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

Abstract

This paper provides maximal function characterizations of anisotropic Triebel–Lizorkin spaces associated to general expansive matrices for the full range of parameters $p \in (0,\infty )$, $q \in (0,\infty ]$ and $\alpha \in {\mathbb {R}}$. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of mild decay, moment, and smoothness conditions.

1 Introduction

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be an expansive matrix; that is, all eigenvalues $\lambda \in \mathbb {C}$ of A satisfy $|\lambda | > 1$. Choose a Schwartz function $\varphi \in {\mathcal {S}} ({\mathbb {R}}^d)$ whose Fourier transform ${\widehat{\varphi }}$ has compact support

$$\begin{aligned} \mathop {\textrm{supp}}\limits {\widehat{\varphi }} = \overline{ \{ \xi \in {\mathbb {R}}^d : {\widehat{\varphi }} (\xi ) \ne 0 \} } \subset {\mathbb {R}}^d \setminus \{ 0 \} \end{aligned}$$

(1.1)

and satisfies

$$\begin{aligned} \sup _{j \in {\mathbb {Z}}} | {\widehat{\varphi }} ((A^*)^j \xi ) | > 0, \quad \xi \in {\mathbb {R}}^d \setminus \{0\}, \end{aligned}$$

(1.2)

where $A^*$ denotes the transpose of A. Following Bownik and Ho [7], we define the (homogeneous) anisotropic Triebel–Lizorkin space $\dot{\textbf{F}}^{\alpha }_{p,q}({\mathbb {R}}^d; A)$, with $p \in (0, \infty )$, $q \in (0,\infty )$ and $\alpha \in {\mathbb {R}}$, as the collection of all tempered distributions $f \in {\mathcal {S}}' ({\mathbb {R}}^d)$ (modulo polynomials) satisfying

$$\begin{aligned} \Vert f \Vert _{\dot{\textbf{F}}^{\alpha }_{p,q}} := \bigg \Vert \bigg ( \sum _{j \in {\mathbb {Z}}} (|\det A|^{j\alpha } |f *\varphi _j |)^q \bigg )^{1/q} \bigg \Vert _{L^p} < \infty , \end{aligned}$$

where $\varphi _j:= |\det A|^j \varphi (A^j \cdot )$. The space $\dot{{\textbf{F}}}^{\alpha }_{p,\infty } ({\mathbb {R}}^d; A)$ is defined via the usual modifications.

The dilation group $\{A^j: j \in {\mathbb {Z}} \} \le \textrm{GL}(d, {\mathbb {R}})$ generated by an expansive matrix A induces the structure of a space of homogeneous type on ${\mathbb {R}}^d$, which differs from the usual isotropic homogeneous structure on ${\mathbb {R}}^d$, unless A is ${\mathbb {C}}$-diagonalizable with all eigenvalues equal in absolute value, [3]. A particular motivation for the study of function spaces defined through such non-isotropic structures is the analysis of mixed homogeneity properties of functions and operators. The scale of spaces $\dot{\textbf{F}}^{\alpha }_{p,q}({\mathbb {R}}^d; A)$ considered here contains, among others, the anisotropic and parabolic Hardy spaces $H^p({\mathbb {R}}^d; A) \cong \dot{{\textbf{F}}}^{0}_{p, 2} ({\mathbb {R}}^d; A)$ for $p \in (0,1]$ and the Lebesgue spaces $L^p({\mathbb {R}}^d) \cong \dot{{\textbf{F}}}^{0}_{p, 2} ({\mathbb {R}}^d; A)$ for $p \in (1,\infty )$; see Sect. 2.5. We refer to Bownik [3,4,5,6,7], Calderón and Torchinsky [13,14,15], and Stein and Wainger [58] for more background and motivation regarding anisotropic dilations and associated function spaces.

The purpose of the present paper is to derive various characterizations of the spaces $\dot{\textbf{F}}^{\alpha }_{p,q}(\mathbb {R}^d; A)$, with $p \in (0,\infty )$ and $q \in (0,\infty ]$, in terms of Peetre-type maximal functions. Our main motivation for such characterizations is that they allow to identify a Triebel–Lizorkin space as a coorbit space [25] associated with a Peetre-type space on an affine-type group. This identification will be used to obtain decompositions of the spaces $\dot{\textbf{F}}^{\alpha }_{p,q}(\mathbb {R}^d; A)$ in which both the analyzing and synthesizing functions are “molecular systems” (see Sect. 1.3); the recent discretization results [53, 64] are used for this purpose.

Similar results for the endpoint case of $p = \infty$ are obtained in the subsequent paper [45].

1.1 Maximal characterizations

Throughout, in addition to A being expansive, we assume that A is exponential, i.e., $A = \exp (B)$ for some $B \in {\mathbb {R}}^{d \times d}$, so that $A^s = \exp (s B)$ is well-defined for all $s \in {\mathbb {R}}$; see Remark 3.6 for additional comments on this assumption. Given $\varphi \in {\mathcal {S}} ({\mathbb {R}}^d)$, $s \in {\mathbb {R}}$ and $\beta > 0$, we define the Peetre-type maximal function of $f \in {\mathcal {S}}' ({\mathbb {R}}^d)$ as

where $\varphi _s:= |\det A|^s \varphi (A^s \cdot )$ and $\rho _A$ is an A-homogeneous quasi-norm on ${\mathbb {R}}^d$; see Sect. 2.

Our first main result (Theorem 3.5) is the following characterization.

Theorem 1.1

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive and exponential. Suppose that $\varphi \in {\mathcal {S}} ({\mathbb {R}}^d)$ has compact Fourier support and satisfies conditions (1.1) and (1.2). Then, for all $p \in (0,\infty )$, $q \in (0, \infty ]$, $\alpha \in {\mathbb {R}}$ and $\beta > \textrm{max}\{1/p, 1/q\}$, the norm equivalences

(1.3)

hold for all $f \in \mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$, with the usual modification for $q = \infty$.

Theorem 1.1 is classical in the setting of isotropic Triebel–Lizorkin spaces, where it has been obtained under varying conditions on the multiplier $\varphi \in {\mathcal {S}} ({\mathbb {R}}^d)$. Among others, it can be found in Triebel [62], Bui, Paluszyński and Taibleson [10, 11], and Rychkov [55, 56]; see Ullrich [63] for a self-contained overview of these characterizations.

In the setting of anisotropic spaces, a maximal characterization of discrete type (i.e., a characterization involving the right-most term in (1.3)) was obtained by Farkas [23] for diagonal dilations $A = \mathop {\textrm{diag}}\limits (2^{ a_1},..., 2^{a_d})$ with anisotropy $(a_1,..., a_d) \in (0,\infty )^d$. For general expansive matrices, a discrete maximal characterization of inhomogeneous anisotropic Triebel–Lizorkin spaces has been obtained by Liu, Yang, and Yuan [48]. However, in contrast to Theorem 1.1, the smoothness parameter $\alpha \in {\mathbb {R}}$ in [48, Theorem 3.4] is restricted to the range $0< \alpha < \infty$. In particular, the results in [48] do not apply to the Lebesgue spaces $L^p$ for $1<p<\infty$ (which correspond to $\alpha = 0$), whereas Theorem 1.1 is applicable to these spaces.

Our proof of Theorem 1.1 is inspired by the approach in Rychkov [56] (see also [63]), which combines Fefferman-Stein vector-valued maximal inequalities with a sub-mean-value property of the convolution products $(f *\varphi _s)_{s \in {\mathbb {R}}}$ for $f \in {\mathcal {S}}' ({\mathbb {R}}^d)$ and $\varphi \in {\mathcal {S}} ({\mathbb {R}}^d)$. This method is a variation of a technique originally due to Strömberg and Torchinsky [59, Chapter V], and is extended here to anisotropic matrix dilations.

In addition to Theorem 1.1, we also provide a maximal characterization for the Triebel–Lizorkin sequence spaces; see Theorem 3.8.

1.2 Wavelet transforms

The continuous maximal characterization provided by Theorem 1.1 can be naturally rephrased in terms of decay properties of wavelet transforms associated to the quasi-regular representation

$$\begin{aligned} \pi (x,s) f = |\det A|^{-s/2} f(A^{-s} (\cdot - x)), \quad (x,s) \in {\mathbb {R}}^d \times {\mathbb {R}}, \; f \in L^2 ({\mathbb {R}}^d), \end{aligned}$$

(1.4)

of the semi-direct product group $G_A = {\mathbb {R}}^d \rtimes _A {\mathbb {R}}$; see Sect. 4 for basic properties.

To be more explicit, given an analyzing vector $\psi \in {\mathcal {S}} ({\mathbb {R}}^d)$, the associated wavelet transform of a distribution $f \in {\mathcal {S}}' ({\mathbb {R}}^d)$ is the function on ${\mathbb {R}}^d \times {\mathbb {R}}$ defined by

$$\begin{aligned} W_{\psi } f: G_A \rightarrow {\mathbb {C}}, \quad (x,s) \mapsto \langle f, \pi (x,s) \psi \rangle . \end{aligned}$$

Here, we use the sesquilinear dual pairing $\langle f, \varphi \rangle := f({\overline{\varphi }})$ for $f \in {\mathcal {S}}'(\mathbb {R}^d)$ and $\varphi \in {\mathcal {S}}(\mathbb {R}^d)$. A function $\psi$ is called admissible if $W_{\psi }: L^2 ({\mathbb {R}}^d) \rightarrow L^{\infty } (G_A)$ defines an isometry into $L^2 (G_A)$. Given a suitable admissible vector $\psi \in {\mathcal {S}}({\mathbb {R}}^d)$, a common procedure for constructing an associated function space is by (formally) defining

$$\begin{aligned} {{\,\textrm{Co}\,}}(Y) = \big \{ f \in {\mathcal {S}}'({\mathbb {R}}^d) / \mathcal {P}(\mathbb {R}^d) \; : \; W_{\psi } f \in Y \big \}, \end{aligned}$$

(1.5)

where Y is an adequate translation-invariant (quasi)-Banach function space on $G_A$. The function spaces such defined form so-called coorbit spaces, see, e.g., [17, 25, 30, 50, 64]. Generally, the definition of abstract coorbit spaces in the quasi-Banach range [50, 64] requires an additional local property of the wavelet transform, but we show that it is automatically satisfied in the concrete setting of the present paper (see Remark 5.12 for details).

In this paper we prove several admissibility properties of functions $\psi \in {\mathcal {S}} ({\mathbb {R}}^d)$ and establish various decay and norm estimates of their associated wavelet transforms. In particular, it is shown in Proposition 5.11 that membership of $f \in {\mathcal {S}}'({\mathbb {R}}^d)$ in the Triebel–Lizorkin space $\dot{\textbf{F}}^{\alpha }_{p,q}$ can be characterized trough decay properties of its wavelet transform $W_{\psi } f$, in the sense that

$$\begin{aligned} \dot{\textbf{F}}^{\alpha }_{p,q}(A)= {{\,\textrm{Co}\,}}_\psi (Y^{\alpha }_{p,q}), \end{aligned}$$

(1.6)

for a Peetre-type function space $Y^{\alpha }_{p,q}$ on $G_A$ and arbitrary $p \in (0, \infty )$, $q \in (0,\infty ]$ and $\alpha \in {\mathbb {R}}$. Such a coorbit realization is new for non-isotropic Triebel–Lizorkin spaces and complements the realizations of anisotropic Besov spaces [1, 4, 16] obtained in [32, 33].

The isotropic Triebel–Lizorkin spaces have been identified as coorbit spaces (1.5) from the very beginning [38]. The function spaces Y used in the identification [38] are the tent spaces of Coifman, Meyer and Stein [18]. It was later shown by Ullrich [47, 63] that alternatively one could use so-called Peetre-type spaces, which allow for a simpler and more transparent treatment (cf. [63, Section 4.1]). Our use of Peetre spaces in Sect. 5 is inspired by [63].

Lastly, it is worth mentioning that the classical papers [25, 38] considered only coorbit spaces associated with Banach spaces, while for treating Triebel–Lizorkin spaces $\dot{\textbf{F}}^{\alpha }_{p,q}$ in the range $\min \{ p,q \} < 1$ it is essential to deal with general quasi-Banach spaces. The framework [49, 50] was used for this purpose in [47]. However, the theory¹ [49, 50] is based on an incorrect convolution relation occurring in [51]; in particular, it does not apply to the affine group (cf. [64, Example 3.13]), although it is used for this purpose in [47]. The present paper uses the framework [64] instead of [50], and it is thus expected that our results in Sect. 4 and Sect. 5 provide a relevant contribution even for isotropic dilations.

1.3 Molecular decompositions

The identification (1.6) of anisotropic Triebel–Lizorkin spaces $\dot{\textbf{F}}^{\alpha }_{p,q}$ as suitable coorbit spaces ${{\,\textrm{Co}\,}}_\psi (Y^{\alpha }_{p,q})$ (cf. Proposition 5.11) enables us to apply general results on the latter spaces to obtain new molecular decompositions of $\dot{\textbf{F}}^{\alpha }_{p,q}$. However, as was already observed in [34], the classical results [25, 38] on coorbit spaces do not guarantee the same form of localization of both the analyzing and synthesizing functions as the decomposition theorems of Triebel–Lizorkin spaces in [28, 29, 34] do. For this reason, the recent results [53, 64] on molecular decompositions will be used, which bridge a gap between [25, 38] and [28, 29, 34].

For $p \in (0,\infty ), q \in (0, \infty ]$, let $r = \min \{1, p, q\}$. Given a countable, discrete set $\Gamma \subset G_A$, a family $(\phi _{\gamma })_{\gamma \in \Gamma }$ of vectors $\phi _{\gamma } \in L^2 ({\mathbb {R}}^d)$ is a (coorbit) molecular system (with respect to the window $\psi$) if there exists an envelope $\Phi \in \mathcal {W}(L^r_w)\subset L^1 (G_A)$ satisfying

$$\begin{aligned} |W_{\psi } \phi _{\gamma } (g)| = | \langle \phi _{\gamma }, \pi (g) \psi \rangle | \le \Phi (\gamma ^{-1} g), \quad \gamma \in \Gamma , \; g \in G_A; \end{aligned}$$

(1.7)

here, $\mathcal {W}(L^r_w)$ denotes a so-called Wiener amalgam space (cf. Sect. 5.3).

This notion of molecules depends on a so-called control weight $w = w^{\alpha }_{p,q}: G_A \rightarrow [1,\infty )$ for the space $Y^{\alpha }_{p,q}$ occurring in (1.6); see Sects. 5.2 and 6.2 for details. Note also that the functions $\phi _{\gamma }$ need not be of the simple form $\pi (\gamma ) \phi$ given by translates and dilates of a fixed function (as in (1.4)); rather, the wavelet transform of $\phi _\gamma$ satisfies appropriate size estimates as if it was obtained in this manner.

Theorem 1.2

Let $A \!\in \! \textrm{GL}(d, {\mathbb {R}})$ be expansive and exponential. For $p \in (0, \infty )$, $q \in (0,\infty ]$ and $\alpha \in {\mathbb {R}}$, let $r \!=\! \min \{1,p,q\}$, $\alpha ' = \alpha + 1/2-1/q$, and $\beta > \textrm{max}\{ 1/p, 1/q \}$.

Suppose $\psi \in L^2 ({\mathbb {R}}^d)$ is an admissible vector satisfying $W_{\psi } \psi \in \mathcal {W}(L^r_w)$ for the standard control weight $w = w^{-\alpha ',\beta }_{p,q}: G_A \rightarrow [1,\infty )$ defined in Lemma 5.7. Moreover, suppose $W_\varphi \psi \in \mathcal {W}(L^r_w)$ for some (thus all) admissible $\varphi \in {\mathcal {S}}_0(\mathbb {R}^d)$. Then there exists a compact unit neighborhood $U \subset G_A$ such that, for any $\Gamma \subset G_A$ satisfying

$$\begin{aligned} G_A = \bigcup _{\gamma \in \Gamma } \gamma U \quad \text {and} \quad \sup _{g \in G_A} \# (\Gamma \cap g U) < \infty , \end{aligned}$$

(1.8)

there exist two molecular systems $(\phi _{\gamma })_{\gamma \in \Gamma } \subset L^2(\mathbb {R}^d)$ and $(f_{\gamma })_{\gamma \in \Gamma } \subset L^2(\mathbb {R}^d)$ such that any $f \in \dot{\textbf{F}}^{\alpha }_{p,q}$ can be represented as

$$\begin{aligned} f = \sum _{\gamma \in \Gamma } \langle f, \pi (\gamma ) \psi \rangle \phi _{\gamma } = \sum _{\gamma \in \Gamma } \langle f, \phi _{\gamma } \rangle \pi (\gamma ) \psi \quad \text {and} \quad f = \sum _{\gamma \in \Gamma } \langle f, f_{\gamma } \rangle f_{\gamma }, \end{aligned}$$

with unconditional convergence in the weak-$*$ topology of ${\mathcal {S}}'({\mathbb {R}}^d) / {\mathcal {P}}({\mathbb {R}}^d)$.

(The dual pairings $\langle f, \pi (\gamma ) \psi \rangle$ and $\langle f, \phi _\gamma \rangle$ are defined suitably; see Definition 6.5.)

The novelty of Theorem 1.2 is that it applies to possibly irregular sets $\Gamma$ —i.e., arising from non-lattice translations—and that both $\{ \pi (\gamma )\psi : \gamma \in \Gamma \}$ and $\{ \phi _{\gamma }: \gamma \in \Gamma \}$ are molecular systems. It resembles the classical results for lattice translations by Frazier and Jawerth [28, Remark 9.17] and Gilbert, Han, Hogan, Lakey, Weiland, and Weiss [34, Theorem 1.5], and the work of Ho [42] for general expansive dilations. In contrast to Theorem 1.2, the notion of molecules used in [7, 28, 34, 42] is defined via explicit smoothness and moment conditions rather than decay estimates of their wavelet transform as in Eq. 1.7. For comparison, we provide explicit smoothness criteria for coorbit molecular systems in Sect. 6.4.

It should be mentioned that for specific vectors $\psi$ and particular construction methods, the validity of wavelet frame expansions in Hardy and Lebesgue spaces have, among others, been obtained by Bui and Laugesen [9] and Cabrelli, Molter and Romero [12]. The results in [9, 12] provide criteria and constructions that work for index sets $\Gamma$ satisfying (1.8) for some neighborhood U, whereas Theorem 1.2 above requires U to be sufficiently small. We mention that even for a molecular frame for $L^2 ({\mathbb {R}}^d)$, the extension of the canonical $L^2$-frame expansions to Hardy and Lebesgue spaces is non-automatic in general, and that such frames might fail to yield decompositions of $L^p$ for $p \ne 2$, see, e.g., Tao [60] and Tchamitchian [61].

Lastly, we complement Theorem 1.2 with a dual result on Riesz sequences. Theorem 1.3 shows that a solution to the interpolation or moment problem in discrete sequence spaces $\dot{{\textbf{p}}}^{-\alpha ', \beta }_{p, q} (\Gamma ) \le {\mathbb {C}}^{\Gamma }$ associated to a discrete $\Gamma \subset G_A$ and the Triebel–Lizorkin spaces $\dot{\textbf{F}}^{\alpha }_{p,q}$ can be obtained using molecular dual Riesz sequences; see Definition 6.1 and Remark 6.2 for details.

Theorem 1.3

Under the same assumptions of Theorem 1.2, the following holds:

There exists a compact unit neighborhood $U \subset G_A$ such that, for any $\Gamma \subset G_A$ satisfying

$$\begin{aligned} \gamma U \cap \gamma ' U = \emptyset , \quad \text {for all } \gamma , \gamma ' \in \Gamma \text { with } \gamma \ne \gamma ', \end{aligned}$$

(1.9)

there exists a molecular system $(\phi _{\gamma })_{\gamma \in \Gamma } \subset \overline{\mathop {\textrm{span}}\limits \{ \pi (\gamma ) \psi : \gamma \in \Gamma \}} \subset L^2(\mathbb {R}^d)$ such that the moment problem

$$\begin{aligned} \langle f, \pi (\gamma ) \psi \rangle = c_{\gamma }, \quad \gamma \in \Gamma , \end{aligned}$$

(1.10)

admits the solution $f: = \sum _{\gamma \in \Gamma } c_{\gamma } \phi _{\gamma } \in \dot{\textbf{F}}^{\alpha }_{p,q}$ for any given $(c_{\gamma })_{\gamma \in \Gamma } \in \dot{{\textbf{p}}}^{- \alpha ', \beta }_{p, q} (\Gamma ) \le {\mathbb {C}}^{\Gamma }.$

Theorem 1.3 seems to be the first result on Riesz sequences in anisotropic Triebel–Lizorkin spaces and it is new even for regular index sets arising from lattice translations. We mention that for regular index sets, the sequence space appearing in Theorem 1.3 coincides with the standard anisotropic Triebel–Lizorkin sequence spaces defined in [7]; see Remark 6.2.

