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 $\dot{\mathbf{F}}^{\alpha}_{p,q}$ for the endpoint case of $p = \infty$ and the full scale of parameters $\alpha \in \mathbb{R}$ and $q \in (0,\infty]$. In particular, a Peetre-type characterization of the anisotropic Besov space $\dot{\mathbf{B}}^{\alpha}_{\infty,\infty} = \dot{\mathbf{F}}^{\alpha}_{\infty,\infty}$ is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $\dot{\mathbf{F}}^{\alpha}_{\infty,q}$.

1.1. 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(sB) 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. Section 2.1).
The following Peetre-type maximal characterizations ofḞ α ∞,q will be proven in Section 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).
For q ∈ (0, ∞) and β > 1/q, the norm equivalences 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 submean-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 Section 4 the coincidenceḞ α ∞,∞ =Ḃ 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 (1.8) A vector ψ ∈ L 2 (R d ) is called admissible if the associated wavelet transform defines an isometry into L 2 (G A ). The existence of admissible vectors and associated Calderóntype 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 , which play an important role in this paper. A (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 Section 6 for further details. It should be mentioned that any family (π(γ)φ) γ∈Γ for suitable φ ∈ L 2 (R d ) defines a molecular system in the sense of (6.1) with Φ = |W ψ φ|, but that generally a molecular system (φ γ ) γ∈Γ does not need to consist of translates and dilates of a fixed function φ. Nevertheless, general molecules (φ γ ) γ∈Γ share many properties with atoms (π(γ)φ) γ∈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 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 Γ ⊂ G A satisfying there exists a molecular system (φ γ ) γ∈Γ such that any f ∈Ḟ α ∞,q admits the expansion where the series converges unconditionally in the weak * -topology of S ′ (R d )/P(R d ).
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∈Γ and molecules (φ γ ) 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. There exists a compact unit neighborhood U ⊂ G A with the following property: For any discrete set Γ ⊂ G A satisfying there exists a molecular system (φ γ ) γ∈Γ in span{π(γ)ψ : γ ∈ Γ} such that the distribution 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) ≤ Cf 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 : R d → C and a matrix A ∈ R d×d , the dilation of 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.
The following lemma collects several basic properties of expansive matrices that will be used in the sequel, see, e.g., [1, Definitions 2.3 and 2.5] and [1, Lemma 2.2]. For a more general background on spaces of homogeneous type on R d , we refer to [7,8].
(i) There exist an ellipsoid Ω = Ω A (that is, Ω = 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 Equation (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) = Ω. Hence, its metric balls are of the form Throughout this paper, given an expansive A ∈ GL(d, R), we will fix an ellipsoid Ω = Ω 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].
For our purposes, it will be convenient to use the metric ball Ω = 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.
The Peetre-type maximal function has the following basic properties.
, the following holds: The symmetry of ρ A and the inequality (2.3) for ν β (y) = (1 + ρ A (y)) β yield that for arbitrary x ∈ R d . Consequently, Since the constants are independent of z ∈ R d , the claim follows easily.
3.2. Sub-mean-value property. The following type of result is often referred to as a "submean-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 Φ := N k=−N ϕ k * ψ k satisfies see, e.g., the proof of [20, Theorem 3.5] for a detailed verification.
Using that Φ 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 Equation (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.
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.
3.3. 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 A = exp(B) for some B ∈ R d×d . Then the power A s = exp(sB) is well-defined for s ∈ R.
For ϕ ∈ S(R d ) and s, β ∈ R with β > 0, the associated Peetre-type maximal function of 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 Peetre-type maximal functions. The result forms an extension of [6, Theorem 1] to possibly anisotropic dilations.
Then, for all q ∈ (0, ∞), α ∈ R and β > 1/q, the norm equivalences , since ϕ has infinitely many vanishing moments and hence P * ϕ s = 0 for every P ∈ P(R d ).) 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 Ḟα ∞,q can be bounded by the middle term of (3.5). This step is modeled on Step 1 of the proof of [20,Theorem 3.5]. By the calculations constituting [20, Equations (3.7)-(3.11)], it follows that, for t ∈ [0, 1], there exists If q < ∞, then raising (3.7) to the q-th power and integrating over t Let w ∈ R d be arbitrary and set Q ℓ = A ℓ Ω + w. By Lemma 2.1, it follows that Q ℓ ⊂ Q ℓ+N . Therefore, averaging over Q ℓ gives Consequently, taking the q-th root and the supremum over ℓ ∈ 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,Equation (3.15)], for t ∈ [0, 1], there exists N = N (A, ϕ) ∈ N such that 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 ℓ ⊂ Q ℓ+N and 1 m(Q ℓ ) = | det A| N 1 m(Q ℓ+N ) in the last step. Taking the supremum over all ℓ ∈ 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 Ḟα We start with using Proposition 3.2 for the exponent 0 < q/r < ∞, where r := √ βq > 1 by assumption. This gives for all x ∈ R d and ℓ ∈ Z For fixed, but arbitrary x ∈ R d and ℓ ∈ 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 ℓ (x) ⊂ Q ℓ+M (w) for all w ∈ R d and ℓ ∈ Z whenever x ∈ Q ℓ (w). Therefore, for all w ∈ R d , ℓ ∈ Z and x ∈ Q ℓ (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 where B = B ρ A (z, s) ranges over all metric balls containing x.
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 ℓ (w) that Thus, averaging over x ∈ Q ℓ (w) and applying the maximal inequalities for L r (R d ) (see, e.g., Lastly, taking the suprema over w ∈ R d and ℓ ∈ 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 ℓ+k+1 (x), where again δ := βq/r − 1 > 0. Note that | det A| −δ(j+ℓ) ≤ 1 for j ≥ −ℓ, which implies that 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 |(f * ϕ j )(y)| q dy, and consequently where we used the index shift ℓ ′ = ℓ + k + 1 in the penultimate estimate. Since the implicit constants are independent of w ∈ R d and ℓ ∈ Z, it follows that Overall, combining the estimates (3.11) and (3.13) finishes the proof.
Proof. The inequality f Ḟα , the norm equivalences of Theorem 3.3 will be used; see also Remark 3.4. For this, suppose β > 1. Then, for all ℓ ∈ Z, w ∈ R d , we see that In particular, the inequality (4.1) implies that, for every ℓ ∈ Z and w ∈ R d there exists Hence, for all z ∈ Q ℓ (w), it holds that Since w∈R d Q ℓ (w) = R d for fixed, but arbitrary, ℓ ∈ Z, we see by combining (4.2) and (4. for all z ∈ R d and ℓ ∈ Z, which showsḞ α ∞,∞ (A, ϕ) =Ḃ α ∞,∞ (A, ϕ) with equivalent norms. An analogous argument using (3.5) , which completes the proof. 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 vector ψ is called admissible if W ψ : 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 : L 2 (R d ) → L 2 (G A ) intertwines the action of π and left translation L h F = F (h −1 ·) on L 2 (G A ), also yields that 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. 1,9,16,28]). Let A ∈ GL(d, R) be an exponential matrix. The following assertions are equivalent:

