Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces F˙p,qα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\textbf{F}}^{\alpha }_{p,q}$$\end{document} for the endpoint case of p=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = \infty $$\end{document} and the full scale of parameters α∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \mathbb {R}$$\end{document} and q∈(0,∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in (0,\infty ]$$\end{document}. In particular, a Peetre-type characterization of the anisotropic Besov space B˙∞,∞α=F˙∞,∞α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\textbf{B}}^{\alpha }_{\infty ,\infty } = \dot{\textbf{F}}^{\alpha }_{\infty ,\infty }$$\end{document} is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in F˙∞,qα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\textbf{F}}^{\alpha }_{\infty ,q}$$\end{document}.


Introduction
In a previous paper [20], we obtained characterizations of anisotropic Triebel-Lizorkin spacesḞ α p,q , with α ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞], in terms of Peetre-type maximal functions and continuous wavelet transforms. In addition, as an application of these characterizations, it was shown that these spaces admit molecular dual frames and Riesz sequences. The purpose of the present paper is to provide analogous results for the endpoint case of p = ∞. For defining the anisotropic Triebel-Lizorkin spaces, let A ∈ GL(d, R) be an expansive matrix, i.e., |λ| > 1 for all λ ∈ σ (A), and let ϕ ∈ S(R d ) be such that it has compact Fourier support (1.1) and satisfies with A * denoting the transpose of A. For j ∈ Z, denote its dilation by ϕ j = | det A| j ϕ(A j ·). The associated (homogeneous) anisotropic Triebel-Lizorkin spacė F α ∞,q =Ḟ α ∞,q (A, ϕ), with α ∈ R and q ∈ (0, ∞], is defined as the collection of all tempered distributions f ∈ S (R d ) (modulo polynomials) that satisfy if q < ∞, and In contrast to the usual quasi-norms defining (anisotropic) Triebel-Lizorkin spaceṡ F α p,q for p < ∞ (see, e.g., [3][4][5]20]), the quantities (1.3) and (1.4) consider only averages over small scales. The quasi-norms (1.3) and (1.4) can therefore be considered as "localized versions"of the immediate analogue of the quasi-norms definingḞ α p,q for p < ∞, which would lead to an unsatisfactory definition ofḞ α ∞,q , see [11,Sect. 5] and the references therein.

Maximal characterizations
As in [20], we assume additionally that the expansive matrix A ∈ GL(d, R) is exponential, in the sense that A = exp(B) for some matrix B ∈ R d×d . The power of A is then defined as A s := exp(s B) for s ∈ R.
For ϕ ∈ S(R d ), s ∈ R and β > 0, the associated Peetre-type maximal function of f ∈ S (R d ) is defined by where ϕ s := | det A| s ϕ(A s ·) and where ρ A : R d → [0, ∞) denotes the step homogeneous quasi-norm associated with A (cf. Sect. 2.1).
The following Peetre-type maximal characterizations ofḞ α ∞,q will be proven in Sect. 3. Here, the notation − Q means the average integral over a measurable set Q ⊆ R d of positive measure. Theorem 1.1 Let A ∈ GL(d, R) be expansive and exponential and let α ∈ R. Assume that ϕ ∈ S(R d ) has compact Fourier support satisfying the support conditions (1.1) and (1.2).
The proof method of Theorem 1.1 is modeled on the proof of the maximal characterizations of Triebel-Lizorkin spacesḞ α p,q with p < ∞ given in [20]. In particular, it combines a sub-mean-value property of the Peetre-type maximal function (Proposition 3.2) with maximal inequalities. See also [22,23,27,31] for similar approaches. Besides the similarities in the approach, the calculations in the proof of Theorem 1.1 differ non-trivially from these in [20,Theorem 3.5] as only averages over small scales appear in the definition ofḞ α ∞,q . As a consequence of Theorem 1.1, we show in Sect. 4 the coincidenceḞ α ∞,∞ = B α ∞,∞ mentioned above.

