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

Authors: Tino Ullrich
Journal: Journal of Function Spaces and Applications
 (2012)

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## 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.


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## Body

## 1. Introduction

This paper deals with Besov-Lizorkin-Triebel spacesḂp,qs(ℝd) and Ḟp,qs(ℝd) on the Euclidean space ℝd 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 AppendixA.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 Section2 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 symbolsℝ,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 ℓ1d-norm of a vector k. For a multi-index α̅ and x∈ℝd, we write(1.1)xα̅=x1α1⋅…⋅xdαdand define the differential operatorsDα̅ and Δ by(1.2)Dα̅=∂|α̅|1∂x1α1⋯∂xdαd,Δ=∑k=1d∂2∂xk2.IfX 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.


## 1.1. Notation

Let us first introduce some basic notation. The symbolsℝ,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 ℓ1d-norm of a vector k. For a multi-index α̅ and x∈ℝd, we write(1.1)xα̅=x1α1⋅…⋅xdαdand define the differential operatorsDα̅ and Δ by(1.2)Dα̅=∂|α̅|1∂x1α1⋯∂xdαd,Δ=∑k=1d∂2∂xk2.IfX 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 onℝd

### 2.1. Vector-Valued Lebesgue Spaces

The spaceLp(ℝd), 0<p≤∞, denotes the collection of complex-valued functions (equivalence classes) with finite (quasi-)norm(2.1)‖f∣Lp(Rd)‖=(∫Rd|f(x)|pdx)1/p,
with the usual modification if p=∞. The Hilbert space L2(ℝd) plays a separate role, see for instance, Section 3. Having a sequence of complex-valued functions {fk}k∈I on ℝd, where I is a countable index set, we put(2.2)‖fk∣lq(Lp(Rd))‖=(∑k∈I‖fk∣Lp(Rd)‖q)1/q,‖fk∣Lp(lq,Rd)‖=‖(∑k∈I|fk(x)|q)1/q∣Lp(Rd)‖,
where we modify appropriately in the case q=∞.


### 2.2. Maximal Functions

For a locally integrable functionf, we denote by Mf(x) the Hardy-Littlewood maximal function defined by(2.3)(Mf)(x)=sup⁡x∈Q1|Q|∫Q|f(y)|dy,x∈Rd,
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.
For1<p<∞ and 1<q≤∞, there exists a constant c>0, such that
(2.4)‖Mfk∣Lp(lq,Rd)‖≤c‖fk∣Lp(lq)‖
holds for all sequences {fk}k∈ℤ of locally Lebesgue-integrable functions on ℝd.Let us recall the classical Peetre maximal operator, introduced in [20]. Given a sequence of function {Ψk}k∈ℕ⊂𝒮(ℝd), a tempered distribution f∈𝒮′(ℝd) and a positive number a>0, we define the system of maximal functions(2.5)(Ψk*f)a(x)=sup⁡y∈Rd|(Ψk*f)(x+y)|(1+2k|y|)a,x∈Rd,k∈Z.Since(Ψk*f)(y) makes sense pointwise (see the following paragraph), everything is well-defined. However, the value “∞’’ is also possible for (Ψk*f)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 Ψ∈𝒮(ℝd), where Ψ0(x) might be given by a separate function. Also continuous dilates are needed. Let the operator 𝒟tLp, t>0, generate the p-normalized dilates of a function Ψ given by 𝒟tLpΨ:=t-d/pΨ(t-1·). If p=1, we omit the super index and use additionally Ψt:=𝒟tΨ:=𝒟tL1Ψ. We define (Ψt*f)a(x) by(2.6)(Ψt*f)a(x)=sup⁡y∈Rd|(Ψt*f)(x+y)|(1+|y|/t)a,x∈Rd,t>0.
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(ℝd) is used for the locally convex space of rapidly decreasing infinitely differentiable functions on ℝd, where the topology is generated by the family of seminorms(2.7)‖φ‖k,l=sup⁡x∈Rd,|α̅|1≤l|Dα̅φ(x)|(1+|x|)k,φ∈S(Rd),k,l∈N0.
The space 𝒮′(ℝd) is called the set of all tempered distributions on ℝd and defined as the topological dual of 𝒮(ℝd). Indeed, a linear mapping f:𝒮(ℝd)→ℂbelongs to 𝒮′(ℝd) if and  only if there  exist numbers k,ℓ∈ℕ0 and a constant c=cf such that(2.8)|f(φ)|≤cfsup⁡x∈Rd,|α̅|1≤l|Dα̅φ(x)|(1+|x|)kfor all φ∈𝒮(ℝd). 𝒮′(ℝd) is equipped with the weak*-topology.The convolutionφ*ψ of two integrable functions φ,ψ is defined via the integral(2.9)(φ*ψ)(x)=∫Rdφ(x-y)ψ(y)dy.
If  φ,ψ∈𝒮(ℝd), then (2.9) still belongs to 𝒮(ℝd). The convolution can be generalized to 𝒮(ℝd)×𝒮′(ℝd) via (φ*f)(x)=f(φ(x-·)), makes sense pointwise, and is a C∞-function in  ℝd of at most polynomial growth.As usual, the Fourier transform defined on both𝒮(ℝd) and 𝒮′(ℝd) is given by (ℱf)(φ):=f(ℱφ), where f∈𝒮′(ℝd),φ∈𝒮(ℝd), and(2.10)Fφ(ξ):=(2π)-d/2∫Rde-ix⋅ξφ(x)dx.
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 subset𝒮0(ℝd)⊂𝒮(ℝd). Following [1, Chapter 5], we put(2.11)S0(Rd)={φ∈S(Rd):Dα̅(Fφ)(0)=0foreverymulti-indexα̅∈N0d}.
The set 𝒮0′(ℝd) denotes the topological dual of 𝒮0(ℝd). If f∈𝒮′(ℝd), the restriction of f to 𝒮0(ℝd) clearly belongs to 𝒮0′(ℝd). Furthermore, if P(x) is an arbitrary polynomial in ℝd, we have (f+P(·))(φ)=f(φ) for every φ∈𝒮0(ℝd). Conversely, if f∈𝒮0′(ℝd), then f can be extended from 𝒮0(ℝd) to 𝒮(ℝd), that is, to an element of 𝒮′(ℝd). However, this fact is not trivial and makes use of the Hahn-Banach theorem in locally convex topological vector spaces. We may identify 𝒮0′(ℝd) with the factor space 𝒮′(ℝd)/𝒫(ℝd), 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) LetΦ(ℝd) be the collection of all systems {φj(x)}j∈ℕ0⊂𝒮(ℝd) with the following properties:(i)
φj(x)=φ(2-jx),j∈ℕ,(ii)
suppφ0⊂{x∈ℝd:|x|≤2},suppφ⊂{x∈ℝd:1/2≤|x|≤2},(iii)
∑j=0∞φj(x)=1 for every x∈ℝd.
(b) Moreover, the systemΦ̇(ℝd) denotes the collection of all systems {φj(x)}j∈ℤ⊂𝒮(ℝd) with the following properties:(i)
φj(x)=φ(2-jx),j∈ℤ,(ii)
suppφ={x∈ℝd:1/2≤|x|≤2},(iii)
∑j=-∞∞φj=1 for every x∈ℝd∖{0}.Remark 2.3.
If we takeφ0∈𝒮(ℝd) satisfying
(2.12)φ0(x)={1:|x|≤10:|x|>2
and define φ(x)=φ0(x)-φ0(2x), then the system {φj(x)}j∈ℕ0 belongs to Φ(ℝd) and the system {φj(x)}j∈ℤ with φ0:=φ belongs to Φ̇(ℝd).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=0∞∈Φ(ℝd) and Φj=ℱ-1φj,j∈ℕ0.Let further -∞<s<∞ and 0<q≤∞.(i)
If0<p≤∞, then(2.13)Bp,qs(Rd)={f∈S′(Rd):‖f∣Bp,qs(Rd)‖=(∑j=0∞2jsq‖Φj*f∣Lp(Rd)‖q)1/q}<∞.(ii)
If0<p<∞, then(2.14)Fp,qs(Rd)={∑j=0∞(∑j=0∞2jsq|(Φj*f)(x)|q)1qf∈S′(Rd):‖f∣Fp,qs(Rd)‖=‖(∑j=0∞2jsq|(Φj*f)(x)|q)1/q∣Lp(Rd)‖<∞(∑j=0∞2jsq|(Φj*f)(x)|q)1q}.
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)}j∈ℤ∈Φ̇(ℝd) and Φj=ℱ-1φj.Let further -∞<s<∞ and 0<q≤∞.(i) If0<p≤∞, then(2.15)Ḃp,qs(Rd)={f∈S0′(Rd):‖f∣Ḃp,qs(Rd)‖=(∑j=-∞∞2jsq‖Φj*f∣Lp(Rd)‖q)1/q<∞}.(ii) If0<p<∞, then(2.16)Ḟp,qs(Rd)={(∑j=-∞∞2jsq|(Φj*f)(x)|q)1qf∈S0′(Rd):‖f∣Ḟp,qs(Rd)‖=‖(∑j=-∞∞2jsq|(Φj*f)(x)|q)1/q∣Lp(Rd)‖<∞‖(∑j=-∞∞2jsq|(Φj*f)(x)|q)1q∣Lp(Rd)‖}.
In case q=∞, we replace the sum by a supremum in both cases.


### 2.5. Inhomogeneous Spaces

Essential for the sequel are functionsΦ0,Φ∈𝒮(ℝd) satisfying(2.17)|(FΦ0)(x)|>0on⁡{|x|<2ε},|(FΦ)(x)|>0on⁡{ε2<|x|<2ε},
for some ε>0, and(2.18)Dα̅(FΦ)(0)=0∀|α̅|≤R.
We will call the functions Φ0 and Φ kernels for local means. Recall that Φk=2kdΦ(2k·), k∈ℕ, and Ψt=𝒟tΨ.Theorem 2.6.
Lets∈ℝ, 0<p<∞,  0<q≤∞,  a>d/min{p,q} and  R+1>s. Let further  Φ0,Φ∈𝒮(ℝd) be given by (2.17) and (2.18). Then the space  Fp,qs(ℝd) can be characterized by
(2.19)Fp,qs(Rd)={f∈S′(Rd):‖f∣Fp,qs(Rd)‖i<∞},i=1,…,5,
where
(2.20)‖f∣Fp,qs‖1=‖Φ0*f∣Lp(Rd)‖+‖(∫01t-sq|(Φt*f)(x)|qdtt)1/q∣Lp(Rd)‖,(2.21)‖f∣Fp,qs‖2=‖Φ0*f∣Lp(Rd)‖+‖(∫01t-sq[sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a]qdtt)1/q∣Lp(Rd)‖,(2.22)‖f∣Fp,qs‖3=‖Φ0*f∣Lp(Rd)‖+‖(∫01t-sq∫|z|<t|(Φt*f)(x+z)|qdttd+1)1/q∣Lp(Rd)‖,(2.23)‖f∣Fp,qs‖4=‖(∑k=0∞2skq[sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a]q)1/q∣Lp(Rd)‖,(2.24)‖f∣Fp,qs‖5=‖(∑k=0∞2skq|(Φk*f)(x)|q)1/q∣Lp(Rd)‖,
with the usual modification in case q=∞. Furthermore, all quantities ∥f∣Fp,qs(ℝd)∥i, i=1,…,5, are equivalent  (quasi-)norms in Fp,qs(ℝd).For the inhomogeneous Besov spaces, we have the following characterizations.Theorem 2.7.
Lets∈ℝ, 0<p,q≤∞,  a>d/p, and  R+1>s. Let further  Φ0,Φ∈𝒮(ℝd) be given by  (2.17) and  (2.18). Then the space  Bp,qs(ℝd) can be characterized by
(2.25)Bp,qs(Rd)={f∈S′(Rd):‖f∣Bp,qs(Rd)‖i<∞},i=1,…,4,
where
(2.26)‖f∣Bp,qs‖1=‖Φ0*f∣Lp(Rd)‖+(∫01t-sq‖(Φt*f)(x)∣Lp(Rd)‖qdtt)1/q,‖f∣Bp,qs‖2=‖Φ0*f∣Lp(Rd)‖+(∫01t-sq‖sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a∣Lp(Rd)‖qdtt)1/q,‖f∣Bp,qs‖3=(∑k=0∞2skq‖sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a∣Lp(Rd)‖q)1/q,‖f∣Bp,qs‖4=(∑k=0∞2skq‖(Φk*f)(x)∣Lp(Rd)‖q)1/q,
with the usual modification if q=∞. Furthermore, all quantities ∥f|Bp,qs(ℝd)∥i, i=1,…,4, are equivalent quasinorms in Bp,qs(ℝd).


### 2.6. Homogeneous Spaces

The homogeneous spaces can be characterized in a similar way. Here we do not have a separate functionΦ0 anymore. We put Φ0=Φ.Theorem 2.8.
Lets∈ℝ, 0<p<∞,  0<q≤∞,  a>d/min{p,q}, and  R+1>s. Let further  Φ∈𝒮(ℝd) be given by  (2.17) and  (2.18). Then the space  Ḟp,qs(ℝd) can be characterized by
(2.27)Ḟp,qs(Rd)={f∈S0′(Rd):‖f∣Ḟp,qs(Rd)‖i<∞},i=1,…,5,
where
(2.28)‖f∣Ḟp,qs‖1=‖(∫0∞t-sq|(Φt*f)(x)|qdtt)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖2=‖(∫0∞t-sq[sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a]qdtt)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖3=‖(∫0∞t-sq∫|z|<t|(Φt*f)(x+z)|qdttd+1)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖4=‖(∑k=-∞∞2skq[sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a]q)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖5=‖(∑k=-∞∞2skq|(Φk*f)(x)|q)1/q∣Lp(Rd)‖
with the usual modification if  q=∞. Furthermore, all quantities  ∥f|Ḟp,qs(ℝd)∥i,  i=1,…,5, are equivalent quasinorms in  Ḟp,qs(ℝd).For the Besov spaces, we obtain the following characterizations.Theorem 2.9.
Lets∈ℝ,0<p,q≤∞,  a>d/p, and  R+1>s. Let further  Φ∈𝒮(ℝd) be given by (2.17) and (2.18). Then the space  Ḃp,qs(ℝd) can be characterized by
(2.29)Ḃp,qs(Rd)={f∈S0′(Rd):‖f∣Ḃp,qs(Rd)‖i<∞},i=1,…,4,
where
(2.30)‖f∣Ḃp,qs‖1=(∫0∞t-sq‖(Φt*f)(x)∣Lp(Rd)‖qdtt)1/q,‖f∣Ḃp,qs‖2=(∫0∞t-sq‖sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a∣Lp(Rd)‖qdtt)1/q,‖f∣Ḃp,qs‖3=(∑k=-∞∞2skq‖sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a∣Lp(Rd)‖q)1/q,‖f∣Ḃp,qs‖4=(∑k=-∞∞2skq‖(Φk*f)(x)∣Lp(Rd)‖q)1/q,
with the usual modification if q=∞. Furthermore, all quantities ∥f∣Ḃp,qs(ℝd)∥i, i=1,…,4, are equivalent quasinorms in Ḃp,qs(ℝd).Remark 2.10.
Observe, that the (quasi-)norms∥·∣Ḟp,qs(ℝd)∥3 and ∥·∣Fp,qs(ℝd)∥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 kernelsΦ0 and Φ, we refer mainly to Triebel [3, Section 3.3].The most prominent nontrivial examples (besides the one given in Remark2.3) 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.
Letp,q,s as in Theorem 2.6. Let further  k0,k0∈𝒮(ℝd) such that
(2.31)Fk0(0),Fk0(0)≠0
and define
(2.32)Φ0=k0,Φ=ΔNk0,
where N∈ℕsuch that 2N>s. Then (2.20), (2.21), (2.22), (2.23), and (2.24) characterize Fp,qs(ℝd).Corollary 2.12.
Letp,q,s as in Theorem 2.6. Let further  Φ0∈𝒮(ℝd) be a radial function such that  ℱΦ0 is non-increasing and atisfying
(2.33)FΦ0(0)≠0,Dα̅FΦ0(0)=0
for 1≤|α̅|1≤R, where R+1>s. Define
(2.34)Φ(x)=Φ0(x)-12dΦ0(x2).
Then (2.23), and (2.24) characterize Fp,qs(ℝd).