1.4 General notation

We write $s^+:= \textrm{max}\{ 0, s \}$ and $s^-:= - \min \{ 0, s \}$ for $s \in {\mathbb {R}}$.

Given functions $f,g: X \rightarrow [0,\infty )$, we write $f \lesssim g$ if there exists $C>0$ satisfying $f(x) \le C g(x)$ for all $x \in X$. We write $f \asymp g$ for $f \lesssim g$ and $g \lesssim f$. The notation $\lesssim _\alpha$ is sometimes used to indicate that the implicit constant depends on a quantity $\alpha$. If G is a group, we write $f^{\vee } (x) = f(x^{-1})$ for $x \in G$. The characteristic function of $\Omega \subset X$ is denoted by $\mathbb {1}_{\Omega }$. For a measurable $\Omega \subset {\mathbb {R}}^d$, its Lebesgue measure is denoted by $\textrm{m}(\Omega )$.

For a matrix $A \in \mathbb {R}^{d \times d}$, its transpose is denoted by $A^*$. The norm $\Vert A\Vert _{\infty }$ denotes the operator norm of the induced map $A: {\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$. The function $\Vert \cdot \Vert : {\mathbb {R}}^d \rightarrow {\mathbb {R}}$ will denote the Euclidean norm on ${\mathbb {R}}^d$.

The space of Schwartz functions will be denoted by ${\mathcal {S}} ({\mathbb {R}}^d)$ and the space of tempered distributions by ${\mathcal {S}}' ({\mathbb {R}}^d)$. Moreover, the set ${\mathcal {P}}({\mathbb {R}}^d)$ denotes the space of all polynomials of d real variables, and $\mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$ denotes the space of equivalence classes of tempered distributions modulo polynomials. The Fourier transform ${\mathcal {F}}: {\mathcal {S}}({\mathbb {R}}^d) \rightarrow {\mathcal {S}}({\mathbb {R}}^d)$ is normalized as ${\widehat{f}} (\xi ) = \int _{{\mathbb {R}}^d} f(x) e^{-2\pi ix \cdot \xi } \; dx$. Its inverse ${\mathcal {F}}^{-1} f:= {\widehat{f}}(- \, \cdot \,)$ will also be denoted by . Similar notations will be used for the unitary Fourier-Plancherel transform ${\mathcal {F}}: L^2 ({\mathbb {R}}^d) \rightarrow L^2 ({\mathbb {R}}^d)$ and its inverse. For $f: \mathbb {R}^d \rightarrow \mathbb {C}$ and $y \in \mathbb {R}^d$, we define $T_y f: \mathbb {R}^d \rightarrow \mathbb {C}, x \mapsto f(x - y)$.

Lastly, if V is a topological vector space consisting of (equivalence classes of) functions such that the conjugation map $V \rightarrow V, \; \varphi \mapsto {\overline{\varphi }}$ is a well-defined, continuous map, then the associated map

$$\begin{aligned} V' \rightarrow V^*, \quad f \mapsto {\underline{f}} \qquad \text {with} \qquad {\underline{f}}(\varphi ):= f({\overline{\varphi }}) \end{aligned}$$

between the dual space $V'$ and the anti-dual space $V^*$ is a canonical isomorphism. In this setting, we will not distinguish between $f \in V'$ and ${\underline{f}} \in V^*$. In particular, the dual pairings $\langle \cdot ,\cdot \rangle = \langle \cdot ,\cdot \rangle _{V',V}$ and $\langle \cdot ,\cdot \rangle = \langle \cdot ,\cdot \rangle _{V^*,V}$ will always be taken to be anti-linear in the second component, i.e., $\langle f,\varphi \rangle := f({\overline{\varphi }})$ for $f \in V'$ and $\langle f, \varphi \rangle := f(\varphi )$ for $f \in V^*$. The two most important cases where this applies is for $V = {\mathcal {S}}(\mathbb {R}^d)$ and $V = {\mathcal {S}}_0(\mathbb {R}^d)$ (cf. Definition 4.3).

2 Expansive matrices and Triebel–Lizorkin spaces

This section provides background on expansive matrices and associated function spaces.

2.1 Expansive matrices

A matrix $A \in \mathbb {R}^{d \times d}$ is called expansive if $\min _{\lambda \in \sigma (A)} |\lambda |>1$, where $\sigma (A) \subset \mathbb {C}$ denotes the spectrum of A. The significance of an expansive matrix is that it induces the structure of a space of homogeneous type on ${\mathbb {R}}^d$; see [19, 20] for background.

The following lemma is collected from [3, Definitions 2.3 and 2.5] and [3, Lemma 2.2].

Lemma 2.1

[3] Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive.

• (i) There exist an ellipsoid $\Omega _A$ (i.e., $\Omega _A$ is the image of the open Euclidean unit ball under an invertible matrix) and $r > 1$ such that

  $$\begin{aligned} \Omega _A \subset r\Omega _A \subset A\Omega _A \end{aligned}$$

  and $\textrm{m}(\Omega _A) = 1$. The map $\rho _A: {\mathbb {R}}^d \rightarrow [0,\infty )$ given by

  $$\begin{aligned} \rho _A (x) = {\left\{ \begin{array}{ll} |\det A|^j, \quad &{} \text {if} \;\; x \in A^{j+1} \Omega _A \setminus A^j \Omega _A, \\ 0, \quad &{} \text {if} \;\; x = 0, \end{array}\right. } \end{aligned}$$

  (2.1)

  is called the step homogeneous quasi norm associated to A. It is measurable and there exists $C \ge 1$ such that it satisfies the following properties:

  $$\begin{aligned}&\rho _A (-x) = \rho _A(x),&\quad x \in {\mathbb {R}}^d,\nonumber \\&\rho _A (x) > 0,&\quad x \in {\mathbb {R}}^d \setminus \{0\}, \nonumber \\&\rho _A (Ax) = |\det A| \rho _A (x),&\quad x \in {\mathbb {R}}^d, \nonumber \\&\rho _A (x+y) \le C \big ( \rho _A (x) + \rho _A (y) \big ),&\quad x,y \in {\mathbb {R}}^d. \end{aligned}$$

  (2.2)

• (ii) Define $d_A: \mathbb {R}^d \times \mathbb {R}^d \rightarrow [0,\infty ), (x,y) \mapsto \rho _A(x-y)$ and let $\textrm{m}$ denote the Lebesgue measure on $\mathbb {R}^d$. Then the triple $({\mathbb {R}}^d, d_A, \textrm{m})$ is a space of homogeneous type.

For $y \in {\mathbb {R}}^d$ and $r>0$, the $d_A$-ball will be denoted by $B_{\rho _A}(y, r):= \{ x \in \mathbb {R}^d :\rho _A (x-y) < r \}$.

The following lemma shows that the homogeneous quasi-norm can be estimated from above and below by (powers of) the Euclidean norm; cf. [3, Equation (2.7) and Lemma 3.2].

Lemma 2.2

[3] Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive. Let $\lambda _-, \lambda _+$ satisfy $1< \lambda _- < \min _{\lambda \in \sigma (A)} |\lambda |$ and $\lambda _+ > \textrm{max}_{\lambda \in \sigma (A)} |\lambda |$. Define

$$\begin{aligned} \zeta _-:= \frac{\ln \lambda _-}{\ln |\det A|} \in \left( 0, \tfrac{1}{d}\right) \quad \text {and} \quad \zeta _+:= \frac{\ln \lambda _+}{\ln |\det A|} \in \left( \tfrac{1}{d}, \infty \right) . \end{aligned}$$

Then there exists $C \ge 1$ such that for every $x \in \mathbb {R}^d$, we have

$$\begin{aligned} C^{-1} [\rho _A (x)]^{\zeta _-}&\le \Vert x \Vert \le C [\rho _A(x)]^{\zeta _+}, \quad \text {if }\rho _A(x) \ge 1, \\ C^{-1} [\rho _A (x)]^{\zeta _+}&\le \Vert x \Vert \le C [\rho _A(x)]^{\zeta _-}, \quad \text {if }\rho _A(x) \le 1. \end{aligned}$$

We will also need the following fact about the integrability of powers of the quasi norm $\rho _A$.

Lemma 2.3

Suppose $A \in \textrm{GL}(d, {\mathbb {R}})$ is expansive. Then for all $\varepsilon > 0$ we have

$$\begin{aligned} \int _{B_{\rho _A}(0,1)} [\rho _A(x)]^{\varepsilon -1} dx< \infty \quad \text {and} \quad \int _{\mathbb {R}^d \setminus B_{\rho _A}(0,1)} [\rho _A(x)]^{-1-\varepsilon } dx < \infty . \end{aligned}$$

Proof

Directly from the definition of $\rho _A$, we see

$$\begin{aligned} \int _{\mathbb {R}^d \setminus B_{\rho _A}(0,1)} [\rho _A(x)]^{-1-\varepsilon } d x&= \sum _{j=0}^\infty |\det A|^{-j(1+\varepsilon )} \textrm{m}(A^{j+1} \Omega _A \setminus A^j \Omega _A) \\&= \sum _{j=0}^\infty |\det A|^{-\varepsilon j} \textrm{m}(A \Omega _A \setminus \Omega _A) < \infty , \end{aligned}$$

since $|\det A| > 1$. The proof for $\int _{B_{\rho _A}(0,1)} [\rho _A(x)]^{\varepsilon -1} dx$ is similar. $\square$

2.2 Exponential matrices

A matrix $A \in \mathbb {R}^{d \times d}$ is called exponential if $A = \exp (B)$ for a matrix $B \in \mathbb {R}^{d \times d}$; here, $\exp (B) = \sum _{n=0}^\infty B^n/n!$ denotes the usual matrix exponential. If A is expansive and has only positive eigenvalues, then A is exponential by [16, Lemma 7.8]. See [21, Theorem 1] for a precise characterization.

For an exponential matrix ${A = \exp (B)}$, the power $A^s = \exp (sB)$ is defined for all $s \in {\mathbb {R}}$. We have $\det A^s = \det (\exp (sB)) = e^{\textrm{tr}(sB)} = (e^{\textrm{tr}(B)})^s = (\det A)^s$, see, e.g., [41, Theorem 2.12]. The family $\{ A^s: s \in {\mathbb {R}} \}$ forms a continuous one-parameter subgroup of $\textrm{GL}(d, {\mathbb {R}})$.

The next lemma provides norm bounds for the powers $A^s$ of an exponential matrix A. For integral powers, these bounds are folklore²; see, e.g., [3, Equations (2.1) and (2.2)].

Lemma 2.4

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive and exponential. Let $\lambda _-, \lambda _+$ be constants such that $1< \lambda _- < \min _{\lambda \in \sigma (A)} |\lambda |$ and $\lambda _+ > \textrm{max}_{\lambda \in \sigma (A)} |\lambda |$. Then there exists $C \ge 1$ such that

$$\begin{aligned}&C^{-1} \, \lambda _-^s \, \Vert x \Vert \le \Vert A^s x \Vert \le C \, \lambda _+^s \, \Vert x \Vert , \quad s \ge 0, \\&C^{-1} \, \lambda _+^s \, \Vert x \Vert \le \Vert A^s x \Vert \le C \, \lambda _-^s \, \Vert x \Vert , \quad s \le 0, \end{aligned}$$

for all $x \in {\mathbb {R}}^d$.

Proof

Since $t \mapsto A^t$ is continuous, there exists $C_A > 0$ such that $\Vert A^t\Vert _\infty \le C_A$ for $t \in [-1,1]$. For $s \ge 0$, we write $s = k + t$ with $k \in \mathbb {N}_0$ and $t \in [0,1)$, and use the result for integral powers [3] to conclude

$$\begin{aligned} \Vert A^sx\Vert = \Vert A^tA^k x\Vert \le \Vert A^t\Vert _{\infty } \, \Vert A^kx\Vert \le C_A \, C \, \lambda _{+}^k \, \Vert x\Vert \le C_A \, C \, \lambda _{+}^s \, \Vert x\Vert . \end{aligned}$$

Similarly,

$$\begin{aligned} C_A \, \Vert A^s x \Vert\ge & {} \Vert A^{-t} \Vert _{\infty } \, \Vert A^s x \Vert \ge \Vert A^{-t} A^s x \Vert \\= & {} \Vert A^k x \Vert \ge C^{-1} \, \lambda _-^k \, \Vert x \Vert \ge (C \, \lambda _-)^{-1} \, \lambda _-^s \, \Vert x \Vert . \end{aligned}$$

The estimate for $s \le 0$ is shown using similar arguments. $\square$

Corollary 2.5

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive and exponential. Then there exists $C \ge 1$ such that

$$\begin{aligned} C^{-1} |\det A|^s \rho _A (x) \le \rho _A (A^s x) \le C |\det A|^s \, \rho _A(x) \quad x \in \mathbb {R}^d, \, s \in \mathbb {R}. \end{aligned}$$

Proof

Due to the A-homogeneity of $\rho _A$, it suffices to verify the claim for ${x \in A \, \Omega _A \setminus \Omega _A}$ (with $\Omega _A$ as in Lemma 2.1) and $s \in [0,1]$. By Lemma 2.4 and by the compactness of ${\overline{A \, \Omega _A {\setminus } \Omega _A} \subset \mathbb {R}^d {\setminus } \{0\}}$, there exist $R_1,R_2 > 0$ such that

$$\begin{aligned} R_1 \le C^{-1} \, \lambda _-^s \, \Vert x \Vert \le \Vert A^s x \Vert \le C \, \lambda _+^s \, \Vert x \Vert \le R_2 \end{aligned}$$

uniformly for all $x \in A\Omega _A {\setminus } \Omega _A$ and $s \in [0,1]$. Furthermore, there exists $k \in \mathbb {N}$ such that $A^{-k}\Omega _A \cap \{y \in \mathbb {R}^d: \Vert y\Vert \ge R_1 \} = \emptyset$. Thus, we see for $s \in [0,1]$ and $x \in A \Omega _A {\setminus } \Omega _A$ that $A^s x \notin A^{-k} \Omega _A$ and hence

$$\begin{aligned} \rho _A(A^sx) \ge |\det A|^{-k} = |\det A|^{-k} \rho _A(x) \ge |\det A|^{-k-1} |\det A|^s \rho _A(x), \end{aligned}$$

where we have used that $\rho _A(x) = 1$ for all $x \in A\Omega _A {\setminus } \Omega _A$. This gives the lower bound with $C:= |\det A|^{k+1} \ge 1$. The upper bound follows by replacing x with $A^{-s}x$. $\square$

An alternative proof of Corollary 2.5 can be obtained by using a homogeous quasi-norm associated to the continuous one-parameter group $\{A^s: s \in {\mathbb {R}} \}$ (cf. [58, Proposition 1-9]) and the equivalence of all homogeneous quasi-norms associated to A (cf. [3, Lemma 2.4]).

2.3 Analyzing vectors

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive. Suppose $\varphi \in {\mathcal {S}} ({\mathbb {R}}^d)$ is such that $\varphi$ has compact Fourier support

$$\begin{aligned} \mathop {\textrm{supp}}\limits {\widehat{\varphi }} := \overline{\{ \xi \in {\mathbb {R}}^d : {\widehat{\varphi }}(\xi ) \ne 0 \}} \subset {\mathbb {R}}^d \setminus \{0\} \end{aligned}$$

(2.3)

and satisfies

$$\begin{aligned} \sup _{j \in {\mathbb {Z}}} \big | {\widehat{\varphi }} ((A^*)^j \xi ) \big | > 0, \quad \xi \in {\mathbb {R}}^d \setminus \{0\}. \end{aligned}$$

(2.4)

Then the function $\psi \in \mathcal {S}(\mathbb {R}^d)$ defined through its Fourier transform as

$$\begin{aligned} {{\widehat{\psi }}}(\xi ) = {\left\{ \begin{array}{ll} \overline{{{\widehat{\varphi }}}(\xi )} / \sum _{k \in \mathbb {Z}} |{\widehat{\varphi }} ((A^*)^k \xi )|^2, \quad &{} \text {if} \;\; \xi \in \mathbb {R}^d \setminus \{0\}, \\ 0, \quad &{} \text {if} \;\; \xi = 0, \end{array}\right. } \end{aligned}$$

is well-defined and satisfies

$$\begin{aligned} \sum _{j \in {\mathbb {Z}}} {\widehat{\varphi }} ((A^*)^j \xi ) \, {\widehat{\psi }} ((A^*)^j \xi ) = 1, \quad \xi \in {\mathbb {R}}^d \setminus \{0\}. \end{aligned}$$

(2.5)

We refer to [7, Lemma 3.6] for more details.

2.4 Triebel–Lizorkin spaces

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive and suppose that $\varphi \in \mathcal {S}(\mathbb {R}^d)$ has compact Fourier support satisfying (2.3) and (2.4). For given $\alpha \in {\mathbb {R}}$, $0< p < \infty$ and $0 < q \le \infty$, the associated (homogeneous) anisotropic Triebel–Lizorkin space $\dot{\textbf{F}}^{\alpha }_{p,q}= \dot{\textbf{F}}^{\alpha }_{p,q}(A,\varphi )$ is defined as in [7] as the set of all $f \in \mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$ for which

$$\begin{aligned} \Vert f \Vert _{\dot{\textbf{F}}^{\alpha }_{p,q}}:= \bigg \Vert \bigg ( \sum _{j \in {\mathbb {Z}}} (|\det A|^{j\alpha } |f *\varphi _j |)^q \bigg )^{1/q} \bigg \Vert _{L^p} < \infty , \end{aligned}$$

where $\varphi _j:= |\det A|^j \, \varphi (A^j \cdot )$, with the usual modification for $q = \infty$.

As shown in [7, Proposition 3.2], the inclusion map $\dot{\textbf{F}}^{\alpha }_{p,q}\hookrightarrow \mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$ is continuous and $\dot{\textbf{F}}^{\alpha }_{p,q}$ is complete with respect to the quasi-norm $\Vert \cdot \Vert _{\dot{\textbf{F}}^{\alpha }_{p,q}}$. Moreover, [7, Corollary 3.7] shows that the space $\dot{\textbf{F}}^{\alpha }_{p,q}$ is independent of the choice of $\varphi$; we will thus simply write $\dot{\textbf{F}}^{\alpha }_{p,q}(A)$ instead of $\dot{\textbf{F}}^{\alpha }_{p,q}(A,\varphi )$.

The sequence space on ${\mathbb {Z}} \times {\mathbb {Z}}^d$ associated to $\dot{\textbf{F}}^{\alpha }_{p,q}$ is defined as the collection of all $c \in {\mathbb {C}}^{{\mathbb {Z}} \times {\mathbb {Z}}^d}$ satisfying

(2.6)

with the usual modification for $q = \infty$.

2.5 Anisotropic Hardy spaces

Denoting by $H^p_A$ the anisotropic Hardy space introduced in [3], it follows by [5, Theorem 7.1] and [3, Remark on p. 16] that

$$\begin{aligned} H^p_A&= \dot{{\textbf{F}}}^{0}_{p, 2} (A), \quad p \in (0,1] \\ L^p = H^p_A&= \dot{{\textbf{F}}}^{0}_{p, 2} (A), \quad p \in (1,\infty ). \end{aligned}$$

Two expansive matrices $A_1, A_2 \in \textrm{GL}(d, {\mathbb {R}})$ are said to be equivalent if $H^p_{A_1} = H^p_{A_2}$ for all $p \in (0,1]$. Given an expansive $A_1$, there exists an equivalent matrix $A_2$ with all eigenvalues positive and such that $\det A_2 = |\det A_1|$; see [16, Lemma 7.7] and [8, Theorem 2.3 and Lemma 3.6]. Recall that such a matrix $A_2$ is exponential (cf. Sect. 2.2).

3 Maximal function characterizations

This section provides maximal function characterizations of Triebel–Lizorkin spaces. In Sect. 3.1 we provide preliminaries on maximal functions. The characterizations of distribution and sequence spaces will be proven in Sects. 3.2 and 3.3, respectively.

3.1 Anisotropic maximal functions

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive. For $f: \mathbb {R}^d \rightarrow \mathbb {C}$ measurable, the (anisotropic) Hardy-Littlewood maximal operator $M_{\rho _A}$ is defined as

$$\begin{aligned} M_{\rho _A} f (x) = \sup _{B \ni x} \frac{1}{\textrm{m}(B)} \int _B |f(y)| \; dy, \qquad x \in {\mathbb {R}}^d, \end{aligned}$$

(3.1)

where the supremum is taken over all $\rho _A$-balls $B = B_{\rho _A}(y, r)$ that contain x.

The following simple observation is central for the remainder of this article.