In addition, if A is an expansive matrix, then the admissible vector
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,   20]). Let ψ ∈ S 0 (R d ) be an admissible vector. Then, for arbitrary f ∈ S ′ 0 (R d ) and ϕ ∈ S 0 (R d ), In particular, the identity W ϕ f = W ψ f * W ϕ ψ holds.

5.2.
Peetre-type spaces. As in [20], we define an auxiliary class of Peetre-type spacesṖ α,β ∞,q on the semi-direct 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 spacesṖ α,β ∞,q will only be defined through averages over small scales.
Definition 5.4. Let A ∈ GL(d, R) be expansive and exponential and let Ω = Ω A be an associated ellipsoid as provided by Lemma 2.1. For ℓ ∈ Z and w ∈ R d , set Q ℓ (w) := A ℓ Ω + w.
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].
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 x = A −t (x − y), w = A −t (w − y), and z = A −t z, as well as s = s − t, yield 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, Ω) ∈ N such that A −t ′ Ω ⊂ A N Ω for all t ′ ∈ [0, 1), and hence Q ℓ−t ( w) ⊂ Q ℓ−k+N ( w). Increasing the upper limit of the inner integral from ℓ − t to ℓ − k + N and substituting ℓ = ℓ − 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 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 ℓ + t ≤ ℓ + k + 1 and clearly Q ℓ (w) = A ℓ Ω + w ⊂ A ℓ+k+1 Ω + w = Q ℓ+k+1 (w) by Lemma 2.1. Hence, and consequently for t > 0, which completes the proof.
Lastly, we mention the following r-norm property of Peetre-type spaces.

5.3.
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 Peetre-type 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.
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.
We split the proof into three steps and consider only q ∈ (0, ∞). The case q = ∞ follows with the usual adjustments. Set α ′ := α + 1/2 − 1/q. The first two steps are essentially identical to the first part of the proof of [20,Proposition 5.11], but included for completeness.
Step 1. Note that ψ * ∈ S 0 (R d ) also satisfies (2.6), where ψ * (t) = ψ(−t). A direct calculation gives W ψ f (x, s) = f, π(x, s)ψ = f, | det A| s/2 T x ψ −s = | det A| s/2 f * ψ * −s (x). 6.1. Molecular systems. In addition to the left-sided local maximal function defined in Equation (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,Section 2] and [32, Section 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.
The following lemma provides an extended duality pairing for S ′ 0 (R d ). See [20, Lemma 6.8] for its proof. Suppose f ∈ S ′ 0 (R d ) satisfies W ψ f ∈ L ∞ 1/w (G A ) and φ ∈ L 2 (R d ) satisfies W ψ φ ∈ L 1 w (G A ). Then the extended pairing defined by f, φ ψ := G A W ψ f (g)W ψ φ(g) dµ G A (g) ∈ C is well-defined and independent of the choice of admissible vector ψ ∈ S 0 (R d ).
6.2. 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 Γ be any family in G A such that sup g∈G A #(Γ ∩ gU ) < ∞.