### 2.8. Proofs

We give the proof for Theorem2.6 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 Fp,qs(ℝd). 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 functions(2.35)Ψ0(t)=Ψ1(t)=e-t2
and, if a>0 is given, the tempered distribution f(t)=|t|n with a<n∈ℕ. Then (Ψk*f)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.
Let0<p,q≤∞ and,  δ>0. Let  {gk}k∈ℕ0 be a sequence of nonnegative measurable functions on  ℝd and put
(2.36)Gl(x)=∑k∈Z2-|k-l|δgk(x),x∈Rd,l∈Z.
Then there is some constant C=C(p,q,δ), such that
(2.37)‖{Gl}l∣lq(Lp(Rd))‖≤C‖{gk}k∣lq(Lp(Rd))‖,‖{Gl}l∣Lp(lq,Rd)‖≤C‖{gk}k∣Lp(lq,Rd)‖
hold true.Proof of Theorem2.6.
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, Fp,qs(ℝd). Indeed, Definition 2.4 can be seen as a special case of (2.24).Step 1.
We are going to prove the following inequalities:(2.38)‖f∣Fp,qs‖2≲‖f∣Fp,qs‖1≲‖f∣Fp,qs‖2
for every f∈𝒮′(ℝd).Substep 1.1.
Putφ0=ℱΦ0 and φℓ=(ℱΦ)(2-ℓ·) if ℓ≥1. Because of (2.17), it is possible to find functions ψ0,ψ∈𝒮(ℝd) with suppψ0⊂{ξ∈ℝd:|ξ|≤2ε}, suppψ⊂{ξ∈ℝd:ε/2≤|ξ|≤2ε} and ψℓ(x)=ψ(2-ℓx) such that
(2.39)∑l∈N0φl(ξ)⋅ψl(ξ)=1.
We need a bit more. Fix a 1≤t≤2. Clearly, we have also
(2.40)∑l∈N0φl(tξ)⋅ψl(tξ)=1
for all ξ∈ℝd. With Ψ0=ℱ-1ψ0 and Ψ=ℱ-1ψ, we obtain then
(2.41)g=∑m∈N0(Ψm)t*(Φm)t*g.
The 2ℓ dilation gives then
(2.42)gl=∑m∈N0(Ψm)t2-l*(Φm)t2-l*gl
for every g∈𝒮′(ℝd), where gℓ(η)=g(η-ℓ), η∈𝒮(ℝd). Obviously, we can rewrite (2.42) to obtain
(2.43)g=∑m∈N0(Ψm)t2-l*(Φm)t2-l*g
for all g∈𝒮′(ℝd). Let us now choose g=(Φℓ)t*f. This gives the final version of the convolution identity
(2.44)(Φl)t*f=∑m∈N0(Φl)t*(Ψm)t2-l*(Φm)t2-l*f.
For m,ℓ∈ℕ0 we define
(2.45)Λm,l(x)={2ldΦ0(2lx):m=0,Φl(x):m>0,x∈Rd.
Clearly, we have
(2.46)(Φl)t*(Φm)t2-l=(Λm,l)t*(Φm+l)t.
Plugging this into (2.44), we end up with the pointwise representation
(2.47)((Φl)t*f)(y)=∑m∈N0((Ψm)2-lt*(Λm,l)t*(Φm+l)t*f)(y)=∑m∈N0[(Ψm)2-lt*(Λm,l)t]*((Φm+l)t*f)(y)=∑m∈N0∫Rd[(Ψm)2-lt*(Λm,l)t](y-z)⋅((Φm+l)t*f)(z)dz
for all y∈ℝd.Substep 1.2.
Let us prove the following important inequality first. For everyr>0 and every N∈ℕ0, we have
(2.48)|((Φl)t*f)(x)|r≤c∑k∈N02-kNr2(k+l)d∫Rd|((Φk+l)t*f)(y)|r(1+2l|x-y|)Nrdy,
where c is independent of f∈𝒮′(ℝd), x∈ℝd, and ℓ∈ℕ0.
The representation (2.47) will be the starting point to prove (2.48). Namely, we have for y∈ℝd(2.49)|((Φl)t*f)(y)|≤∑m∈N0∫Rd|((Ψm)t2-l*(Λm,l)t)(y-z)|⋅|((Φm+l)t*f)(z)|dz≤∑m∈N0Sm,l,t∫Rd|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz,
where
(2.50)Sm,l,t=sup⁡x∈Rd|[(Ψm)2-lt*(Λm,l)t](x)|⋅(1+2l|x|)N.
Elementary properties of the convolution yield (compare with (2.84))
(2.51)Sm,l,t=2ldtdsup⁡x∈Rd|(Ψm*(Λm,l)2l)(x2lt)|⋅(1+2l|x|)N=2ldtdsup⁡x∈Rd|(ψm*ηm,l)(x)|⋅(1+|tx|)N,
where
(2.52)ηm,l(x)={Φ(x):l≥0,m>0,Φ0(x):otherwise.
Lemma A.3 yields
(2.53)Sm,l,t≤cN2ld2-mN,
which we put into (2.49) to obtain
(2.54)|((Φl)t*f)(y)|≤CN∑m∈N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz.
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 replacingℓ by k+ℓ in (2.54) and multiply on both sides with 2-kN. Then we can estimate
(2.55)2-kN|((Φk+l)t*f)(y)|≤CN∑m∈N02-kN2-mN∫Rd2(m+k+l)d|((Φm+k+l)t*f)(z)|(1+2k+l|y-z|)Ndz≤CN∑m∈N02-(m+k)N∫Rd2(m+k+l)d|((Φm+k+l)t*f)(z)|(1+2l|y-z|)Ndz=CN∑m∈k+N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz,(2.56)≤CN∑m∈N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz.
Next, we apply the elementary inequalities
(2.57)(1+2l|y-z|)⋅(1+2l|x-y|)≥(1+2l|x-z|),|((Φm+l)t*f)(z)|≤|((Φm+l)t*f)(z)|r(1+2l|x-z|)N(1-r)×sup⁡y∈Rd|((Φm+l)t*f)(y)|1-r(1+2l|x-y|)N(1-r),
where 0<r≤1. We define the maximal function
(2.58)Ml,N(x,t)=sup⁡k∈N0sup⁡y∈Rd2-kN|((Φk+l)t*f)(y)|(1+2l|x-y|)N,x∈Rd,
and estimate
(2.59)Ml,N(x,t)≤CN∑m∈N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|x-z|)Ndz(2.60)≤CN∑m∈N02-mNr(2-mNsup⁡y∈Rd|((Φm+l)t*f)(y)|(1+2l|x-y|)N)1-r×∫Rd2(m+l)d|((Φm+l)t*f)(z)|r(1+2l|x-z|)Nrdz.
Observe that we can estimate the term (⋯)1-r in the right-hand side of (2.60) by Mℓ,N(x,t)1-r. Hence, if Mℓ,N(x,t)<∞ we obtain from (2.60)
(2.61)Ml,N(x,t)r≤CN∑m∈N02-mNr∫Rd2(m+l)d|((Φm+l)t*f)(z)|r(1+2l|x-z|)Nrdz,
where CN is independent of x, f, ℓ, and t∈[1,2]. We claim that there exists Nf∈ℕ0 such that Mℓ,N(x,t)<∞ for all N≥Nf. Indeed, we use that f∈𝒮′(ℝd), that is, there is an M∈ℕ0 and cf>0 such that
(2.62)|((Φk+l)t*f)(y)|≤cfsup⁡|α̅|1≤Msup⁡z∈Rd|Dα̅Φk+l(z)|⋅(1+|y-z|)M.
Assuming N>M, we estimate as follows:
(2.63)|((Φl)t*f)(x)|≤Ml,N(x,t)≤csup⁡k∈N0sup⁡y∈Rd2-kN|((Φk+l)t*f)(y)|(1+|x-y|2)N/2≤csup⁡k∈N0sup⁡y∈Rd2-kN2(k+l)(M+d)sup⁡z∈Rdsup⁡|α̅|1≤M|Dα̅γk+l(z)|⋅(1+|y-z|)M(1+|x-y|)N≤c2l(M+d)sup⁡k∈N0sup⁡z∈Rdsup⁡|α̅|1≤M|Dα̅γk+l(z)|(1+|x-z|)N,
where we again used the inequality (compare with (2.57))
(2.64)1+|y-z|≤(1+|x-y|)(1+|x-z|)
and have set
(2.65)γl(t)={Φ0(t):Φ(t):l=0,l>0.
Hence γk+ℓ gives us only two different functions from 𝒮(ℝd). This implies the boundedness of Mℓ,N(x,t) for x∈ℝd if N>M=Nf. Therefore, (2.61) together with (2.63) yield (2.48) with c=CN, independent of x, f, and ℓ, for all N≥Nf. 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 allN∈ℕ0 but with c=c(f)=CNf  depending onf. 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 Mℓ,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 Mℓ,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<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(2.66)(Φ2-lt*f)a(x)r≤c∑k∈N02-kNr2(k+l)d∫Rd|((Φk+l)t*f)(y)|r(1+2l|x-y|)ardy.
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
(2.67)(∫12|⋅|q/rdtt)r/q,
on both sides and use Minkowski’s inequality for integrals, which yields
(2.68)(∫12|(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02-kNr2(k+l)d∫Rd(∫12|((Φk+l)t*f)(y)|q(dt/t))r/q(1+2l|x-y|)ardy.
If ar>d,then we have
(2.69)gl(y)=2dl(1+2l|y|)ar∈L1(Rd),
and we observe
(2.70)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02-kNr2kd2lsr[gl*(∫12|2ls((Φk+l)t*f)(⋅)|qdtt)r/q](x).
Now we use the majorant property of the Hardy-Littlewood maximal operator (see Section 2.2 and [25, Chapter 2]) and continue estimating
(2.71)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02lrs2k(-Nr+d)M[(∫12|((Φk+l)t*f)(⋅)|qdtt)r/q](x).
An index shift on the right-hand side gives
(2.72)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q=c∑k∈l+N02lrs2(k-l)(-Nr+d)M[(∫12|((Φk)t*f)(⋅)|qdtt)r/q](x)=c∑k∈l+N02(l-k)(Nr-d+rs)2krsM[(∫12|((Φk)t*f)(⋅)|qdtt)r/q](x).
Choose now 1/a<r<min{p,q}, N>max{0,-s}+a and put
(2.73)δ=N+s-dr>0.
We obtain for ℓ∈ℕ(2.74)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N2-δr|l-k|2krsM[(∫12|((Φk)t*f)(⋅)|qdtt)r/q](x).
Now we apply Lemma 2.13 in Lp/r(ℓq/r,ℝd), which yields
(2.75)‖(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q∣Lp/r(lq/r)‖≤c‖M[(∫12|2ks((Φk)t*f)(⋅)|qdtt)r/q]∣Lp/r(lq/r)‖.
The Fefferman-Stein inequality (see Section 2.2 /Theorem 2.1, having in mind that p/r,q/r>1) gives
(2.76)‖(∫12|2ls(Φ2-lt*f)a(x)|qdtt)1/q∣Lp(lq)‖r=‖M[(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q]∣Lp/r(lq/r)‖≲‖(∫12|2ks((Φk)t*f)(⋅)|qdtt)r/q∣lq/r(lq/r)‖=‖(∫12|2ks((Φk)t*f)(⋅)|qdtt)1/q∣Lp(lq)‖r.
Hence, we obtain
(2.77)‖(∫01|λ-sq(Φλ*f)a(x)|qdλλ)1/q∣Lp(Rd)‖≍‖(∑l=1∞∫12|2ls(Φ2-lt*f)a(x)|qdtt)1/q∣Lp‖=‖(∫12|2ls(Φ2-lt*f)a(x)|qdtt)1/q∣Lp(lq)‖≲‖(∫01|λ-sq(Φλ*f)(x)|qdλλ)1/q∣Lp(Rd)‖.
This proves ∥f∣Fp,qs(ℝd)∥2≲∥f∣Fp,qs(ℝd)∥1. Since the reverse inequality is trivial, this finishes Step 1.Step 2.
LetΨ0,Ψ∈𝒮′(ℝd) 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(2.78)‖f∣Fp,qs(Rd)‖4Ψ≲‖f∣Fp,qs(Rd)‖2Φ
for all f∈𝒮′(ℝd).We decompose f similar as in Step 1. Exploiting the property (2.17) for the system Φ, we find 𝒮(ℝd)-functions λ0,λ∈𝒮(ℝd)such that suppλ0⊂{ξ∈ℝd:|ξ|≤2ε} and suppλ⊂{ξ∈ℝd:ε/2≤|ξ|≤2ε} and
(2.79)∑l∈N0λl(tx)⋅φl(tx)=1
for x∈ℝd and t∈[1,2] fix. Putting Λ0=ℱλ0 and Λ=ℱΛ, we obtain the decomposition
(2.80)g=∑l∈N0(Λl)t*(Φl)t*g
for every g∈𝒮′(ℝd). We put g=Ψℓ*f and see
(2.81)Ψl*f=∑k∈N0Ψl*(Λk)t*(Φk)t*f.
Now, we estimate as follows:
(2.82)|((Ψl*(Λk)t)*((Φk)t*f))(y)|≤∫Rd|(Ψl*(Λk)t)(z)|⋅|((Φk)t*f)(y-z)|dz≤(Φ2-kt*f)a(y)∫Rd|(Φl*(Λk)t)(z)|⋅(1+2k|z|)adz≤(Φ2-kt*f)a(y)Jl,k,
where
(2.83)Jl,k=∫Rd|(Ψl*(Λk)t)(z)|(1+2k|z|)adz.
We first observe that for x∈ℝd and functions μ,η∈𝒮(ℝd), the following identity holds true for u,v>0(2.84)(μu*ηv)(z)=1ud[μ*ηv/u](zu)=1vd[μu/v*η](zv).
This yields in case ℓ≥k (with a minor change if k=0 or ℓ=0)
(2.85)Jl,k=∫Rd|[(Ψl-k)1/t*Λ](z)|(1+|tz|)a≲sup⁡z∈Rd|[(Ψl-k)1/t*Λ](z)|(1+|z|)a+d+1≲2(k-l)(R+1),
where we used Lemma A.3 for the last estimate.
Ifk>l, we change the roles of Ψ and Λ to obtain again with Lemma A.3(2.86)Jl,k=∫Rd|[Ψ*(Λk-l)](z)|(1+|2k-lz|)adx≲2(k-l)asup⁡z∈Rd|[Ψ*(Λk-l)t](z)|(1+|z|)a+d+1≲2(l-k)(L+1-a),
where L can be chosen arbitrary large since Λ satisfies (ML) for every L∈ℕaccording to its construction. Let us further use the estimate
(2.87)(Ψk*f)a(y)≤(Ψk*f)a(x)(1+2k|x-y|)a≲(Ψk*f)a(x)(1+2l|x-y|)amax⁡{1,2(k-l)a}.
Consequently,(2.88)sup⁡y∈Rd2sl|(Ψl*(Λk)t*((Φk)t*f))(y)|(1+2l|x-y|)a≲2ks(Φ2-kt*f)a(x)2(l-k)smax⁡{1,2(k-l)a}Jl,k≤2ks(Φ2-kt*f)a(x){2(l-k)(L+1-a+s):k>l,2(k-l)(R+1-s):l≥k.
Plugging this into (2.81) and choosing L≥a+|s| and δ=min{1,R+1-s}, we obtain the inequality
(2.89)2ls(Ψl*f)a(x)≲∑k∈N02-|k-l|δ2ks(Φ2-kt*f)a(x)
for all x∈ℝd and all t∈[1,2]. Suppose first that q≥1. Then we take on both sides (∫12|·|qdt/t)1/q, which gives
(2.90)2ls(Ψl*f)a(x)≲(∑k∈N02-|k-l|δ2ks∫12|(Φ2-kt*f)a(x)|qdtt)1/q.
Applying Lemma 2.13 yields
(2.91)‖2ls(Ψl*f)a(x)Lp(lq,Rd)‖≲‖(∑k=1∞2ksq∫12|(Φ2-kt*f)a(x)|qdtt)1/q∣Lp(Rd)‖,
which gives the desired result.
In caseq<1, we argue as follows. The quantity (∫12|·|qdt/t)1/q is not longer a norm, but a q-norm. This gives
(2.92)2ls(Ψl*f)a(x)q≲∑k∈N02-|k-l|δq2ksq∫12|(Φ2-kt*f)a(x)|qdtt.
Notice that the right-hand side is nothing more than a convolution (γ*β)ℓ of the sequences
(2.93)γk=2-|k|δq,βk=2ksq∫12|(Φ2-kt*f)a(x)|qdtt.
Now we apply the ℓ1-norm to both sides and get for all x∈ℝd(2.94)‖2ls(Ψl*f)a(x)∣lq‖q≤‖γ∣l1‖⋅‖β∣l1‖≲∑k=1∞2ksq∫12|(Φ2-kt*f)a(x)|qdtt.
We take both sides to the power (⋯)1/q and apply the Lp(ℝd)-norm. This gives (2.78).Substep 2.2.
With similar arguments and obvious modifications of Substep2.1, we obtain for all f∈𝒮′(ℝd)(2.95)‖f∣Fp,qs(Rd)‖2Ψ≲‖f∣Fp,qs(Rd)‖4Φ.Step 3.
Choosingt=1 in Step 1 and omitting the integration over t, we see immediately
(2.96)‖f∣Fp,qs(Rd)‖5≲‖f∣Fp,qs(Rd)‖4≲‖f∣Fp,qs(Rd)‖5.Step 4.
What remains is to show that (2.22) is equivalent to the rest.Substep 4.1.
Let us prove(2.97)‖f∣Fp,qs(Rd)‖2≲‖f∣Fp,qs(Rd)‖3.
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:
(2.98)(Φ2-lt*f)a(x)r≤CN∑k∈N02-kNr2(k+l)d∫Rd|((Φk+l)t*f)(y+z)|r(1+2l|x-y|)ardy.
Indeed, we have
(2.99)1+2l|x-y|≤1+2l(|x-(y+z)|+|z|)≲1+2l(|x-(y+z)|)+2-k≲1+2l(|x-(y+z)|),
where the last estimate follows from the fact that k∈ℕ0 in the sum. Instead of the integral (∫12|·|q/rdt/t)r/q (see Step 3), we take now on both sides of (2.98) the norm
(2.100)(∫12∫|z|<t|⋅|q/rdzdttd+1)r/q.
The integration over z does not influence the left-hand side. Instead of (2.68), we obtain
(2.101)(∫12|(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02-kNs2(k+l)d∫Rd(∫12∫|z|<t|((Φk+l)t*f)(y)|qdz(dt/td+1))r/q(1+2l|x-y|)ardy.
We continue with analogous arguments as after (2.68) and end up with (2.97).Substep 4.2.
We prove∥f∣Fp,qs(ℝd)∥3≲∥f∣Fp,qs(ℝd)∥2. Indeed, it is easy to see, that we have for all t>0(2.102)1td∫|z|<t|(Φt*f)(x+z)|dz≲sup⁡|z|<t(|(Φt*f)(x+z)|(1+1/t|z|)a≲(Φt*f)a(x),
and we are done. The proof is complete.Proof of Theorem2.8.
The proof of Theorem2.8 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 Λm,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 Λm,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∈𝒮0′(ℝd). The rest is clear.Proof of Corollaries2.11 and 2.12.
Corollary2.11 is more or less clear. We know that ΔN gives (∑k=1d|ξk|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 Corollary2.12, 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 Fp,qs(ℝd). For the remaining estimates we apply Theorem 2.6 with the system (Φ0,Φ̃), where
(2.103)Φ̃=Φ0(x)-12kdΦ0(x2k)
and k∈ℕis chosen in such a way that (2.17) is satisfied. What remains is a simple consequence of the fact that
(2.104)Φ̃=Φ+Φ-1+⋯+Φ-(k-1)
and the triangle inequality. This type of argument is due to Triebel [3, Section 3.3.3].