Molecular decompositions
For an expansive and exponential matrix A ∈ GL(d, R), denote by G A = R d A R the associated semi-direct product group. Then G A acts unitarily on L 2 (R d ) via the quasi-regular representation π , defined by defines an isometry into L 2 (G A ). The existence of admissible vectors and associated Calderón-type reproducing formulae for this representation have been studied, among others, in [9,12,17,21]. The assumption that A is expansive is essential for the existence of admissible vectors ψ ∈ S(R d ) satisfying ψ ∈ C ∞ c (R d \ {0}) (cf. Lemma 5.2), which play an important role in this paper.
A countable family (φ γ ) γ ∈ of functions φ γ ∈ L 2 (R d ) parametrized by a discrete set ⊂ G A is called a molecular system if there exists a function ∈ W(L r (1.9) The space W(L r w ) denotes a weighted Wiener amalgam space for r = min{1, q} and the standard control weight w = w α ,β ∞,q : G A → [1, ∞); see Sect. 6 for further details. It should be mentioned that any family (π(γ )φ) γ ∈ for suitable φ ∈ L 2 (R d ) defines a molecular system in the sense of (6.1) with = |W ψ φ|, but that generally a molecular system (φ γ ) γ ∈ does not need to consist of translates and dilates of a fixed function φ. Nevertheless, general molecules (φ γ ) γ ∈ share many properties with atoms (π(γ )φ) γ ∈ , see, e.g., [18,32].
The following theorem provides decomposition theorems ofḞ α ∞,q in terms of molecules. Theorem 1.2 Let A ∈ GL(d, R) be expansive and exponential. For α ∈ R, q ∈ (0, ∞], let r := min{1, q} and let α = α + 1/2 − 1/q if q < ∞ and α = α + 1/2, otherwise. Let β > 1/q if q < ∞ and β > 1 otherwise. Lemma 5.8. Additionally, suppose that W ϕ ψ ∈ W(L r w ) for some (equivalently, all) admissible ϕ ∈ S 0 (R d ). Then there exists a compact unit neighborhood U ⊂ G A with the following property: For any discrete set ⊂ G A satisfying there exists a molecular system (φ γ ) γ ∈ such that any f ∈Ḟ α ∞,q admits the expansion where the series converges unconditionally in the weak * -topology of Atomic decompositions of the anisotropic spacesḞ α ∞,q have been obtained earlier by Bownik [3]. However, Theorem 1.2 provides a frame decomposition of all elements f ∈Ḟ α ∞,q in terms of the atoms (π(γ )ψ) γ ∈ and molecules (φ γ ) γ ∈ , whereas the atoms in [3,Theorem 5.7] depend on the element f ∈Ḟ α ∞,q that is represented. For anisotropic Triebel-Lizorkin spacesḞ α p,q with p < ∞, decompositions as in Theorem 1.2 were obtained in [19,20], but they appear to be new for the case of p = ∞. In fact, Theorem 1.2 seems even valuable for merely isotropic dilations, where the state-of-the-art [13] excludes the case p = ∞. Theorem 1.2 will be obtained from the recent results on dual molecules [26,32] through the identification ofḞ α ∞,q with a coorbit space; see Proposition 5. 10. This identification appears to be new for the full scale ofḞ α ∞,q with α ∈ R and q ∈ (0, ∞], even for isotropic dilations.
In addition to the existence of dual molecular frames, we also obtain a corresponding result for Riesz sequences. Here, the spaceṗ −α ,β ∞,q denotes a sequence space associated toḞ α ∞,q ; see Definition 6.3 for its precise definition.
Riesz sequences inḞ α ∞,q seem not to have appeared in the literature before, which makes Theorem 1.3 new even for isotropic dilations. Similarly to Theorem 1.2, we obtain Theorem 1.3 by applying results of [26,32] to the coorbit realization of the Triebel-Lizorkin spacesḞ α ∞,q .

Notation
We denote by s + := max{0, s} and s − := − min{0, s} the positive and negative part of s ∈ R. If f 1 , f 2 are positive functions on a common base set X , the notation f 1 f 2 is used to denote the existence of a constant C > 0 such that f 1 (x) ≤ C f 2 (x) for all x ∈ X . The notation f 1 f 2 is used whenever both f 1 f 2 and f 2 f 1 . We will sometimes use α to indicate that the implicit constant depends on a quantity α.
For a function f : The class of Schwartz functions on R d will be denoted by S(R d ). Its dual space is simply denoted by S (R d ). Moreover, the notation P(R d ) will be used for the collection of polynomials on R d , and we write S (R d )/P(R d ) for the quotient space of tempered distributions modulo polynomials. The Fourier transform F : . Similar notations will be used for the extension of the Fourier transforms to L 2 (R d ).
The Lebesgue measure on R d is denoted by m. For a measurable set Q ⊂ R d of finite, positive measure, it will be written If G is a group, then the left and right translation of a function F : G → C by h ∈ G will be denoted by L h F = F(h −1 ·) and R h F = F(· h), respectively. In addition, we write F ∨ (x) = F(x −1 ).

