Classification of anisotropic Triebel-Lizorkin spaces

This paper provides a characterization of expansive matrices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in \textrm{GL}(d, {\mathbb {R}})$$\end{document}A∈GL(d,R) generating the same anisotropic homogeneous Triebel–Lizorkin space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\textbf{F}}^{\alpha }_{p,q}(A)$$\end{document}F˙p,qα(A) for \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}α∈R and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q \in (0,\infty ]$$\end{document}p,q∈(0,∞]. It is shown that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\textbf{F}}^{\alpha }_{p,q}(A) = \dot{\textbf{F}}^{\alpha }_{p,q}(B)$$\end{document}F˙p,qα(A)=F˙p,qα(B) if and only if the homogeneous quasi-norms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _A, \rho _B$$\end{document}ρA,ρB associated to the matrices A, B are equivalent, except for the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\textbf{F}}^0_{p, 2} = L^p$$\end{document}F˙p,20=Lp with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in (1,\infty )$$\end{document}p∈(1,∞). The obtained results complement and extend the classification of anisotropic Hardy spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^p(A) = \dot{\textbf{F}}^{0}_{p,2}(A)$$\end{document}Hp(A)=F˙p,20(A), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in (0,1]$$\end{document}p∈(0,1], in Bownik (Mem Am Math Soc 164(781):vi+122, 2003).