## 2.1. Vector-Valued Lebesgue Spaces

The spaceLp(ℝd), 0<p≤∞, denotes the collection of complex-valued functions (equivalence classes) with finite (quasi-)norm(2.1)‖f∣Lp(Rd)‖=(∫Rd|f(x)|pdx)1/p,
with the usual modification if p=∞. The Hilbert space L2(ℝd) plays a separate role, see for instance, Section 3. Having a sequence of complex-valued functions {fk}k∈I on ℝd, where I is a countable index set, we put(2.2)‖fk∣lq(Lp(Rd))‖=(∑k∈I‖fk∣Lp(Rd)‖q)1/q,‖fk∣Lp(lq,Rd)‖=‖(∑k∈I|fk(x)|q)1/q∣Lp(Rd)‖,
where we modify appropriately in the case q=∞.


## 2.2. Maximal Functions

For a locally integrable functionf, we denote by Mf(x) the Hardy-Littlewood maximal function defined by(2.3)(Mf)(x)=sup⁡x∈Q1|Q|∫Q|f(y)|dy,x∈Rd,
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.
For1<p<∞ and 1<q≤∞, there exists a constant c>0, such that
(2.4)‖Mfk∣Lp(lq,Rd)‖≤c‖fk∣Lp(lq)‖
holds for all sequences {fk}k∈ℤ of locally Lebesgue-integrable functions on ℝd.Let us recall the classical Peetre maximal operator, introduced in [20]. Given a sequence of function {Ψk}k∈ℕ⊂𝒮(ℝd), a tempered distribution f∈𝒮′(ℝd) and a positive number a>0, we define the system of maximal functions(2.5)(Ψk*f)a(x)=sup⁡y∈Rd|(Ψk*f)(x+y)|(1+2k|y|)a,x∈Rd,k∈Z.Since(Ψk*f)(y) makes sense pointwise (see the following paragraph), everything is well-defined. However, the value “∞’’ is also possible for (Ψk*f)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 Ψ∈𝒮(ℝd), where Ψ0(x) might be given by a separate function. Also continuous dilates are needed. Let the operator 𝒟tLp, t>0, generate the p-normalized dilates of a function Ψ given by 𝒟tLpΨ:=t-d/pΨ(t-1·). If p=1, we omit the super index and use additionally Ψt:=𝒟tΨ:=𝒟tL1Ψ. We define (Ψt*f)a(x) by(2.6)(Ψt*f)a(x)=sup⁡y∈Rd|(Ψt*f)(x+y)|(1+|y|/t)a,x∈Rd,t>0.
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(ℝd) is used for the locally convex space of rapidly decreasing infinitely differentiable functions on ℝd, where the topology is generated by the family of seminorms(2.7)‖φ‖k,l=sup⁡x∈Rd,|α̅|1≤l|Dα̅φ(x)|(1+|x|)k,φ∈S(Rd),k,l∈N0.
The space 𝒮′(ℝd) is called the set of all tempered distributions on ℝd and defined as the topological dual of 𝒮(ℝd). Indeed, a linear mapping f:𝒮(ℝd)→ℂbelongs to 𝒮′(ℝd) if and  only if there  exist numbers k,ℓ∈ℕ0 and a constant c=cf such that(2.8)|f(φ)|≤cfsup⁡x∈Rd,|α̅|1≤l|Dα̅φ(x)|(1+|x|)kfor all φ∈𝒮(ℝd). 𝒮′(ℝd) is equipped with the weak*-topology.The convolutionφ*ψ of two integrable functions φ,ψ is defined via the integral(2.9)(φ*ψ)(x)=∫Rdφ(x-y)ψ(y)dy.
If  φ,ψ∈𝒮(ℝd), then (2.9) still belongs to 𝒮(ℝd). The convolution can be generalized to 𝒮(ℝd)×𝒮′(ℝd) via (φ*f)(x)=f(φ(x-·)), makes sense pointwise, and is a C∞-function in  ℝd of at most polynomial growth.As usual, the Fourier transform defined on both𝒮(ℝd) and 𝒮′(ℝd) is given by (ℱf)(φ):=f(ℱφ), where f∈𝒮′(ℝd),φ∈𝒮(ℝd), and(2.10)Fφ(ξ):=(2π)-d/2∫Rde-ix⋅ξφ(x)dx.
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 subset𝒮0(ℝd)⊂𝒮(ℝd). Following [1, Chapter 5], we put(2.11)S0(Rd)={φ∈S(Rd):Dα̅(Fφ)(0)=0foreverymulti-indexα̅∈N0d}.
The set 𝒮0′(ℝd) denotes the topological dual of 𝒮0(ℝd). If f∈𝒮′(ℝd), the restriction of f to 𝒮0(ℝd) clearly belongs to 𝒮0′(ℝd). Furthermore, if P(x) is an arbitrary polynomial in ℝd, we have (f+P(·))(φ)=f(φ) for every φ∈𝒮0(ℝd). Conversely, if f∈𝒮0′(ℝd), then f can be extended from 𝒮0(ℝd) to 𝒮(ℝd), that is, to an element of 𝒮′(ℝd). However, this fact is not trivial and makes use of the Hahn-Banach theorem in locally convex topological vector spaces. We may identify 𝒮0′(ℝd) with the factor space 𝒮′(ℝd)/𝒫(ℝd), 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) LetΦ(ℝd) be the collection of all systems {φj(x)}j∈ℕ0⊂𝒮(ℝd) with the following properties:(i)
φj(x)=φ(2-jx),j∈ℕ,(ii)
suppφ0⊂{x∈ℝd:|x|≤2},suppφ⊂{x∈ℝd:1/2≤|x|≤2},(iii)
∑j=0∞φj(x)=1 for every x∈ℝd.
(b) Moreover, the systemΦ̇(ℝd) denotes the collection of all systems {φj(x)}j∈ℤ⊂𝒮(ℝd) with the following properties:(i)
φj(x)=φ(2-jx),j∈ℤ,(ii)
suppφ={x∈ℝd:1/2≤|x|≤2},(iii)
∑j=-∞∞φj=1 for every x∈ℝd∖{0}.Remark 2.3.
If we takeφ0∈𝒮(ℝd) satisfying
(2.12)φ0(x)={1:|x|≤10:|x|>2
and define φ(x)=φ0(x)-φ0(2x), then the system {φj(x)}j∈ℕ0 belongs to Φ(ℝd) and the system {φj(x)}j∈ℤ with φ0:=φ belongs to Φ̇(ℝd).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=0∞∈Φ(ℝd) and Φj=ℱ-1φj,j∈ℕ0.Let further -∞<s<∞ and 0<q≤∞.(i)
If0<p≤∞, then(2.13)Bp,qs(Rd)={f∈S′(Rd):‖f∣Bp,qs(Rd)‖=(∑j=0∞2jsq‖Φj*f∣Lp(Rd)‖q)1/q}<∞.(ii)
If0<p<∞, then(2.14)Fp,qs(Rd)={∑j=0∞(∑j=0∞2jsq|(Φj*f)(x)|q)1qf∈S′(Rd):‖f∣Fp,qs(Rd)‖=‖(∑j=0∞2jsq|(Φj*f)(x)|q)1/q∣Lp(Rd)‖<∞(∑j=0∞2jsq|(Φj*f)(x)|q)1q}.
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)}j∈ℤ∈Φ̇(ℝd) and Φj=ℱ-1φj.Let further -∞<s<∞ and 0<q≤∞.(i) If0<p≤∞, then(2.15)Ḃp,qs(Rd)={f∈S0′(Rd):‖f∣Ḃp,qs(Rd)‖=(∑j=-∞∞2jsq‖Φj*f∣Lp(Rd)‖q)1/q<∞}.(ii) If0<p<∞, then(2.16)Ḟp,qs(Rd)={(∑j=-∞∞2jsq|(Φj*f)(x)|q)1qf∈S0′(Rd):‖f∣Ḟp,qs(Rd)‖=‖(∑j=-∞∞2jsq|(Φj*f)(x)|q)1/q∣Lp(Rd)‖<∞‖(∑j=-∞∞2jsq|(Φj*f)(x)|q)1q∣Lp(Rd)‖}.
In case q=∞, we replace the sum by a supremum in both cases.


## 2.5. Inhomogeneous Spaces

Essential for the sequel are functionsΦ0,Φ∈𝒮(ℝd) satisfying(2.17)|(FΦ0)(x)|>0on⁡{|x|<2ε},|(FΦ)(x)|>0on⁡{ε2<|x|<2ε},
for some ε>0, and(2.18)Dα̅(FΦ)(0)=0∀|α̅|≤R.
We will call the functions Φ0 and Φ kernels for local means. Recall that Φk=2kdΦ(2k·), k∈ℕ, and Ψt=𝒟tΨ.Theorem 2.6.
Lets∈ℝ, 0<p<∞,  0<q≤∞,  a>d/min{p,q} and  R+1>s. Let further  Φ0,Φ∈𝒮(ℝd) be given by (2.17) and (2.18). Then the space  Fp,qs(ℝd) can be characterized by
(2.19)Fp,qs(Rd)={f∈S′(Rd):‖f∣Fp,qs(Rd)‖i<∞},i=1,…,5,
where
(2.20)‖f∣Fp,qs‖1=‖Φ0*f∣Lp(Rd)‖+‖(∫01t-sq|(Φt*f)(x)|qdtt)1/q∣Lp(Rd)‖,(2.21)‖f∣Fp,qs‖2=‖Φ0*f∣Lp(Rd)‖+‖(∫01t-sq[sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a]qdtt)1/q∣Lp(Rd)‖,(2.22)‖f∣Fp,qs‖3=‖Φ0*f∣Lp(Rd)‖+‖(∫01t-sq∫|z|<t|(Φt*f)(x+z)|qdttd+1)1/q∣Lp(Rd)‖,(2.23)‖f∣Fp,qs‖4=‖(∑k=0∞2skq[sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a]q)1/q∣Lp(Rd)‖,(2.24)‖f∣Fp,qs‖5=‖(∑k=0∞2skq|(Φk*f)(x)|q)1/q∣Lp(Rd)‖,
with the usual modification in case q=∞. Furthermore, all quantities ∥f∣Fp,qs(ℝd)∥i, i=1,…,5, are equivalent  (quasi-)norms in Fp,qs(ℝd).For the inhomogeneous Besov spaces, we have the following characterizations.Theorem 2.7.
Lets∈ℝ, 0<p,q≤∞,  a>d/p, and  R+1>s. Let further  Φ0,Φ∈𝒮(ℝd) be given by  (2.17) and  (2.18). Then the space  Bp,qs(ℝd) can be characterized by
(2.25)Bp,qs(Rd)={f∈S′(Rd):‖f∣Bp,qs(Rd)‖i<∞},i=1,…,4,
where
(2.26)‖f∣Bp,qs‖1=‖Φ0*f∣Lp(Rd)‖+(∫01t-sq‖(Φt*f)(x)∣Lp(Rd)‖qdtt)1/q,‖f∣Bp,qs‖2=‖Φ0*f∣Lp(Rd)‖+(∫01t-sq‖sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a∣Lp(Rd)‖qdtt)1/q,‖f∣Bp,qs‖3=(∑k=0∞2skq‖sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a∣Lp(Rd)‖q)1/q,‖f∣Bp,qs‖4=(∑k=0∞2skq‖(Φk*f)(x)∣Lp(Rd)‖q)1/q,
with the usual modification if q=∞. Furthermore, all quantities ∥f|Bp,qs(ℝd)∥i, i=1,…,4, are equivalent quasinorms in Bp,qs(ℝd).