Anisotropic Triebel-Lizorkin spaces with p = ∞
This section provides preliminaries on expansive matrices and Triebel-Lizorkin spaces.

Expansive matrices
The following lemma collects several basic properties of expansive matrices that will be used in the sequel, see, e.g., [ [7,8].
(i) There exist an ellipsoid = A (that is, = P(B 1 (0)) for some P ∈ GL(d, R)) and r > 1 such that is Borel measurable and forms a quasi-norm, i.e., there exists C ≥ 1 such that For an expansive matrix A ∈ GL(d, R), a function ρ A : R d → [0, ∞) defined by Eq. (2.1) will be called a step homogeneous quasi-norm associated to A. Given y ∈ R d and r > 0, its associated metric ball will be denoted by It is readily verified that B ρ A (0, 1) = . Hence, its metric balls are of the form Throughout this paper, given an expansive A ∈ GL(d, R), we will fix an ellipsoid = A as appearing in Lemma 2.1 (i). This choice is not unique. Any other choice of ellipsoid will yield an equivalent quasi-norm, see, e.g., [1, Lemma 2.4].

Analyzing vectors
Let A ∈ GL(d, R) be expansive. Choose a function ϕ ∈ S(R d ) with compact Fourier support satisfying, in addition, Then the function ψ ∈ S(R d ) defined by is well-defined and satisfies For more details and further properties, see, e.g., [5, Lemma 3.6]
For our purposes, it will be convenient to use the metric ball = B ρ A (0, 1) instead of the cube [0, 1] d in definingḞ α ∞,q . The independence of this choice is guaranteed by the following lemma, whose simple proof follows from a standard covering argument and is hence omitted.
with implicit constants only depending on d, A.

Maximal function characterizations
Throughout this section, A ∈ GL(d, R) will denote an expansive matrix.

Peetre-type maximal function
Let ϕ ∈ S(R d ). For j ∈ Z and β > 0, the associated Peetre-type maximal function of The Peetre-type maximal function has the following basic properties.
, the following holds: Consequently, Since the constants are independent of z ∈ R d , the claim follows easily.

Sub-mean-value property
The following type of result is often referred to as a "sub-mean-value property"and will play an essential role in deriving the main results. It forms an anisotropic analogue of the isotropic result [30,Theorem 5].
Step 1. (The case q ∈ [1, ∞)). By the compact Fourier support condition (2.5) and the identity (2.7), it follows that there exists N ∈ N (depending on ϕ and A) such that the function : see, e.g., the proof of [20, Theorem 3.5] for a detailed verification. Using that j = N k=−N ϕ j+k * ψ j+k for j ∈ Z, the equality (3.3) gives for all x, z ∈ R d , j ∈ Z and β > 0. Combining Hölder's inequality for 1 q + 1 q = 1 and the translation invariance of L q (R d ) yields If q ∈ (1, ∞), applying the transformation A j y →ỹ in the L q -norm above gives Using the seminorms defined in Eq. (2.4) and the fact that ϕ, ψ ∈ S(R d ), the last integral can be estimated by a constant C k = C k (A, ϕ, q, β) > 0. Combining the above gives with implicit constant depending on A, β, q, ϕ. Taking the supremum over z ∈ R d and the q-th power yields the claim for q ∈ (1, ∞). The case q = 1 follows by the same arguments with the usual modifications.
Step 2. (The case q ∈ (0, 1)). For f ∈ S (R d ), the estimate obtained in Step 1 (for q = 1) gives For the case β < N , we use the already proven result for N to obtain where the second inequality used that The right-hand side being independent of z ∈ R d , taking the supremum yields the claim for β < N . Overall, this completes the proof.

Maximal function characterizations
In this section, the matrix A ∈ GL(d, R) is often additionally assumed to be exponential, i.e., it is assumed that A admits the form The following theorem forms a main result of this paper. It characterizes the anisotropic Triebel-Lizorkin spacesḞ α ∞,q , with 0 < q ≤ ∞, in terms of Peetretype maximal functions. The result forms an extension of [6, Theorem 1] to possibly anisotropic dilations.
is expansive and exponential. Assume that ϕ ∈ S(R d ) has compact Fourier support and satisfies (2.5) and (2.6).