Introduction
Let A ∈ GL(d, R) be an expansive matrix and consider an analyzing vector ϕ ∈ S(R d ) for A, that is, a Schwartz function ϕ : where A * denotes the transpose of A. Denote its L 1 -normalized dilation by ϕ i := | det A| i ϕ(A i •) for i ∈ Z.For α ∈ R and p, q ∈ (0, ∞], the associated anisotropic homogeneous Triebel-Lizorkin space Ḟα p,q (A) on R d is defined to consist of all tempered distributions f ∈ S ′ (R d ) (modulo polynomials) with finite quasi-norm f Ḟα p,q (A) , defined by , p ∈ (0, ∞), with the usual modifications for q = ∞, and For the scalar dilation matrix A = 2 • I d , the spaces Ḟα p,q (A) defined above coincide with the usual homogeneous Triebel-Lizorkin spaces on R d as studied in, e.g., [15,16,22].For this particular case, the Triebel-Lizorkin spaces provide a unifying scale of function spaces that encompasses, among others, the Lebesgue, Sobolev, Hardy and BMO spaces.The anisotropic Triebel-Lizorkin spaces Ḟα p,q (A) associated to a general expansive matrix A were first introduced in [6] and further studied in, e.g., [1,4,5,8,[18][19][20].These anisotropic spaces are useful for the analysis of mixed homogeneity properties of functions and operators as the dilation structure allows different directions to be scaled by different dilation factors.Among others, the anisotropic Triebel-Lizorkin spaces include Lebesgue spaces and various anisotropic/parabolic versions of Hardy and BMO spaces as studied in, e.g., [2,7,[9][10][11]14].See these papers (and the references therein) for further motivation for considering anisotropic function spaces.
In the present paper, the main objective is to characterize when two expansive matrices induce the same anisotropic Triebel-Lizorkin space.For the special case of anisotropic Hardy spaces H p (A) (= Ḟ0 p,2 (A)) with p ∈ (0, 1], a full solution to this problem has been obtained in [2].Explicitly, it is shown in [2, Section 10] that H p (A) = H p (B) for some (equivalently, all) p ∈ (0, 1] if and only if two homogeneous quasi-norms ρ A , ρ B : R d → [0, ∞) associated to the expansive matrices A, B are equivalent, in the usual sense of quasi-norms.See also [7] for a slightly corrected version and [13] for an extension of the classification result of [2] to Hardy spaces with variable anisotropy.Analogous to these results on Hardy spaces, a classification of anisotropic Besov spaces [3] has more recently been obtained in [12].The aim of this paper is to provide a complementary characterization for the scale of Triebel-Lizorkin spaces.
1.1.Main results.The first key result obtained in this paper is the following rigidity theorem.Here, as well as below, two expansive matrices A and B are called equivalent if they have equivalent homogeneous quasi-norms; see Sections 2.1 and 2.2 for precise definitions.
The following theorem provides a converse to Theorem 1.1.
A combination of Theorem 1.1 and Theorem 1.2 provides a full characterization of two expansive matrices inducing the same anisotropic Triebel-Lizorkin space.This characterization extends the classification of anisotropic Hardy spaces [2] to the full scale of Triebel-Lizorkin spaces, while complementing the classification of anisotropic Besov spaces [12] with a counterpart for Triebel-Lizorkin spaces.
In effect, the aforementioned classification theorems translate the problem of comparing function spaces into the comparison of homogeneous quasi-norms.For this latter problem, explicit and verifiable criteria in terms of spectral properties of the involved dilation matrices can be given, see, e.g., [2,Section 10], [12,Section 7] and [7,Section 4].
As an illustration of Theorem 1.1, we note that a matrix B ∈ GL(d, R) is equivalent to the scalar dilation A = 2 • I d if and only if B is diagonalizable over C with all eigenvalues equal in absolute value, see, e.g., [2,Example,p.7].Combined with Theorem 1.1, this shows that for matrices B that are not of this special form, Ḟα p,q (A) = Ḟα p,q (B), unless α = 0, p ∈ (1, ∞) and q = 2.In particular, the (homogeneous) Sobolev spaces L p α (= Ḟα p,2 (A)) with 1 < p < ∞ and α = 0 do not coincide with Ḟα p,2 (B) for non-diagonalizible matrices B.
Lastly, let us mention an application of Theorem 1.2.In [18,19], we proved continuous maximal characterizations of anisotropic Triebel-Lizorkin spaces Ḟα p,q (A) and obtained new results on their molecular decomposition.These results were obtained under the additional assumption that the expansive matrix A is exponential, in the sense that A = exp(C) for some matrix C ∈ R d×d .Theorem 1.2 implies that this additional assumption does not restrict the scale of anisotropic Triebel-Lizorkin spaces.Indeed, since there always exists an expansive and exponential matrix B that is equivalent to the given expansive matrix A (cf. [12,Section 7]), it follows by Theorem 1.2 that Ḟα p,q (A) = Ḟα p,q (B) for all α ∈ R and p, q ∈ (0, ∞]. The formulation (1.1) of the equivalence of matrices A and B is what is actually used in the proofs of our main results, as we expand upon next.
Necessary conditions.In the proof of Theorem 1.1, we show the asserted equivalence of two matrices A and B by showing that the criterion (1.1) holds.For this, we first carefully construct auxiliary functions in Ḟα p,q (A) = Ḟα p,q (B) whose Fourier supports are contained in finitely many of the sets (A * ) i k Q and (B * ) j k P , where i k , j k ∈ Z, of appropriate homogeneous covers (A * ) i Q i∈Z and (B * ) j P j∈Z .Then, using adequate estimates of the norms of these auxiliary functions (see Section 4.2), it is shown directly that (1.1) must hold for the case α = 0, in which case A and B must be equivalent.The proof strategy for the case α = 0 is similar, but requires some additional arguments and tools.For p < ∞, it is shown using the Khintchine inequality that necessarily q = 2 whenever A and B are not equivalent.For p = ∞, we use dual norm characterizations of Triebel-Lizorkin norms to conclude that A and B must be equivalent.
Sufficient conditions.In the proof of Theorem 1.2, the criterion (1.1) is used to control the overlap of the Fourier supports of the A-dilates and B-dilates of the analyzing vectors ϕ and ψ, respectively, that are used to define the spaces Ḟα p,q (A) and Ḟα p,q (B).Combined with our maximal characterizations of Triebel-Lizorkin spaces obtained in [18,19], this allows to conclude that the analyzing vectors ϕ and ψ for A respectively B define the same space Ḟα p,q (A) = Ḟα p,q (B).
As mentioned above, the used criterion (1.1) for equivalent matrices stems from [12], where it was used for the purpose of classifying anisotropic Besov spaces.For the actual comparison of function spaces, the approach of [12] consists of showing that an anisotropic Besov space can be identified with a (Besov-type) decomposition space [24], which allows to apply the embedding theory [24] developed by the third named author.In contrast, the Triebel-Lizorkin spaces considered in this paper cannot be directly treated in the framework [24]; in particular, our main theorems cannot be easily deduced from [24].Some of our arguments for proving Theorem 1.1 are, however, inspired by ideas used in [24], most notably the use of the Khintchine inequality.Nevertheless, all of our calculations and estimates differ non-trivially from corresponding arguments in [24] as the latter concerns Besov-type norms, which are technically easier to deal with than the Triebel-Lizorkin norms considered in this paper.1.3.Organization.The overall structure of this paper is as follows: Section 2 collects various notions and results related to expansive matrices and associated homogeneous covers.The essential background on anisotropic Triebel-Lizorkin spaces is contained in Section 3. Theorem 1.1 is proven in Section 4, whereas Section 5 provides the proof of Theorem 1.2.Lastly, some technical auxiliary results are postponed to two appendices.The Schwartz space on R d is denoted by S(R d ) and S ′ (R d ) denotes its dual, the space of tempered distributions.For f ∈ S ′ (R d ) and g ∈ S(R d ), we define f, g := f (g), so that the dual pairing •, • is sesquilinear, in agreement with the inner product on L 2 (R d ).The subspace of S(R d ) consisting of functions with all moments vanishing (i.e., ´xα f (x) dx = 0 for all α ∈ N d 0 ) is denoted by S 0 (R d ).The dual space S ′ 0 (R d ) is often identified with the quotient S ′ /P of S ′ (R d ) and the space of polynomials P(R d ).Lastly, the space of smooth compactly supported functions on an open set U ⊆ R d is as usual denoted by C ∞ c (U ).For a function f : R d → C, its translation T y f and modulation M y f by y ∈ R d are defined by The notation Ff := f is also sometimes used.
For two functions f, g : X → [0, ∞) on a set X, we write f g whenever there exists C > 0 such that f (x) ≤ Cg(x) for all x ∈ X.We simply use the notation f ≍ g whenever f g and g f .We also write A B for the inequality A ≤ CB, where C > 0 is constant independent of A and B. In case the implicit constant in depends on a quantity α, we also sometimes write α .