Lemma 3.1

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive. For $f: \mathbb {R}^d \rightarrow \mathbb {C}$ measurable, it holds

$$\begin{aligned} M_{\rho _A} [f \circ A^j] = [M_{\rho _A} f] \circ A^j, \qquad j \in \mathbb {Z}. \end{aligned}$$

(3.2)

Proof

For $z \in \mathbb {R}^d$, the property $A^j z \in B_{\rho _A} (y, r)$ is equivalent to $z \in B_{\rho _A}(A^{-j} y, r /$ $|\det A|^j)$. Hence, the substitutions $z = A^{-j} y$ and $s = r / |\det A|^j$ and the change-of-variable ${v = A^{-j} w}$ show

$$\begin{aligned} (M_{\rho _A} f)(A^j x)= & {} \sup _{\begin{array}{c} y \in \mathbb {R}^d, r> 0 \\ A^j x \in B_{\rho _A} (y,r) \end{array}} \frac{1}{\textrm{m}(B_{\rho _A}(y,r))} \int _{B_{\rho _A}(y,r)} |f(w)| \, d w \\= & {} \sup _{\begin{array}{c} z \in \mathbb {R}^d, s> 0 \\ x \in B_{\rho _A}(z, s) \end{array}} \frac{1}{\textrm{m}(B_{\rho _A}(A^j z, |\det A|^j s))} \int _{B_{\rho _A}(A^j z,|\det A|^j s)} |f(w)| \, d w \\= & {} \sup _{\begin{array}{c} z \in \mathbb {R}^d, s > 0 \\ x \in B_{\rho _A}(z, s) \end{array}} \frac{|\det A|^j}{\textrm{m}(B_{\rho _A}(A^j z, |\det A|^j s))} \int _{B_{\rho _A}(z,s)} |f(A^j v)| \, d v \\= & {} (M_{\rho _A} [f \circ A^j]) (x) , \end{aligned}$$

as desired. $\square$

A further central property is the vector-valued Fefferman-Stein inequality [24], in the form stated in the following theorem. It follows, e.g., from [37, Theorem 1.2], by using that $(\mathbb {R}^d, d_A, \textrm{m})$ is a space of homogeneous type.

Theorem 3.2

[37] Let $A \in {\text {GL}}(d,\mathbb {R})$ be expansive. For $p \in (1, \infty ), q \in (1, \infty ]$, there exists $C=C(p,q,A,d) > 0$ such that

$$\begin{aligned} \bigg \Vert \bigg ( \sum _{i \in {\mathbb {N}}} [ M_{\rho _A} f_i ]^q \bigg )^{1/q} \bigg \Vert _{L^p } \le C \bigg \Vert \bigg ( \sum _{i \in {\mathbb {N}}} |f_i |^q \bigg )^{1/q} \bigg \Vert _{L^p} \end{aligned}$$

for any sequence of measurable functions $f_i: \mathbb {R}^d \rightarrow \mathbb {C}$, $i \in \mathbb {N}$, with the usual modification for $q = \infty$.

The following majorant property of the anisotropic maximal operator can be found in [2, Lemma 3.1] in a slightly different setting. Nevertheless, the proof given in [2] applies verbatim in our setting.

Lemma 3.3

[2] Let $\theta : [0,\infty ) \rightarrow [0,\infty )$ be non-increasing, and assume that ${\Theta : \mathbb {R}^d \rightarrow [0,\infty )}$ given by $\Theta (x) = \theta (\rho _A (x))$ is integrable. Suppose that $g \in L^1 ({\mathbb {R}}^d)$ satisfies $|g(x)| \le \Theta (x)$ for almost all $x \in \mathbb {R}^d$. Then, for $f \in L^1 ({\mathbb {R}}^d)$,

$$\begin{aligned} |(f *g) (x) | \le \Vert \Theta \Vert _{L^1} M_{\rho _A} f(x) \end{aligned}$$

for all $x \in {\mathbb {R}}^d$.

Given an exponential matrix $A \in \textrm{GL}(d, {\mathbb {R}})$ and $s \in {\mathbb {R}}$, we define the dilation of a function $\varphi : \mathbb {R}^d \rightarrow \mathbb {C}$ by $\varphi _s (x):= |\det A|^s \varphi (A^s x).$ For $\beta > 0$, the Peetre-type maximal function of $f \in {\mathcal {S}}' ({\mathbb {R}}^d)$ with respect to $\varphi \in \mathcal {S}(\mathbb {R}^d)$ is defined as

(3.3)

see Lemma A.1 for the validity of the second equality for the step homogeneous quasi-norm $\rho _A$. If A is not exponential, we define also by (3.3), but only for $s \in \mathbb {Z}$.

The Peetre-type maximal function and the Hardy-Littlewood operator are related by Peetre’s inequality, cf. [7, Lemma 3.4] for a proof.

Lemma 3.4

(Anisotropic Peetre inequality) Let $K \subset {\mathbb {R}}^d$ be compact and $\beta > 0$. There exists $C = C(K,\beta ,A) > 0$ such that for any $g \in {\mathcal {S}}'({\mathbb {R}}^d)$ with $\mathop {\textrm{supp}}\limits {\widehat{g}} \subset K$, we have

$$\begin{aligned} \sup _{z \in {\mathbb {R}}^d} \frac{|g (x-z)|}{(1+\rho _{A} (z))^{\beta }} \le C \bigl [(M_{\rho _A} |g|^{1/\beta })(x)\bigr ]^{\beta } \end{aligned}$$

(3.4)

for all $x \in {\mathbb {R}}^d$.

The expression g(x) in (3.4) makes sense, since every tempered distribution with compact Fourier support is given by (integration against) a smooth function, cf. [54, Theorem 7.23].

3.2 Function spaces

The following theorem is one of the main results of this paper. It provides an anisotropic extension of corresponding results in [11, 62, 63].

Theorem 3.5

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive and exponential. Assume that $\varphi \in \mathcal {S}(\mathbb {R}^d)$ has compact Fourier support and satisfies (2.3) and (2.4). Then, for all $p \in (0,\infty )$, $q \in (0, \infty ]$, $\alpha \in {\mathbb {R}}$ and $\beta > \textrm{max}\{1/p, 1/q\}$, the norm equivalences

(3.5)

hold for all $f \in \mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$, with the usual modifications for $q = \infty$.

(The function is well-defined for $f \in \mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$, since $\varphi$ has infinitely many vanishing moments and hence $P *\varphi _s = 0$ for every $P \in \mathcal {P}(\mathbb {R}^d)$.)

Remark 3.6

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive.

• (a) The proof of Theorem 3.5 shows that the characterization

  

  does not require A to be exponential. Instead, it holds for arbitrary expansive matrices: the estimate “$\lesssim$” is trivial, whereas Step 3 of the proof shows “$> rsim$”.

• (b) For anisotropic Hardy spaces $H^p_A$ with $p \in (0,\infty )$, the matrix A may be assumed to be exponential by the discussion in Sect. 2.5.

Proof of Theorem 3.5

As seen in Sect. 2.3, there exists $\psi \in \mathcal {S}(\mathbb {R}^d)$ with $\mathop {\textrm{supp}}\limits {\widehat{\psi }} \subset \mathop {\textrm{supp}}\limits {\widehat{\varphi }}$ and such that

$$\begin{aligned} \sum _{j \in {\mathbb {Z}}} {\widehat{\varphi }}\bigl ((A^*)^j \xi \bigr ) \, {\widehat{\psi }}\bigl ((A^*)^j \xi \bigr ) = 1, \quad \; \xi \in {\mathbb {R}}^d \setminus \{0\}. \end{aligned}$$

Note that with A, also $A^*$ is expansive and exponential. By Lemma 2.4, it follows that there exist $0< R_1 \le R_2 < \infty$ such that

$$\begin{aligned} R_1 \le \Vert (A^*)^{-t} \xi \Vert \le R_2, \qquad t \in [-1,1], \quad \xi \in \mathop {\textrm{supp}}\limits {\widehat{\varphi }}. \end{aligned}$$

Choose $N > 0$ such that $(A^*)^j \mathop {\textrm{supp}}\limits {\widehat{\varphi }} \cap \{ \xi \in {\mathbb {R}}^d: R_1 \le \Vert \xi \Vert \le R_2 \} = \emptyset$ for $|j|\ge N$, and define $\Phi \in \mathcal {S}(\mathbb {R}^d)$ via its Fourier transform as

$$\begin{aligned} {\widehat{\Phi }}(\xi ):= \sum _{\ell = -N}^N {\widehat{\varphi }} \bigl ((A^*)^\ell \xi \bigr ) \, {\widehat{\psi }} \bigl ((A^*)^\ell \xi \bigr ), \end{aligned}$$

noting that ${\widehat{\Phi }}(\xi ) = 1$ for $R_1 \le \Vert \xi \Vert \le R_2$. A direct calculation based on the preceding observations and using the convolution theorem shows that

$$\begin{aligned} \varphi _{k} *\Phi _{{k+t}} = \varphi _{k} \quad \text {and} \quad \varphi _{k+t} *\Phi _k = \varphi _{k+t}, \quad \; k \in {\mathbb {Z}}, \; t \in [0,1]. \end{aligned}$$

(3.6)

The remainder of the proof is split into three steps. For notational simplicity, we write throughout $\nu _{\beta } (y):= (1+\rho _{A} (y))^{\beta }$ for $y \in {\mathbb {R}}^d$. By Eq. (2.2), it follows that $\nu _{\beta }$ satisfies $\nu _{\beta }(x + y) \lesssim \nu _{\beta }(x) \; \nu _{\beta }(y)$ for $x, y \in {\mathbb {R}}^d$, with implicit constant only depending on $A, \beta$.

Let $f \in \mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$ be arbitrary. We prove the equivalences in (3.5) in several steps.

Step 1. In this step we show that $\Vert f \Vert _{\dot{\textbf{F}}^{\alpha }_{p,q}}$ can be estimated by the middle term of (3.5). For arbitrary, but fixed $t \in [0,1]$, a direct calculation using (3.6) gives

$$\begin{aligned}&\Big \Vert \! \Big ( |\det A|^{\alpha j} \, |(f *\varphi _j)(x)| \Big )_{j \in \mathbb {Z}} \Big \Vert _{\ell ^q} \! = \Big \Vert \Big ( |\det A|^{\alpha j} |(f *\Phi _{{j + t}} *\varphi _{j}) (x) | \Big )_{j \in {\mathbb {Z}}} \Big \Vert _{\ell ^q} \nonumber \\&\quad \lesssim \!\! \sum _{\ell = - N}^N \! \Big \Vert \! \Big ( |\det A|^{\alpha j} \, | ( f *\varphi _{{j + \ell + t}} *\psi _{{j + \ell + t}} *\varphi _{j} ) (x) | \Big )_{\!j \in {\mathbb {Z}}} \Big \Vert _{\ell ^q} . \end{aligned}$$

(3.7)

To estimate (3.7), note that for arbitrary $x \in {\mathbb {R}}^d$,

(3.8)

where is as in (3.3). We can estimate the integral in (3.8) by change-of-variables as

$$\begin{aligned}&\int _{{\mathbb {R}}^d} \nu _{\beta } (A^{j+\ell +t} y) |(\psi _{{j + \ell + t}} *\varphi _{j}) (-y) | \; dy \nonumber \\&\quad \le \int _{{\mathbb {R}}^d} \nu _{\beta } (A^{j+\ell +t} y) \int _{\mathbb {R}^d} |\det A|^{{j + \ell + t}} \; |\psi ( A^{j + \ell + t} w)| \; |\det A|^j \; | \varphi ( A^j( -y -w ) ) | \; dw \; dy \nonumber \\&\quad = \int _{{\mathbb {R}}^d} \int _{\mathbb {R}^d} \nu _{\beta } (A^{\ell +t} z) \; |\psi ( v) | \; | \varphi ( -z - A^{-(\ell + t)} v ) | \; d z \; d v \nonumber \\&\quad = \int _{{\mathbb {R}}^d} \int _{\mathbb {R}^d} \nu _{\beta } (A^{\ell +t} y - v) \; |\psi (v) | \; | \varphi ( -y ) | \; d y \; d v \nonumber \\&\quad \lesssim \int _{{\mathbb {R}}^d} \nu _{\beta } (A^{\ell +t} y) \; | \varphi ( -y ) |\; dy \int _{\mathbb {R}^d} \nu _{\beta }(v) \; |\psi (v) | \; d v . \end{aligned}$$

(3.9)

By Corollary 2.5 and Lemma 2.2, we see for $-N \le \ell \le N$ and $t \in [0,1]$ that

$$\begin{aligned} \nu _{\beta } (A^{\ell + t} y) \lesssim |\det A|^{\beta (N+1)} (1+\rho _A(y))^{\beta } \lesssim |\det A|^{\beta (N+1)} (1 + \Vert y \Vert )^{\beta / \zeta _-}. \nonumber \\ \end{aligned}$$

(3.10)

The integrals in (3.9) can therefore be bound independently of $-N \le \ell \le N$ and $t \in [0,1]$. Thus, (3.8) implies

where we can estimate $|\det A|^{-\alpha (\ell +t)} \lesssim 1$ with implicit constants independent of $\ell , t$. Combining this with (3.7) gives

(3.11)

Lastly, the left-hand side of (3.11) being independent of t, we average over $t \in [0,1]$. For this, let us assume $q < \infty$. Taking the q-th power of (3.11) and integrating gives

and thus

The case $q = \infty$ follows by the usual modifications.

Step 2. This step will show that the middle term can be bounded by the right-most term in (3.5). Using the convolution identity (3.6), we calculate for $x, z \in \mathbb {R}^d$, $j \in \mathbb {Z}$, and $t \in [0,1]$,

$$\begin{aligned}&\frac{|(f *\varphi _{{j+t}})(x+z)|}{\nu _{\beta } (A^{j+t} z)} \quad \le \sum _{\ell = -N}^N \int _{{\mathbb {R}}^d} \frac{| (f *\varphi _{{j+\ell }}) (x+y+z) |}{\nu _{\beta } (A^{j+t} z)} |(\psi _{j+\ell } *\varphi _{{j+t}}) (-y)| \; dy \nonumber \\&\quad \le \sum _{\ell = -N}^N \sup _{w \in {\mathbb {R}}^d} \frac{|(f *\varphi _{{j+\ell }} ) (x+w)|}{\nu _{\beta } (A^{j+\ell }w)} \int _{{\mathbb {R}}^d} \frac{\nu _{\beta } (A^{j + \ell } (z+y))}{\nu _{\beta } (A^{j + t} z)} | (\psi _{j+\ell } *\varphi _{{j+t}})(-y)| \; dy. \end{aligned}$$

(3.12)

To estimate the integral in (3.12), note that the essential submultiplicativity of $\nu _{\beta }$ and a change-of-variable gives

$$\begin{aligned}&\int _{{\mathbb {R}}^d} \frac{\nu _{\beta } (A^{j + \ell } (z+y))}{\nu _{\beta } (A^{j + t} z)} | (\psi _{j+\ell } *\varphi _{{j+t}}) (-y)| \; dy \nonumber \\&\quad \le \!\int _{{\mathbb {R}}^d} \frac{\nu _{\beta } (A^{j \!+\! \ell } (z+y))}{\nu _{\beta } (A^{j \!+\! t} z)} \int _{\mathbb {R}^d} |\det A|^{j\!+\!\ell } |\psi (A^{j+\ell }w)| \!\, |\det A|^{j+t} |\varphi (A^{j\!+\!t}(-y-w))| \; dw \; dy\nonumber \\&\quad \le \int _{{\mathbb {R}}^d} \int _{\mathbb {R}^d} \frac{\nu _{\beta } (A^{j + \ell } z+ A^{\ell - t} \zeta )}{\nu _{\beta }(A^{j + t} z)} \; |\psi (v)| \; |\varphi (- \zeta - A^{t -\ell } v)| \; d v \; d \zeta \nonumber \\&\quad \lesssim \int _{{\mathbb {R}}^d} \int _{\mathbb {R}^d} \frac{\nu _{\beta } (A^{j + \ell }z)}{\nu _{\beta }(A^{j + t} z)} \; \nu _{\beta }(A^{\ell - t} \zeta ) \; |\psi (v)| \; |\varphi (-\zeta )| \; dv \; d\zeta . \end{aligned}$$

(3.13)

Next, by Corollary 2.5, we have $\rho _A (A^t z) > rsim \rho _A (z)$ for $t \in [0,1]$ and $z \in {\mathbb {R}}^d$. Therefore, we see for $-N \le \ell \le N$ and $t \in [0,1]$ that

$$\begin{aligned} \frac{1+ \rho _A (A^{\ell } z)}{1+\rho _A (A^t z)} \le |\det A|^N \frac{1+ \rho _A (z)}{1+\rho _A (A^t z)} \lesssim 1, \end{aligned}$$

(3.14)

with an implicit constant independent of $j, \ell , t$ and z. Combining (3.14) with (3.10), we then see that the integral (3.13) can be estimated independently of $j, \ell , t$. Therefore, (3.12) shows for $q < \infty$ that

(3.15)

The right-hand side of (3.15) being independent of t, integrating (3.15) over [0, 1] shows that

(3.16)

and thus

The case $q = \infty$ follows by the usual modifications.

Step 3. This final step will show that the right-most term in (3.5) can be estimated by $\Vert f \Vert _{\dot{\textbf{F}}^{\alpha }_{p,q}}$. Note first that

(3.17)

where the symmetry of $\rho _A$ is used. In order to estimate (3.17), we apply Peetre’s inequality in Lemma 3.4 to $g_j:= (f *\varphi _j) \circ A^{-j}$. To this end, note with the (bilinear) dual pairing $\langle \cdot ,\cdot \rangle _{\mathcal {S}',\mathcal {S}}$ that

$$\begin{aligned} \langle \widehat{g_j} , \phi \rangle _{\mathcal {S}', \mathcal {S}}&= \big \langle f *\varphi _j , \,\, |\det A^j| \, {\widehat{\phi }} \circ A^j \big \rangle _{\mathcal {S}',\mathcal {S}} = \big \langle \widehat{f *\varphi _j}, \,\, \phi \circ (A^*)^{-j} \big \rangle _{\mathcal {S}',\mathcal {S}} \\&= \big \langle {\widehat{f}}, \,\, ({\widehat{\varphi }} \circ \! (A^*)^{-j}) \cdot (\phi \circ \! (A^*)^{-j}) \big \rangle _{\mathcal {S}',\mathcal {S}} = 0 \end{aligned}$$

for all $\phi \in \mathcal {S}(\mathbb {R}^d)$ with $\mathop {\textrm{supp}}\limits \phi \subset \mathbb {R}^d \!{\setminus }\! \mathop {\textrm{supp}}\limits {\widehat{\varphi }}$. Thus, $\mathop {\textrm{supp}}\limits \widehat{g_j} \subseteq \mathop {\textrm{supp}}\limits {\widehat{\varphi }}$ is contained in the same compact set for all $j \in {\mathbb {Z}}$. An application of Lemma 3.4 therefore provides a uniform constant $C > 0$ such that, for all $j \in {\mathbb {Z}}$,

where $M_{\rho _A}$ is as in (3.1). Therefore, the right-hand side of (3.5) can be estimated using Lemma 3.1 and the vector-valued Fefferman-Stein inequality (Theorem 3.2) as follows:

The last step used that $p\beta , q \beta > 1$, so that Theorem 3.2 is applicable. $\square$

3.3 Sequence spaces

This section provides a maximal function characterization of the sequence spaces defined in Sect. 2.4. We start with a simple lemma.

Lemma 3.7

Let $A \in {\text {GL}}(d,\mathbb {R})$ be expansive, let $K \subset \mathbb {R}^d$ be bounded and measurable with positive measure, and let $\beta \ge 0$. For $\ell \in \mathbb {Z}$ and $z \in \mathbb {R}^d$, set $K_{\ell ,z}:= A^{-\ell } (K + z)$. Then

$$\begin{aligned} \bigl (1 + \rho _A (A^\ell x - z)\bigr )^{-\beta } \lesssim \bigg ( \mathbb {1}_{K_{\ell ,z}} *\frac{|\det A|^\ell }{(1 + \rho _A (A^\ell \cdot ))^\beta } \bigg ) (x) \qquad \, x \in \mathbb {R}^d, \end{aligned}$$

where the implied constant only depends on $K, \beta , A$.