Remark 3.4 The proof of Theorem 3.3 shows that the discrete characterizations
also hold without the assumption that A ∈ GL(d, R) is exponential.

Proof of Theorem 3.3
Only the cases q ∈ (0, ∞) will be treated; the case q = ∞ follows by the arguments for q = 1, with the usual modification to accommodate the supremum. The proof is split into three steps and for some parts we refer to calculations from the proof of [20,Theorem 3.5].
Throughout the proof, we will make use of the equivalent norms provided by Lemma 2.2.
Step 1. In this step it will be shown that f Ḟ α ∞,q can be bounded by the middle term of (3. If q < ∞, then raising (3.7) to the q-th power and integrating over t Let w ∈ R d be arbitrary and set Q = A + w. By Lemma 2.1, it follows that Q ⊂ Q +N . Therefore, averaging over Q gives Consequently, taking the q-th root and the supremum over ∈ Z and w ∈ R d yields the desired estimate.
Step 2. In this step we estimate the middle term by the right-most term of (3.5). This requires discretizing the inner-most integral, which works analogously to Step 2 in the proof of [20,Theorem 3.5]. By [20,Eq. (3.15)], for t ∈ [0, 1], there exists Starting with the inner-most integral of the middle term in (3.5), we use a simple periodization argument and (3.8) to obtain

Taking the averaged integral over
where we used Q ⊂ Q +N and 1 m(Q ) = | det A| N 1 m(Q +N ) in the last step. Taking the supremum over all ∈ Z and w ∈ R d and the q-th root yields the claim for q ∈ (0, ∞).
Step 3. Lastly, it will be shown that the right-most term of (3.5) can be bounded by f Ḟ α ∞,q . We start with using Proposition 3.2 for the exponent 0 < q/r < ∞, where r := √ βq > 1 by assumption. This gives for all For fixed, but arbitrary x ∈ R d and ∈ Z, we partition Combining this with the simple fact that (a + b) r a r + b r for a, b ≥ 0 yields In the remainder, the series defining S 1 and S 2 will be estimated.
Step 3.1. We look at the sum S 1 first. Note that since | det A| > 1, there exists M ∈ N such that | det A| M ≥ 2C, where C > 0 denotes the constant in the triangle-inequality for ρ A (cf. Lemma 2.1). A straightforward computation shows that Q (x) ⊂ Q +M (w) for all w ∈ R d and ∈ Z whenever x ∈ Q (w). Therefore, for all w ∈ R d , ∈ Z and x ∈ Q (w), it follows that To estimate the terms I 1 and I 2 , we will use the (anisotropic) Hardy-Littlewood maximal operator for locally integrable f : R d → C given by

s) ranges over all metric balls containing x.
For estimating I 1 , note that For estimating I 2 , note that ρ A (A j (x − y)) = | det A| m for y ∈Q m− j+1 (x) by definition of ρ A (see (2.1)). This and setting δ := βq/r − 1 implies where the last inequality used that δ = βq/r − 1 > 0. Here, the implicit constant only depends on A, q and β.
Combining (3.9) and (3.10) shows for x ∈ Q (w) that Thus, averaging over x ∈ Q (w) and applying the maximal inequalities for L r (R d ) (see, e.g., [ Lastly, taking the suprema over w ∈ R d and ∈ Z yields with implicit constant depending on A, ϕ, q, and β. Step 3.2. In this step, we deal with the sum S 2 . Recall again that, for y ∈Q +k+1 (x), ρ A ((A j (x − y))) = | det A| j+k+ . Hence, where we used Minkowski's integral inequality (see, e.g., [29,Appendix 1]) to obtain the last line. An application of Jensen's inequality to the integral yields and consequently where we used the index shift = + k + 1 in the penultimate estimate. Since the implicit constants are independent of w ∈ R d and ∈ Z, it follows that Overall, combining the estimates (3.11) and (3.13) finishes the proof.
∞,∞ , the norm equivalences of Theorem 3.3 will be used; see also Remark 3.4. For this, suppose β > 1. Then, for all ∈ Z, w ∈ R d , we see that In particular, the inequality (4.1) implies that, for every ∈ Z and w ∈ R d there exists Since w∈R d Q (w) = R d for fixed, but arbitrary, ∈ Z, we see by combining (4.2) and (4. An analogous argument using (3.5) gives f Ḃ α ∞,∞ f Ḟ α ∞,q , which completes the proof.