Expansive matrices and homogeneous covers
This section collects background on expansive matrices and homogeneous quasi-norms.A standard reference for most of the presented material is [2].

Expansive matrices.
In the sequel, we will primarily work with the so-called step homogeneous quasi-norm ρ A associated to A, defined as where  As a corollary of the previous lemma (see also [12,Remark 4.9]), we see that equivalence of expansive matrices is preserved under taking transposes.
Corollary 2.2.Two expansive matrices A and B are equivalent if and only if A * and B * are equivalent.

Homogeneous covers. Let
is called a homogeneous cover induced by A. Given two homogeneous covers (A i Q) i∈Z and (B j P ) j∈Z induced by A, B ∈ GL(d, R), we define for fixed i, j ∈ Z.
The index sets defined in Equation (2.4) can be used for characterizing the equivalence of two expansive matrices as the following lemma shows.See [12,Lemma 6.2] for a proof.

Lemma 2.3 ([12]
).Let A, B ∈ GL(d, R) be expansive and let (A i Q) i∈Z and (B j P ) j∈Z be associated induced covers of R d \ {0}.Then the step homogeneous quasi-norms ρ A and ρ B are equivalent if and only if sup In addition to Lemma 2.3, we will also make use of more refined estimates on the cardinalities of the index sets defined in Equation (2.4).We provide the required estimates in the following two lemmata.The provided proofs follow arguments in the proof of Lemma 2.3 (cf.[12,Lemma 6.2]) closely, but are included here for completeness.Lemma 2.4.Let A, B ∈ GL(d, R) be two equivalent expansive matrices and Q, P ⊆ R d open such that Q, P are compact in R d \ {0}.Then there exists C > 0 such that Hence, by homogeneity of ρ A , ρ B and the assumption of their equivalence, it follows that where max x∈P {ρ B (x)}/ min x∈Q {ρ A (x)} is finite by Equation (2.1) as Q, P are compact in R d \ {0}.The left inequality of (2.5) follows analogously by using that which completes the proof.
We also need the following estimates involving parameters α, β ∈ R.
then there exists N ∈ N such that, for all i, j ∈ Z, Setting |α| ln(| det A|) + 1, it follows that The desired inclusion for J i is obtained analogously with N 2 :=  Proof.This follows from Lemma 2.4 and Lemma 2.5 with α = β = 1.
Lastly, for a single homogeneous cover (A i Q) i∈Z , we also define the index set Note that N i (A) coincides with the index sets in (2.4) for the choice A = B and Q = P .Therefore, the following is a direct consequence of Corollary 2.6.
Then there exists N ∈ N such that, for all i ∈ Z,

Anisotropic Triebel-Lizorkin spaces
Throughout this section, let A ∈ GL(d, R) be expansive and Ω A be an associated ellipsoid.

3.2.
Triebel-Lizorkin spaces.Let ϕ ∈ S(R d ) be a fixed A-analyzing vector.For i ∈ Z, with the usual modifications for q = ∞.The space Ḟα ∞,q (A) consists of all f ∈ S ′ /P such that In [6] and the introduction of this paper, the spaces Ḟα ∞,q (A), q ∈ (0, ∞), are alternatively defined using the cube [0, 1] d instead of an expansive ellipsoid Ω A .However, it is easily seen that both conditions define the same space, see, e.g., [19,Lemma 2.2].See also Theorem 3.1 below for the equivalent norm on Ḟα ∞,∞ (A) used in the introduction.Each space Ḟα p,q (A) is continuously embedded into S ′ /P and is complete with respect to the quasi-norm • Ḟα p,q .In addition, Ḟα p,q (A) is independent of the choice of A-analyzing vector ϕ, with equivalent quasi-norms for different choices.See [6, Section 3] and [4, Section 3.3] for details.We will often simply write • Ḟα p,q (A) for • Ḟα p,q (A;ϕ) whenever the precise choice of analyzing vector ϕ does not play a role in our arguments.
The following theorem provides characterizations of Triebel-Lizorkin spaces in terms of Peetre-type maximal functions and will play a key role in Section 5. See [18,19] for proofs.