## 2.6. Homogeneous Spaces

The homogeneous spaces can be characterized in a similar way. Here we do not have a separate functionΦ0 anymore. We put Φ0=Φ.Theorem 2.8.
Lets∈ℝ, 0<p<∞,  0<q≤∞,  a>d/min{p,q}, and  R+1>s. Let further  Φ∈𝒮(ℝd) be given by  (2.17) and  (2.18). Then the space  Ḟp,qs(ℝd) can be characterized by
(2.27)Ḟp,qs(Rd)={f∈S0′(Rd):‖f∣Ḟp,qs(Rd)‖i<∞},i=1,…,5,
where
(2.28)‖f∣Ḟp,qs‖1=‖(∫0∞t-sq|(Φt*f)(x)|qdtt)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖2=‖(∫0∞t-sq[sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a]qdtt)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖3=‖(∫0∞t-sq∫|z|<t|(Φt*f)(x+z)|qdttd+1)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖4=‖(∑k=-∞∞2skq[sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a]q)1/q∣Lp(Rd)‖,‖f∣Ḟp,qs‖5=‖(∑k=-∞∞2skq|(Φk*f)(x)|q)1/q∣Lp(Rd)‖
with the usual modification if  q=∞. Furthermore, all quantities  ∥f|Ḟp,qs(ℝd)∥i,  i=1,…,5, are equivalent quasinorms in  Ḟp,qs(ℝd).For the Besov spaces, we obtain the following characterizations.Theorem 2.9.
Lets∈ℝ,0<p,q≤∞,  a>d/p, and  R+1>s. Let further  Φ∈𝒮(ℝd) be given by (2.17) and (2.18). Then the space  Ḃp,qs(ℝd) can be characterized by
(2.29)Ḃp,qs(Rd)={f∈S0′(Rd):‖f∣Ḃp,qs(Rd)‖i<∞},i=1,…,4,
where
(2.30)‖f∣Ḃp,qs‖1=(∫0∞t-sq‖(Φt*f)(x)∣Lp(Rd)‖qdtt)1/q,‖f∣Ḃp,qs‖2=(∫0∞t-sq‖sup⁡z∈Rd|(Φt*f)(x+z)|(1+|z|/t)a∣Lp(Rd)‖qdtt)1/q,‖f∣Ḃp,qs‖3=(∑k=-∞∞2skq‖sup⁡z∈Rd|(Φk*f)(x+z)|(1+2k|z|)a∣Lp(Rd)‖q)1/q,‖f∣Ḃp,qs‖4=(∑k=-∞∞2skq‖(Φk*f)(x)∣Lp(Rd)‖q)1/q,
with the usual modification if q=∞. Furthermore, all quantities ∥f∣Ḃp,qs(ℝd)∥i, i=1,…,4, are equivalent quasinorms in Ḃp,qs(ℝd).Remark 2.10.
Observe, that the (quasi-)norms∥·∣Ḟp,qs(ℝd)∥3 and ∥·∣Fp,qs(ℝd)∥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 kernelsΦ0 and Φ, we refer mainly to Triebel [3, Section 3.3].The most prominent nontrivial examples (besides the one given in Remark2.3) 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.
Letp,q,s as in Theorem 2.6. Let further  k0,k0∈𝒮(ℝd) such that
(2.31)Fk0(0),Fk0(0)≠0
and define
(2.32)Φ0=k0,Φ=ΔNk0,
where N∈ℕsuch that 2N>s. Then (2.20), (2.21), (2.22), (2.23), and (2.24) characterize Fp,qs(ℝd).Corollary 2.12.
Letp,q,s as in Theorem 2.6. Let further  Φ0∈𝒮(ℝd) be a radial function such that  ℱΦ0 is non-increasing and atisfying
(2.33)FΦ0(0)≠0,Dα̅FΦ0(0)=0
for 1≤|α̅|1≤R, where R+1>s. Define
(2.34)Φ(x)=Φ0(x)-12dΦ0(x2).
Then (2.23), and (2.24) characterize Fp,qs(ℝd).


## 2.8. Proofs

We give the proof for Theorem2.6 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 Fp,qs(ℝd). 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 functions(2.35)Ψ0(t)=Ψ1(t)=e-t2
and, if a>0 is given, the tempered distribution f(t)=|t|n with a<n∈ℕ. Then (Ψk*f)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.
Let0<p,q≤∞ and,  δ>0. Let  {gk}k∈ℕ0 be a sequence of nonnegative measurable functions on  ℝd and put
(2.36)Gl(x)=∑k∈Z2-|k-l|δgk(x),x∈Rd,l∈Z.
Then there is some constant C=C(p,q,δ), such that
(2.37)‖{Gl}l∣lq(Lp(Rd))‖≤C‖{gk}k∣lq(Lp(Rd))‖,‖{Gl}l∣Lp(lq,Rd)‖≤C‖{gk}k∣Lp(lq,Rd)‖
hold true.Proof of Theorem2.6.
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, Fp,qs(ℝd). Indeed, Definition 2.4 can be seen as a special case of (2.24).Step 1.
We are going to prove the following inequalities:(2.38)‖f∣Fp,qs‖2≲‖f∣Fp,qs‖1≲‖f∣Fp,qs‖2
for every f∈𝒮′(ℝd).Substep 1.1.
Putφ0=ℱΦ0 and φℓ=(ℱΦ)(2-ℓ·) if ℓ≥1. Because of (2.17), it is possible to find functions ψ0,ψ∈𝒮(ℝd) with suppψ0⊂{ξ∈ℝd:|ξ|≤2ε}, suppψ⊂{ξ∈ℝd:ε/2≤|ξ|≤2ε} and ψℓ(x)=ψ(2-ℓx) such that
(2.39)∑l∈N0φl(ξ)⋅ψl(ξ)=1.
We need a bit more. Fix a 1≤t≤2. Clearly, we have also
(2.40)∑l∈N0φl(tξ)⋅ψl(tξ)=1
for all ξ∈ℝd. With Ψ0=ℱ-1ψ0 and Ψ=ℱ-1ψ, we obtain then
(2.41)g=∑m∈N0(Ψm)t*(Φm)t*g.
The 2ℓ dilation gives then
(2.42)gl=∑m∈N0(Ψm)t2-l*(Φm)t2-l*gl
for every g∈𝒮′(ℝd), where gℓ(η)=g(η-ℓ), η∈𝒮(ℝd). Obviously, we can rewrite (2.42) to obtain
(2.43)g=∑m∈N0(Ψm)t2-l*(Φm)t2-l*g
for all g∈𝒮′(ℝd). Let us now choose g=(Φℓ)t*f. This gives the final version of the convolution identity
(2.44)(Φl)t*f=∑m∈N0(Φl)t*(Ψm)t2-l*(Φm)t2-l*f.
For m,ℓ∈ℕ0 we define
(2.45)Λm,l(x)={2ldΦ0(2lx):m=0,Φl(x):m>0,x∈Rd.
Clearly, we have
(2.46)(Φl)t*(Φm)t2-l=(Λm,l)t*(Φm+l)t.
Plugging this into (2.44), we end up with the pointwise representation
(2.47)((Φl)t*f)(y)=∑m∈N0((Ψm)2-lt*(Λm,l)t*(Φm+l)t*f)(y)=∑m∈N0[(Ψm)2-lt*(Λm,l)t]*((Φm+l)t*f)(y)=∑m∈N0∫Rd[(Ψm)2-lt*(Λm,l)t](y-z)⋅((Φm+l)t*f)(z)dz
for all y∈ℝd.Substep 1.2.
Let us prove the following important inequality first. For everyr>0 and every N∈ℕ0, we have
(2.48)|((Φl)t*f)(x)|r≤c∑k∈N02-kNr2(k+l)d∫Rd|((Φk+l)t*f)(y)|r(1+2l|x-y|)Nrdy,
where c is independent of f∈𝒮′(ℝd), x∈ℝd, and ℓ∈ℕ0.
The representation (2.47) will be the starting point to prove (2.48). Namely, we have for y∈ℝd(2.49)|((Φl)t*f)(y)|≤∑m∈N0∫Rd|((Ψm)t2-l*(Λm,l)t)(y-z)|⋅|((Φm+l)t*f)(z)|dz≤∑m∈N0Sm,l,t∫Rd|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz,
where
(2.50)Sm,l,t=sup⁡x∈Rd|[(Ψm)2-lt*(Λm,l)t](x)|⋅(1+2l|x|)N.
Elementary properties of the convolution yield (compare with (2.84))
(2.51)Sm,l,t=2ldtdsup⁡x∈Rd|(Ψm*(Λm,l)2l)(x2lt)|⋅(1+2l|x|)N=2ldtdsup⁡x∈Rd|(ψm*ηm,l)(x)|⋅(1+|tx|)N,
where
(2.52)ηm,l(x)={Φ(x):l≥0,m>0,Φ0(x):otherwise.
Lemma A.3 yields
(2.53)Sm,l,t≤cN2ld2-mN,
which we put into (2.49) to obtain
(2.54)|((Φl)t*f)(y)|≤CN∑m∈N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz.
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 replacingℓ by k+ℓ in (2.54) and multiply on both sides with 2-kN. Then we can estimate
(2.55)2-kN|((Φk+l)t*f)(y)|≤CN∑m∈N02-kN2-mN∫Rd2(m+k+l)d|((Φm+k+l)t*f)(z)|(1+2k+l|y-z|)Ndz≤CN∑m∈N02-(m+k)N∫Rd2(m+k+l)d|((Φm+k+l)t*f)(z)|(1+2l|y-z|)Ndz=CN∑m∈k+N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz,(2.56)≤CN∑m∈N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|y-z|)Ndz.
Next, we apply the elementary inequalities
(2.57)(1+2l|y-z|)⋅(1+2l|x-y|)≥(1+2l|x-z|),|((Φm+l)t*f)(z)|≤|((Φm+l)t*f)(z)|r(1+2l|x-z|)N(1-r)×sup⁡y∈Rd|((Φm+l)t*f)(y)|1-r(1+2l|x-y|)N(1-r),
where 0<r≤1. We define the maximal function
(2.58)Ml,N(x,t)=sup⁡k∈N0sup⁡y∈Rd2-kN|((Φk+l)t*f)(y)|(1+2l|x-y|)N,x∈Rd,
and estimate
(2.59)Ml,N(x,t)≤CN∑m∈N02-mN∫Rd2(m+l)d|((Φm+l)t*f)(z)|(1+2l|x-z|)Ndz(2.60)≤CN∑m∈N02-mNr(2-mNsup⁡y∈Rd|((Φm+l)t*f)(y)|(1+2l|x-y|)N)1-r×∫Rd2(m+l)d|((Φm+l)t*f)(z)|r(1+2l|x-z|)Nrdz.
Observe that we can estimate the term (⋯)1-r in the right-hand side of (2.60) by Mℓ,N(x,t)1-r. Hence, if Mℓ,N(x,t)<∞ we obtain from (2.60)
(2.61)Ml,N(x,t)r≤CN∑m∈N02-mNr∫Rd2(m+l)d|((Φm+l)t*f)(z)|r(1+2l|x-z|)Nrdz,
where CN is independent of x, f, ℓ, and t∈[1,2]. We claim that there exists Nf∈ℕ0 such that Mℓ,N(x,t)<∞ for all N≥Nf. Indeed, we use that f∈𝒮′(ℝd), that is, there is an M∈ℕ0 and cf>0 such that
(2.62)|((Φk+l)t*f)(y)|≤cfsup⁡|α̅|1≤Msup⁡z∈Rd|Dα̅Φk+l(z)|⋅(1+|y-z|)M.
Assuming N>M, we estimate as follows:
(2.63)|((Φl)t*f)(x)|≤Ml,N(x,t)≤csup⁡k∈N0sup⁡y∈Rd2-kN|((Φk+l)t*f)(y)|(1+|x-y|2)N/2≤csup⁡k∈N0sup⁡y∈Rd2-kN2(k+l)(M+d)sup⁡z∈Rdsup⁡|α̅|1≤M|Dα̅γk+l(z)|⋅(1+|y-z|)M(1+|x-y|)N≤c2l(M+d)sup⁡k∈N0sup⁡z∈Rdsup⁡|α̅|1≤M|Dα̅γk+l(z)|(1+|x-z|)N,
where we again used the inequality (compare with (2.57))
(2.64)1+|y-z|≤(1+|x-y|)(1+|x-z|)
and have set
(2.65)γl(t)={Φ0(t):Φ(t):l=0,l>0.
Hence γk+ℓ gives us only two different functions from 𝒮(ℝd). This implies the boundedness of Mℓ,N(x,t) for x∈ℝd if N>M=Nf. Therefore, (2.61) together with (2.63) yield (2.48) with c=CN, independent of x, f, and ℓ, for all N≥Nf. 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 allN∈ℕ0 but with c=c(f)=CNf  depending onf. 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 Mℓ,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 Mℓ,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<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(2.66)(Φ2-lt*f)a(x)r≤c∑k∈N02-kNr2(k+l)d∫Rd|((Φk+l)t*f)(y)|r(1+2l|x-y|)ardy.
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
(2.67)(∫12|⋅|q/rdtt)r/q,
on both sides and use Minkowski’s inequality for integrals, which yields
(2.68)(∫12|(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02-kNr2(k+l)d∫Rd(∫12|((Φk+l)t*f)(y)|q(dt/t))r/q(1+2l|x-y|)ardy.
If ar>d,then we have
(2.69)gl(y)=2dl(1+2l|y|)ar∈L1(Rd),
and we observe
(2.70)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02-kNr2kd2lsr[gl*(∫12|2ls((Φk+l)t*f)(⋅)|qdtt)r/q](x).
Now we use the majorant property of the Hardy-Littlewood maximal operator (see Section 2.2 and [25, Chapter 2]) and continue estimating
(2.71)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02lrs2k(-Nr+d)M[(∫12|((Φk+l)t*f)(⋅)|qdtt)r/q](x).
An index shift on the right-hand side gives
(2.72)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q=c∑k∈l+N02lrs2(k-l)(-Nr+d)M[(∫12|((Φk)t*f)(⋅)|qdtt)r/q](x)=c∑k∈l+N02(l-k)(Nr-d+rs)2krsM[(∫12|((Φk)t*f)(⋅)|qdtt)r/q](x).
Choose now 1/a<r<min{p,q}, N>max{0,-s}+a and put
(2.73)δ=N+s-dr>0.
We obtain for ℓ∈ℕ(2.74)(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N2-δr|l-k|2krsM[(∫12|((Φk)t*f)(⋅)|qdtt)r/q](x).
Now we apply Lemma 2.13 in Lp/r(ℓq/r,ℝd), which yields
(2.75)‖(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q∣Lp/r(lq/r)‖≤c‖M[(∫12|2ks((Φk)t*f)(⋅)|qdtt)r/q]∣Lp/r(lq/r)‖.
The Fefferman-Stein inequality (see Section 2.2 /Theorem 2.1, having in mind that p/r,q/r>1) gives
(2.76)‖(∫12|2ls(Φ2-lt*f)a(x)|qdtt)1/q∣Lp(lq)‖r=‖M[(∫12|2ls(Φ2-lt*f)a(x)|qdtt)r/q]∣Lp/r(lq/r)‖≲‖(∫12|2ks((Φk)t*f)(⋅)|qdtt)r/q∣lq/r(lq/r)‖=‖(∫12|2ks((Φk)t*f)(⋅)|qdtt)1/q∣Lp(lq)‖r.
Hence, we obtain
(2.77)‖(∫01|λ-sq(Φλ*f)a(x)|qdλλ)1/q∣Lp(Rd)‖≍‖(∑l=1∞∫12|2ls(Φ2-lt*f)a(x)|qdtt)1/q∣Lp‖=‖(∫12|2ls(Φ2-lt*f)a(x)|qdtt)1/q∣Lp(lq)‖≲‖(∫01|λ-sq(Φλ*f)(x)|qdλλ)1/q∣Lp(Rd)‖.
This proves ∥f∣Fp,qs(ℝd)∥2≲∥f∣Fp,qs(ℝd)∥1. Since the reverse inequality is trivial, this finishes Step 1.Step 2.
LetΨ0,Ψ∈𝒮′(ℝd) 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(2.78)‖f∣Fp,qs(Rd)‖4Ψ≲‖f∣Fp,qs(Rd)‖2Φ
for all f∈𝒮′(ℝd).We decompose f similar as in Step 1. Exploiting the property (2.17) for the system Φ, we find 𝒮(ℝd)-functions λ0,λ∈𝒮(ℝd)such that suppλ0⊂{ξ∈ℝd:|ξ|≤2ε} and suppλ⊂{ξ∈ℝd:ε/2≤|ξ|≤2ε} and
(2.79)∑l∈N0λl(tx)⋅φl(tx)=1
for x∈ℝd and t∈[1,2] fix. Putting Λ0=ℱλ0 and Λ=ℱΛ, we obtain the decomposition
(2.80)g=∑l∈N0(Λl)t*(Φl)t*g
for every g∈𝒮′(ℝd). We put g=Ψℓ*f and see
(2.81)Ψl*f=∑k∈N0Ψl*(Λk)t*(Φk)t*f.
Now, we estimate as follows:
(2.82)|((Ψl*(Λk)t)*((Φk)t*f))(y)|≤∫Rd|(Ψl*(Λk)t)(z)|⋅|((Φk)t*f)(y-z)|dz≤(Φ2-kt*f)a(y)∫Rd|(Φl*(Λk)t)(z)|⋅(1+2k|z|)adz≤(Φ2-kt*f)a(y)Jl,k,
where
(2.83)Jl,k=∫Rd|(Ψl*(Λk)t)(z)|(1+2k|z|)adz.
We first observe that for x∈ℝd and functions μ,η∈𝒮(ℝd), the following identity holds true for u,v>0(2.84)(μu*ηv)(z)=1ud[μ*ηv/u](zu)=1vd[μu/v*η](zv).
This yields in case ℓ≥k (with a minor change if k=0 or ℓ=0)
(2.85)Jl,k=∫Rd|[(Ψl-k)1/t*Λ](z)|(1+|tz|)a≲sup⁡z∈Rd|[(Ψl-k)1/t*Λ](z)|(1+|z|)a+d+1≲2(k-l)(R+1),
where we used Lemma A.3 for the last estimate.
Ifk>l, we change the roles of Ψ and Λ to obtain again with Lemma A.3(2.86)Jl,k=∫Rd|[Ψ*(Λk-l)](z)|(1+|2k-lz|)adx≲2(k-l)asup⁡z∈Rd|[Ψ*(Λk-l)t](z)|(1+|z|)a+d+1≲2(l-k)(L+1-a),
where L can be chosen arbitrary large since Λ satisfies (ML) for every L∈ℕaccording to its construction. Let us further use the estimate
(2.87)(Ψk*f)a(y)≤(Ψk*f)a(x)(1+2k|x-y|)a≲(Ψk*f)a(x)(1+2l|x-y|)amax⁡{1,2(k-l)a}.
Consequently,(2.88)sup⁡y∈Rd2sl|(Ψl*(Λk)t*((Φk)t*f))(y)|(1+2l|x-y|)a≲2ks(Φ2-kt*f)a(x)2(l-k)smax⁡{1,2(k-l)a}Jl,k≤2ks(Φ2-kt*f)a(x){2(l-k)(L+1-a+s):k>l,2(k-l)(R+1-s):l≥k.
Plugging this into (2.81) and choosing L≥a+|s| and δ=min{1,R+1-s}, we obtain the inequality
(2.89)2ls(Ψl*f)a(x)≲∑k∈N02-|k-l|δ2ks(Φ2-kt*f)a(x)
for all x∈ℝd and all t∈[1,2]. Suppose first that q≥1. Then we take on both sides (∫12|·|qdt/t)1/q, which gives
(2.90)2ls(Ψl*f)a(x)≲(∑k∈N02-|k-l|δ2ks∫12|(Φ2-kt*f)a(x)|qdtt)1/q.
Applying Lemma 2.13 yields
(2.91)‖2ls(Ψl*f)a(x)Lp(lq,Rd)‖≲‖(∑k=1∞2ksq∫12|(Φ2-kt*f)a(x)|qdtt)1/q∣Lp(Rd)‖,
which gives the desired result.
In caseq<1, we argue as follows. The quantity (∫12|·|qdt/t)1/q is not longer a norm, but a q-norm. This gives
(2.92)2ls(Ψl*f)a(x)q≲∑k∈N02-|k-l|δq2ksq∫12|(Φ2-kt*f)a(x)|qdtt.
Notice that the right-hand side is nothing more than a convolution (γ*β)ℓ of the sequences
(2.93)γk=2-|k|δq,βk=2ksq∫12|(Φ2-kt*f)a(x)|qdtt.
Now we apply the ℓ1-norm to both sides and get for all x∈ℝd(2.94)‖2ls(Ψl*f)a(x)∣lq‖q≤‖γ∣l1‖⋅‖β∣l1‖≲∑k=1∞2ksq∫12|(Φ2-kt*f)a(x)|qdtt.
We take both sides to the power (⋯)1/q and apply the Lp(ℝd)-norm. This gives (2.78).Substep 2.2.
With similar arguments and obvious modifications of Substep2.1, we obtain for all f∈𝒮′(ℝd)(2.95)‖f∣Fp,qs(Rd)‖2Ψ≲‖f∣Fp,qs(Rd)‖4Φ.Step 3.
Choosingt=1 in Step 1 and omitting the integration over t, we see immediately
(2.96)‖f∣Fp,qs(Rd)‖5≲‖f∣Fp,qs(Rd)‖4≲‖f∣Fp,qs(Rd)‖5.Step 4.
What remains is to show that (2.22) is equivalent to the rest.Substep 4.1.
Let us prove(2.97)‖f∣Fp,qs(Rd)‖2≲‖f∣Fp,qs(Rd)‖3.
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:
(2.98)(Φ2-lt*f)a(x)r≤CN∑k∈N02-kNr2(k+l)d∫Rd|((Φk+l)t*f)(y+z)|r(1+2l|x-y|)ardy.
Indeed, we have
(2.99)1+2l|x-y|≤1+2l(|x-(y+z)|+|z|)≲1+2l(|x-(y+z)|)+2-k≲1+2l(|x-(y+z)|),
where the last estimate follows from the fact that k∈ℕ0 in the sum. Instead of the integral (∫12|·|q/rdt/t)r/q (see Step 3), we take now on both sides of (2.98) the norm
(2.100)(∫12∫|z|<t|⋅|q/rdzdttd+1)r/q.
The integration over z does not influence the left-hand side. Instead of (2.68), we obtain
(2.101)(∫12|(Φ2-lt*f)a(x)|qdtt)r/q≤c∑k∈N02-kNs2(k+l)d∫Rd(∫12∫|z|<t|((Φk+l)t*f)(y)|qdz(dt/td+1))r/q(1+2l|x-y|)ardy.
We continue with analogous arguments as after (2.68) and end up with (2.97).Substep 4.2.
We prove∥f∣Fp,qs(ℝd)∥3≲∥f∣Fp,qs(ℝd)∥2. Indeed, it is easy to see, that we have for all t>0(2.102)1td∫|z|<t|(Φt*f)(x+z)|dz≲sup⁡|z|<t(|(Φt*f)(x+z)|(1+1/t|z|)a≲(Φt*f)a(x),
and we are done. The proof is complete.Proof of Theorem2.8.
The proof of Theorem2.8 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 Λm,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 Λm,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∈𝒮0′(ℝd). The rest is clear.Proof of Corollaries2.11 and 2.12.
Corollary2.11 is more or less clear. We know that ΔN gives (∑k=1d|ξk|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 Corollary2.12, 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 Fp,qs(ℝd). For the remaining estimates we apply Theorem 2.6 with the system (Φ0,Φ̃), where
(2.103)Φ̃=Φ0(x)-12kdΦ0(x2k)
and k∈ℕis chosen in such a way that (2.17) is satisfied. What remains is a simple consequence of the fact that
(2.104)Φ̃=Φ+Φ-1+⋯+Φ-(k-1)
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 Wgf, 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 groupℍ and the short-time Fourier transform and leads to the well-known modulation spaces as coorbits of weighted Lp(ℍ) spaces, see [11, Section 7.1] and also [27].