Proof

Define $\nu (x):= (1 + \rho _A(x))^{-\beta }$. Note that

$$\begin{aligned} \mathbb {1}_{K_{\ell ,z}} (x) = |\det A|^{-\ell } (\mathbb {1}_K)_\ell (x - A^{-\ell } z) = |\det A|^{-\ell } \big [ T_{A^{-\ell } z} (\mathbb {1}_K)_\ell \big ] (x). \end{aligned}$$

By applying similar manipulations to the left-hand side of the target estimate, and multiplying both sides of the target estimate by $|\det A|^\ell$, it is easily seen that the claim is equivalent to

$$\begin{aligned} T_{A^{-\ell } z }\nu _\ell \lesssim [T_{A^{-\ell } z} (\mathbb {1}_K)_\ell ] *\nu _\ell . \end{aligned}$$

Since convolution commutes with translation, we can assume that $z = 0$, i.e., we need to show that $\nu _\ell \lesssim (\mathbb {1}_K)_\ell *\nu _\ell$. Furthermore, using the identity $(f \circ A) *(g \circ A) = |\det A|^{-1} \cdot (f *g) \circ A$, it follows that it suffices to prove $\nu \lesssim \mathbb {1}_K *\nu$. For this, note that since $\rho _A$ is bounded on K, we have $1 + \rho _A(x - y) \lesssim 1 + \rho _A (x) + \rho _A(-y) \lesssim 1 + \rho _A(x)$ and hence $\bigl (1 + \rho _A(x-y)\bigr )^{-\beta } > rsim \bigl (1 + \rho _A(x)\bigr )^{-\beta }$ for $x \in \mathbb {R}^d$ and $y \in K$. Therefore,

$$\begin{aligned} \mathbb {1}_K *\nu (x) = \int _{K} \bigl (1 + \rho _A (x-y)\bigr )^{-\beta } \, d y > rsim \int _K \bigl (1 + \rho _A(x)\bigr )^{-\beta } \, d y = \textrm{m}(K) \cdot \nu (x), \end{aligned}$$

which completes the proof. $\square$

The following is a discrete counterpart of Theorem 3.5 and will be used in Sect. 6.1.

Theorem 3.8

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive and exponential. Then, for all $p \in (0, \infty )$, $q \in (0,\infty ]$, $\alpha \in {\mathbb {R}}$ and $\beta > \textrm{max}\{1/p,1/q\}$, the (quasi)-norm equivalence

holds for all $c = (c_{\ell , k})_{\ell \in {\mathbb {Z}}, k \in {\mathbb {Z}}^d} \in {\mathbb {C}}^{{\mathbb {Z}} \times {\mathbb {Z}}^d},$ with the usual modifications for $q = \infty$.

Proof

We only prove the case $q < \infty$; the case $q = \infty$ can be proven by the usual modifications. For $\ell \in {\mathbb {Z}}$ and $k \in {\mathbb {Z}}^d$, define $Q_{\ell ,k}:= A^{-\ell } ([-1,1)^d + k)$ and $P_{\ell ,k}:= A^{-\ell } ([0,1)^d + k)$. Given $c = (c_{\ell ,k})_{\ell \in \mathbb {Z}, k \in \mathbb {Z}^d} \in \mathbb {C}^{\mathbb {Z}\times \mathbb {Z}^d}$, let $F: {\mathbb {R}}^d \times {\mathbb {R}} \rightarrow [0,\infty ]$ be defined by

$$\begin{aligned} F(x,s):= \sum _{\ell \in {\mathbb {Z}}, k \in {\mathbb {Z}}^d} |c_{\ell , k}| \, \mathbb {1}_{Q_{\ell , k}} (x) \, \mathbb {1}_{-\ell + [-1, 1)} (s), \quad (x,s) \in {\mathbb {R}}^d \times {\mathbb {R}}. \end{aligned}$$

Then we can re-write

$$\begin{aligned} I&:= \!\int _{{\mathbb {R}}} \!\bigg ( \mathop {\mathrm {ess\,sup}}\limits _{z \!\in \!{\mathbb {R}}^d} \frac{|\det A|^{-(\alpha \!+\! 1/2) s}}{(1\!+\!\rho _A (A^{-s} z))^{\beta }} \sum _{\ell \!\in \!{\mathbb {Z}}, k \in {\mathbb {Z}}^d} |c_{\ell , k}| \mathbb {1}_{A^{-\ell } ([-1,1)^d + k)} (\cdot +z) \mathbb {1}_{-\ell \! +\! [-1, 1)} (s) \bigg )^q ds \nonumber \\&= \int _{{\mathbb {R}}} \bigg ( |\det A|^{-(\alpha + 1/2)s } \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{|F(\cdot +z, s)|}{(1+\rho _A (A^{-s} z))^{\beta }} \bigg )^q ds \nonumber \\&= \sum _{j \in {\mathbb {Z}}} \int _{(0,1)} \bigg ( |\det A|^{(\alpha + 1/2) (j+t)} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{|F(\cdot +z, -(j+t))|}{(1+\rho _A (A^{j+t} z))^{\beta }} \bigg )^q dt . \end{aligned}$$

(3.18)

Note that for $j \in {\mathbb {Z}}$ and $t \in (0,1)$, we have $F(x+z,-( j+t)) \le \sum _{\ell =j}^{j+1} \sum _{k \in {\mathbb {Z}}^d} |c_{\ell ,k} | \mathbb {1}_{Q_{\ell ,k}} (x+z)$ for $x,z \in {\mathbb {R}}^d$. Moreover, for fixed $j \in \mathbb {Z}$, each $y \in \mathbb {R}^d$ belongs to at most a fixed number of sets from the family $(Q_{j,k})_{k \in \mathbb {Z}^d}$; thus,

$$\begin{aligned} \sum _{k \in \mathbb {Z}^{d}} |c_{j,k}| \, \mathbb {1}_{P_{j,k}}(x+z)\lesssim & {} \bigl | F(x+z, -( j+t)) \bigr |^q \nonumber \\\lesssim & {} \sum _{\ell =j}^{j+1} \sum _{k \in {\mathbb {Z}}^d} |c_{\ell ,k} |^q \; \mathbb {1}_{Q_{\ell ,k}} (x+z). \end{aligned}$$

(3.19)

Therefore,

$$\begin{aligned} I&\lesssim \sum _{m=0}^1 \sum _{j \in {\mathbb {Z}}} \int _{(0,1)} |\det A|^{(\alpha + 1/2) (j+m-m+t)q} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{ \sum _{k \in {\mathbb {Z}}^d} |c_{j+m,k} |^q \mathbb {1}_{Q_{j+m,k}} (\cdot +z) }{(1+\rho _A (A^{j+m-m+t} z))^{\beta q}} \, d t\nonumber \\&\lesssim \sum _{\ell \in {\mathbb {Z}}} |\det A|^{(\alpha + 1/2) \ell q} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{ \sum _{k \in {\mathbb {Z}}^d} |c_{\ell ,k} |^q \mathbb {1}_{Q_{\ell ,k}} (\cdot +z)}{(1+\rho _A (A^{\ell } z))^{\beta q}} , \end{aligned}$$

(3.20)

where the last step follows by using Corollary 2.5 and noting that $|\det A|^{t-m}, |\det A|^{m-t}$ $\lesssim 1$, with implicit constants independent of $t \in (0,1)$ and $m \in \{ 0,1 \}$.

Next, since $\beta \min \{p,q\} > 1$, we can choose $r \in (0,\beta )$ such that $r \, \min \{p,q\} > 1$, and estimate

$$\begin{aligned} I \lesssim \sum _{j \in {\mathbb {Z}}} \sum _{k \in {\mathbb {Z}}^d} |\det A|^{(\alpha + 1/2) j q} \, |c_{j,k} |^q \, \bigg ( \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{ \mathbb {1}_{Q_{j,k}} (\cdot +z)}{(1+\rho _A (A^j z))^{\beta / r}} \bigg )^{q r} . \end{aligned}$$

(3.21)

To estimate (3.21) further, note that $x + z \in Q_{j,k}$ for $x \in {\mathbb {R}}^d$, implies $A^j(x+z) - k \in [-1,1]^d$, hence

$$\begin{aligned} 1 + \rho _A (A^j x - k)&= 1 + \rho _A \bigl (A^j x + A^j z - k + (-A^j z)\bigr ) \\&\lesssim (1 + \rho _A(A^j (x + z) - k)) (1 + \rho _A (-A^j z)) \\&\lesssim 1 + \rho _A (A^j z) . \end{aligned}$$

Therefore, for arbitrary $x \in {\mathbb {R}}^d$,

$$\begin{aligned} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{ \mathbb {1}_{Q_{j,k}} (x+z)}{(1+\rho _A (A^{j} z))^{\beta / r}}&\lesssim \frac{1}{(1+\rho _A (A^j x - k))^{\beta / r}} \nonumber \\&\lesssim \bigg ( \mathbb {1}_{P_{j, k}} *\frac{|\det A|^j}{(1+\rho _A (A^j \cdot ))^{\beta / r}} \bigg ) (x), \end{aligned}$$

(3.22)

where the last inequality follows from Lemma 3.7. The function $g_j:= |\det A|^j (1+\rho _A (A^j \cdot ))^{-\beta / r}$ is in $L^1 ({\mathbb {R}}^d)$ by Lemma 2.3. Moreover, we have $\Vert g_j \Vert = \Vert g_0 \Vert _{L^1}$ for every $j \in {\mathbb {Z}}$. Therefore, noting that $g_j(x) = |\det A|^j (1 + |\det A|^j \, \rho _A(x))^{-\beta / r}$ and applying the majorant property of the Hardy-Littlewood maximal function (see Lemma 3.3) to the right-hand side of (3.22) gives

$$\begin{aligned} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{\mathbb {1}_{Q_{j,k}} (x+z)}{(1+\rho _A (A^{j} z))^{\beta / r}} \lesssim M_{\rho _A} \mathbb {1}_{P_{j,k}} (x), \quad x \in {\mathbb {R}}^d. \end{aligned}$$

(3.23)

Combining (3.21) and (3.23) yields

$$\begin{aligned} I&\lesssim \sum _{j \in {\mathbb {Z}}} \sum _{k \in {\mathbb {Z}}^d} |\det A|^{(\alpha + 1/2) j q} |c_{j,k} |^q \Big ( M_{\rho _A} \mathbb {1}_{P_{j,k}} (\cdot ) \Big )^{q r} \\&= \sum _{j \in {\mathbb {Z}}} \sum _{k \in {\mathbb {Z}}^d} \bigg ( M_{\rho _A} \Big [ |\det A|^{(\alpha + 1/2) j / r} |c_{j,k} |^{1/r} \mathbb {1}_{P_{j,k}} \Big ] (\cdot ) \bigg )^{q r}. \end{aligned}$$

This, together with an application of the Fefferman-Stein inequality of Theorem 3.2, gives

The reverse estimate follows easily by combining the lower bound

$$\begin{aligned} F(x,s) \lesssim \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{|F(x+z, s)|}{(1+\rho _A (A^{-s} z))^{\beta }}, \quad (x, s) \in {\mathbb {R}}^d \times {\mathbb {R}}, \end{aligned}$$

(see Lemma B.1) with (3.18) and (3.19). $\square$

4 Admissible Schwartz functions and wavelet coefficient decay

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be an exponential matrix. Define the associated semi-direct product

$$\begin{aligned} G_A = {\mathbb {R}}^d \rtimes _A {\mathbb {R}} = \{ (x, s): x \in {\mathbb {R}}^d, s \in {\mathbb {R}} \} \end{aligned}$$

(4.1)

with multiplication $(x,s) (y,t) = (x + A^sy, s+t)$ and inversion $(x,s)^{-1} = (-A^{-s} x, -s)$. Left Haar measure on $G_A$ is given by $d \mu _{G_A} (x,s) = |\det A|^{-s} ds dx$, and the modular function on $G_A$ is $\Delta _{G_A} (x,s) = |\det A|^{-s}$. To ease notation, we often simply write $\mu := \mu _{G_A}$.

For $p \in (0, \infty )$, the Lebesgue space on $G_A$ is denoted by $L^p (G_A) = L^p (G_A, \mu _{G_A})$. The left and right translation by $h \in G_A$ of a function $F: G_A \rightarrow {\mathbb {C}}$ are defined by

$$\begin{aligned} L_h F = F(h^{-1} \cdot ) \qquad \text {and} \qquad R_h F = F (\cdot \, h) \end{aligned}$$

respectively.

4.1 Admissible vectors

The quasi-regular representation $(\pi , L^2 ({\mathbb {R}}^d))$ of $G_A = {\mathbb {R}}^d \rtimes _A {\mathbb {R}}$ is given by

$$\begin{aligned} \pi (x,s) f = |\det A|^{-s/2} f (A^{-s} (\, \cdot \, - x)), \quad f \in L^2 ({\mathbb {R}}^d). \end{aligned}$$

For fixed $\psi \in L^2 ({\mathbb {R}}^d)$, the associated wavelet transform is defined as

$$\begin{aligned} W_{\psi }: \quad L^2 ({\mathbb {R}}^d) \rightarrow L^{\infty } (G_A), \quad W_{\psi } f (x,s) = \langle f, \pi (x,s) \psi \rangle , \quad (x,s) \in G_A, \end{aligned}$$

and $\psi$ is admissible if $W_{\psi }$ defines an isometry into $L^2 (G_A)$. This implies ${W_\psi ^*W_\psi = \textrm{id}_{L^2(\mathbb {R}^d)}}$, which gives rise to the reconstruction formula

$$\begin{aligned} f = W_\psi ^*W_\psi f = \int _{G_A} W_\psi f (g) \pi (g) \psi \, d \mu _{G_A} (g), \qquad \, f \in L^2(\mathbb {R}^d), \end{aligned}$$

(4.2)

with the integral interpreted in the weak sense. Furthermore, the reproducing formula

$$\begin{aligned} W_{\varphi }f = W_{\psi }f *W_{\varphi }\psi , \quad f,\varphi \in L^2(\mathbb {R}^d) \end{aligned}$$

(4.3)

follows directly from the isometry of $W_\psi$ and the intertwining property $W_\psi [\pi (g) f] = L_g [W_\psi f]$.

Admissibility of a vector can be conveniently characterized in terms of its Fourier transform, see, e.g., [46, Theorem 1.1] and [31, Theorem 1].

Lemma 4.1

[31, 46] A vector $\psi \in L^2 ({\mathbb {R}}^d)$ is admissible if, and only if,

$$\begin{aligned} \int _{{\mathbb {R}}} \bigl |{\widehat{\psi }} ((A^*)^s \xi )\bigr |^2 ds = 1, \quad \text {a.e.} \; \xi \in {\mathbb {R}}^d. \end{aligned}$$

(4.4)

The significance of A being expansive is that this guarantees the existence of admissible vectors with convenient additional properties:

Theorem 4.2

[3, 22, 39, 44] Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be an exponential matrix. Then the following assertions are equivalent:

• (i) Either A or $A^{-1}$ is expansive.

• (ii) There exists an admissible vector $\psi \in L^2 ({\mathbb {R}}^d)$ such that ${\widehat{\psi }} \in C_c^{\infty } ({\mathbb {R}}^d)$.

If A is expansive, there exists an admissible $\varphi \in \mathcal {S}(\mathbb {R}^d)$ satisfying ${\widehat{\varphi }} \in C_c^\infty (\mathbb {R}^d {\setminus } \{ 0 \})$. In addition, it can be chosen to satisfy the support condition (2.4).

Proof

The claimed equivalence is proven in [39, 44], see also [57, p. 319]. The final claim easily follows from [22, Proposition 10] or [3, Chapter II, Theorem 4.2] and their proofs. $\square$

In the sequel, a matrix $A \in \textrm{GL}(d, {\mathbb {R}})$ will be assumed to be expansive and exponential.

4.2 Decay estimates

This section concerns decay properties of the wavelet transform. The derived decay estimates will play an important role in the subsequent sections, but are also of independent interest.

We recall the following Fréchet space of Schwartz functions with all moments vanishing.

Definition 4.3

Let ${\mathcal {S}}_0 ({\mathbb {R}}^d)$ denote the space of all $\varphi \in {\mathcal {S}} ({\mathbb {R}}^d)$ satisfying

$$\begin{aligned} \int _{{\mathbb {R}}^d} \varphi (x) x^{\alpha } dx = 0 \end{aligned}$$

for all multi-indices $\alpha \in {\mathbb {N}}_0^d$. The space ${\mathcal {S}}_0 ({\mathbb {R}}^d)$ will be equipped with the subspace topology coming from $\mathcal {S}(\mathbb {R}^d)$. Its (topological) dual space will be denoted by ${\mathcal {S}}_0' ({\mathbb {R}}^d)$.

The dual space ${\mathcal {S}}_0' ({\mathbb {R}}^d)$ can be identified with $\mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$; see, e.g., [36, Proposition 1.1.3].

The following lemma will be helpful in establishing decay of the wavelet transform. It is a generalization to the anisotropic setting of a well-known estimate, see, e.g.,  [36, Appendix B.1].

Lemma 4.4

If $s \ge 0$ and $L > 1$, then

$$\begin{aligned} \int _{\mathbb {R}^d} \bigl (1 + \rho _A(y)\bigr )^{-L} \bigl (1 + \rho _A (A^{-s} (y - x))\bigr )^{-L} \, d y \lesssim _{d, A, L} \bigl (1 + \rho _A (A^{-s} x)\bigr )^{-L} \end{aligned}$$

for all $x \in \mathbb {R}^d$.

Proof

Since $L > 1$, an application of Lemma 2.3 shows $\int _{\mathbb {R}^d} (1 + \rho _A (y))^{-L} \, d y \lesssim 1$. Therefore, if $\rho _A (A^{-s} x) \le 1$, then

$$\begin{aligned} \int _{\mathbb {R}^d} \bigl (1 + \rho _A(y)\bigr )^{-L} \bigl (1 + \rho _A (A^{-s} (y - x))\bigr )^{-L} \, d y&\le \int _{\mathbb {R}^d} (1 + \rho _A (y))^{-L} \, d y \\&\lesssim (1 + \rho _A (A^{-s} x))^{-L}. \end{aligned}$$

In the remainder of the proof, it may therefore be assumed that $\rho _A (A^{-s} x) > 1$.

Let $C_1 \ge 1$ with $\rho _A (x + y) \le C_1 (\rho _A (x) + \rho _A (y))$, and let $C_2 \ge 1$ denote the constant in Corollary 2.5, so that $\rho _A(A^s x) \le C_2 \, |\det A|^s \rho _A(x)$ for all $x,y \in \mathbb {R}^d$ and $s \in \mathbb {R}$. Define

$$\begin{aligned} U:= \bigl \{ y \in \mathbb {R}^d \, :\rho _A(y) \ge (2 C_1C_2)^{-1} |\det A|^s \rho _A(A^{-s} x) \bigr \} \end{aligned}$$

and $V:= \bigl \{ y \in \mathbb {R}^d \, :\rho _A (A^{-s} (y-x)) \ge \rho _A (A^{-s} x) / (2 C_1) \bigr \}.$ Then $\mathbb {R}^d = U \cup V$; otherwise,

$$\begin{aligned} \rho _A (A^{-s} x)&\le C_1 \big ( \rho _A (A^{-s} (x-y)) + \rho _A (A^{-s} y) \big ) \\&\le C_1 \big ( \rho _A (A^{-s} (x-y)) + C_2 \, |\det A|^{-s} \rho _A (y) \big ) \\&< C_1 \big ( \rho _A(A^{-s} x) / (2C_1) + C_2 \, |\det A|^{-s} (2 C_1 C_2)^{-1} |\det A|^s \rho _A(A^{-s} x) \big ) \\&= \rho _A(A^{-s} x), \end{aligned}$$

for any $y \in {\mathbb {R}}^d \setminus (U \cup V)$.

On the one hand, it follows by $\rho _A(A^{-s} x) \ge 1$ and a change-of-variable that

$$\begin{aligned}&\int _U \bigl (1 + \rho _A (y)\bigr )^{-L} \, \bigl (1 + \rho _A (A^{-s} (y - x))\bigr )^{-L} \, d y \\&\quad \le \frac{(2 C_1C_2)^L \cdot |\det A|^{-L s}}{\rho _A (A^{-s} x)^L} \int _{\mathbb {R}^d} \big ( 1 + \rho _A (A^{-s} (y - x)) \big )^{-L} \, d y \\&\quad \le \frac{(4 C_1C_2)^L \, |\det A|^{-(L-1)s}}{(1 + \rho _A(A^{-s} x))^L} \int _{\mathbb {R}^d} (1 + \rho _A(z))^{-L} \, d z \\&\quad \lesssim \frac{1}{(1 + \rho _A(A^{-s} x))^L} , \end{aligned}$$

where the last inequality uses Lemma 2.3 and ${|\det A|^{-(L-1)s} \le 1}$ since $L > 1$ and $s \ge 0$. On the other hand, if $y \in V$, then $1 + \rho _A(A^{-s} (y - x)) \ge (2C_1)^{-1} (1 + \rho _A(A^{-s} x)).$ Therefore,

$$\begin{aligned}&\int _V \bigl (1 + \rho _A(y)\bigr )^{-L} \bigl (1 + \rho _A (A^{-s} (y - x))\bigr )^{-L} \, d y \lesssim \frac{1}{(1 + \rho _A(A^{-s} x))^L} \end{aligned}$$

by Lemma 2.3. Combining these estimates yields the claim. $\square$

Lemma 4.5

Let $f_1,f_2 \in L^2(\mathbb {R}^d)$.

• (i) If $|f_i(\cdot )| \le C_i (1 + \rho _A(\cdot ))^{-L}$ a.e. for some $L > 1$ and all $i \in \{ 1, 2 \}$, then

  $$\begin{aligned} |W_{f_1} f_2 (x,s)| \lesssim C_1 C_2 \, |\det A|^{-|s|/2} \big (1 + \rho _A (A^{- s^+} x)\big )^{-L} \end{aligned}$$

  (4.5)

  for all $s \in \mathbb {R}$, where the implied constant only depends on d, L, A.