Necessary conditions
This section is devoted to the proof of the following theorem involving necessary conditions for coincidence of two Triebel-Lizorkin spaces.This theorem corresponds to Theorem 1.1 in the introduction.
In the proof of Theorem 4.1, we will often actually use the norm equivalence rather than the coincidence of the spaces Ḟα . By a standard density argument, the norm equivalence (4.1) is equivalent to the same condition being satisfied for all elements in a dense subspace.Both facts are contained in the following simple lemma, which will often be used without further mentioning.
For the second part of the lemma, recall that n=1 is a Cauchy sequence in Ḟβ p 2 ,q 2 (B) converging to some g ∈ Ḟβ p 2 ,q 2 (B).Since convergence in Ḟα p 1 ,q 1 (A), respectively Ḟβ p 2 ,q 2 (B), implies weak convergence in S ′ /P (cf.Section 3.2), it follows that f = g ∈ Ḟβ p 2 ,q 2 (B).This shows Ḟα p 1 ,q 1 (A) ⊆ Ḟβ p 2 ,q 2 (B).The reverse inclusion is shown similarly.4.1.Preparations and notation.This section sets up some essential objects and notation that will be used for the proof of Theorem 4.1.This notation will be kept throughout Section 4.
Let A, B ∈ GL(d, R) be expansive matrices.Fix analyzing vectors ϕ ∈ S(R d ) and ψ ∈ S(R d ) satisfying Equation (3.3) for A and B, respectively.Then are open, relatively compact sets in R d \ {0}.In the following, we mainly consider the covers (A * ) i Q i∈Z and (B * ) j P j∈Z of R d \{0}.In particular, we will take the sets I j and J i defined in Equation (2.4) to be defined with respect to these two coverings; explicitly, this means for i, j ∈ Z. Furthermore, for i ∈ Z, we will use the index sets As shown in Corollary 2.7, there exists N = N (A, B, Q, P ) ∈ N satisfying Throughout, we fix such an N and define the functions In view of Equation (3.3) and because (A ), it follows that Φ ≡ 1 on Q and Ψ ≡ 1 on P .In particular, Φ and Ψ satisfy the analyzing vector conditions (3.1) and (3.2) for A and B, respectively.
In addition to the above, we fix throughout a non-zero function φ ∈ S(R d ) satisfying φ ≥ 0 and supp φ ⊆ B 1 (0).For δ > 0, define Then φ δ (ξ) = φ(ξ/δ) and thus supp φ δ ⊆ B δ (0).4.2.Norm estimates for auxiliary functions.This subsection consists of two estimates of the Triebel-Lizorkin norms of functions with specific Fourier support.These functions play an essential role in our proof of Theorem 4.1 and will be used in the following subsections.
with an implicit constant independent of i 0 and f .
Proof.With notation as in Section 4.1, we start by collecting some basic facts about the convolutions f * ϕ i and f * Φ i 0 for f as in the statement of the proposition.First, note that since For the convolution f * Φ i 0 observe that Φ i 0 ≡ 1 on (A * ) i 0 Q by construction, and therefore In the remainder of this proof, we deal with the cases p < ∞, p = ∞ and q < ∞, and p = q = ∞ separately.
Case 2: p = ∞, q ∈ (0, ∞).As in the previous case, we use Equation (4.6) for the upper estimate.This yields For the lower bound, we use the continuous embedding Ḟα A;Φ) , and Equation (3.5) to obtain where the final step follows from Equation (4.7).
| det A| αi 0 f L ∞ has been shown in the previous case already.For the reverse, observe that (3.5) and (4.6) yield which completes the proof.
The following simple consequence is what actually will be used in obtaining necessary conditions for the coincidence of Triebel-Lizorkin spaces.
Using the estimates of Proposition 4.3 for f δ Ḟα p 1 ,q 1 (A) and f δ Ḟβ , with implicit constants independent of i, j, δ, δ 0 .
The following proposition provides a more technical version of Proposition 4.3 and involves a linear combination of functions with Fourier supports in (A * ) i k Q for suitable points i k ∈ Z.The proof strategy resembles the one of Proposition 4.3, but requires various technical modifications.
Proof.Using the notation from Section 4.1, we first state some basic observations for f * ϕ i and f * Φ i k with f as in the statement.First, note that by assumption (a) it follows that supp ).Furthermore, note that for fixed i ∈ Z, there can be at most one point i k such that |i − i k | ≤ N due to the pairwise minimal distance between the chosen points i 1 , ..., i K .This implies that The remainder of the proof is divided into three cases and deals with p < ∞, p = ∞ and q < ∞, and p = q = ∞ separately.
Case 1: p ∈ (0, ∞).For the upper bound in Equation (4.8), set M = d p + 1.Then, in view of Equation (4.9), an application of Lemma A.3 with ℓ = i k shows that , with the usual modification of the argument for q = ∞.Consequently, this yields where the last step used that M > d p , so that ´Rd (1 + |x|) −M p dx < ∞.
Proof.We prove the three assertions separately.
(i) Since ϕ, ψ ∈ S(R d ) are analyzing vectors for A resp.B, it follows that Hence, there exist i 0 , j 0 ∈ Z such that (A * ) i 0 Q ∩ (B * ) j 0 P = ∅.By Corollary 4.4, this implies the existence of some δ 0 > 0 such that for, all 0 < δ ≤ δ 0 , with implicit constant independent of δ.In turn, this implies which is only possible for p 1 = p 2 .
(ii) Under the assumption p 1 = p 2 = p, we show that where the implied constant is independent of K and c.This easily implies q 1 = q 2 .Let K ∈ N be arbitrary and let N ∈ N be as chosen in Equation (4.4).Recall the identity (4.11) and note that each image set (A * ) i Q, (B * ) j P for i, j ∈ Z is relatively compact and hence bounded.Therefore, it is not hard to see that there exist points η 1 , . . ., η K ∈ R d and increasing sequences (i k ) K k=1 and ( Since the sets Q, P are open, there exists δ 1 > 0 such that Additionally, by continuity of φ ∈ S(R d ), there exists δ 2 > 0 such that In combination, this shows that the assumptions of Proposition 4.5 are met for Ḟα p,q 1 (A; ϕ) with δ 0 := min{δ 1 , δ 2 }, η 1 , . . ., η K and (i k ) K k=1 , as well as for Ḟβ p,q 2 (B; ψ) with (j k ) K k=1 replacing the sequence (i k ) K k=1 .For showing the claim (4.12), let c ∈ C K and 0 < δ ≤ δ 0 be fixed.Then defining (iii) Assuming p 1 = p 2 = p, it follows by Corollary 4.4 that there exists C ≥ 1 such that We consider the cases α = 0 or β = 0, and α = 0 = β.Case 1: α = 0 or β = 0. Suppose first that α = 0.As a consequence of Equation (4.11), for all j ∈ Z, there needs to exist i ∈ Z such that (A * ) i Q ∩ (B * ) j P = ∅.Equation (4.15) implies therefore that | det B| βj ≤ C as α = 0. Since this holds for all j ∈ Z, and | det B| = 0, it follows that necessarily also β = 0.If β = 0, then also α = 0 by symmetry.
Case 2: α = 0 = β.Suppose that α = 0 = β.Then, by Equation (4.15), the assumptions of Lemma 2.5 are satisfied for (A * ) i Q i∈Z and (B * ) j P j∈Z .Hence, there exists M ∈ N such that with J i , I j as defined in Equation (4.3), we have where c = c(A, B) := ln | det A|/ ln | det B|.In particular, this implies that Therefore, an application of Lemma 2.3 implies that A * and B * are equivalent, and hence so are A and B by Corollary 2.2.
It remains to show that α = β.To see this, note that, for all j ∈ Z, it holds that Since | det B| = 0, this is only possible for β α = 1, and hence α = β as claimed.