Wavelet coefficient decay and Peetre-type spaces
Throughout this section, let A ∈ GL(d, R) be an exponential matrix. Define the associated semi-direct product Left Haar measure on G A is given by dμ G A (x, s) = | det A| −s ds dx and the modular function on G A is G A (x, s) = | det A| −s . To ease notation, we will often write μ := μ G A .

Wavelet transforms
The group G A = R d A R acts unitarily on L 2 (R d ) by means of the quasi-regular representation π , defined by For a fixed vector ψ ∈ L 2 (R d )\{0}, the associated wavelet transform W ψ : The following lemma is a special case of [ The significance of an admissible vector ψ is that W * ψ W ψ = I L 2 (R d ) , and hence that the weak-sense integral formula holds for every f ∈ L 2 (R d ). This, combined with fact that W ψ : intertwines the action of π and left translation L h F = F(h −1 ·) on L 2 (G A ), also yields that 2) for all f , ϕ ∈ L 2 (R d ). The identity (5.2) will be referred to as a reproducing formula.
Henceforth, it will always be assumed that A ∈ GL(d, R) is both exponential and expansive. This is essential for the existence of admissible vectors ψ ∈ S(R d ) with compact Fourier support, as the following result shows.
We will also need to use wavelet transforms of distributions. For this, consider the subspace of S(R d ) given by and equip it with the subspace topology of S(R d ). Its topological dual space will be denoted by S 0 (R d ) and will often be identified with S (R d )/P(R d ), see, e.g., [14,Proposition 1.1.3]. The dual bracket between S 0 and S 0 (R d ) is denoted by Note that this pairing is conjugate-linear in the second variable. For a fixed ψ ∈ S 0 (R d ) \ {0}, the extended wavelet transform of f ∈ S 0 (R d ) is defined as By the continuity of the map (x, The reproducing formula (5.2) can be naturally extended to S 0 (R d ). See [20, Lemma 4.7 and Lemma 4.8] for a proof of the following result.

Peetre-type spaces
As in [20], we define an auxiliary class of Peetre-type spacesṖ α,β ∞,q on the semidirect product G A = R d R. These spaces are an essential ingredient for identifying Triebel-Lizorkin spaces with associated coorbit spaces [10,32].
In contrast to the spacesṖ α,β p,q defined in [20, Definition 5.1] for p < ∞, the spaceṡ P α,β ∞,q will only be defined through averages over small scales. A ∈ GL(d, R) be expansive and exponential and let = A be an associated ellipsoid as provided by Lemma 2.1. For ∈ Z and w ∈ R d , set

Definition 5.4 Let
For α ∈ R, β > 0, and q ∈ (0, ∞), the Peetre-type spaceṖ α,β ∞,q (G A ) is defined as the space of all (equivalence classes of a.e. equal) measurable F : The following lemma collects some basic properties ofṖ α,β ∞,q (G A ) and gives explicit estimates for the operator norm of left and right translation. The estimates involve the following weight function v : The function v is measurable and submultiplicative by [20, Lemma 5.2].

is left-and right-translation invariant and there exists N =
if q < ∞, and otherwise. Here, the operator norm is written |||·||| := · Ṗ α,β Proof Most of the quasi-norm properties for · Ṗ α,β ∞,q can be easily verified from the definition, while the positive definiteness follows from [20,Lemma B.1]. It is clear that · Ṗ α,β ∞,q is solid and for q ≥ 1 a norm. The completeness follows from · Ṗ α,β ∞,q satisfying the Fatou property (see [34,Sect.  , which is easily verified by a straightforward computation using Fatou's lemma, see, e.g., the proof of [20,Lemma 5.3]. It remains to prove the translation invariance and associated norm estimates. We will only consider q ∈ (0, ∞), the arguments for q = ∞ are analogous. To this end, let F ∈Ṗ α,β ∞,q (G A ) and (y, t) ∈ R d × R be arbitrary. Then the substitutions To estimate this further, we decompose t = k + t with k ∈ Z and t ∈ [0, 1). By [20,Lemma 2.4], there exists N = N (A, ) ∈ N such that A −t ⊂ A N for all t ∈ [0, 1), and hence Q −t ( w) ⊂ Q −k+N ( w). Increasing the upper limit of the inner integral from − t to − k + N and substituting = − k + N ∈ Z gives For the right-translation, a direct calculation using the substitutions z = z + A s y and s = s + t shows that where we used the fact that 1 For t ≤ 0, the claimed estimate follows immediately. For t > 0, we again write t = k + t with k ∈ N 0 and t ∈ [0, 1). Then + t ≤ + k + 1 and clearly for t > 0, which completes the proof.
Lastly, we mention the following r -norm property of Peetre-type spaces.