### 3.1. Function Spaces on𝒢

Integration on𝒢 will always be with respect to the left Haar measure dμ(x). The Haar module on 𝒢 is denoted by Δ. We define further LxF(y)=F(x-1y) and RxF(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 L1loc(𝒢),(ii)Y is invariant under left and right translation Ly and Ry, 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 weightw 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 Lpw(𝒢) of functions F on the group 𝒢 is defined via the norm(3.1)‖F∣Lpw(G)‖=(∫G|F(x)w(x)|pdμ(x))1/p,
where 1≤p≤∞ (modification if p=∞). If w≡1, then we simply write Lp(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.
LetX={xi}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∈IxiV.(ii)X is called relatively separated if for all compact sets K⊂𝒢,there exists a constant CK such that
(3.2)sup⁡j∈I♯{i∈I:xiK∩xjK≠∅}≤CK.(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 familyX={xi}i∈I which is V-well-spread with respect to a relatively compact neighborhood V of e∈𝒢 we define the sequence space Yb and Y♯ associated to Y as
(3.3)Yb={{λi}i∈I:‖{λi}i∈I∣Yb‖=‖∑i∈I|λi||xiV|-1χxiV∣Y‖<∞},Y♯={{λi}i∈I:‖{λi}i∈I∣Y♯‖=‖∑i∈I|λi|χxiV∣Y‖<∞}.Remark 3.3.
For a well-spread familyX, the spaces Yb and Y♯ do not depend on the choice of V, that is, different sets V define equivalent norms on Yb and Y♯, respectively. For more facts on these sequence spaces, confer [12].


### 3.3. Coorbit Spaces

The starting point is a Hilbert spaceℋ and an integrable, irreducible, unitary and continuous representation π of 𝒢 on ℋ. Then the general voice transform Vgf is a function on the group 𝒢 given by(3.4)Vgf(x)=〈π(x)g,f〉,
where the brackets denote the inner product in ℋ.The continuous wavelet transformWgf (appendix) is a voice transform for the situation ℋ=L2(ℝd) and the ax+b-group.Definition 3.4.
For a weightw(·)≥1 on 𝒢, we define the space Aw⊂ℋ of admissible vectors by
(3.5)Aw={g∈H:Vgg∈L1w(G)}.
If Aw≠{0} and g∈Aw, we define further
(3.6)Hw1(Rd)={f∈H:‖f∣Hw1‖=‖Vgf∣L1w(G)‖<∞}.
Finally, we denote by (ℋw1)~ the canonical antidual of ℋw1, that is, the space of conjugate linear functionals on ℋw1.We see immediately thatAw⊂ℋw1⊂ℋ. The voice transform (3.4) can now be extended to ℋw×(ℋw1)~ by the usual dual pairing. The space ℋw1 can be considered as the space of test functions, whereas the space (ℋw1)~ serves as reservoir, that is, distributions.Let nowY be a space on 𝒢 such that (i)–(iii) in Section 3.1 hold true. We define further(3.7)wY(x)=max⁡{‖Lx‖,‖Lx-1‖,‖Rx‖,Δ(x-1)‖Rx-1‖},x∈G,
where the operator norms are considered from Y to Y.Definition 3.5.
LetY be a space on 𝒢 satisfying (i)–(iii) and wY(x) be given by (3.7). Let further g∈Aw. We define the space CoY, called coorbit space of Y, through
(3.8)CoY={f∈(Hw1)~:Vgf∈Y}with‖f∣CoY‖=‖Vgf∣Y‖.
The following result is of crucial importance. See [14] for details.Lemma 3.6 (Correspondence principle).
LetG=Vgg∈L1w(𝒢) for a fixed analyzing vector  g∈Aw with proper normalization. Then a function  F∈Y is of the form  Vgf for some  f∈CoY if and only if
(3.9)F(x)=(F*G)(x):=∫GF(y)LyG(x)dμ(y),x∈G.
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 spaceCoY is independent of the analyzing vector g∈Aw.
(ii) The definition of the spaceCoY is independent of the reservoir in the following sense. Assume that S⊂ℋw1 is a nontrivial locally convex vector space, which is invariant under π. Assume further that there exists a nonzero vector g∈S∩Aw for which the reproducing formula
(3.10)Vgf=Vgg*Vgf
holds for all f∈S~. Then we have
(3.11)CoY={f∈(Hw1)~:Vgf∈Y}={f∈S~:Vgf∈Y}.