4.4.
The case α = 0 and p < ∞.In this section, we prove the following theorem, showing that if two Triebel-Lizorkin spaces coincide and the matrices are not equivalent, then necessarily q = 2.The only shortcoming of this theorem is that it only applies when p < ∞.We will deal with the case p = ∞ in the following subsection.

.16)
If A and B are not equivalent, then q = 2.
In particular, if Ḟ0 p,q (A) = Ḟ0 p,q (B) and A and B are not equivalent, then q = 2.
The following observation will be key in proving Theorem 4.8.It provides a condition under which the hypotheses of Proposition 4.5 are satisfied for Ḟα p,q (A).
Lemma 4.9.Let A, B ∈ GL(d, R) be expansive and suppose that sup j∈Z |I j | = ∞, with I j as defined in Equation (4.3).Then, for every K ∈ N, there exist δ 0 > 0, j 0 ∈ Z, points η 1 , . . .η K ∈ R d , and a (strictly) increasing sequence i 1 , . . ., i K ∈ Z with |i k − i k ′ | > 2N for k = k ′ , where N ∈ N as in (4.4), such that the following assertions hold: In particular, the assumptions (a) and (b) of Proposition 4.5 are satisfied for Ḟα p,q (A).
We will now provide the proof of Theorem 4.8.