• (ii) If $f_1 \in C^N(\mathbb {R}^d)$ satisfies $|\partial ^\alpha f_1 (x)| \le C_3$ for all $\alpha \in \mathbb {N}_0^d$ such that $|\alpha | \le N$, and

  $$\begin{aligned} \int _{\mathbb {R}^d} \Vert x\Vert ^N \, |f_2(x)| \, d x \le C_4, \quad \text {and} \quad \int _{\mathbb {R}^d} x^\alpha \, f_2(x) \, d x = 0 \text { for } |\alpha | < N, \end{aligned}$$

  then

  $$\begin{aligned} |W_{f_1}f_2 (x,s)| \lesssim C_3 C_4 \, |\det A|^{-s/2} \, \Vert A^{-s} \Vert _{\infty }^N, \end{aligned}$$

  (4.6)

  for all $s \in \mathbb {R}$, where the implied constant only depends on d, N.

Proof

(i) $s \ge 0$, Lemma 4.4 implies

$$\begin{aligned} |W_{f_1} f_2(x,s)|&\le C_1 C_2 \int _{\mathbb {R}^d} (1 + \rho _A(y))^{-L} \, |\det A|^{-s/2} \, \big (1 + \rho _A(A^{-s} (y-x))\big )^{-L} \, d y \\&\lesssim C_1 C_2 \, |\det A|^{-s/2} \, \big (1 + \rho _A(A^{-s} x)\big )^{-L}, \end{aligned}$$

as claimed. For $s \le 0$, note that

$$\begin{aligned} \big | W_{f_1} f_2 (x, s) \big |&= \big | W_{f_2} f_1 (-A^{-s} x, - s) \big | \\&\lesssim C_1 C_2 |\det A|^{-|-s|/2} \big ( 1 + \rho _A(-A^{-(-s)^+} A^{-s} x) \big )^{-L} \\&= C_1 C_2 |\det A|^{-|s|/2} \big ( 1 + \rho _A (A^{-s^+} x) \big )^{-L} . \end{aligned}$$

(ii) By Taylor’s theorem, there exists a polynomial $P_x$ of degree $N-1$ that satisfies

$$\begin{aligned} |f_1(x + z) - P_x(z)| \lesssim C_3 \Vert z\Vert ^N \quad \text {for all} \quad z \in \mathbb {R}^d, \end{aligned}$$

with implied constant only depending on d, N. Since $\int _{\mathbb {R}^d} P(y) f_2(y) \, d y = 0$ for any polynomial P with degree at most $N-1$, it follows that

$$\begin{aligned} |W_{f_1}f_2(x,s)|&= \Big | \int _{\mathbb {R}^d} f_2(y) |\det A|^{-s/2} \, \overline{f_1(A^{-s} (y - x))} \, d y \Big | \\&= \Big | \int _{\mathbb {R}^d} f_2(y) |\det A|^{-s/2} \overline{ [f_1(A^{-s}y - A^{-s}x) - P_{-A^{-s}x}(A^{-s}y)] } \, d y \Big | \\&\lesssim C_3 \, |\det A|^{-s/2} \int _{\mathbb {R}^d} |f_2(y)| \Vert A^{-s}y\Vert ^N \, d y \\&\le C_3 C_4 \, |\det A|^{-s/2} \, \Vert A^{-s}\Vert _{\infty }^N, \end{aligned}$$

as required. $\square$

The following consequence is what we will actually use in most applications.

Corollary 4.6

Let $\psi , \varphi \in \mathcal {S}_0(\mathbb {R}^d)$ and $1< \lambda _{-} < \min _{\lambda \in \sigma (A)}|\lambda |$ be as in Lemma 2.4. Then, for every $L, N \in \mathbb {N}$,

$$\begin{aligned} |W_{\psi }\varphi (x,s)| \lesssim \bigl (1 + \rho _A(x)\bigr )^{-L} \lambda _{-}^{- |s| N} \Vert \psi \Vert \, \Vert \varphi \Vert , \end{aligned}$$

(4.7)

where $\Vert \,\cdot \,\Vert$ is a suitable continuous Schwartz semi-norm. The implied constant and the choice of the semi-norms depend only on $L,N, A, d,\lambda _-$.

Proof

Note that $\psi , \varphi \in \mathcal {S}_0(\mathbb {R}^d)$ guarantees that all assumptions of Lemma 4.5 are satisfied and the bounds $C_1, \dots , C_4$ can be replaced by suitable Schwartz semi-norms of $\psi$ or $\varphi$.

We first use the estimate (4.5). Note that $\rho _A(A^{-s^+}x) > rsim |\det A|^{- s^+} \rho _A(x)$ by Corollary 2.5. Therefore, we see for any $K > 1$ that

$$\begin{aligned} |W_{\psi }\varphi (x,s)|&\lesssim \Vert \psi \Vert \, \Vert \varphi \Vert \, |\det A|^{-|s|/2} \bigl ( 1 + |\det A|^{- s^+} \rho _A(x) \bigr )^{-K} \nonumber \\&\lesssim \Vert \psi \Vert \, \Vert \varphi \Vert \, |\det A|^{-|s|/2} \, \textrm{max}\bigl \{1, |\det A|^{K \, s^+}\bigr \} \, \bigl (1 + \rho _A(x)\bigr )^{-K} \nonumber \\&\lesssim \Vert \psi \Vert \, \Vert \varphi \Vert \, |\det A|^{K \, s^+} \bigl (1 \!+\! \rho _A(x)\bigr )^{-K} \nonumber \\&\le \Vert \psi \Vert \, \Vert \varphi \Vert \, |\det A|^{|s| K} \bigl (1 \!+\! \rho _A(x)\bigr )^{-K} , \end{aligned}$$

(4.8)

where $\Vert \,\cdot \,\Vert$ is a suitable Schwartz semi-norm depending on K, A.

We now show for arbitrary $M \in \mathbb {N}$ that

$$\begin{aligned} |W_{\psi }\varphi (x,s)| \lesssim \lambda _{-}^{- |s| M} \Vert \psi \Vert \, \Vert \varphi \Vert . \end{aligned}$$

(4.9)

Indeed, if $s \ge 0$, then $\Vert A^{-s}\Vert _{\infty } \lesssim \lambda _{-}^{-s}$ by Lemma 2.4 and the claim follows immediately from (4.6). The claim for $s \le 0$ follows from the case $s \ge 0$ via $W_{\psi }\varphi (x,s)= \overline{W_{\varphi }\psi (-A^{-s}x,-s)}$.

Finally, we interpolate between (4.8) and (4.9). To this end, note that a priori the seminorms in (4.8) and (4.9) are distinct, but that we can assume that they are equal by possibly enlarging them. Now, since $\lambda _{-} > 1$, we can choose $H = H(A,\lambda _-) \in \mathbb {N}$ such that $\lambda _{-}^H \ge |\det A|$. Taking $K = 2\,L$ and $M= 2(H L + N)$ yields that

$$\begin{aligned} |W_{\psi }\varphi (x,s)|&= |W_{\psi }\varphi (x,s)|^{1/2} |W_{\psi }\varphi (x,s)|^{1/2} \\&\lesssim \Vert \psi \Vert \, \Vert \varphi \Vert \, |\det A|^{|s| L} \, \bigl (1 + \rho _A(x)\bigr )^{-L} \, \lambda _{-}^{- |s| (H L + N)} \\&\lesssim \Vert \psi \Vert \, \Vert \varphi \Vert (1 + \rho _A(x))^{-L} \lambda _{-}^{- |s| N} , \end{aligned}$$

as claimed. $\square$

4.3 Extended wavelet transform

The wavelet transform can be extended via duality to $\mathcal {S}_0'(\mathbb {R}^d) \cong \mathcal {S}' (\mathbb {R}^d) / \mathcal {P} (\mathbb {R}^d)$. Throughout, we will use the dual bracket defined by

$$\begin{aligned} \langle \cdot , \cdot \rangle : \quad \mathcal {S}_0'(\mathbb {R}^d) \times \mathcal {S}_0(\mathbb {R}^d) \rightarrow \mathbb {C}, \qquad \langle f, \varphi \rangle := f({\overline{\varphi }}). \end{aligned}$$

(4.10)

The bracket is a sesquilinear form naturally extending the $L^2$-inner product.

If $\psi \in \mathcal {S}_0(\mathbb {R}^d)$, then the (extended) wavelet transform

$$\begin{aligned} W_{\psi }: \quad \mathcal {S}_0'(\mathbb {R}^d) \rightarrow C (G_A), \quad W_{\psi } f (x,s) = \langle f, \pi (x,s) \psi \rangle , \quad (x,s) \in {\mathbb {R}}^d \times {\mathbb {R}}, \nonumber \\ \end{aligned}$$

(4.11)

is well-defined. Here, we implicitly use the continuity of $\mathbb {R}^d \times \mathbb {R}\rightarrow \mathcal {S}(\mathbb {R}^d), (x,s) \mapsto \pi (x,s) \psi$. In addition to the wavelet transform, we also extend the representation $\pi$ to $\mathcal {S}_0'(\mathbb {R}^d)$ by defining

$$\begin{aligned} \langle \pi (h)f, \varphi \rangle := \langle f, \pi (h^{-1})\varphi \rangle \qquad \text {for } f \in \mathcal {S}_0'(\mathbb {R}^d) \text { and } \varphi \in \mathcal {S}_0(\mathbb {R}^d). \end{aligned}$$

The following lemma extends the reconstruction formula (4.2) to all of $\mathcal {S}_0'(\mathbb {R}^d)$.

Lemma 4.7

Let $\psi \in \mathcal {S}_0(\mathbb {R}^d)$ be admissible. Then

$$\begin{aligned} \int _{G_A} W_\psi f (g) \, \overline{W_\psi \varphi (g)} \, d \mu _{G_A}(g) = \langle f, \varphi \rangle \end{aligned}$$

(4.12)

for all $f \in \mathcal {S}_0'(\mathbb {R}^d)$ and $\varphi \in \mathcal {S}_0(\mathbb {R}^d)$.

Proof

The proof follows [32, Lemma 2.11] and [33, Lemma 40], with suitable modifications.

For $M, N \in \mathbb {N}_{\ge d+1}$, let $\mathcal {S}_{M,N}(\mathbb {R}^d)$ denote the space of all functions $f \in C^N(\mathbb {R}^d)$ satisfying

$$\begin{aligned} \Vert f\Vert _{M,N}:= \textrm{max}_{\beta \in \mathbb {N}_0^d, |\beta | \le N} \sup _{x \in \mathbb {R}^d} (1+\Vert x\Vert )^{M} |\partial ^{\beta } f(x)| < \infty . \end{aligned}$$

(4.13)

The function space $\mathcal {S}_{M,N}(\mathbb {R}^d)$ equipped with the norm in (4.13) is a Banach space. Furthermore, $\mathcal {S}(\mathbb {R}^d) \hookrightarrow \mathcal {S}_{M,N}(\mathbb {R}^d)$. Since $G_A \rightarrow \mathcal {S}(\mathbb {R}^d), g \mapsto \pi (g) \psi$ is continuous and $G_A$ is $\sigma$-compact, this implies that the map

$$\begin{aligned} G_A \rightarrow \mathcal {S}_{M,N}(\mathbb {R}^d), \quad g \mapsto W_{\psi }\varphi (g) \, \pi (g) \psi \end{aligned}$$

(4.14)

is continuous and has a $\sigma$-compact (and hence separable) range. Moreover, the decay estimates of Corollary 4.6 show ${\int _{G_A} |W_\psi \varphi (g)| \, \Vert \pi (g) \psi \Vert _{M,N} \, d \mu _{G_A}(g) < \infty }$. Overall, this shows that the map in (4.14) is Bochner integrable, for arbitrary $M,N \in \mathbb {N}_{\ge d+1}$.

The reconstruction formula (4.2) shows for $\varphi \in \mathcal {S}_0(\mathbb {R}^d) \subset L^2(\mathbb {R}^d) \cap \mathcal {S}_{M,N}(\mathbb {R}^d)$ that

$$\begin{aligned} \varphi = \int _{G_A} W_{\psi }\varphi (g) \, [\pi (g)\psi ] \, d\mu _{G_A}(g) \end{aligned}$$

(4.15)

where the integral is understood in the weak sense in $L^2(\mathbb {R}^d)$. As shown above, the right-hand side also exists as a Bochner integral in $\mathcal {S}_{M,N} (\mathbb {R}^d)$. Since $M \ge d+1$, we have $\mathcal {S}_{M,N} \hookrightarrow L^2(\mathbb {R}^d)$. Furthermore, if $\varphi \in \mathcal {S}_{M,N}$ satisfies $\langle \varphi , f \rangle = 0$ for all $f \in L^2(\mathbb {R}^d)$, then $\varphi \equiv 0$. Hence the identity (4.15) also holds in $\mathcal {S}_{M,N}(\mathbb {R}^d)$.

Lastly, if $f \in \mathcal {S}_0'(\mathbb {R}^d)$, then f extends to a continuous linear functional on $\mathcal {S}(\mathbb {R}^d)$ by [36, Proposition 1.1.3]. Hence, there are $M,N \in \mathbb {N}_{\ge d+1}$, such that the restriction of f to $\mathcal {S}_0(\mathbb {R}^d)$ is continuous with respect to $\Vert \cdot \Vert _{M,N}$; see [35, Proposition 2.3.4]. Using the Hahn-Banach theorem, we can extend f to a bounded linear functional ${\widetilde{f}}$ on $\mathcal {S}_{M,N}(\mathbb {R}^d)$. In view of (4.15), and using that the Bochner-integral can be interchanged with bounded linear functionals by [66, V.5, Corollary 2], we obtain that

$$\begin{aligned} \langle f, \varphi \rangle= & {} {\widetilde{f}}({\overline{\varphi }}) = {\widetilde{f}} \Big ( \int _{G_A} \overline{W_{\psi }\varphi (g)} \overline{\pi (g)\psi } \, d\mu _{G_A}(g) \Big ) \\= & {} \int _{G_A} \overline{W_{\psi }\varphi (g)} \langle f, \pi (g) \psi \rangle \, d\mu _{G_A}(g) \end{aligned}$$

for any $\varphi \in \mathcal {S}_0(\mathbb {R}^d)$. $\square$

Corollary 4.8

(Reproducing formula) Let $\psi \in \mathcal {S}_0(\mathbb {R}^d)$ be admissible. Then

$$\begin{aligned} W_{\varphi }f = W_{\psi }f *W_{\varphi }\psi \end{aligned}$$

(4.16)

holds for all $f \in \mathcal {S}_0'(\mathbb {R}^d)$ and $\varphi \in \mathcal {S}_0(\mathbb {R}^d)$.

Proof

Replacing $\varphi$ by $\pi (h) \varphi$ in Lemma 4.7 easily yields the claim. $\square$

5 Coorbit spaces associated to Peetre-type spaces

This section is devoted to characterizations of anisotropic Triebel–Lizorkin spaces in terms of wavelet transforms. Explicitly, it will be shown that Triebel–Lizorkin spaces can be identified with coorbit spaces associated to so-called Peetre-type spaces.

5.1 Peetre-type spaces

For $p,q \in (0,\infty ]$, the mixed-norm Lebesgue space $L^{p,q}(G_A)$ consists of all (equivalence classes of a.e. equal) measurable functions ${F: G_A \rightarrow \mathbb {C}}$ satisfying

$$\begin{aligned} \Vert F \Vert _{L^{p,q}} := \big \Vert x \mapsto \Vert F(x,\cdot ) \Vert _{L^q(\nu )} \big \Vert _{L^p(\mathbb {R}^d)} < \infty , \end{aligned}$$

(5.1)

relative to the Borel measure $\nu$ on ${\mathbb {R}}$ defined by $\nu (M) = \int _M \frac{d s}{|\det A|^s}$. The weighted space is given by $L_w^{p,q}(G_A) = \{ F: G_A \rightarrow \mathbb {C}:w \cdot F \in L^{p,q}(G_A) \}$, with norm $\Vert F \Vert _{L_w^{p,q}}:= \Vert w \cdot F \Vert _{L^{p,q}}$.

Definition 5.1

For $\alpha \in {\mathbb {R}}, \beta > 0$, and $p \in (0, \infty )$ and $q \in (0, \infty ]$, the Peetre-type space on $G_A$ is defined as the collection of all (equivalence classes of a.e. equal) measurable $F: G_A \rightarrow {\mathbb {C}}$ satisfying

with the usual modification for $q = \infty$.

An essential property of the Peetre-type spaces for our purposes is their two-sided translation invariance. For proving this, the following lemma will be used. Its proof is deferred to Appendix 1.

Lemma 5.2

The weight function

$$\begin{aligned} v: \quad G_A \rightarrow [0,\infty ), \quad (y,t) \mapsto \sup _{(z, u) \in G_A} \frac{1 + \rho _A (A^{-u} z)}{1 + \rho _A (A^{-u} A^t z - y)} \end{aligned}$$

(5.2)

is well-defined, measurable, and submultiplicative. Furthermore, we have

$$\begin{aligned} v(y,t)&\asymp&\textrm{max}\bigl \{ 1, |\det A|^{-t} \bigr \} \bigl (1 + \min \{ \rho _A (y), \rho _A (A^{-t} y) \}\bigr )\nonumber \\&\asymp&1 + |\det A|^{-t} + \quad \rho _A (A^{-t} y). \end{aligned}$$

(5.3)

The basic properties of Peetre-type spaces are collected in the following lemma.

Lemma 5.3

Let $\alpha \in {\mathbb {R}}, \beta > 0$, and $p \in (0, \infty )$ and $q \in (0, \infty ]$. Then the Peetre-type space is a solid quasi-Banach function space (Banach function space if $p,q \ge 1$). Furthermore, the operator norms of the translation operators $L_g$ and $R_g$ acting on can be bounded by

where and v is the weight function defined in Lemma 5.2.

Proof

It is easy to see that is a solid quasi-norm, as defined in [65, Chapter 2], and a solid norm if $p,q \ge 1$. The positive definiteness of follows from Lemma B.1.

For the completeness of , suppose that $(F_n)_{n \in {\mathbb {N}}}$ satisfies , and let be such that $|F(x, s)| \le \liminf _{n \rightarrow \infty } |F_n (x, s)|$ for a.e. $(x,s) \in {\mathbb {R}}^d \times {\mathbb {R}}$. Then it follows directly from Fatou’s lemma and the definition of that

(5.4)

and thus satisfies the so-called Fatou property, which in particular implies that is complete; see [67, Section 65, Theorem 1] and [65, Lemma 2.2.15].

We show the translation-invariance for $q \in (0, \infty )$. Let and ${(y,t) \in {\mathbb {R}}^d \times {\mathbb {R}}}$ be arbitrary. Then a direct calculation using the substitutions ${\widetilde{x}} = x - y$ and $x = A^{-t} {\widetilde{x}}$, as well as ${\widetilde{z}} = A^{-t} z$ shows

For the right-translation, the substitutions ${\widetilde{z}} = z + A^s y$ and ${\widetilde{s}} = s + t$ show that

By Lemma 5.2, $\bigl (1 + \rho _A (A^{-s} A^t z - y)\bigr )^{-1} \le \frac{v(y,t)}{1 + \rho _A (A^{-s} z)}$ for all $(z,s), (y,t) \in \mathbb {R}^d \times \mathbb {R}$, showing the desired estimate. The case $q = \infty$ follows via the usual modifications. $\square$

Lastly, the following simple observation allows to apply results of [64] in the remainder.

Lemma 5.4

Let $\alpha \in {\mathbb {R}}, \beta > 0$. For $p \in (0, \infty )$, $q \in (0, \infty ]$, let $r:= \min \{1, p, q \}$. The quasi-norm is an r-norm, i.e.,

Proof

The case $p,q \ge 1$ follows directly by Lemma 5.3, so let $p, q < 1$ throughout the proof. For with $i = 1, 2$, define

$$\begin{aligned} H_i (x, s) = |\det A|^{\alpha s} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{| F_i (x+z, s) |}{(1 + \rho _A (A^{-s} z))^{\beta }}, \quad (x,s) \in {\mathbb {R}}^d \times {\mathbb {R}}. \end{aligned}$$

Using this notation and the inequalities $r = \min \{1, p, q\} < 1$ and $q/r, p/r \ge 1$, a direct calculation yields

where $\nu$ denotes the Borel measure on ${\mathbb {R}}$ given by $\nu (M) = \int _M \frac{d s}{|\det A|^s}$ as in Eq. 5.1. $\square$

5.2 Standard envelope and control weight

The notion of a control weight plays an essential role in coorbit theory, see, e.g., [25, 30, 38, 64]. For the study of control weights in the setting of the present paper, the class of functions will be useful.