### 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(xi)g}i∈I, where {xi}i∈I⊂𝒢 represents a discrete subset, whereas g denotes a fixed admissible analyzing vector.Definition 3.8.
A family{gi}i∈I in a Banach space B is called an atomic decomposition for B if there exists a family of bounded linear functionals {λi}i∈I⊂B′ (not necessary unique) and a Banach sequence space B♯=B♯(I) such that(a)
{λi(f)}i∈I∈B♯ for all f∈B and there exists a constant C1>0 with(3.12)‖{λi(f)}i∈I∣B♯‖≤C1‖f∣B‖.(b)
For allf∈B, we have
(3.13)f=∑i∈Iλi(f)gi
in some suitable topology.(c)
If{λi}i∈I∈B♯, then ∑i∈Iλigi∈B and there exists a constant C2>0 such that(3.14)‖∑i∈Iλigi∣B‖≤C2‖{λi}i∈I∣B♯‖.Definition 3.9.
A family{hi}i∈I⊂B′ is called a Banach frame for B if there exists a Banach sequence space Bb=Bb(I) and a linear bounded reconstruction operator Θ:Bb→B such that(a)
{hi(f)}i∈I∈Bb for all f∈B and there exist constants C1,C2 such that(3.15)C1‖f∣B‖≤‖{hi(f)}i∈I∣Bb‖≤C2‖f∣B‖,(b)
Θ({hi(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 inCoY is due to Feichtinger and Gröchenig (see [12, Theorem 6.1]).Theorem 3.11.
LetY be a function space on the group  𝒢 which satisfies the hypotheses  (i)–(iii). Let further
(3.16)wY(x):=max⁡{‖Lx‖,‖Lx-1‖,‖Rx‖,Δ(x-1)‖Rx-1‖},
where the operator norms are taken from Y toY, and g∈Aw such that
(3.17)∫G(sup⁡y∈xV|〈π(y)g,g〉|)w(x,t)dμ(x)<∞.
Then there exists a neighborhood U of e∈𝒢 and constants C0,C1>1 such that for every U-well-spread discrete set X={xi}i∈I⊂𝒢, the following is true.(i) (Analysis) Everyf∈CoY has a representation
(3.18)f=∑i∈Iλiπ(xi)g
with coefficients {λi}i∈I depending linearly on f and satisfying the estimate
(3.19)‖{λi}i∈I∣Y♯‖≤C0‖f∣Y‖.(ii) (Synthesis) Conversely, for any sequence{λi}i∈I∈Y♯, the element f=∑i∈Iλiπ(xi)g is in CoY and one has
(3.20)‖f∣CoY‖≤C1‖{λi}i∈I∣Y♯‖.
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 (ℋw1)~ otherwise.Remark 3.12.
According to Definition3.8, the family {π(xi)g}i∈I represents an atomic decomposition for CoY.Theorem 3.13.
Under the same conditions as in Theorem3.11, the system  {π(xi)g}i∈I represents a Banach frame for  CoY, that is,
(3.21)‖f∣CoY‖≍‖{〈π(xi)g,f〉}∣Yb‖.The following powerful result goes back to Gröchenig [14] and was generalized by Rauhut [17].Theorem 3.14.
Suppose that the functionsgr,γr,  r=1,…,n, satisfy (3.17). Let  X={xi}i∈I be a well-spread set such that
(3.22)f=∑r=1n∑i∈I〈π(xi)γr,f〉π(xi)gr
for all f∈H. Then expansion (3.22) extends to all f∈CoY. Moreover, f∈(ℋw1)~ belongs to CoY if and only if {〈π(xi)γr,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(yi)g}i∈I by Theorem 3.11 with a certain g satisfying (3.17) and a corresponding sequence of points Z={yi}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.


## 3.1. Function Spaces on𝒢

Integration on𝒢 will always be with respect to the left Haar measure dμ(x). The Haar module on 𝒢 is denoted by Δ. We define further LxF(y)=F(x-1y) and RxF(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 L1loc(𝒢),(ii)Y is invariant under left and right translation Ly and Ry, 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 weightw 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 Lpw(𝒢) of functions F on the group 𝒢 is defined via the norm(3.1)‖F∣Lpw(G)‖=(∫G|F(x)w(x)|pdμ(x))1/p,
where 1≤p≤∞ (modification if p=∞). If w≡1, then we simply write Lp(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.
LetX={xi}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∈IxiV.(ii)X is called relatively separated if for all compact sets K⊂𝒢,there exists a constant CK such that
(3.2)sup⁡j∈I♯{i∈I:xiK∩xjK≠∅}≤CK.(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 familyX={xi}i∈I which is V-well-spread with respect to a relatively compact neighborhood V of e∈𝒢 we define the sequence space Yb and Y♯ associated to Y as
(3.3)Yb={{λi}i∈I:‖{λi}i∈I∣Yb‖=‖∑i∈I|λi||xiV|-1χxiV∣Y‖<∞},Y♯={{λi}i∈I:‖{λi}i∈I∣Y♯‖=‖∑i∈I|λi|χxiV∣Y‖<∞}.Remark 3.3.
For a well-spread familyX, the spaces Yb and Y♯ do not depend on the choice of V, that is, different sets V define equivalent norms on Yb and Y♯, respectively. For more facts on these sequence spaces, confer [12].


## 3.3. Coorbit Spaces

The starting point is a Hilbert spaceℋ and an integrable, irreducible, unitary and continuous representation π of 𝒢 on ℋ. Then the general voice transform Vgf is a function on the group 𝒢 given by(3.4)Vgf(x)=〈π(x)g,f〉,
where the brackets denote the inner product in ℋ.The continuous wavelet transformWgf (appendix) is a voice transform for the situation ℋ=L2(ℝd) and the ax+b-group.Definition 3.4.
For a weightw(·)≥1 on 𝒢, we define the space Aw⊂ℋ of admissible vectors by
(3.5)Aw={g∈H:Vgg∈L1w(G)}.
If Aw≠{0} and g∈Aw, we define further
(3.6)Hw1(Rd)={f∈H:‖f∣Hw1‖=‖Vgf∣L1w(G)‖<∞}.
Finally, we denote by (ℋw1)~ the canonical antidual of ℋw1, that is, the space of conjugate linear functionals on ℋw1.We see immediately thatAw⊂ℋw1⊂ℋ. The voice transform (3.4) can now be extended to ℋw×(ℋw1)~ by the usual dual pairing. The space ℋw1 can be considered as the space of test functions, whereas the space (ℋw1)~ serves as reservoir, that is, distributions.Let nowY be a space on 𝒢 such that (i)–(iii) in Section 3.1 hold true. We define further(3.7)wY(x)=max⁡{‖Lx‖,‖Lx-1‖,‖Rx‖,Δ(x-1)‖Rx-1‖},x∈G,
where the operator norms are considered from Y to Y.Definition 3.5.
LetY be a space on 𝒢 satisfying (i)–(iii) and wY(x) be given by (3.7). Let further g∈Aw. We define the space CoY, called coorbit space of Y, through
(3.8)CoY={f∈(Hw1)~:Vgf∈Y}with‖f∣CoY‖=‖Vgf∣Y‖.
The following result is of crucial importance. See [14] for details.Lemma 3.6 (Correspondence principle).
LetG=Vgg∈L1w(𝒢) for a fixed analyzing vector  g∈Aw with proper normalization. Then a function  F∈Y is of the form  Vgf for some  f∈CoY if and only if
(3.9)F(x)=(F*G)(x):=∫GF(y)LyG(x)dμ(y),x∈G.
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 spaceCoY is independent of the analyzing vector g∈Aw.
(ii) The definition of the spaceCoY is independent of the reservoir in the following sense. Assume that S⊂ℋw1 is a nontrivial locally convex vector space, which is invariant under π. Assume further that there exists a nonzero vector g∈S∩Aw for which the reproducing formula
(3.10)Vgf=Vgg*Vgf
holds for all f∈S~. Then we have
(3.11)CoY={f∈(Hw1)~:Vgf∈Y}={f∈S~:Vgf∈Y}.


## 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(xi)g}i∈I, where {xi}i∈I⊂𝒢 represents a discrete subset, whereas g denotes a fixed admissible analyzing vector.Definition 3.8.
A family{gi}i∈I in a Banach space B is called an atomic decomposition for B if there exists a family of bounded linear functionals {λi}i∈I⊂B′ (not necessary unique) and a Banach sequence space B♯=B♯(I) such that(a)
{λi(f)}i∈I∈B♯ for all f∈B and there exists a constant C1>0 with(3.12)‖{λi(f)}i∈I∣B♯‖≤C1‖f∣B‖.(b)
For allf∈B, we have
(3.13)f=∑i∈Iλi(f)gi
in some suitable topology.(c)
If{λi}i∈I∈B♯, then ∑i∈Iλigi∈B and there exists a constant C2>0 such that(3.14)‖∑i∈Iλigi∣B‖≤C2‖{λi}i∈I∣B♯‖.Definition 3.9.
A family{hi}i∈I⊂B′ is called a Banach frame for B if there exists a Banach sequence space Bb=Bb(I) and a linear bounded reconstruction operator Θ:Bb→B such that(a)
{hi(f)}i∈I∈Bb for all f∈B and there exist constants C1,C2 such that(3.15)C1‖f∣B‖≤‖{hi(f)}i∈I∣Bb‖≤C2‖f∣B‖,(b)
Θ({hi(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 inCoY is due to Feichtinger and Gröchenig (see [12, Theorem 6.1]).Theorem 3.11.
LetY be a function space on the group  𝒢 which satisfies the hypotheses  (i)–(iii). Let further
(3.16)wY(x):=max⁡{‖Lx‖,‖Lx-1‖,‖Rx‖,Δ(x-1)‖Rx-1‖},
where the operator norms are taken from Y toY, and g∈Aw such that
(3.17)∫G(sup⁡y∈xV|〈π(y)g,g〉|)w(x,t)dμ(x)<∞.
Then there exists a neighborhood U of e∈𝒢 and constants C0,C1>1 such that for every U-well-spread discrete set X={xi}i∈I⊂𝒢, the following is true.(i) (Analysis) Everyf∈CoY has a representation
(3.18)f=∑i∈Iλiπ(xi)g
with coefficients {λi}i∈I depending linearly on f and satisfying the estimate
(3.19)‖{λi}i∈I∣Y♯‖≤C0‖f∣Y‖.(ii) (Synthesis) Conversely, for any sequence{λi}i∈I∈Y♯, the element f=∑i∈Iλiπ(xi)g is in CoY and one has
(3.20)‖f∣CoY‖≤C1‖{λi}i∈I∣Y♯‖.
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 (ℋw1)~ otherwise.Remark 3.12.
According to Definition3.8, the family {π(xi)g}i∈I represents an atomic decomposition for CoY.Theorem 3.13.
Under the same conditions as in Theorem3.11, the system  {π(xi)g}i∈I represents a Banach frame for  CoY, that is,
(3.21)‖f∣CoY‖≍‖{〈π(xi)g,f〉}∣Yb‖.The following powerful result goes back to Gröchenig [14] and was generalized by Rauhut [17].Theorem 3.14.
Suppose that the functionsgr,γr,  r=1,…,n, satisfy (3.17). Let  X={xi}i∈I be a well-spread set such that
(3.22)f=∑r=1n∑i∈I〈π(xi)γr,f〉π(xi)gr
for all f∈H. Then expansion (3.22) extends to all f∈CoY. Moreover, f∈(ℋw1)~ belongs to CoY if and only if {〈π(xi)γr,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(yi)g}i∈I by Theorem 3.11 with a certain g satisfying (3.17) and a corresponding sequence of points Z={yi}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 theax+b-Group

Let𝒢=ℝd⋊ℝ+* the d-dimensional ax+b-group. Its multiplication is given by(4.1)(x,t)(y,s)=(x+ty,st).The left Haar measureμ on 𝒢 is given by dt/td+1, the Haar module is Δ(x,t)=t-d. Giving a function F on 𝒢, the left and right translation Ly=L(y,r) and Ry=R(y,r) are given by(4.2)L(y,r)F(x,t)=F((y,r)-1(x,t))=F(x-yr,tr),R(y,r)F(x,t)=F((x,t)(y,r))=F(x+ty,rt).


### 4.1. Peetre-Type Spaces on𝒢

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 group𝒢. 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.
Lets∈ℝ,0<p,q≤∞,and a>0.We define the spaces L̇p,qs(𝒢),Ṫp,qs(𝒢),and Ṗp,qs,a(𝒢) on the group 𝒢 via the finiteness of the following (quasi-)norms:
(4.3)‖F∣L̇p,qs(G)‖=(∫0∞t-sq‖F(⋅,t)∣Lp(Rd)‖qdttd+1)1/q,‖F∣Ṫp,qs(G)‖=‖(∫0∞t-sq∫B(0,t)|F(x+z,t)|qdzdttd+1)1/q∣Lp(Rd)‖,‖F∣Ṗp,qs,a(G)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖,
using the usual modification in case q=∞.
Proposition 4.2.
The spacesL̇p,qs(𝒢),Ṫp,qs(𝒢) and  Ṗp,qs(𝒢), are left and right translation invariant. Precisely, we have
(4.4)‖L(z,r):L̇p,qs(G)⟶L̇p,qs(G)‖=rd(1/p-1/q)-s,‖R(z,r):L̇p,qs(G)⟶L̇p,qs(G)‖=rs+d/q,‖L(z,r):Ṫp,qs(G)⟶Ṫp,qs(G)‖=rd/q-s,‖R(z,r):Ṫp,qs(G)⟶Ṫp,qs(G)‖≤Crd/q+smax⁡{1,r-b(1+|z|)b},
where b>0 is a constant depending on d, p, and q, and
(4.5)‖L(z,r):Ṗp,qs,a(G)⟶Ṗp,qs,a(G)‖=rd(1/p-1/q)-s,‖R(z,r):Ṗp,qs,a(G)⟶Ṗp,qs,a(G)‖≤rs+d/qmax⁡{1,r-a}(1+|z|)a.Proof.
Step 1.
The left and right translation invariance ofL̇p,qs(𝒢) and Ṫp,qs(𝒢) was shown in [28, Lemma. 4.7.10].Step 2.
Let us considerṖp,qs,a(𝒢). Clearly, we have for F∈Ṗp,qs,a(𝒢)(4.6)‖L(z,r)F∣Ṗp,qs(G)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F((x+y-z)/r,t/r)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖=rd/p‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t/r)|(1+r|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖=rd(1/p-1/q)-s‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖.
Hence, we obtain
(4.7)‖L(z,r):Ṗp,qs,a(G)⟶Ṗp,qs(G)‖=rd(1/p-1/q)-s.
The right translation invariance is obtained by
(4.8)‖R(z,r)F∣Ṗp,qs,a(G)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F(x+tz+y,tr)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,tr)|(1+|y-tz|/t)a]qdttd+1)1/q∣Lp(Rd)‖=rs+d/q‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t)|(1+|y-tz|r/t)a]qdttd+1)1/q∣Lp(Rd)‖.
Observe that
(4.9)sup⁡y∈Rd|F(x+y,t)|(1+|y-tz|r/t)a=sup⁡y∈Rd[|F(x+y,t)|(1+|y|/t)a⋅(1+|y|/t)a(1+|y-tz|r/t)a],(1+|y|/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.
This yields
(4.10)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
and consequently
(4.11)‖R(z,r):Ṗp,qs,a(G)⟶Ṗp,qs,a(G)‖≤rs+d/qmax⁡{1,r-a}(1+|z|)a.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 ofṖp,qs(𝒢).This gives room for generalizations, that is, replacing the space Lp(ℝd) by some weighted Lebesgue space Lp(ℝd,ω) in the definition.