Proof of
By exchanging the roles of A and B if necessary, it may be assumed that sup j∈Z |I j | = ∞, so that the assumption of Lemma 4.9 is satisfied.Using Lemma 4.9, it will be shown that where the implied constant is independent of K and c.This easily implies q = 2.
The finer analysis in the case where α = 0 and q = 2 can be performed by using that Ḟ0 p,2 (A) coincides with the anisotropic Hardy space H p (A) and using the classification results of [2, Section 10].The details are as follows: Proof.Let p ∈ (0, ∞) and denote by H p (A) the anisotropic Hardy space introduced in [2].By [4,Theorem 7.1], it follows that Ḟ0 p,2 (A) = H p (A).Hence, if Ḟ0 p,2 (A) = Ḟ0 p,2 (B), then H p (A) = H p (B).

4.5.
The case α = 0 and p = ∞.This section provides the following theorem, which finishes the necessary conditions of Theorem 4.1.
B), then A and B are equivalent.
The following lemma will reduce the proof of Theorem 4.13 to the case q ≥ 1.
B) and the matrices A and B are not equivalent, then q ≥ 1.
Proof.The claim is trivial for q = ∞; therefore, we can assume that q < ∞.Since A and B are not equivalent, Corollary 2.2 and Lemma 2.3 again imply for the covers (A * ) i Q i∈Z and (B * ) j P j∈Z that sup where we may assume sup j∈Z |I j | = ∞ by interchanging A and B if necessary.
By duality, we now provide a proof of Theorem 4.13.
Proof of Theorem 4.13.Arguing by contradiction, we assume that A and B are not equivalent.Then Lemma 4.14 implies that q ≥ 1.First, suppose that q ∈ (1, ∞], so that its conjugate exponent q ′ satisfies q ′ ∈ [1, ∞).Then [5,Theorem 4.8] shows that Ḟ0 ∞,q (A) is the dual space of Ḟ0 1,q ′ (A) (with equivalent norms).Likewise, it follows that Ḟ0 ∞,q (B) is the dual space of Ḟ0 1,q ′ (B) (with equivalent norms).By the first part of Lemma 4.2, we have for Therefore, by the usual dual characterization of the norm, it holds that Second, if q = 1, then it follows directly from Proposition B.6 that In combination, for any q ∈ [1, ∞], this yields g Ḟ0 1,q ′ (A) ≍ g Ḟ0 1,q ′ (B) for all g ∈ S 0 (R d ).Since A and B are not equivalent, an application of Theorem 4.8 shows that q ′ = 2 and hence q = 2.But for p = 1, q = 2, the above norm equivalence holds on a common dense subset, hence Ḟ0 1,2 (A) = Ḟ0 1,2 (B) by the second part of Lemma 4.2.Now Theorem 4.11 implies that A and B need to be equivalent, a contradiction.

Sufficient conditions
This section is devoted to the sufficient conditions of Theorem 1.2 and consists of the proof of the following theorem.A key ingredient used in the proof is the maximal characterization of Triebel-Lizorkin spaces (see Theorem 3.1).Theorem 5.1.Let A, B ∈ GL(d, R) be two expansive matrices.If A and B are equivalent, then Ḟα p,q (A) = Ḟα p,q (B) for all p, q ∈ (0, ∞] and α ∈ R. Proof.Let A, B ∈ GL(d, R) be two equivalent expansive matrices.Suppose ϕ, ψ ∈ S(R d ) are analyzing vectors for A respectively B satisfying additionally Equation (3.3), i.e., so that Then (A * ) i Q i∈Z and (B * ) j P j∈Z are covers of R d \{0}.Furthermore, a straightforward calculation yields ϕ i = ϕ((A * ) −i •) and ψ j = ψ((B * ) −j •), and hence ϕ i ≡ 0 outside of (A * ) i Q and ψ j ≡ 0 outside of (B * ) j P .Since A and B are equivalent, so are A * and B * (cf.Corollary 2.2.)For fixed i ∈ Z, define Ψ i ∈ S(R d ) as where We will use (5.1) to obtain a pointwise estimate of the convolution products f * ϕ i , i ∈ Z, in terms of the Peetre-type maximal function ψ * * j,β f for a fixed β > max{1/p, 1/q}, defined by To bound the integral in (5.2), we note that, since ρ A , ρ B are equivalent, we have it follows that for every N ∈ N, there exists . Combining these observations with (2.1) easily yields with implicit constant independent of i ∈ Z and j ∈ J i .Using | det A| i ≍ | det B| j for j ∈ J i once again, it follows thus that for all i ∈ Z.
The remainder of the proof is split into three cases dealing with p < ∞, p = ∞ and q < ∞, and p = q = ∞ separately.
Case 1: p ∈ (0, ∞).We only prove this case for q ∈ (0, ∞), since analogous arguments using suprema yield the case for q = ∞.Hence, for q < ∞, raising (5.3) to the q-th power and summing over i ∈ Z results in where we used in the last step that sup i∈Z |J i | < ∞ by Lemma 2.3.Since Lemma 2.3 also implies sup j∈Z |I j | < ∞ for I j := {i ∈ Z : (A * ) i Q ∩ (B * ) j P = ∅}, it follows that Note that N 1 + 1 ≤ N 2 and hence ℓ 1 ≤ ℓ 2 .Therefore, we obtain where we used in the last step that Taking the q-th root and the supremum over ℓ 2 , ℓ ∈ Z and w ∈ R d yields where the last equivalence follows again from the maximal characterizations of Theorem 3.1.
Exchanging the roles of A and B yield the converse norm estimate, and therefore it yields that Ḟα ∞,q (A) = Ḟα ∞,q (B).
Case 3: p = q = ∞.By Equation ( 5.3), it follows that Combining this with Equation (3.5) yields where it is used that sup , and completes the proof.