Definition 5.5

For $\sigma = (\sigma _1, \sigma _2) \in (0,\infty )^2$ and $L \in \mathbb {R}$, define $\eta _L: G_A \rightarrow (0,\infty )$ and $\theta _\sigma : \mathbb {R}\rightarrow (0,\infty )$ by

$$\begin{aligned} \eta _L(x,s):= \big ( 1 + \min \{ \rho _A (x), \rho _A(A^{-s} x) \} \big )^{-L} \qquad \text {and} \qquad \theta _\sigma (s):= {\left\{ \begin{array}{ll} \sigma _1^s, &{} \text {if } s \ge 0, \\ \sigma _2^s, &{} \text {if } s < 0. \end{array}\right. } \end{aligned}$$

The standard envelope $\Xi _{\sigma ,L}: G_A \rightarrow (0,\infty )$ is given by $\Xi _{\sigma ,L}(x,s):= \theta _\sigma (s) \eta _L(x,s)$.

Lemma 5.6

For each $L \in \mathbb {R}$, we have $\eta _L (x,s) \asymp \bigl (1 + \rho _A (A^{-s^+} x)\bigr )^{-L}$ for all $(x,s) \in G_A$.

Proof

Corollary 2.5 shows $\rho _A(A^{-s} x) \asymp |\det A|^{-s} \rho _A(x)$. Because of $|\det A| > 1$, this implies

$$\begin{aligned} \min \{ \rho _A(x), \rho _A(A^{-s} x) \}&\asymp&\min \{ \rho _A(x), |\det A|^{-s} \rho _A(x) \}\\&\asymp&|\det A|^{-s^+} \rho _A(x) \asymp \rho _A (A^{-s^+} x), \end{aligned}$$

where Corollary 2.5 was again used in the last step. This estimate easily implies the claim. $\square$

The next lemma provides the existence of a so-called control weight for and shows how to estimate it by a standard envelope.

Lemma 5.7

Let $\alpha \in \mathbb {R}$, and $\beta > 0$. For $p \in (0, \infty )$, $q \in (0, \infty ]$, let $r:= \min \{1, p, q \}$. As in Lemma 5.3, write . There exists a continuous, submultiplicative weight $w = w^{\alpha ,\beta }_{p,q}: G_A \rightarrow [1,\infty )$ such that

with implicit constant depending on $A,\beta$. The weight w is called a standard control weight.

Furthermore, define $\sigma _1:= |\det A|^{1/r + |\alpha +1/p-1/q|}$ and $\sigma _2:= |\det A|^{-|\alpha +1/p-1/q|}$, as well as

$$\begin{aligned} \kappa _1:= {\left\{ \begin{array}{ll} |\det A|^{1/r+\alpha +\beta -1/q} &{} \text {if } \alpha \ge -\frac{1/r+\beta -2/q}{2}, \\ |\det A|^{-(\alpha -1/q)} &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \kappa _2:= {\left\{ \begin{array}{ll} |\det A|^{-(\alpha +\beta -1/q)} &{} \text {if } \alpha \ge -\frac{1/r+\beta -2/q}{2}, \\ |\det A|^{1/r + \alpha - 1/q} &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Then the standard control weight w satisfies $w \asymp \Xi _{\sigma , 0} + \Xi _{\kappa , -\beta }.$

Proof

The weight $v: G_A \rightarrow [0,\infty )$ constructed in Lemma 5.2 is submultiplicative, measurable, and locally bounded; see Eq. 5.3. Furthermore, $v \ge 1$. Thus, v is a weight function in the sense of [52, Definition 3.7.1] and by the proof of [52, Theorem 3.7.5], there exists a continuous, submultiplicative function $v_0: G_A \rightarrow [1,\infty )$ satisfying $v \asymp v_0$.

Let $\tau \in \mathbb {R}$ and set $a_{\tau }(g) = a_{\tau } (x,s):= |\det A|^{s \tau }$ for $g = (x,s) \in G_A$. Note that $a_{\tau }$ is multiplicative and that $\Delta = a_{-1}$. For $\gamma , \delta \in \mathbb {R}$, define the function $w_{\gamma , \delta }: G_A \rightarrow [1,\infty )$ by

$$\begin{aligned} w_{\gamma , \delta }:= \textrm{max}\big \{ 1, \,\,\, a_{1/r}, \,\,\, a_{\gamma }, \,\,\, a_{-\gamma }, \,\,\, a_{\gamma + 1/r}, \,\,\, a_{1/r - \gamma }, \,\,\, a_{\delta + 1/r} \cdot (v_0^\vee )^\beta , \,\,\, a_{-\delta } \cdot v_0^\beta \big \}. \end{aligned}$$

Then $w_{\gamma , \delta }$ is again continuous and submultiplicative. Since $a_{\tau }^{\vee } = a_{-\tau }$, it follows easily that $(\Delta ^{1/r})^{\vee } \cdot w^{\vee }_{\gamma , \delta } = w_{\gamma , \delta }.$ Choosing $\gamma := \alpha + 1/p - 1/q$ and $\delta := \alpha - 1/q$ and setting $w= w^{\alpha , \beta }_{p,q}:=w_{\gamma , \delta }$ yields, by Lemma 5.3, that and

For proving the second part of the lemma, note that $w \asymp w_1 + w_2$ for the weights given by ${w_1:= \textrm{max}\{ a_0, a_{1/r}, a_\gamma , a_{-\gamma }, a_{1/r+\gamma }, a_{1/r-\gamma } \}}$ and $w_2:= \textrm{max}\bigl \{ a_{\delta + 1/r} \cdot (v_0^{\vee })^\beta , \,\, a_{-\delta } \cdot v_0^\beta \bigr \}$. It remains therefore to show that $w_1 \asymp \Xi _{\sigma ,0}$ and $w_2 \asymp \Xi _{\kappa ,-\beta }$, with $\kappa$ and $\sigma$ as in the statement of the lemma. To estimate $w_1$, note that if $I = \{ 0, 1/r, \gamma , -\gamma , 1/r + \gamma , 1/r - \gamma \}$, then

$$\begin{aligned} \textrm{max}_{\tau \in I} a_{\tau } (x,s)&= {\left\{ \begin{array}{ll} |\det A|^{s \cdot \textrm{max}I}, &{} \text {if } s \ge 0, \\ |\det A|^{s \cdot \min I}, &{} \text {if } s< 0, \\ \end{array}\right. } \\&= {\left\{ \begin{array}{ll} |\det A|^{s \cdot (1/r + |\gamma |)}, &{} \text {if } s \ge 0, \\ |\det A|^{- s |\gamma |}, &{} \text {if } s < 0. \\ \end{array}\right. } \end{aligned}$$

Hence, by the choice of $\gamma$ and $\sigma$, this yields $w_1 (x,s) = \textrm{max}_{\tau \in I} a_{\tau } (x,s) = \theta _\sigma (s) = \Xi _{\sigma ,0}(x,s)$. Lastly, for estimating $w_2$, note that the estimate for v in Lemma 5.2 implies

$$\begin{aligned} v_0^\vee (x,s)&\asymp \textrm{max}\{ 1, |\det A|^s \} \big ( 1 + \min \{ \rho _A (-A^{-s} x), \,\, \rho _A(-A^s A^{-s} x) \} \big ) \\&= |\det A|^{s^+} \big ( 1 + \min \{ \rho _A(x), \rho _A(A^{-s} x) \} \big ) \\&= |\det A|^{s^+} \eta _{-1} (x,s) . \end{aligned}$$

Similarly, one can show that $v_0(x,s) \asymp |\det A|^{s^-} \, \eta _{-1} (x,s)$. In case of $s \ge 0$, this gives

$$\begin{aligned} w_2(x,s)&\asymp \big ( \eta _{-1}(x,s) \big )^{\beta } \textrm{max}\big \{ |\det A|^{(1/r + \delta + \beta ) s} , |\det A|^{-\delta s} \big \} \\&= \eta _{-\beta }(x,s) \kappa _1^s \\&= \Xi _{\kappa ,-\beta } (x,s) , \end{aligned}$$

since $\textrm{max}\{ 1/r + \delta + \beta , -\delta \} = \textrm{max}\{ 1/r + \alpha + \beta - 1/q, -\alpha + 1/q \} = 1/r + \alpha + \beta - 1/q$ if and only if $\alpha \ge - \frac{1/r+\beta -2/q}{2}$. The estimate for $s < 0$ follows similarly. $\square$

5.3 Norm estimates

Let $Q \subset G_A$ be a relatively compact unit-neighborhood. The two-sided local maximal function $M_Q F$ of a measurable function $F \!:\! G_A \! \rightarrow \! \mathbb {C}$ is defined by

$$\begin{aligned} M_Q F (g):= \mathop {\mathrm {ess\,sup}}\limits _{u,v \in Q} |F(u g v)|. \end{aligned}$$

(5.5)

Two properties of this maximal function that will be used below are its measurability (see, e.g., [43, Lemma B.4]) and the estimate $|F| \le M_Q F$ a.e. (see, e.g., [65, Lemma 2.3.3]).

For $p \in (0, \infty )$, $q \in (0, \infty ]$, let $r:= \min \{ 1, p, q\}$. The (weighted) Wiener amalgam space $\mathcal {W}(L^r_w)$ is defined by

$$\begin{aligned} \mathcal {W}(L^r_w):= {\mathcal {W}}_Q (L_w^r):= \bigg \{ F \in C (G_A): M_QF \in L^r_w (G_A) \bigg \}, \end{aligned}$$

where $w: G_A \rightarrow [1,\infty )$ is a standard control weight for as provided by Lemma 5.7.

The space $\mathcal {W}(L^r_w)$ is independent of the choice of Q.³ In particular, this implies that

$$\begin{aligned} F \in \mathcal {W}(L^r_w)\quad \text {if and only if} \quad F^{\vee } \in \mathcal {W}(L^r_w); \end{aligned}$$

(5.6)

since the condition $w(g) = \Delta ^{1/r} (g^{-1}) w(g^{-1})$ in Lemma 5.7 implies $\Vert F^\vee \Vert _{L_w^r} = \Vert F \Vert _{L_w^r}$, and by choosing Q to be symmetric it follows that $M_Q (\Phi ^{\vee }) = (M_{Q} \Phi )^{\vee }$.

The following norm estimate will be used repeatedly in the remainder.

Lemma 5.8

Let $Q \subset G_A$ be a relatively compact unit neighborhood. Let $\psi \in \mathcal {S}_0(\mathbb {R}^d)$ and let $w: G_A \rightarrow [0, \infty )$ be any weight such that $w \lesssim \Xi$, where $\Xi$ is a linear combination of standard envelopes (see Definition 5.5).

Then, for all $p,q \in (0,\infty ]$, there exists a continuous Schwartz seminorm $\Vert \cdot \Vert$ such that

$$\begin{aligned} \Vert W_\psi \varphi \Vert _{L_w^{p,q}} \le \Vert M_Q (W_\psi \varphi ) \Vert _{L_w^{p,q}} \lesssim \Vert \varphi \Vert \end{aligned}$$

for all $\varphi \in \mathcal {S}_0(\mathbb {R}^d)$; in particular, $W_{\psi } \varphi \in \mathcal {W}(L^r_w)$ for all $r \in (0, \infty ]$.

Proof

Let $1< \lambda _- < \min _{\lambda \in \sigma (A)} |\lambda |$. By Corollaries 4.6 and 2.5 and Lemma 5.6, it follows that for all $L,N \in \mathbb {N}$ and $\varphi \in \mathcal {S}_0(\mathbb {R}^d)$,

$$\begin{aligned} |W_{\psi }\varphi (x,s)|&\lesssim \Vert \varphi \Vert (1 + \rho _A(x))^{-L} \, \lambda _{-}^{-|s| N} \lesssim \Vert \varphi \Vert \bigl (1 + |\det A|^{-s^+} \, \rho _A(x)\bigr )^{-L} \lambda _{-}^{-|s| N} \\&\lesssim \Vert \varphi \Vert \Xi _{L, \tau } (x,s), \end{aligned}$$

where $\tau := (\lambda _-^{-N}, \lambda _-^{N})$ and a suitable continuous Schwartz seminorm $\Vert \cdot \Vert$, depending on L, N. Lemma B.2 yields $M_Q \Xi _{L,\tau } \lesssim \Xi _{L,\tau }$, and hence $M_Q [W_\psi \varphi ] \lesssim \Vert \varphi \Vert \Xi _{L,\tau }$. In addition, Lemma 5.6 shows that $\eta _L (x,s) \asymp \bigl (1 + |\det A|^{-s^+} \rho _A(x)\bigr )^{-L} \le |\det A|^{|s| L} (1 + \rho _A (x))^{-L}.$ Therefore,

$$\begin{aligned} M_Q [W_\psi \varphi ] (x,s) \lesssim \Vert \varphi \Vert (1 + \rho _A(x))^{-L} (|\det A|^{|L|} / \lambda _{-}^N)^{|s|}, \end{aligned}$$

(5.7)

where $L,N \in \mathbb {N}$ are arbitrary and $\Vert \cdot \Vert$ is a continuous Schwartz semi-norm depending on N, L.

It clearly suffices to prove the claim for the case where $w = \theta _{\sigma } \cdot \eta _M$ is a standard envelope, with $\sigma = (\sigma _1, \sigma _2) \in (0, \infty )^2$ and $M \in \mathbb {R}$. Define ${\widetilde{\sigma }}:= \textrm{max}\{ \sigma _1, \sigma _2^{-1} \}$; then $\theta _{\sigma }(s) \le {\widetilde{\sigma }}^{|s|}$ for all $s \in \mathbb {R}$. Since $\eta _M \le 1 = \eta _0$ for $M \ge 0$, we may assume that $M \le 0$. Then, Lemma 5.6 and Corollary 2.5 imply

$$\begin{aligned} \eta _M(x,s) \lesssim \bigl (1 + \rho _A(A^{-s^+} x)\bigr )^{-M} \lesssim \bigl (1 + |\det A|^{-s^+} \rho _A(x) \bigr )^{|M|} \le (1 + \rho _A(x))^{|M|}. \end{aligned}$$

Since $|\det A|^{-\frac{s}{q}} \le |\det A|^{|s|/q}$, we thus have

$$\begin{aligned} |\det A|^{-\frac{s}{q}} w(x,s) M_Q [W_\psi \varphi ] (x,s) \lesssim \Vert \varphi \Vert \bigl (1 + \rho _A(x)\bigr )^{|M|-L} \bigl ( |\det A|^{|L| + \frac{1}{q}} \, {\widetilde{\sigma }} / \lambda _{-}^N \bigr )^{|s|}. \end{aligned}$$

(5.8)

Therefore, choosing L, N sufficiently large, it is an easy consequence of Lemma 2.3 that

$$\begin{aligned} \Vert M_Q [W_\psi \varphi ] \Vert _{L_w^{p,q}} = \big \Vert |\det A|^{-\frac{s}{q}} \, w(\cdot ) \, M_Q[W_\psi \varphi ] (\cdot ) \big \Vert _{L^{p,q}} \lesssim \Vert \varphi \Vert , \end{aligned}$$

which completes the proof. $\square$

5.4 Coorbit spaces

This section proves wavelet characterizations of anisotropic Triebel–Lizorkin spaces by identifying them with so-called coorbit spaces (cf. [25, 64]).

For technical reasons, coorbit spaces associated with quasi-Banach function spaces are commonly defined in terms of merely left local maximal functions. For a function $F \in L^{\infty }_{\mathop {\textrm{loc}}\limits } (G_A)$, its left maximal function is defined by

$$\begin{aligned} M_Q^L F (g) = \mathop {\mathrm {ess\,sup}}\limits _{u \in Q} |F(g u)|, \quad g \in G_A, \end{aligned}$$

where $Q \subset G_A$ is a relatively compact unit neighborhood.

Definition 5.9

Let $p \in (0,\infty ),q \in (0,\infty ]$, $\alpha \in \mathbb {R}$, and $\beta > 0$. Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be expansive exponential and let $Q \subset G_A$ be a relatively compact, symmetric unit neighborhood.

For an admissible vector $\psi \in \mathcal {S}_0({\mathbb {R}}^d)$, the coorbit space is the collection of all $f \in \mathcal {S}_0'(\mathbb {R}^d)$ satisfying

and equipped with the norm .

In the above definition, note that there exist admissible vectors $\psi \in \mathcal {S}_0(\mathbb {R}^d)$ by Theorem 4.2.

Remark 5.10

The space defined in Definition 5.9 can be identified with the abstract coorbit spaces defined in [64, Definition 4.7]. In particular, several basic properties of coorbit spaces, such as independence of the defining vector $\psi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ and neighborhood Q, follow directly from the theory [64]. See Lemma D.1 for details on the identification.

Anisotropic Triebel–Lizorkin spaces are identified with coorbit spaces by the following proposition.

Proposition 5.11

Let $p \in (0, \infty )$, $q \in (0, \infty ]$, $\alpha \in {\mathbb {R}}$, and $\beta > \textrm{max}\{ 1/p, 1/q \}$. Then

Proof

Throughout, let $\psi \in \mathcal {S}_0(\mathbb {R}^d)$ be admissible (4.4) with compact Fourier support in $\mathbb {R}^d {\setminus } \{0\}$ satisfying the support condition (2.4). The existence of such vectors is guaranteed by Theorem 4.2. Furthermore, let $Q:= [-1, 1)^d \times [-1,1)$. The space is independent of these choices by Remark 5.10. To ease notation, set $\alpha ':= \alpha +1/2-1/q$.

The proof is split into three steps.

Step 1. Let $\psi \in \mathcal {S}_0(\mathbb {R}^d)$ be an admissible vector (4.4) with compact Fourier support in $\mathbb {R}^d {\setminus } \{0\}$ satisfying (2.4). Then also $\psi ^*\in \mathcal {S}_0(\mathbb {R}^d)$ satisfies (2.4), where $\psi ^*(t):= {\overline{\psi }}(-t)$. Since

$$\begin{aligned} \pi (x,s)\psi = |\det A|^{-s/2} \, \psi (A^{-s}( \,\cdot - x)) = |\det A|^{s/2} \, T_x \psi _{-s}, \end{aligned}$$

it follows that

$$\begin{aligned} W_{\psi }f(x,s) = \langle f, \pi (x,s) \psi \rangle = \langle f , |\det A|^{s/2} T_x \psi _{-s} \rangle = |\det A|^{s/2} f *\psi ^*_{-s}(x) . \end{aligned}$$

(5.9)

Therefore, using $\psi ^*$ as the analyzing vector in Theorem 3.5 yields

(5.10)

for any $f \in \mathcal {S}_0'(\mathbb {R}^d)$.

Step 2. Since $|F| \le M^L_Q F$ a.e. on $G_A$ for $F \in L^1_{\mathop {\textrm{loc}}\limits } (G_A)$ (see, e.g., [65, Lemma 2.3.3]), it follows by Step 1 that

for $f \in \mathcal {S}_0'(\mathbb {R}^d)$.

Step 3. This step will show the remaining estimate for $f \in \mathcal {S}_0'(\mathbb {R}^d)$. Parts of the used arguments resemble Step 2 in the proof of Theorem 3.5 and will for this reason only be sketched.

First, a direct calculation using the involved definitions and a change-of-variable yields that

$$\begin{aligned} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{|M^L_Q (W_\psi f) (x+z,s)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }}&= \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \\ (y,t) \in Q \end{array}} \frac{|W_\psi f (x + z + A^s y,s+t)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }} \nonumber \\&= \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \\ (y,t) \in Q \end{array}} \frac{|W_\psi f (x + z,s+t)|}{(1 + \rho _{A}(A^{-s} z - y))^{\beta }} \nonumber \\&\lesssim _{A, \beta } \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \\ t \in [-1,1) \end{array}} \frac{|W_\psi f (x + z,s+t)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }}, \end{aligned}$$

(5.11)

where the last inequality follows from $(1+ \rho _A (A^s z - y))^{-\beta } \lesssim (1+ \rho _A (A^s z))^{-\beta }$ for $y \in [-1,1)^d$.

