼|𝐷^𝛼−𝛽ℱΦ₀(𝜉)|. Choosing 0<𝜃<1 such that 𝐿^=(1−𝜃)⌊𝐿^⌋+𝜃(⌊𝐿^⌋+1), we obtain by a kind of interpolation argument LHS(A.9)=LHS(A.9)^1−𝜃LHS(A.9)^𝜃≤𝑐^′_𝛼𝑡^𝐿′′1+||𝜉||^𝐿′′𝐺(𝜉).(A.14) In particular, we obtain instead of (A.9) ||𝐷^𝛼ℱ𝒟_𝑡Φ∗Φ₀(𝜉)||≤𝑐′_𝛼𝑡^𝐾1+||𝜉||^𝐾_𝛽≤𝛼|||𝐷_𝛼−𝛽ℱΦ₀(𝜉)|||,𝜉∈ℝ^𝑑.(A.15) We exploit property (𝑆_𝐾) for Φ₀ and proceed analogously as above. This proves (A.6).
Step 2. The estimate in (ii) is an immediate consequence of (A.6) and the fact 𝑊_ΦΨ(𝑥,𝑡)=𝑊_ΨΦ−𝑡𝑥,1𝑡.(A.16) This completes the proof.

Corollary A.4. Let Φ,Ψ belong to the Schwartz space 𝒮₀(ℝ^𝑑). By Lemma A.3/(ii) for every 𝐿,𝑁∈ℕthere is a constant 𝐶_𝐿,𝑁>0 such that ||𝑊_ΦΨ(𝑥,𝑡)||≤𝐶_𝐿,𝑁𝑡^𝐿+𝑑/2(1+𝑡)^2𝐿+𝑑1+1|𝑥+|𝑡^−𝑁,𝑥∈ℝ^𝑑,𝑡>0.(A.17) Additionally, we obtain for Φ∈𝒮₀(ℝ^𝑑) and Φ₀∈𝒮(ℝ^𝑑) such that ||𝑊_ΦΦ₀(𝑥,𝑡)||≤𝐶_𝐿,𝑁𝑡^𝐿+𝑑/2(1+|𝑥|)^𝑁,𝑥∈ℝ^𝑑,0<𝑡<1.(A.18)

A.2. Orthonormal Wavelet Bases

The following Lemma is proved in Wojtaszczyk [34, Section 5.1].

Lemma A.5. Suppose we have a multiresolution analysis in 𝐿₂(ℝ) with scaling functions Ψ⁰(𝑡) and associated wavelets Ψ¹(𝑡). Let 𝐸={0,1}^𝑑⧵{(0,…,0)}. For 𝑐=(𝑐₁,…,𝑐_𝑑)∈𝐸, let Ψ^𝑐=⊗^𝑑𝑗=1Ψ^𝑐𝑗. Then the system 2^𝑗𝑑/2Ψ^𝑐2^𝑗𝑥−𝑘_{𝑐∈𝐸,𝑗∈ℤ,𝑘∈ℤ𝑑}(A.19) is an orthonormal basis in 𝐿₂(ℝ^𝑑).

Spline Wavelets
As a main example, we will consider the spline wavelet system. The normalized cardinal B-spline of order 𝑚+1 is given by 𝒩_𝑚+1(𝑥)∶=𝒩_𝑚∗𝒳(𝑥),𝑥∈ℝ,𝑚∈ℕ,(A.20) beginning with 𝒩₁=𝒳, the characteristic function of the interval (0,1). By 𝜑_𝑚(𝑥)∶=1√2𝜋ℱ⁻¹⎡⎢⎢⎢⎣ℱ𝒩_𝑚(𝜉)∑^{∞𝑘=−∞}||ℱ𝒩_𝑚(𝜉+2𝜋𝑘)||²^1/2⎤⎥⎥⎥⎦(𝑥),𝑥∈ℝ,(A.21) we obtain an orthonormal scaling function, which is again a spline of order 𝑚. Finally, by 𝜓_𝑚(𝑥)∶=^∞_𝑘_=−∞𝜑^𝑚2𝑡,𝜑^𝑚(𝑡−𝑘)(−1)^𝑘𝜑_𝑚(2𝑥+𝑘+1)(A.22) the generator of an orthonormal wavelet system is defined. For 𝑚=1, it is easily checked that −𝜓₁(𝑥−1) is the Haar wavelet. In general, these functions 𝜓_𝑚 have the following properties:(i)𝜓_𝑚 restricted to intervals [𝑘/2,(𝑘+1)/2], 𝑘∈ℤ, is a polynomial of degree at most 𝑚−1.(ii)𝜓_𝑚∈𝐶^𝑚−2(ℝ) if 𝑚≥2.(iii)𝜓^{𝑚(𝑚−2)} is uniformly Lipschitz continuous on ℝif 𝑚≥2.(iv)The function 𝜓_𝑚 satisfies a moment condition of order 𝑚−1, that is,^∞−∞𝑥^ℓ𝜓_𝑚(𝑥)𝑑𝑥=0,ℓ=0,1,…,𝑚−1.(A.23)
In particular, 𝜓_𝑚 satisfies (𝐷), (𝑀_𝐿) for 0<𝐿≤𝑚 and (𝑆_𝐾) for 𝐾<𝑚−1.

Acknowledgments

The author would like to thank Holger Rauhut, Martin Schäfer, Benjamin Scharf, and Hans Triebel for valuable discussions, a critical reading of preliminary versions of this paper and for several hints how to improve it.