Appendix A. Miscellaneous results
This section contains two results used in the proofs of the main theorems.The most important such result is the following convolution relation, which is [22, Proposition in Section 1.5.1] with the implied constant written out explicitly.A proof can also be found in [24,Theorem 3.4].
Proposition A. 1 ([22]).Let K 1 , K 2 ⊆ R d be compact and p ∈ (0, 1].If f, ψ ∈ S(R d ) satisfy supp ψ ⊆ K 1 and supp f ⊆ K 2 , then the following quasi-norm estimate holds: Corollary A.2. Let A ∈ GL(d, R) be expansive, let K ⊆ R d be compact, and let N ∈ N and p ∈ (0, 1).Then there exists a constant C = C(A, K, N, p) > 0 with the following property: Proof.By compactness of K ⊆ R d , there exists R = R(A, K, N ) > 0 such that Hence, an application of Proposition A.1 easily yields the claim.
The second result is the following technical estimate.

By rearranging, this shows
for all x ∈ R d .Hence, This easily implies the claim of the lemma.
This section provides a dual characterization for the norm of Ḟ0 1,∞ (A), which is used in the proof of Theorem 4.13.Its proof hinges on associated Triebel-Lizorkin sequence spaces for which we recall the basic objects first.
Let A ∈ GL(d, R) be an expansive matrix and let D A be the collection of all dilated cubes The following lemma provides a convenient cover for the union of elements of a tent and will be used in two proofs below.
Lemma B.1.There exists N = N (A, d) ∈ N such that for all D ∈ D A , we have where the inequality used that Using Equation (B.3), it follows therefore that as required.
The Triebel-Lizorkin sequence spaces ḟ 0 1,∞ (A) and ḟ 0 ∞,1 (A) are defined as the collections of all complex-valued sequences c = (c D ) D∈D A satisfying respectively.
The following simple characterization of f 0 ∞,1 (A) will be used below.This equivalence is already claimed in [4, Remark 3.5], but a short proof is included for the sake of completeness.

Lemma B.2. For all complex-valued sequences
where T (D ′ ) denotes the tent over D ′ ∈ D A .
Proof.First, note that interchanging the sum and integral in Equation (B.4) yields that which easily implies the claimed inequality in Equation (B.5).
For the reverse inequality, let which completes the proof.
For obtaining the actual dual characterization of the spaces ḟ 0 1,∞ (A) and ḟ 0 ∞,1 (A), the following lemma will be used.It is [17,Proposition 1.4] applied to the special case of dilated cubes; see also [23,Theorem 4] for the case of isotropic dilations.

Lemma B.3 ([17]
).Let a = (a D ) D∈D A be a fixed but arbitrary sequence of non-negative reals.Then for every C > 0, the following assertions are equivalent: for every subcollection D ′ A of the dilated cubes D A .(ii) For every sequence b = (b D ) D∈D A of non-negative reals, the estimate The significance of a Carleson sequence (B.7) for the purpose of the present paper is that it characterizes membership of ḟ 0 ∞,1 (A).Although this fact is well-known for isotropic dilations (cf.[17,23]), the anisotropic version requires some additional arguments due to the fact that dilated cubes are not necessarily nested.The details are provided in the next lemma.
Lemma B.4.Let A ∈ GL(d, R) be expansive and let (c D ) D∈D A be a complex-valued sequence.Then c ∈ ḟ 0 ∞,1 (A) if, and only if, there exists C > 0 such that with implicit constant independent of c.
Proof.First, it will be shown that if (c D ) D∈D A satisfies (B.8), then (c D ) D∈D A ∈ ḟ 0 ∞,1 (A).For this, let D ′ ∈ D A be arbitrary.Then for any C > 0 satisfying (B.8), we have, by Lemma B.1, C by Lemma B.2.This also implies in Equation (B.9).