For fixed $s \in {\mathbb {R}}$ and $t \in [-1,1)$, the identity (5.9) and Corollary 2.5 allows to estimate

$$\begin{aligned} |\det A|^{-s/2} \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \end{array}} \frac{ |W_{\psi } f (x + z, s + t)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }}&\lesssim _A \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \end{array}} \frac{ |(f *\psi ^*_{-(s+t)}) (x + z)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }} \\&\lesssim _{A, \beta } \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \end{array}} \frac{ |(f *\psi ^*_{-(s+t)}) (x + z)|}{(1 + \rho _{A}(A^{-(s+t)} z))^{\beta }} \\&= (\psi ^*)^{**}_{-(s+t), \beta } f(x). \end{aligned}$$

This, combined with (5.11) and $|\det A|^{-\alpha t} \lesssim _{A,\alpha } 1$ for $t \in [-1,1)$, yields that

$$\begin{aligned}&|\det A|^{-s/2} \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{|M^L_Q (W_\psi f) (x+z,s)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }} \\&\quad \lesssim _{A,\beta } |\det A|^{-s/2} \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \\ t \in [-1,1) \end{array}} \frac{|W_\psi f (x + z,s+t)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }} \\&\quad \lesssim _{A,\beta } \mathop {\mathrm {ess\,sup}}\limits _{t \in [-1,1)} (\psi ^*)^{**}_{-(s+t), \beta } f(x) \\&\quad \lesssim _{A, \alpha } \mathop {\mathrm {ess\,sup}}\limits _{t \in [-1,1)} |\det A|^{-\alpha t} (\psi ^*)^{**}_{-(s+t), \beta } f(x). \end{aligned}$$

Now let $q < \infty$. Then arguments similar to proving (3.15) yield $N \in {\mathbb {N}}$ such that

$$\begin{aligned} \bigg ( |\det A|^{-\alpha (s+t)} (\psi ^*)^{**}_{-(s+t), \beta } f(x) \bigg )^q \lesssim \sum _{\ell = - N}^N \bigg ( |\det A|^{- \alpha (s+\ell )} (\psi ^*)^{**}_{-(s+\ell ), \beta } f(x) \bigg )^q. \end{aligned}$$

The right-hand side being independent of $t \in [-1,1)$, it follows that

$$\begin{aligned}&\bigg (|\det A|^{-(\alpha +1/2)s} \mathop {\mathrm {ess\,sup}}\limits _{\begin{array}{c} z \in {\mathbb {R}}^d \end{array}} \frac{ | M^L_Q(W_{\psi } f) (x + z, s)|}{(1 + \rho _{A}(A^{-s} z))^{\beta }} \bigg )^q \nonumber \\&\quad \lesssim \!\mathop {\mathrm {ess\,sup}}\limits _{t \in [-1,1)} \bigg (|\det A|^{-\alpha (s+t)} (\psi ^*)^{**}_{-(s+t), \beta } f(x) \bigg )^q \nonumber \\&\quad \lesssim \sum _{\ell = - N}^N \bigg ( |\det A|^{- \alpha (s+\ell )} (\psi ^*)^{**}_{-(s+\ell ), \beta } f (x) \bigg )^q. \end{aligned}$$

(5.12)

Combining this estimate with Theorem 3.5 gives

The case $q = \infty$ follows from (5.12) by similar arguments. $\square$

Remark 5.12

For $p \in [1,\infty )$ and $q \in [1,\infty ]$, the coorbit spaces of Definition 5.9 are genuine Banach spaces, which are well-known to admit the equivalent norm

see, e.g., [26, Theorem 8.3] and [64, Proposition 4.10]. The proof of Proposition 5.11 shows that the same holds for the Peetre-type spaces in the quasi-Banach range $\min \{ p,q \} < 1$.

6 Molecular characterizations

This section provides new molecular characterizations of anisotropic Triebel–Lizorkin spaces. The results will be obtained from [53, 64] by exploiting the coorbit identification of Triebel–Lizorkin spaces provided by Proposition 5.11.

6.1 Peetre-type sequence space

Let $\Gamma \subset G_A$ be arbitrary and let $U \subset G_A$ be a relatively compact unit neighborhood. The set $\Gamma$ is relatively separated if

$$\begin{aligned} \sup _{g \in G} \# \big ( \Gamma \cap g U) = \sup _{g \in G} \sum _{\gamma \in \Gamma } \mathbb {1}_{\gamma U^{-1}} (g) < \infty \end{aligned}$$

and is called U-dense if $G = \bigcup _{\gamma \in \Gamma } \gamma U$. The set $\Gamma$ is U-separated in G if $\mu _{G_A} (\gamma U \cap \gamma ' U) = 0$ for all $\gamma , \gamma ' \in \Gamma$ with $\gamma \ne \gamma '$ and is separated if it is U-separated for some unit neighborhood U. Any separated set is relatively separated. Furthermore, the notion of being relatively separated is independent of the choice of the relatively compact unit neighborhood U.

Definition 6.1

Let $\Gamma \subset G_A$ be relatively separated and let $U \subset G_A$ be a relatively compact unit neighborhood. For $p \in (0,\infty ),q \in (0,\infty ]$, $\alpha \in \mathbb {R}$, and $\beta > 0$, the sequence space associated to the Peetre-type space is defined as the set of all $c = (c_{\gamma })_{\gamma \in \Gamma } \in {\mathbb {C}}^{\Gamma }$ such that

and equipped with the (quasi)-norm .

The sequence space is a well-defined quasi-Banach space, independent of the choice of the defining neighborhood U; see, e.g., [25, 51] or [65, Lemma 2.3.16].

Remark 6.2

The Triebel–Lizorkin space defined in (2.6) can be identified with a sequence space via Theorem 3.8. To be more explicit, if $\Gamma = \{(A^{-j} k, -j): j \in {\mathbb {Z}}, k \in {\mathbb {Z}}^d \}$, then the map

is an isomorphism of (quasi)-Banach spaces, for any $p \in (0,\infty )$, $q \in (0,\infty ]$, $\alpha \in {\mathbb {R}}$ and $\beta > \textrm{max}\{1/p, 1/q \}$.

6.2 Molecular systems and the extended pairing

Following [40, 53, 64], the notion of molecular systems used in this paper is defined through properties of the associated wavelet transform.

Definition 6.3

Let $\Gamma \subset G_A$ be relatively separated and let $\psi \in L^2 ({\mathbb {R}}^d)$ be an admissible vector such that $W_{\psi } \psi \in \mathcal {W}(L^r_w)$, where $w = w_{p,q}^{\alpha , \beta }: G_A \rightarrow [1, \infty )$ is the standard control weight of Lemma 5.7.

A family $(\phi _{\gamma })_{\gamma \in \Gamma }$ of vectors $\phi _{\gamma } \in L^2 ({\mathbb {R}}^d)$ is an $L^r_w$-molecular system if there exists an envelope $\Phi \in \mathcal {W}(L^r_w)$ such that

$$\begin{aligned} |W_{\psi } \phi _{\gamma } (x) | \le L_{\gamma } \Phi (x), \end{aligned}$$

(6.1)

for $x \in G_A$ and $\gamma \in \Gamma$.

Remark 6.4

The condition (6.1) is independent of the choice of the window $\psi$ in the following sense: If $\psi ,\varphi \in L^2(\mathbb {R}^d)$ are both admissible satisfying $W_\psi \varphi , W_\psi \psi , W_\varphi \varphi \in \mathcal {W}(L^r_w)$, then $(\phi _\gamma )_{\gamma \in \Gamma } \subset L^2(\mathbb {R}^d)$ is a molecular system with respect to the window $\psi$ if and only if the same holds with respect to the window $\varphi$; see [64, Lemma 6.3].

In order to treat molecular systems consisting of general vectors in $L^2 ({\mathbb {R}}^d)$ in a meaningful manner, we define the following extended dual pairing.

Definition 6.5

Let $\psi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ be admissible. For $f \in {\mathcal {S}}'_0 ({\mathbb {R}}^d)$ and $\phi \in L^2 ({\mathbb {R}}^d)$, define the extended pairing as

$$\begin{aligned} \langle f, \phi \rangle _{\psi }:= \int _{G_{A}} \langle f, \pi (x,s) \psi \rangle \,\, \overline{\langle \phi , \pi (x,s) \psi \rangle _{L^2}} \,\, d\mu _{G_A} (x,s), \end{aligned}$$

provided that the integral converges.

Remark 6.6

Let $\psi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ be admissible.

• (a) If $f \in {\mathcal {S}}'_0 ({\mathbb {R}}^d)$ and $\phi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$, then the extended pairing $\langle f, \phi \rangle _{\psi }$ coincides with the standard conjugate linear pairing $\langle f, \phi \rangle := f ({\overline{\phi }})$ by Lemma 4.7.

• (b) If both $f, \phi \in L^2 ({\mathbb {R}}^d)$, then $\langle f, \phi \rangle _{\psi }$ coincides with the $L^2$-inner product $\langle f, \phi \rangle$ by Eq. (4.2).

For showing that the extended pairing defined in Definition 6.5 is well-defined, in the sense that it does not depend on the choice of admissible vectors, the following approximation property will be used.

Lemma 6.7

Let $f \in {\mathcal {S}}_0' ({\mathbb {R}}^d)$ and let $\psi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ be admissible with $W_\psi f \in L_{1/w}^\infty (G_A)$, where $w: G \rightarrow [1,\infty )$ denotes the standard control weight provided by Lemma 5.7.

There exists a sequence $(f_n)_{n \in \mathbb {N}}$ of functions $f_n \in L^2(\mathbb {R}^d)$ with the following properties:

• (i) As $n \rightarrow \infty$, $f_n \rightarrow f$ with weak-$*$-convergence in ${\mathcal {S}}_0' ({\mathbb {R}}^d)$;

• (ii) For each $\varphi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$, there is a constant $C = C(\varphi ,\psi ,f) > 0$ such that

  $$\begin{aligned} |W_\varphi f_n (g)| \le C w(g), \quad g \in G_A.\end{aligned}$$

Proof

For $n \in \mathbb {N}$, define $\Omega _n:= [-n,n]^d \times [-n,n]$ and $F_n:= W_\psi f \cdot \mathbb {1}_{\Omega _n}$. Note that since w is continuous and $\Omega _n \subset G_A$ is compact, for each $n \in \mathbb {N}$ there is $C_n > 0$ satisfying $w(x) \le C_n$ for all $x \in \Omega _n$. This implies $|F_n(\cdot )| \le C_n \Vert W_\psi f \Vert _{L_{1/w}^\infty } \mathbb {1}_{\Omega _n}(\cdot ) \in L^1(G_A).$ Since $g \mapsto \pi (g) \psi$ is continuous from $G_A$ into $L^2(\mathbb {R}^d)$ and $\Vert F_n(\cdot ) \, \pi (\cdot )\psi \Vert _{L^2} \le \Vert \psi \Vert _{L^2} |F_n(\cdot )| \in L^1(G_A)$, this shows that ${f_n:= \int _{G_A} F_n(g) \, \pi (g) \psi \, d \mu _{G_A}(g) \in L^2(\mathbb {R}^d)}$ is well-defined as a Bochner integral.

Let $\varphi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ be arbitrary. For $h \in G_A$, a direct calculation gives

$$\begin{aligned} |W_\varphi f_n (h)|&\le \int _{G_A} |F_n(g)| |\langle \pi (g) \psi , \pi (h) \varphi \rangle | \, d \mu _{G_A}(g) \\&\le \int _{G_A} |W_\psi f(g)| |W_\psi \varphi (h^{-1} g)| \, d \mu _{G_A}(g) \\&\le \Vert W_\psi f \Vert _{L_{1/w}^{\infty }} \! \int _{G_A} \!\!\!\! w(h) \, w(h^{-1} \! g) \, |W_\psi \varphi (h^{-1} g)| \, d \mu _{G_A}(g) \\&\le w(h) \, \Vert W_\psi f \Vert _{L_{1/w}^{\infty }} \, \Vert W_\psi \varphi \Vert _{L_w^1} , \end{aligned}$$

where $\Vert W_\psi \varphi \Vert _{L_w^1} < \infty$ by Lemma 5.8. This proves (ii).

To prove (i), applying the dominated convergence theorem and Corollary 4.8 gives

$$\begin{aligned} \lim _{n \rightarrow \infty } W_\psi f_n (h)&= \lim _{n \rightarrow \infty } \int _{G_A} F_n (g) \langle \pi (g) \psi , \pi (h) \psi \rangle \, d \mu _{G_A} (g) \\&= (W_\psi f *W_\psi \psi )(h) = W_\psi f (h). \end{aligned}$$

As shown above, $W_\psi f_n \rightarrow W_\psi f$ pointwise and $|W_\psi f_n (g)| \le C w(g)$. On the other hand, given $\varphi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$, Lemma 5.8 shows that $W_\psi \varphi \in L_w^1(G_A)$. Therefore, a combination of Corollary 4.8 with the dominated convergence theorem shows that

$$\begin{aligned} \langle f, \varphi \rangle = \lim _{n \rightarrow \infty } \int _{G_A} W_\psi f_n (g) \overline{W_\psi \varphi (g)} \, d \mu _{G_A} (g) = \lim _{n \rightarrow \infty } \langle f_n, \varphi \rangle , \end{aligned}$$

proving that $f_n \rightarrow f$ with respect to the weak-$*$-topology on ${\mathcal {S}}'_0 (\mathbb {R}^d)$. $\square$

Lemma 6.8

Let $w: G_A \rightarrow [1,\infty )$ be a standard control weight as in Lemma 5.7. Let $\psi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ be admissible.

If $f \in {\mathcal {S}}_0'({\mathbb {R}}^d)$ satisfies $W_\psi f \in L_{1/w}^\infty (G_A)$ and $\phi \in L^2(\mathbb {R}^d)$ satisfies $W_\psi \phi \in L_w^1(G_A)$, then $\langle f, \phi \rangle _{\psi }$ is well-defined and independent of the choice of $\psi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$.

Proof

We first show for $Y = L_{1/w}^\infty (G_A)$ or $Y = L_w^1(G_A)$ that if $f \in {\mathcal {S}}_0' ({\mathbb {R}}^d)$ satisfies $W_\psi f \in Y$, then $W_\varphi f \in Y$ for every $\varphi \in {\mathcal {S}}_0({\mathbb {R}}^d)$. For this, first note that ${W_\varphi f = W_\psi f *W_\varphi \psi }$; see Corollary 4.8. If $Y = L_w^1(G_A)$, the submultiplicativity of w implies $Y *L_w^1(G_A) \subset L_w^1(G_A)$, while Lemma 5.8 shows that $W_\varphi \psi \in L_w^1(G_A)$. Thus, $W_\varphi f \in L^1_{w} (G_A)$. In case of $Y = L_{1/w}^\infty (G_A)$, note that $|(W_\varphi \psi ) (h)| = |(W_\psi \varphi ) (h^{-1})|$, and

$$\begin{aligned} \frac{1}{w(g)} \, \bigl | W_\varphi f (g) \bigr |&\le \int _{G_A} \frac{1}{w(h)} \, \bigl |W_\psi f (h)\bigr | \frac{w(h)}{w(g)} \, \bigl |W_\psi \varphi (g^{-1} h)\bigr | \, d \mu _{G_A} (h) \\&\le \Vert W_\psi f \Vert _{L_{1/w}^\infty } \Vert W_\psi \varphi \Vert _{L_w^1} \end{aligned}$$

for all $g \in G_A$; here, we used that $\frac{w(h)}{w(g)} \le \frac{w(g)w(g^{-1}h)}{w(g)} = w(g^{-1} h).$ Thus, also $W_\varphi f \in L_{1/w}^\infty (G_A)$.

Since $W_\psi f \in L_{1/w}^\infty (G_A)$ and $W_\psi \phi \in L_w^1(G_A)$ by assumption, it is clear that $\langle f, \phi \rangle _{\psi } \in \mathbb {C}$ is well-defined. Next, let $\varphi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ be admissible, and let $(f_n)_{n \in \mathbb {N}} \subset L^2(\mathbb {R}^d)$ as provided by Lemma 6.7. Note that $W_\phi f_n (g) = \langle f_n, \pi (g) \phi \rangle \rightarrow \langle f, \pi (g) \phi \rangle = W_\phi f (g)$ for all $\phi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ and $g \in G_A$. Therefore, an application of the dominated convergence theorem implies

$$\begin{aligned} \langle f, \phi \rangle _{\psi }&= \langle W_\psi f, W_\psi \phi \rangle = \lim _{n \rightarrow \infty } \langle W_\psi f_n , W_\psi \phi \rangle _{L^2}\\&= \lim _{n \rightarrow \infty } \langle W_\varphi f_n , W_\varphi \phi \rangle _{L^2} = \langle f, \phi \rangle _{\varphi } , \end{aligned}$$

where we used the isometry of $W_\varphi , W_\psi : L^2(\mathbb {R}^d) \rightarrow L^2(G_A)$. $\square$

6.3 Molecular decompositions

This section provides the proofs of Theorems 1.2 and 1.3.

We first show the following auxiliary claim which is implicit in the statements of Theorems 1.2 and 1.3.

Lemma 6.9

Let $p \in (0,\infty )$, $q \in (0,\infty ]$, $\alpha \in \mathbb {R}$, and $\beta > 0$. Let $w = w_{p,q}^{\alpha ,\beta }: G_A \rightarrow [1,\infty )$ be a standard control weight as defined in Lemma 5.7 and let $r = \min \{ 1,p,q \}$. If $\psi \in L^2(\mathbb {R}^d)$ and if $\varphi \in {\mathcal {S}}_0(\mathbb {R}^d)$ is admissible with $W_{\varphi } \psi \in \mathcal {W}(L^r_w)$, then $W_\phi \psi \in \mathcal {W}(L^r_w)$ for all $\phi \in {\mathcal {S}}_0 (\mathbb {R}^d)$.

Proof

By Eq. 4.3, the identity $W_{\phi } \psi = W_{\varphi } \psi *W_{\phi } \varphi$ holds. Note that $W_\phi \varphi \in \mathcal {W}(L^r_w)$ by Lemma 5.8 and $W_\varphi \psi \in \mathcal {W}(L^r_w)$ by assumption. The weight $w:= w_{p,q}^{\alpha ,\beta }$ is continuous and submultiplicative with $w \ge 1$ and satisfies $w(g) = w(g^{-1}) \Delta ^{1/r}(g^{-1})$, meaning that it is an r-weight in the terminology of [64, Definition 3.1]. Therefore, using the convolution relation $\mathcal {W}(L^r_w)*\mathcal {W}(L^r_w)\hookrightarrow \mathcal {W}(L^r_w)$ from [64, Corollary 3.9], we see that $W_\phi \psi \in \mathcal {W}(L^r_w)$, as claimed. $\square$

Proof of Theorem 1.2

Let $\varphi \in {\mathcal {S}} (\mathbb {R}^d)$ be an admissible vector satisfying ${\widehat{\varphi }} \in C_c^{\infty } ({\mathbb {R}}^d {\setminus } \{0\})$ and the support condition (2.4); see Theorem 4.2. Then an application of Proposition 5.11 yields that . Furthermore, since $\psi , \varphi \in L^2 ({\mathbb {R}}^d)$ are admissible and satisfy $W_\psi \psi , W_{\varphi } \varphi , W_{\varphi } \psi \in \mathcal {W}(L^r_w)$ (see Lemma 5.8 and 6.9), it follows by Lemma D.1 that $\dot{\textbf{F}}^{\alpha }_{p,q}$ can be identified with the abstract coorbit space used in [64].

An application of [64, Theorem 6.14] yields a compact unit neighborhood $U \subset G_A$ such that for any $\Gamma \subset G_A$ satisfying condition (1.8), there exist molecular systems $(\phi _\gamma )_{\gamma \in \Gamma } \subset L^2(\mathbb {R}^d)$ and $(f_\gamma )_{\gamma \in \Gamma } \subset L^2(\mathbb {R}^d)$, such that every can be represented as

$$\begin{aligned} f \!=\! \sum _{\gamma \in \Gamma } \langle f, \pi (\gamma ) \psi \rangle _{{\mathcal {R}}_w, {\mathcal {H}}_w^1} \!\, \phi _\gamma = \sum _{\gamma \in \Gamma } \langle f, \phi _\gamma \rangle _{{\mathcal {R}}_w, {\mathcal {H}}_w^1} \!\, \pi (\gamma ) \psi \!=\! \sum _{\gamma \in \Gamma } \langle f, f_\gamma \rangle _{{\mathcal {R}}_w, {\mathcal {H}}_w^1} \, f_\gamma , \nonumber \\ \end{aligned}$$

(6.2)

with unconditional convergence of the series in the weak-$*$-topology on the space ${\mathcal {R}}_w = {\mathcal {R}}_w (\psi )$ introduced in Sect. 1. By Lemma D.1, any $f \in \dot{\textbf{F}}^{\alpha }_{p,q}$ can be extended uniquely to an element . Since $(\phi _\gamma )_{\gamma \in \Gamma }$ and $(f_\gamma )_{\gamma \in \Gamma }$ are molecules with respect to $\psi$, they also satisfy the molecule condition with respect to $\varphi$ by Remark 6.4. Therefore, Eq. D.2 shows that the dual pairings occurring in Eq. 6.2 coincide with the extended dual pairing from Definition 6.5. Lastly, applying Eq. 6.2 to ${\widetilde{f}}$, using Eq. D.2, and restricting the domain of both sides of Eq. 6.2 to $\mathcal {S}_0(\mathbb {R}^d) \subset {\mathcal {H}}_w^1(\varphi ) = {\mathcal {H}}_w^1(\psi )$, we see that Eq. 6.2 holds for all $f \in \dot{\textbf{F}}^{\alpha }_{p,q}$, with unconditional convergence of the series in the weak-$*$-topology on $\mathcal {S}'(\mathbb {R}^d) / \mathcal {P}(\mathbb {R}^d) = \mathcal {S}_0 ' (\mathbb {R}^d) \hookrightarrow {\mathcal {R}}_w (\psi )$. $\square$

Proof of Theorem 1.3

As in the proof of Theorem 1.2, the Triebel–Lizorkin space $\dot{\textbf{F}}^{\alpha }_{p,q}$ can be identified with of Lemma D.1. An application of [64, Theorem 6.15] yields a compact unit neighborhood $U \subset G_A$ such that for any $\Gamma \subset G_A$ satisfying condition (1.9), there exists a molecular systems $(\phi _{\gamma })_{\gamma \in \Gamma }$ in $\overline{\mathop {\textrm{span}}\limits \{ \pi (\gamma ) \psi : \gamma \in \Gamma \} }$ such that, given $(c_{\gamma })_{\gamma \in \Gamma } \in \dot{{\textbf{p}}}^{- \alpha ', \beta }_{p, q}$, the vector satisfies

$$\begin{aligned} \langle {\widetilde{f}}, \pi (\gamma ) \psi \rangle _{{\mathcal {R}}_w, {\mathcal {H}}^1_w} = c_{\gamma }, \quad \gamma \in \Gamma . \end{aligned}$$

(6.3)

Arguing as in the proof of Theorem 1.2, another application of Lemma D.1 yields that the restriction $f = {\widetilde{f}}|_{{\mathcal {S}}_0} \in \dot{\textbf{F}}^{\alpha }_{p,q}$ satisfies $\langle f, \pi (\gamma ) \psi \rangle _{\varphi } = \langle {\widetilde{f}}, \pi (\gamma ) \psi \rangle _{{\mathcal {R}}_w, {\mathcal {H}}^1_w}$ for all $\gamma \in \Gamma$. $\square$

6.4 Explicit criteria

This section provides explicit criteria for coorbit molecules. The proof relies on the following lemma concerning the standard envelope from Definition 5.5.