Standard envelope and control weight
We recall the definition of a standard envelope given in [20,Definition 5.5]. A central notion in the theory of coorbit spaces [10,32] is that of a so-called control weight. In the following lemma, we show the existence of such a weight for Peetretype spaces and show that it can be estimated by standard envelopes as defined in Definition 5.7. The construction of the control weight follows [20,Lemma 5.7], but besides the slightly different parameters, the case distinction for the right translation needs to be accommodated with a few extra terms. The details are as follows.

Coorbit spaces
The aim of this section is to show that Triebel-Lizorkin spacesḞ α ∞,q can be identified with so-called coorbit spaces [10,32] by use of Theorem 3.3.
The definition of coorbit spaces requires the notion of a local maximal function. For a function F ∈ L ∞ loc (G A ), its (left-sided) maximal function is defined by where Q ⊂ G A is a relatively compact unit neighborhood.
The space Co(Ṗ α,β ∞,q ) as defined in Definition 5.9 is complete with respect to the quasi-norm · Co(Ḟ α ∞,q ) . In addition, its definition is independent of the chosen defining vector ψ and unit neighborhood Q, with equivalent norms for different choices. These basic properties follow from the general theory [32]; see [20,Remark 5.10] for details and references.
The following is the key result of this section.
for any f ∈ S 0 (R d ).
Step 2. The estimate |W ψ f | ≤ M L Q W ψ f a.e. implies that Step 3. We prove the remaining estimate f In [20,Equation (5.12)], we already showed that with implicit constant depending on A, α, β, q and N , where N ∈ N depends on the support of ψ. Combining this with Theorem 3.3 yields where we used the fact that m(A +N ) m(A ) = | det A| N N 1 in the last step.

Molecular systems
In addition to the left-sided local maximal function defined in Eq. (5.3), we will also need a two-sided version, defined by M Q F(g) = ess sup u,v∈Q |F(ugv)|, g ∈ G A , for F ∈ L ∞ loc (G A ), with Q ⊂ G A being a fixed relatively compact unit neighborhood. For r = min{1, q} with q ∈ (0, ∞] and the standard control weight w = w α,β ∞,q : G A → [1, ∞) provided by Lemma 5.8, define the associated Wiener amalgam space W(L r w ) by The space W(L r w ) is independent of the choice of neighborhood Q and is complete with respect to the quasi-norm F W(L r w ) := M Q F L r w ; see, e.g., [24,Sect. 2] and [32,Sect. 2].
The space W(L r w ) provides the class of envelopes that will be used for defining molecules. Definition 6.1 Let ψ ∈ L 2 (R d ) be a non-zero vector such that W ψ ψ ∈ W(L r w ) for the standard control weight w : G A → [1, ∞) defined in Lemma 5.8.
Then the extended pairing defined by is well-defined and independent of the choice of admissible vector ψ ∈ S 0 (R d ).

Sequence spaces
The following definition provides a class of sequence spaces that will be used in the molecular decomposition of anisotropic Triebel-Lizorkin spaces.

Definition 6.3
Let U ⊂ G A be a relatively compact unit neighborhood with non-void interior and let be any family in G A such that sup g∈G A #( ∩ gU ) < ∞.

Proofs of Theorems 1.2 and 1.3
Theorems 1.2 and 1.3 will be obtained from corresponding results for abstract coorbit spaces [32].
Step 1. Under the assumptions, it follows by an application of Proposition 5.10 thatḞ α ∞,q = Co ϕ (Ṗ −α ,β ∞,q ). For applying the relevant results of [32], it will next be shown that Co ϕ (Ṗ −α ,β ∞,q ) can be identified with the abstract coorbit spaces used in [32]. To this end, note thatṖ −α ,β ∞,q is a solid, translation invariant quasi-Banach space by Lemma 5.5, which satisfies the r -norm property by Lemma 5.6. The standard control weight w = w in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.