Conversely, let D ′
A ⊆ D A be any subcollection.Note first that if, for all N ∈ N, there exists some D ′ ∈ D where the equality can be shown using the solidity of the associate norms and choosing sequences c ′ with appropriate (complex) signs; see also [26, Section 69, Theorem 1] for details.
In the following, we consider • (1) and • (2) in more detail.Starting with • (1) , we interpret the supremum as an infimum over all upper bounds.Then the characterizations of Lemma B. The final result of this section is the desired dual norm characterization of Ḟ0 1,∞ (A).
Proposition B.6.Let A ∈ GL(d, R) be expansive.Then, for all g ∈ S 0 (R d ), Proof.By [4, Theorem 3.12], there exists a function ψ ∈ S(R d ) with compact Fourier support such that the operator C ψ f = ( f, ψ D ) D∈D is bounded from Ḟ0 p,q (A) into ḟ 0 p,q (A) and furthermore the operator D ψ c = D∈D c D ψ D is bounded from ḟ 0 p,q (A) into Ḟ0 p,q (A) for all p, q ∈ (0, ∞].Moreover, their composition D ψ • C ψ is the identity on Ḟ0 p,q (A).Here, for Combining both facts with the estimate (B.10), it follows that For the reverse inequality, first note that since D ψ • C ψ is the identity on Ḟ0 1,∞ (A), and since these operators are bounded, we have and thus C ψ g ḟ 0 1,∞ (A) ≍ g Ḟ0 1,∞ (A) .Next, note that by Corollary B.5, there exists a sequence (c (n) ) n∈N in ḟ 0 ∞,1 (A) with c (n) ḟ 0 ∞,1 (A) ≤ 1 such that lim sup

1. 4 .
Notation.For a measurable set Ω ⊆ R d , we denote its Lebesgue measure by m(Ω) and the indicator function of Ω by 1 Ω .The notation | • | : R d → [0, ∞) is used for the Euclidean norm.The open Euclidean ball of radius r > 0 and center x ∈ R d is denoted by B r (x).The closure of a set Ω ⊆ R d will be denoted by Ω.

Corollary 2 . 6 .
Let A, B ∈ GL(d, R) be equivalent expansive matrices and Q, P ⊆ R d open such that Q, P are compact in R d \ {0}.Then there exists N ∈ N such that, for all i, j ∈ Z, J i ⊆ {j ∈ Z : |j − ⌊c i⌋| ≤ N } and I j ⊆ {i ∈ Z : |i − ⌊j/c⌋| ≤ N }, where c = c(A, B) := ln | det A|/ ln | det B|.

Lemma A. 3 .
Let A ∈ GL(d, R) be expansive, let M > 0, N ∈ N, and Q ⊆ R d be bounded.Further, let ϕ, φ as in Section 4.1.Then there exists a constant C = C(d, M, N, Q, φ, ϕ, A) > 0 with the following property: If i, ℓ ∈ Z and δ > 0 are such that |i − ℓ| ≤ N and B associated to A. The scale of a dilated cube D = A i ([0, 1] d +k) ∈ D A is defined as scale(D) = i; alternatively, scale(D) = log | det A| m(D).The tent over D ∈ D A is defined as T (D) := D ′ ∈ D A : m(D ′ ∩ D) > 0 and scale(D ′ ) ≤ scale(D) .
[12,Methods.An essential ingredient in our proof of Theorem 1.1 and Theorem 1.2 is a simple characterization of the equivalence of two expansive matrices A and B in terms of properties of the associated covers (A * ) i Q i∈Z and (B * ) j P j∈Z of R d \ {0}, where P, Q ⊆ R d \ {0} are suitable relatively compact sets; see[12, Lemma 6.2] and Section 2.3.Explicitly, this criterion asserts that two expansive matrices A, B are equivalent if and only if the associated homogeneous covers (A * ) i Q i∈Z and (B * ) j P j∈Z satisfy sup i∈Z Ω A is the fixed expansive ellipsoid (2.2); see [2, Definition 2.5].Two expansive matrices A, B ∈ GL(d, R) are called equivalent if the associated step homogeneous quasi-norms ρ A and ρ B are equivalent.Note that, by Equation (2.3), two expansive matrices are equivalent if and only if all of their associated quasi-norms are equivalent.The following characterization is [2, Lemma 10.2].
Theorem 4.8.If A and B are not equivalent, then neither are A * and B * (cf.Corollary 2.2).Hence, an application of Lemma 2.3 implies for (A * ) i Q i∈Z and (B * ) j P j∈Z that sup