### 4.2. New Old Coorbit Spaces

We start withℋ=L2(ℝd) and the representation(4.12)π(x,t)=TxDtL2,
which is unitary continuous on ℋ. This representation is not irreducible. However, if we restrict to radial functions g∈L2(ℝd), then span{π(x,t)g:(x,t)∈𝒢} is dense in L2(ℝd). 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 transformWgf, see Appendix A.1 in the appendix. Recall the abstract definition of the space ℋw1 and 𝒜w 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 ℋw1 is nontrivial.Lemma 4.4.
If the weight functionw(x,t)≥1 satisfies the condition
(4.13)w(x,t)≤(1+|x|)r(ts+t-s′)
for some r,s,s′≥0, then
(4.14)S0(Rd)↪Hw1.This is a kind of minimal condition which we need in order to define coorbit spaces in a reasonable way. Instead of(Hw1)~, one may use 𝒮0′(ℝd) as reservoir and a radial g∈𝒮0(ℝd) 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 wY in (3.16))(iv)(4.15)wY(x,t)≲(1+|x|)r(ts+t-s′)for somer,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) For1≤p,q≤∞, and s∈ℝ,we have Ḃp,qs(ℝd)=CoL̇p,qs+d/2-d/q(𝒢),
(ii) for1≤p<∞, 1≤q≤∞ and s∈ℝwe have
(4.16)Ḟp,qs(Rd)=CoṪp,qs+d/2(G)
(iii) and if additionallya>d/min{p,q}, we obtain
(4.17)Ḟp,qs(Rd)=CoṖp,qs+d/2-d/q,a(G).Proof.
Theorem4.5 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 Ṫp,qs(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 spacesṪp,qs(𝒢) obsolete for this issue. We showed that Y=Pp,qs,a(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 in𝒢 given by 𝒰=[0,α]d×[β-1,1], where α>0 and 1<β. Furthermore, we consider the discrete set of points(4.18){xj,k=(αkβ-j,β-j):j∈Z,k∈Zd}.
Then the family {xj,k𝒰}j∈ℤ,k∈ℤd defines a partition of 𝒢. Indeed,(4.19)xj,kU=Qj,k×[β-(j+1),β-j],where(4.20)Qj,k=[αk1β-j,α(k1+1)β-j]×⋯×[αkdβ-j,α(kd+1)β-j].Note that in this case the spacesY♯ and Yb coincide. We will further use the notation(4.21)χj,k(x)={1:x∈Qj,k0:otherwise.Definition 4.7.
LetY be a function space on 𝒢 satisfying Section 3/(i), (ii), (iii). We put
(4.22)Y♯(α,β)={{λj,k}j,k:‖{λj,k}j,k∣Y♯(α,β)‖<∞},
where
(4.23)‖{λj,k}j,k∣Y♯(α,β)‖=‖∑j,k|λj,k|χj,k(x)χ[βj,βj+1](t)∣Y‖.Theorem 4.8.
Let1≤p,q≤∞, s∈ℝ,and a>d/min{p,q}. Then
(4.24)‖{λj,k}j,k∣(Ṗp,qs,a)♯(α,β)‖∽‖(∑l∈Z∑k∈Zdβl(s+d/q)q|λl,k|qχl,k(x))1/q∣Lp(Rd)‖,‖{λj,k}j,k∣(L̇p,qs)♯(α,β)‖∽(∑l∈Zβl(s+d/q-d/p)q(∑k∈Zd|λl,k|p)q/p)1/q.Proof.
We prove the first statement. The proof for the second one is even simpler. Let(4.25)F(x,t)=∑j,k|λj,k|χj,k(x)⋅χ[β-(j+1),β-j](t).
Discretizing the integral over t by t~β-ℓ, we obtain
(4.26)‖F∣Y‖=‖(∫0∞t-sq[sup⁡w|F(x+w,t)|(1+|w|/t)a]qdttd+1)1/q∣Lp(Rd)‖≍‖(∑l∈Zβl(s+d/q)q∫β-(l+1)β-l[sup⁡w|F(x+w,t)|(1+βl|w|)a]qdtt)1/q∣Lp(Rd)‖.
With t∈[β-(ℓ+1),β-ℓ], we observe
(4.27)F(x,t)=∑k|λl,k|χl,k(x)
and estimate
(4.28)‖F∣Y‖≤‖(∑l∈Zβl(s+d/q)qsup⁡w∈Rd1(1+βl|w|)a[∑k|λl,k|χl,k(x+w)]q)1/q∣Lp(Rd)‖.
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:
(4.29)‖F∣Y‖≤‖(∑l∈Z[∑kβl(s+d/q)r|λl,k|rsup⁡w∈Rdχl,k(x+w)(1+βl|w|)ar]q/r)1/q∣Lp(Rd)‖.
The next observation is the useful estimate(4.30)sup⁡w|χl,k(x+w)|(1+βl|w|)ar≲1(1+βl|x-kβ-l|)ar≲(χl,k(⋅)*βld(1+βl|⋅|)ar)(x).
Indeed, the first estimate is obvious. Let us establish the second one
(4.31)(χl,k(⋅)*1(1+βl|⋅|)ar)(x)=∫|yi-kiβ-l|≤β-li=1,…,d1(1+βl|x-y|)ardy≳∫|y|≤cβ-l1(1+βl|x-kβ-l-y|)ardy≳∫|y|≤cβ-l1(1+βl|x-kβ-l|+βl|y|)ardy≳β-ld∫01ud-1(1+βl|x-kβ-l|+u)ardu≳β-ld(1+βl|x-kβ-l|)ar.
Note, that the functions
(4.32)gl(x)=βld(1+βl|⋅|)ar
belong to L1(ℝd) with uniformly bounded norm where we need ar>d. Putting (4.31) and (4.30) into (4.29), we obtain
(4.33)‖F∣Y‖r≤‖(∑l∈Z[gl*∑k∈Zdβl(s+d/q)r|λl,k|rχl,k)(x)]q/r)r/q∣Lp/r(Rd)‖.
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(ℝd)-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 Lp/r(ℓq/r) situation. This gives
(4.34)‖F∣Y‖r≲‖(∑l∈Z[∑k∈Zdβl(s+d/q)r|λl,k|rχl,k(x)]q/r)r/q∣Lp/r(Rd)‖=‖(∑l∈Z∑k∈Zdβl(s+d/q)q|λl,k|qχl,k(x))1/q∣Lp(Rd)‖r
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
(4.35)sup⁡w|F(x+w,t)|(1+βl|w|)a≥|F(x,t)|,
which results in
(4.36)‖F∣Y‖≳‖(∑l∈Zβl(s+d/q)q∫β-(l+1)β-l|F(x,t)|qdtt)1/q∣Lp(Rd)‖.
A further use of (4.27) gives finally
(4.37)‖F∣Y‖≳‖(∑l∈Z∑k∈Zdβl(s+d/q)q|λl,k|qχl,k(x))1/q∣Lp(Rd)‖.
The proof is complete.


### 4.4. Atomic Decompositions

The following theorem is a direct consequence of the abstract results in Theorems3.11, and 3.13.Theorem 4.9.
Let1≤p,q≤∞,  a>d/min{p,q}, and  s∈ℝ. Let further  g∈𝒮0(ℝd) 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
(4.38){gj,k}j∈Z,k∈Zd={TαkβjDβjL2g}
has the following properties.(i)
{gj,k}j∈ℤ,k∈ℤd forms a Banach frame for CoL̇p,qs(𝒢) and CoṖp,qs,a(𝒢), that is, we have a dual frame {ej,k}j∈ℤ,k∈ℤd⊂𝒮0(ℝd)with f=∑j∈ℤ,k∈ℤd〈gj,k,f〉ej,k and the norm equivalences
(4.39)‖f∣CoL̇p,qs(G)‖≍‖〈gj,k,f〉∣(L̇p,qs(G))♯(α,β)‖,f∈CoL̇p,qs(G)
as well as
(4.40)‖f∣CoṖp,qs,a(G)‖≍‖〈gj,k,f〉∣(Ṗp,qs,a(G))♯(α,β)‖,f∈CoṖp,qs,a(G).(ii)
{gj,k}j∈ℤ,k∈ℤd is an atomic decomposition, that is, for f∈CoṖp,qs,a(𝒢), we have a (not necessary unique) decomposition ∑j∈ℤ,k∈ℤdλj,k(f)gj,k such that
(4.41)‖{λj,k(f)}j,k∣(Ṗp,qs,a(G))♯(α,β)‖≲‖f∣CoṖp,qs,a(G)‖.
Conversely, if {λj,k}j∈ℤ,k∈ℤd∈(Ṗp,qs,a(𝒢))♯(α,β), then f=∑j∈ℤ,k∈ℤdλj,kgj,k converges and belongs to CoṖp,qs,a(𝒢) and moreover,
(4.42)‖f∣CoṖp,qs,a(G)‖≲‖{λj,k}j,k∣(Ṗp,qs,a(G))♯(α,β)‖
(analogously for CoL̇p,qs(𝒢)). Convergence is considered in the strong topology if the finite sequences are dense in (Ṗp,qs,a(𝒢))♯(α,β) and in the weak*-topology otherwise.Remark 4.10.
(i) Since the analyzing function or atomg can be chosen arbitrarily, we allow more flexibility here than in the results given inFrazier/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 onℝd 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 Ψc, c∈E. Our aim is to specify, that is, give sufficient conditions to Ψ0, Ψ1, such that (A.19) represents an unconditional basis in Bp,qs(ℝd)and Fp,qs(ℝd). We will apply the abstract Theorem 3.14.In order to do so we need to have (3.17) for all functions Ψc. 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.
LetL∈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  (SK) and  Ψ1 is supposed to satisfy  (D),  (SK), and  (ML-1). For r1,r2∈ℝ,the weight  w(x,t) is given by
(4.43)w(x,t)=(1+|x|)v(tr2+t-r1),(x,t)∈G.
If now
(4.44)r1<min⁡⁡{L,K}-d2,r2<min⁡⁡{L,K}+d2-v,
then we have
(4.45)∫Rd∫0∞sup⁡(y,s)∈(x,t)V|〈π(y,s)Ψc,Ψc〉|w(x,t)dttd+1dx<∞.Proof.
With LemmaA.3, we obtain for WΨ1Ψ1 the following estimates:
(4.46)|(WΨ1Ψ1)(s,t)|≲tmin⁡{L,K}+1/2(1+t)2min⁡{L,K}+1⋅1(1+|s|/(1+t))N.
And in addition
(4.47)|(WΨiΨi)(s,t)|≲t1/2(t+1)1(1+|s|/(1+t))N,i=1,2.
Hence, for any c∈E, the tensor product structure gives (assume without restriction that cd=1)
(4.48)|WΨcΨc(x,t)|≲tmin⁡{L,K}(1+t)2min⁡{L,K}⋅td/2(1+t)d∏i=1d1(1+|xi|/(1+t))N.
The expression sup(y,s)∈(x,t)V|WΨcΨc(y,s)| can be estimated similar
(4.49)sup⁡(y,s)∈(x,t)V|WΨcΨc(y,s)|=sup⁡|yi-xi|≤tt/2≤s≤t|WΨcΨc(y,s)|≲tmin⁡{L,K}(1+t)2min⁡{L,K}td/2(1+t)d∏i=1d1(1+|xi|/(1+t))N≲tmin⁡{L,K}(1+t)2min⁡{L,K}td/2(1+t)d1(1+|x|/(1+t))N.
Fubini’s theorem and a change of variable yields
(4.50)∫Rd∫0∞sup⁡(y,s)∈(x,t)V|〈π(y,s)Ψc,Ψc〉|w(x,t)dttd+1dx≲∫0∞tmin⁡{L,K}(1+t)2min⁡{L,K}td/2⋅(tr2+t-r1)dttd+1.
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.
LetL∈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  (SK), and  Ψ1 is supposed to satisfy  (D),  (SK) and  (ML-1). (a) If1≤p,q≤∞ and
(4.51)-min⁡⁡{L,K}+dp<s<min⁡⁡{L,K}-d(1-1p),
then (A.19) is a Banach frame for Ḃp,qs(ℝd) in the sense of (3.22).(b) If1≤p,q<∞ and
(4.52)-min⁡⁡{L,K}+2dmax⁡⁡{1p,1q}<s<min⁡⁡{L,K}-dmax⁡{1p,1q,1-1p},
then (A.19) is a Banach frame for Ḟp,qs(ℝd)in the sense of (3.22).Proof.
Let us prove (a). First of all, we apply Theorem4.5/(i). Afterwards, we use Proposition 4.2 in order to estimate the weight wY(y,t) for Y=L̇p,qs+d/2-d/q(𝒢). We obtain
(4.53)wY(x,t)=max⁡⁡{td(1/p-1/2)-s,ts-d(1/p-1/2),ts+d/2,t-s+d/2}≤{t-r1:0<t<1,tr2:t≥1.
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
(4.54)0≤s<min⁡⁡{L,K}-d(1-1/p).
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
(4.55)-min⁡⁡{L,K}+d/p<s<0.
Finally (4.54), (4.55), and Theorem 3.14 yield (a).
Step 2.
We prove (b). We apply Theorem4.5/(iii) and afterwards Proposition 4.2 and obtain for Y=Ṗp,qs+d/2-d/q,a(𝒢)(4.56)wY(x,t)=max⁡{td(1/p-1/2)-s,ts-d(1/p-1/2),ts+d/2max⁡{1,t-a}(1+|x|a),t-s+d/2max⁡{t-a,ta}(1+|x|)a}≤(1+|x|)a{t-r1:0<t<1,tr2:t≥1.
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
(4.57)0≤s<min⁡{L,K}-max⁡⁡{a,d(1-1p)}
which can be rewritten to
(4.58)0≤s<min⁡{L,K}-dmax⁡{1p,1q,1-1p}
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
(4.59)-min⁡⁡{L,K}+2a<s<0
and can be rewritten to
(4.60)-min⁡{L,K}+2dmax⁡{1/p,1/q}<s<0.
This yields the lower bound in (b) and we are done.The following corollary is a direct consequence of Theorem4.12 and the facts in  Appendix A.2.Corollary 4.13.
Letm>0 and  (Ψ0,Ψ1)=(φm,ψm) the spline wavelet system of order  m. Then (a) If1≤p,q≤∞ and -m+1+dp<s<m-1-d(1-1p),
then (A.19) is a Banach frame for Ḃp,qs(ℝd) in the sense of (3.22);
(b) If1≤p,q≤∞ and
(4.61)-m+1+2dmax⁡{1p,1q}<s<m-1-dmax⁡{1p,1q,1-1p},
then (A.19) is a Banach frame for Ḟp,qs(ℝd) 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∈Cu(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 orderN 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.


## 4.1. Peetre-Type Spaces on𝒢

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 group𝒢. 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.
Lets∈ℝ,0<p,q≤∞,and a>0.We define the spaces L̇p,qs(𝒢),Ṫp,qs(𝒢),and Ṗp,qs,a(𝒢) on the group 𝒢 via the finiteness of the following (quasi-)norms:
(4.3)‖F∣L̇p,qs(G)‖=(∫0∞t-sq‖F(⋅,t)∣Lp(Rd)‖qdttd+1)1/q,‖F∣Ṫp,qs(G)‖=‖(∫0∞t-sq∫B(0,t)|F(x+z,t)|qdzdttd+1)1/q∣Lp(Rd)‖,‖F∣Ṗp,qs,a(G)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖,
using the usual modification in case q=∞.
Proposition 4.2.
The spacesL̇p,qs(𝒢),Ṫp,qs(𝒢) and  Ṗp,qs(𝒢), are left and right translation invariant. Precisely, we have
(4.4)‖L(z,r):L̇p,qs(G)⟶L̇p,qs(G)‖=rd(1/p-1/q)-s,‖R(z,r):L̇p,qs(G)⟶L̇p,qs(G)‖=rs+d/q,‖L(z,r):Ṫp,qs(G)⟶Ṫp,qs(G)‖=rd/q-s,‖R(z,r):Ṫp,qs(G)⟶Ṫp,qs(G)‖≤Crd/q+smax⁡{1,r-b(1+|z|)b},
where b>0 is a constant depending on d, p, and q, and
(4.5)‖L(z,r):Ṗp,qs,a(G)⟶Ṗp,qs,a(G)‖=rd(1/p-1/q)-s,‖R(z,r):Ṗp,qs,a(G)⟶Ṗp,qs,a(G)‖≤rs+d/qmax⁡{1,r-a}(1+|z|)a.Proof.
Step 1.
The left and right translation invariance ofL̇p,qs(𝒢) and Ṫp,qs(𝒢) was shown in [28, Lemma. 4.7.10].Step 2.
Let us considerṖp,qs,a(𝒢). Clearly, we have for F∈Ṗp,qs,a(𝒢)(4.6)‖L(z,r)F∣Ṗp,qs(G)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F((x+y-z)/r,t/r)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖=rd/p‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t/r)|(1+r|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖=rd(1/p-1/q)-s‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖.
Hence, we obtain
(4.7)‖L(z,r):Ṗp,qs,a(G)⟶Ṗp,qs(G)‖=rd(1/p-1/q)-s.
The right translation invariance is obtained by
(4.8)‖R(z,r)F∣Ṗp,qs,a(G)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F(x+tz+y,tr)|(1+|y|/t)a]qdttd+1)1/q∣Lp(Rd)‖=‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,tr)|(1+|y-tz|/t)a]qdttd+1)1/q∣Lp(Rd)‖=rs+d/q‖(∫0∞t-sq[sup⁡y∈Rd|F(x+y,t)|(1+|y-tz|r/t)a]qdttd+1)1/q∣Lp(Rd)‖.
Observe that
(4.9)sup⁡y∈Rd|F(x+y,t)|(1+|y-tz|r/t)a=sup⁡y∈Rd[|F(x+y,t)|(1+|y|/t)a⋅(1+|y|/t)a(1+|y-tz|r/t)a],(1+|y|/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.
This yields
(4.10)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
and consequently
(4.11)‖R(z,r):Ṗp,qs,a(G)⟶Ṗp,qs,a(G)‖≤rs+d/qmax⁡{1,r-a}(1+|z|)a.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 ofṖp,qs(𝒢).This gives room for generalizations, that is, replacing the space Lp(ℝd) by some weighted Lebesgue space Lp(ℝd,ω) in the definition.


## 4.2. New Old Coorbit Spaces

We start withℋ=L2(ℝd) and the representation(4.12)π(x,t)=TxDtL2,
which is unitary continuous on ℋ. This representation is not irreducible. However, if we restrict to radial functions g∈L2(ℝd), then span{π(x,t)g:(x,t)∈𝒢} is dense in L2(ℝd). 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 transformWgf, see Appendix A.1 in the appendix. Recall the abstract definition of the space ℋw1 and 𝒜w 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 ℋw1 is nontrivial.Lemma 4.4.
If the weight functionw(x,t)≥1 satisfies the condition
(4.13)w(x,t)≤(1+|x|)r(ts+t-s′)
for some r,s,s′≥0, then
(4.14)S0(Rd)↪Hw1.This is a kind of minimal condition which we need in order to define coorbit spaces in a reasonable way. Instead of(Hw1)~, one may use 𝒮0′(ℝd) as reservoir and a radial g∈𝒮0(ℝd) 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 wY in (3.16))(iv)(4.15)wY(x,t)≲(1+|x|)r(ts+t-s′)for somer,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) For1≤p,q≤∞, and s∈ℝ,we have Ḃp,qs(ℝd)=CoL̇p,qs+d/2-d/q(𝒢),
(ii) for1≤p<∞, 1≤q≤∞ and s∈ℝwe have
(4.16)Ḟp,qs(Rd)=CoṪp,qs+d/2(G)
(iii) and if additionallya>d/min{p,q}, we obtain
(4.17)Ḟp,qs(Rd)=CoṖp,qs+d/2-d/q,a(G).Proof.
Theorem4.5 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 Ṫp,qs(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 spacesṪp,qs(𝒢) obsolete for this issue. We showed that Y=Pp,qs,a(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 in𝒢 given by 𝒰=[0,α]d×[β-1,1], where α>0 and 1<β. Furthermore, we consider the discrete set of points(4.18){xj,k=(αkβ-j,β-j):j∈Z,k∈Zd}.
Then the family {xj,k𝒰}j∈ℤ,k∈ℤd defines a partition of 𝒢. Indeed,(4.19)xj,kU=Qj,k×[β-(j+1),β-j],where(4.20)Qj,k=[αk1β-j,α(k1+1)β-j]×⋯×[αkdβ-j,α(kd+1)β-j].Note that in this case the spacesY♯ and Yb coincide. We will further use the notation(4.21)χj,k(x)={1:x∈Qj,k0:otherwise.Definition 4.7.
LetY be a function space on 𝒢 satisfying Section 3/(i), (ii), (iii). We put
(4.22)Y♯(α,β)={{λj,k}j,k:‖{λj,k}j,k∣Y♯(α,β)‖<∞},
where
(4.23)‖{λj,k}j,k∣Y♯(α,β)‖=‖∑j,k|λj,k|χj,k(x)χ[βj,βj+1](t)∣Y‖.Theorem 4.8.
Let1≤p,q≤∞, s∈ℝ,and a>d/min{p,q}. Then
(4.24)‖{λj,k}j,k∣(Ṗp,qs,a)♯(α,β)‖∽‖(∑l∈Z∑k∈Zdβl(s+d/q)q|λl,k|qχl,k(x))1/q∣Lp(Rd)‖,‖{λj,k}j,k∣(L̇p,qs)♯(α,β)‖∽(∑l∈Zβl(s+d/q-d/p)q(∑k∈Zd|λl,k|p)q/p)1/q.Proof.
We prove the first statement. The proof for the second one is even simpler. Let(4.25)F(x,t)=∑j,k|λj,k|χj,k(x)⋅χ[β-(j+1),β-j](t).
Discretizing the integral over t by t~β-ℓ, we obtain
(4.26)‖F∣Y‖=‖(∫0∞t-sq[sup⁡w|F(x+w,t)|(1+|w|/t)a]qdttd+1)1/q∣Lp(Rd)‖≍‖(∑l∈Zβl(s+d/q)q∫β-(l+1)β-l[sup⁡w|F(x+w,t)|(1+βl|w|)a]qdtt)1/q∣Lp(Rd)‖.
With t∈[β-(ℓ+1),β-ℓ], we observe
(4.27)F(x,t)=∑k|λl,k|χl,k(x)
and estimate
(4.28)‖F∣Y‖≤‖(∑l∈Zβl(s+d/q)qsup⁡w∈Rd1(1+βl|w|)a[∑k|λl,k|χl,k(x+w)]q)1/q∣Lp(Rd)‖.
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:
(4.29)‖F∣Y‖≤‖(∑l∈Z[∑kβl(s+d/q)r|λl,k|rsup⁡w∈Rdχl,k(x+w)(1+βl|w|)ar]q/r)1/q∣Lp(Rd)‖.
The next observation is the useful estimate(4.30)sup⁡w|χl,k(x+w)|(1+βl|w|)ar≲1(1+βl|x-kβ-l|)ar≲(χl,k(⋅)*βld(1+βl|⋅|)ar)(x).
Indeed, the first estimate is obvious. Let us establish the second one
(4.31)(χl,k(⋅)*1(1+βl|⋅|)ar)(x)=∫|yi-kiβ-l|≤β-li=1,…,d1(1+βl|x-y|)ardy≳∫|y|≤cβ-l1(1+βl|x-kβ-l-y|)ardy≳∫|y|≤cβ-l1(1+βl|x-kβ-l|+βl|y|)ardy≳β-ld∫01ud-1(1+βl|x-kβ-l|+u)ardu≳β-ld(1+βl|x-kβ-l|)ar.
Note, that the functions
(4.32)gl(x)=βld(1+βl|⋅|)ar
belong to L1(ℝd) with uniformly bounded norm where we need ar>d. Putting (4.31) and (4.30) into (4.29), we obtain
(4.33)‖F∣Y‖r≤‖(∑l∈Z[gl*∑k∈Zdβl(s+d/q)r|λl,k|rχl,k)(x)]q/r)r/q∣Lp/r(Rd)‖.
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(ℝd)-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 Lp/r(ℓq/r) situation. This gives
(4.34)‖F∣Y‖r≲‖(∑l∈Z[∑k∈Zdβl(s+d/q)r|λl,k|rχl,k(x)]q/r)r/q∣Lp/r(Rd)‖=‖(∑l∈Z∑k∈Zdβl(s+d/q)q|λl,k|qχl,k(x))1/q∣Lp(Rd)‖r
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
(4.35)sup⁡w|F(x+w,t)|(1+βl|w|)a≥|F(x,t)|,
which results in
(4.36)‖F∣Y‖≳‖(∑l∈Zβl(s+d/q)q∫β-(l+1)β-l|F(x,t)|qdtt)1/q∣Lp(Rd)‖.
A further use of (4.27) gives finally
(4.37)‖F∣Y‖≳‖(∑l∈Z∑k∈Zdβl(s+d/q)q|λl,k|qχl,k(x))1/q∣Lp(Rd)‖.
The proof is complete.


## 4.4. Atomic Decompositions

The following theorem is a direct consequence of the abstract results in Theorems3.11, and 3.13.Theorem 4.9.
Let1≤p,q≤∞,  a>d/min{p,q}, and  s∈ℝ. Let further  g∈𝒮0(ℝd) 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
(4.38){gj,k}j∈Z,k∈Zd={TαkβjDβjL2g}
has the following properties.(i)
{gj,k}j∈ℤ,k∈ℤd forms a Banach frame for CoL̇p,qs(𝒢) and CoṖp,qs,a(𝒢), that is, we have a dual frame {ej,k}j∈ℤ,k∈ℤd⊂𝒮0(ℝd)with f=∑j∈ℤ,k∈ℤd〈gj,k,f〉ej,k and the norm equivalences
(4.39)‖f∣CoL̇p,qs(G)‖≍‖〈gj,k,f〉∣(L̇p,qs(G))♯(α,β)‖,f∈CoL̇p,qs(G)
as well as
(4.40)‖f∣CoṖp,qs,a(G)‖≍‖〈gj,k,f〉∣(Ṗp,qs,a(G))♯(α,β)‖,f∈CoṖp,qs,a(G).(ii)
{gj,k}j∈ℤ,k∈ℤd is an atomic decomposition, that is, for f∈CoṖp,qs,a(𝒢), we have a (not necessary unique) decomposition ∑j∈ℤ,k∈ℤdλj,k(f)gj,k such that
(4.41)‖{λj,k(f)}j,k∣(Ṗp,qs,a(G))♯(α,β)‖≲‖f∣CoṖp,qs,a(G)‖.
Conversely, if {λj,k}j∈ℤ,k∈ℤd∈(Ṗp,qs,a(𝒢))♯(α,β), then f=∑j∈ℤ,k∈ℤdλj,kgj,k converges and belongs to CoṖp,qs,a(𝒢) and moreover,
(4.42)‖f∣CoṖp,qs,a(G)‖≲‖{λj,k}j,k∣(Ṗp,qs,a(G))♯(α,β)‖
(analogously for CoL̇p,qs(𝒢)). Convergence is considered in the strong topology if the finite sequences are dense in (Ṗp,qs,a(𝒢))♯(α,β) and in the weak*-topology otherwise.Remark 4.10.
(i) Since the analyzing function or atomg can be chosen arbitrarily, we allow more flexibility here than in the results given inFrazier/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 onℝd 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 Ψc, c∈E. Our aim is to specify, that is, give sufficient conditions to Ψ0, Ψ1, such that (A.19) represents an unconditional basis in Bp,qs(ℝd)and Fp,qs(ℝd). We will apply the abstract Theorem 3.14.In order to do so we need to have (3.17) for all functions Ψc. 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.
LetL∈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  (SK) and  Ψ1 is supposed to satisfy  (D),  (SK), and  (ML-1). For r1,r2∈ℝ,the weight  w(x,t) is given by
(4.43)w(x,t)=(1+|x|)v(tr2+t-r1),(x,t)∈G.
If now
(4.44)r1<min⁡⁡{L,K}-d2,r2<min⁡⁡{L,K}+d2-v,
then we have
(4.45)∫Rd∫0∞sup⁡(y,s)∈(x,t)V|〈π(y,s)Ψc,Ψc〉|w(x,t)dttd+1dx<∞.Proof.
With LemmaA.3, we obtain for WΨ1Ψ1 the following estimates:
(4.46)|(WΨ1Ψ1)(s,t)|≲tmin⁡{L,K}+1/2(1+t)2min⁡{L,K}+1⋅1(1+|s|/(1+t))N.
And in addition
(4.47)|(WΨiΨi)(s,t)|≲t1/2(t+1)1(1+|s|/(1+t))N,i=1,2.
Hence, for any c∈E, the tensor product structure gives (assume without restriction that cd=1)
(4.48)|WΨcΨc(x,t)|≲tmin⁡{L,K}(1+t)2min⁡{L,K}⋅td/2(1+t)d∏i=1d1(1+|xi|/(1+t))N.
The expression sup(y,s)∈(x,t)V|WΨcΨc(y,s)| can be estimated similar
(4.49)sup⁡(y,s)∈(x,t)V|WΨcΨc(y,s)|=sup⁡|yi-xi|≤tt/2≤s≤t|WΨcΨc(y,s)|≲tmin⁡{L,K}(1+t)2min⁡{L,K}td/2(1+t)d∏i=1d1(1+|xi|/(1+t))N≲tmin⁡{L,K}(1+t)2min⁡{L,K}td/2(1+t)d1(1+|x|/(1+t))N.
Fubini’s theorem and a change of variable yields
(4.50)∫Rd∫0∞sup⁡(y,s)∈(x,t)V|〈π(y,s)Ψc,Ψc〉|w(x,t)dttd+1dx≲∫0∞tmin⁡{L,K}(1+t)2min⁡{L,K}td/2⋅(tr2+t-r1)dttd+1.
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.
LetL∈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  (SK), and  Ψ1 is supposed to satisfy  (D),  (SK) and  (ML-1). (a) If1≤p,q≤∞ and
(4.51)-min⁡⁡{L,K}+dp<s<min⁡⁡{L,K}-d(1-1p),
then (A.19) is a Banach frame for Ḃp,qs(ℝd) in the sense of (3.22).(b) If1≤p,q<∞ and
(4.52)-min⁡⁡{L,K}+2dmax⁡⁡{1p,1q}<s<min⁡⁡{L,K}-dmax⁡{1p,1q,1-1p},
then (A.19) is a Banach frame for Ḟp,qs(ℝd)in the sense of (3.22).Proof.
Let us prove (a). First of all, we apply Theorem4.5/(i). Afterwards, we use Proposition 4.2 in order to estimate the weight wY(y,t) for Y=L̇p,qs+d/2-d/q(𝒢). We obtain
(4.53)wY(x,t)=max⁡⁡{td(1/p-1/2)-s,ts-d(1/p-1/2),ts+d/2,t-s+d/2}≤{t-r1:0<t<1,tr2:t≥1.
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
(4.54)0≤s<min⁡⁡{L,K}-d(1-1/p).
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
(4.55)-min⁡⁡{L,K}+d/p<s<0.
Finally (4.54), (4.55), and Theorem 3.14 yield (a).
Step 2.
We prove (b). We apply Theorem4.5/(iii) and afterwards Proposition 4.2 and obtain for Y=Ṗp,qs+d/2-d/q,a(𝒢)(4.56)wY(x,t)=max⁡{td(1/p-1/2)-s,ts-d(1/p-1/2),ts+d/2max⁡{1,t-a}(1+|x|a),t-s+d/2max⁡{t-a,ta}(1+|x|)a}≤(1+|x|)a{t-r1:0<t<1,tr2:t≥1.
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
(4.57)0≤s<min⁡{L,K}-max⁡⁡{a,d(1-1p)}
which can be rewritten to
(4.58)0≤s<min⁡{L,K}-dmax⁡{1p,1q,1-1p}
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
(4.59)-min⁡⁡{L,K}+2a<s<0
and can be rewritten to
(4.60)-min⁡{L,K}+2dmax⁡{1/p,1/q}<s<0.
This yields the lower bound in (b) and we are done.The following corollary is a direct consequence of Theorem4.12 and the facts in  Appendix A.2.Corollary 4.13.
Letm>0 and  (Ψ0,Ψ1)=(φm,ψm) the spline wavelet system of order  m. Then (a) If1≤p,q≤∞ and -m+1+dp<s<m-1-d(1-1p),
then (A.19) is a Banach frame for Ḃp,qs(ℝd) in the sense of (3.22);
(b) If1≤p,q≤∞ and
(4.61)-m+1+2dmax⁡{1p,1q}<s<m-1-dmax⁡{1p,1q,1-1p},
then (A.19) is a Banach frame for Ḟp,qs(ℝd) 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∈Cu(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 orderN 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.



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*Source: 163213-2012-03-07.xml*