Lemma 6.10

Let $r \in (0,1]$. If $\sigma \in (0,\infty )^2$ satisfies $\sigma _1 < 1$, $\sigma _2 > |\det A|^{1/r}$ and if $L > 1/r$, then $\Xi _{\sigma ,L} \in L^r(G_A)$.

Proof

Using Lemma 5.6, a change-of-variable yields

$$\begin{aligned} \int _{\mathbb {R}^d} |\eta _{L} (x,s)|r \, d x&\asymp \!\int _{\mathbb {R}^d} \bigl (1 \!+\! \rho _A (A^{-s^+} \!\, x)\bigr )^{-Lr} \, d x \!=\! |\det A|^{s^+} \int _{\mathbb {R}^d} \bigl (1 \!+\! \rho _A (y) \bigr )^{-Lr} \, \!d y \\&\asymp |\det A|^{s^+}, \end{aligned}$$

where the last step follows from Lemma 2.3 and the assumption $Lr > 1$. Therefore,

$$\begin{aligned} \Vert \Xi _{\sigma ,L} \Vert _{L^r(G_A)}&= \int _{\mathbb {R}} |\det A|^{-s} (\theta _\sigma (s))^r \int _{\mathbb {R}^d} (\eta _L (x,s))^r \, d x \, d s \\&\asymp \int _{\mathbb {R}} |\det A|^{s^+ - s} (\theta _\sigma (s))^r \, d s \\&= \int _0^\infty \sigma _1^{rs} \, d s + \int _{-\infty }^0 |\det A|^{-s} \sigma _2^{rs} \, d s \\&= \int _0^\infty \! e^{s r \ln \sigma _1} d s + \int _{-\infty }^0 e^{s(r \ln \sigma _2 - \ln |\det A|)} \, d s < \infty \end{aligned}$$

since $\ln \sigma _1 < 0$ and $r \ln \sigma _2 > \ln |\det A|$ by assumption. $\square$

Theorem 6.11

For $p \in (0, \infty )$, $q \in (0, \infty ]$, let $r = \min \{p,q, 1\}$. Let $\alpha \in \mathbb {R}$, $\beta > 0$. Choose constants $L > 1$, $N \in \mathbb {N}_0$ and $\delta \in (0,1)$ such that $L \cdot (1 - \delta ) > 1/r + \beta$ and

$$\begin{aligned} \lambda _-^{\delta N} > \textrm{max}\bigg \{ |\det A|^{\frac{1}{r} - \frac{1}{2} + |\alpha +\frac{1}{p}-\frac{1}{q}|}, \; |\det A|^{-\frac{1}{2} + \frac{1}{r} + \alpha + \beta - \frac{1}{q}}, \; |\det A|^{-\frac{1}{2} - (\alpha - \frac{1}{q})} \bigg \}, \end{aligned}$$

(6.4)

where $\lambda _{-} \in {\mathbb {R}}$ satisfies $1< \lambda _{-} < \min _{\lambda \in \sigma (A)} | \lambda |$ as in Sect. 2.1.

Suppose $f \in L^2(\mathbb {R}^d) \cap C^N(\mathbb {R}^d)$ satisfies

$$\begin{aligned}{} & {} |f(x)| \lesssim (1 + \rho _A(x))^{-L}, \quad \int _{\mathbb {R}^d} \Vert x\Vert ^N |f(x)| \, d x< \infty , \quad \text {and} \quad \textrm{max}_{|\alpha | \le N} \sup _{x \in \mathbb {R}^d} |\partial ^\alpha f (x)| < \infty ,\nonumber \\ \end{aligned}$$

(6.5)

as well as

$$\begin{aligned} \int _{\mathbb {R}^d} x^\alpha \, f(x) \, d x = 0 \quad \text {for all } \alpha \in \mathbb {N}_0^d \text { with } |\alpha | < N. \end{aligned}$$

(6.6)

Then $W_f f \in \mathcal {W}(L^r_w)$ for the control weight $w = w^{\alpha ,\beta }_{p,q}$ provided by Lemma 5.7.

Proof

We need to show that $M_Q ( W_f f) \in L_w^r (G_A)$. The proof is split into two steps.

Step 1. In this step, we show that $|W_ff(x,s)| \lesssim \Xi _{\tau ,L(1-\delta )}(x,s)$, where $\tau = (\tau _1, \tau _2)$ with $\tau _1: =|\det A|^{-1/2} \lambda _-^{-N \delta }$ and $\tau _2:=|\det A|^{1/2} \lambda _-^{N \delta }$. Assumptions (6.5) and (6.6) together with Lemma 4.5 imply that

$$\begin{aligned} |W_ff(x,s)| \lesssim |\det A|^{-|s|/2} \big (1+\rho _A(A^{-s^+}x)\big )^{-L}, \quad x \in \mathbb {R}^d, s \in \mathbb {R}, \end{aligned}$$

(6.7)

and $|W_ff(x,s)| \lesssim |\det A|^{-s/2} \Vert A^{-s}\Vert _{\infty }^N \lesssim |\det A|^{-s/2} \lambda _-^{-sN}$ for $x \in \mathbb {R}^d, s \ge 0$, by Lemma 2.4. Applying this latter estimate to $W_ff(x,s) = \overline{ W_ff(-A^{-s}x,-s)}$ for $s<0$ yields immediately that $|W_ff(x,s)| = |W_ff(-A^{-s}x,-s)| \lesssim |\det A|^{s/2} \lambda _-^{sN}$ and therefore

$$\begin{aligned} |W_ff(x,s)| \lesssim |\det A|^{-|s|/2} \lambda _-^{-|s|N}, \quad x \in \mathbb {R}^d, s \in \mathbb {R}. \end{aligned}$$

(6.8)

Combining (6.7) and (6.8) gives

$$\begin{aligned} |W_ff(x,s)|&= |W_ff(x,s)|^{\delta } |W_ff(x,s)|^{1-\delta } \\&\lesssim |\det A|^{-|s|/2} \lambda _-^{-|s|N\delta } \big (1+\rho _A(A^{-s^+}x)\big )^{-L(1-\delta )}, \end{aligned}$$

as desired.

Step 2. Step 1 and Lemma B.2 yield $w M_Q (W_f f) \lesssim w M_Q (\Xi _{\tau ,L(1-\delta )}) \lesssim w \Xi _{\tau ,L(1-\delta )}.$ Recall from Lemma 5.7 that the standard control weight satisfies $w \asymp \Xi _{\sigma ,0} + \Xi _{\kappa ,-\beta }$, where $\sigma , \kappa \in (0,\infty )^2$ are as in the statement of Lemma 5.7. Denote by $\tau \odot \sigma :=(\tau _1 \sigma _1, \tau _2 \sigma _2)$ component-wise multiplication. Then

$$\begin{aligned} w M_Q (W_f f) \lesssim w \Xi _{\tau ,L(1-\delta )} \lesssim \Xi _{\tau \odot \sigma ,L(1-\delta )} + \Xi _{\tau \odot \kappa ,L(1-\delta )-\beta }. \end{aligned}$$

(6.9)

It remains to verify the integrability conditions for standard envelopes of Lemma 6.10 for the right-hand side of (6.9). The assumption $L \cdot (1 - \delta ) > 1/r + \beta$ guarantees that

$$\begin{aligned} \min \{L(1-\delta ), L(1-\delta )-\beta \} > 1/r, \end{aligned}$$

while the assumption (6.4) implies that both of the conditions $\textrm{max}\{\tau _1 \sigma _1, \tau _1 \kappa _1\} < 1$ and $\min \{\tau _2 \sigma _2, \tau _2 \kappa _2\} > |\det A|^{1/r}$ are satisfied. An application of Lemma 6.10 therefore yields $M_Q (W_f f) \in L_w^ r(G_A)$. $\square$

Appendices

Appendix A. The Peetre-type maximal function

Lemma A.1

Let $A \in {\text {GL}}(d,\mathbb {R})$ be expansive and $\beta > 0$. Let either $s \in \mathbb {Z}$ or let $s \in \mathbb {R}$ and assume that A is exponential. Let $f: \mathbb {R}^d \rightarrow [0,\infty )$ be continuous. Then

$$\begin{aligned} \sup _{z \in \mathbb {R}^d} \frac{f(z)}{(1 + \rho _A (A^s z))^\beta } = \mathop {\mathrm {ess\,sup}}\limits _{z \in \mathbb {R}^d} \frac{f(z)}{(1 + \rho _A (A^s z))^\beta }. \end{aligned}$$

Proof

First, we see from the definition of $\rho _A$ (see Eq. (2.1)) for arbitrary $\lambda \in \mathbb {R}$ that

$$\begin{aligned} \{ x \in \mathbb {R}^d :\rho _A (x)< \lambda \} = {\left\{ \begin{array}{ll} \emptyset , &{} \text {if } \lambda \le 0, \\ A^{k+1} \Omega _A, &{} \text {if } |\det A|^k < \lambda \le |\det A|^{k+1} \text { for } k \in \mathbb {Z}. \end{array}\right. }\nonumber \\ \nonumber \\ \end{aligned}$$

(A.1)

Since $\Omega _A \subset \mathbb {R}^d$ is open, this shows that $\{ x \in \mathbb {R}^d :\rho _A(x) < \lambda \}$ is always open.

We now show for arbitrary $\theta \in \mathbb {R}$ that $W:= \bigl \{ z \in \mathbb {R}^d :f(z) / (1 + \rho _A(A^s z))^\beta > \theta \bigr \}$ is open; this then easily implies the claim of the lemma, since every non-empty open set has positive Lebesgue measure. First, if $\theta < 0$, then $W = \mathbb {R}^d$ is open. Next, if $\theta = 0$, then $W = \{ z \in \mathbb {R}^d :f(z) \ne 0 \}$ is open. Finally, let $\theta > 0$, $z_0 \in W$, $a:= f(z_0)$, and $b:= (1 + \rho _A(A^{s} z_0))^\beta$. Then $a/b> \theta > 0$ and hence $a > 0$. We can thus choose $0< a' < a$ with $a'/b > \theta$, meaning that $\rho _A (A^s z_0) < (a'/\theta )^{1/\beta } - 1$. Note that $U:= \{ z \in \mathbb {R}^d :f(z) > a'\}$ is an open neighborhood of $z_0$. Similar, the considerations from the beginning of the proof show that $V:= \{ z \in \mathbb {R}^d :\rho _A(A^s z) < (a'/\theta )^{1/\beta } - 1 \}$ is an open neighborhood of $z_0$. Finally, note that $U \cap V \subset W$. Since $z_0 \in W$ was arbitrary, this shows that $W \subset \mathbb {R}^d$ is indeed open. $\square$

Appendix B. Peetre-type maximal function on ${\mathbb {R}}^d \times {\mathbb {R}}$

Let $A \in \textrm{GL}(d, {\mathbb {R}})$ be an expansive exponential matrix and $\beta > 0$. For any measurable function $F: {\mathbb {R}}^d \times {\mathbb {R}} \rightarrow {\mathbb {C}}$, define

$$\begin{aligned} (F^{**})_{\beta }(x,s):= \mathop {\mathrm {ess\,sup}}\limits _{z \in {\mathbb {R}}^d} \frac{|F(x+z, s)|}{(1 + \rho _A (A^{-s} z))^{\beta }} \end{aligned}$$

for $(x, s) \in {\mathbb {R}}^d \times {\mathbb {R}}$. We use the following basic properties repeatedly.

Lemma B.1

If $F: {\mathbb {R}}^d \times {\mathbb {R}} \rightarrow {\mathbb {C}}$ is measurable, then $(F^{**})_{\beta }: {\mathbb {R}}^d \times {\mathbb {R}} \rightarrow [0, \infty ]$ is measurable. Furthermore, there is a constant $C = C(\beta ,A) \ge 1$ such that for each $s \in \mathbb {R}$, there is a null-set $N_s \subset \mathbb {R}^d$ such that for all $x, w \in \mathbb {R}^d$ with $x + w \notin N_s$, we have

$$\begin{aligned} \frac{|F(x+w, s)|}{(1 + \rho _A (A^{-s} w))^\beta } \le C (F^{**})_{\beta }(x,s). \end{aligned}$$

In particular, if $(F^{**})_{\beta }= 0$ almost everywhere, then $F = 0$ almost everywhere.

Proof

Since the map $H: (x,z,s) \mapsto \frac{|F(x+z, s)|}{(1 + \rho _A(A^{-s} z))^\beta }$ is measurable, it is well-known that $(F^{**})_{\beta }(x,s) = \mathop {\mathrm {ess\,sup}}\limits _{z \in \mathbb {R}^d} H(x,z,s)$ is measurable as well; see, e.g., [43, Lemma B.4].

Let us fix $s \in \mathbb {R}$ and write $F_s (x):= F(x,s)$ and $Q_s:= A^s \Omega _A$, with $\Omega _A \subset \mathbb {R}^d$ as in Lemma 2.1. For the open unit neighborhood $Q_s \subset \mathbb {R}^d$, we consider the local maximal function $M^L_{Q_s} f (x) = \mathop {\mathrm {ess\,sup}}\limits _{q \in Q_s} |f(x + q)|$ of a measurable $f: \mathbb {R}^d \rightarrow \mathbb {C}$.

Note that if $x \in \Omega _A {\setminus } \{ 0 \}$, then $x \in A^j \Omega _A$ for some minimal $j \in \mathbb {Z}$, which necessarily satisfies $j \le 0$. Hence, $x \in A^{(j-1)+1} \Omega _A {\setminus } A^{j-1} \Omega$, so that $\rho _A (x) = |\det A|^{j-1} \le |\det A|^{-1} \le 1$, by definition of $\rho _A$. Furthermore, Lemma 2.1 yields $C' \ge 1$ with $\rho _A(x+y) \le C' [\rho _A(x) + \rho _A(y)]$ for all $x,y \in \mathbb {R}^d$. For $q \in Q_s$, we then have $A^{-s} q \in \Omega _A$ and hence $\rho _A(A^{-s} q) \le 1$. Thus, $1 + \rho _A (A^{-s} (z + q)) \le (1 + C') (1 + \rho _A (A^{-s} z))$ for $q \in Q_s$ and $z \in \mathbb {R}^d$.

Now, by definition of $(F^{**})_{\beta }$, given $x \in \mathbb {R}^d$ and $s \in \mathbb {R}$, there is a null-set $N_{s,x} \subset \mathbb {R}^d$ with

$$\begin{aligned} \frac{|F(x+z, s)|}{(1 + \rho _A(A^{-s} z))^\beta } \le (F^{**})_{\beta }(x,s), \qquad \, z \in \mathbb {R}^d \setminus N_{s,x}. \end{aligned}$$

Fix $x,z \in \mathbb {R}^d$ and $s \in \mathbb {R}$. Then, for $q \in Q_s {\setminus } (N_{s,x} - z)$, we have $z + q \in \mathbb {R}^d {\setminus } N_{s,x}$, and hence

$$\begin{aligned} (F^{**})_{\beta }(x,s) \ge \frac{|F_s(x + z + q)|}{(1 + \rho _A(A^{-s} (z+q)))^\beta } \ge (1 + C')^{-\beta } \frac{|F_s (x + z + q)|}{(1 + \rho _A (A^{-s} z))^\beta }. \end{aligned}$$

Therefore, $M^L_{Q_s} F_s (x + z) \le C (1 + \rho _A(A^{-s} z))^\beta (F^{**})_{\beta }(x,s)$ for all $x,z \in \mathbb {R}^d$ and $s \in \mathbb {R}$, with $C:= (1 + C')^\beta$.

Lastly, it follows by [65, Lemma 2.3.3] that there exists a null-set $N_s = N_{s,F} \subset \mathbb {R}^d$ with $|F_s(x)| \le M^L_{Q_s} F_s (x)$ for all $x \in \mathbb {R}^d {\setminus } N_s$. For $x,w \in \mathbb {R}^d$ with $x + w \notin N_s$, we then see that

$$\begin{aligned} |F(x+w, s)|&= |F_s (x + w)| \le M^L_{Q_s} F_s (x + w) \\&\le C (1 + \rho _A (A^{-s} w))^\beta \, (F^{**})_{\beta }(x, s), \end{aligned}$$

which completes the proof. $\square$

The following lemma allows to estimate maximal functions of the standard envelope defined in Definition 5.5.

Lemma B.2

For arbitrary $\sigma \in (0,\infty )^2$ and $L \ge 0$, let $\Xi _{\sigma ,L}: G_A \rightarrow (0,\infty )$ denote the standard envelope defined in Definition 5.5. Then for any relatively compact unit-neighborhood $Q \subset G_A$, there exists a constant $C = C(Q,\sigma , L, A) > 0$ such that $M_Q \Xi _{\sigma ,L} \le C \cdot \Xi _{\sigma ,L}$.

Proof

Since $Q \subset G_A$ is relatively compact, we have $Q \subset [-N,N]^d \times [-N,N]$ for some $N \ge 1$. Recall that $\Xi _{\sigma ,L}(x,s)= \theta _\sigma (s) \cdot \eta _L(x,s)$ with

$$\begin{aligned} \eta _L(x,s) \asymp \big ( 1 + \rho _A(A^{-s^{+}} x) \big )^{-L} \qquad \text {and} \qquad \theta _\sigma (s) = {\left\{ \begin{array}{ll} \sigma _1^s, &{} \text {if } s \ge 0, \\ \sigma _2^s, &{} \text {if } s < 0, \end{array}\right. } \end{aligned}$$

where the first estimate is due to Lemma 5.6 and $s^{+}:= \textrm{max}\{0,s\}$.

We split the proof into several steps and treat the two factors separately.

Step 1. We show that, for all $s \in \mathbb {R}$ and $t \in [-N,N]$, we have $\theta _{\sigma } (s + t) \le c^4 \theta _{\sigma } (s),$ where $c:= \textrm{max}\bigl \{ \sigma _1^N, \sigma _1^{-N}, \sigma _2^N, \sigma _2^{-N} \bigr \} \in [1,\infty ).$ Note that if $s > N$, then $s + t > 0$, and hence $\theta _{\sigma } (s+t) = \sigma _1^{s+t} = \sigma _1^t \theta _\sigma (s) \le c \theta _\sigma (s).$ Likewise, if $s < -N$, then $s + t < 0$ and thus $\theta _\sigma (s+t) = \sigma _2^{s+t} = \sigma _2^t \theta _\sigma (s) \le c \theta _\sigma (s).$ Lastly, if ${s \in [-N,N]}$, then $s, s + t \in [-2N,2N]$. But for $x \in [-2N,2N]$, it follows that ${c^{-2} \le \theta _\sigma (x) \le c^2}$, and hence $\theta _\sigma (s+t) \le c^{4} \theta _\sigma (s)$.

Step 2. We show that $M_Q^R \eta _{L} \lesssim \eta _{L}$, where $M_Q^R F (g):= \mathop {\mathrm {ess\,sup}}\limits _{u \in Q} |F(u g)|$ for any measurable $F: G_A \rightarrow \mathbb {C}$. Let ${(x,s) \in G_A}$ and $(y,t) \in Q$ be arbitrary. Then Corollary 2.5 implies

$$\begin{aligned} \rho _A(A^{-s^+} x)&\asymp |\det A|^{-s^+ -t} \rho _A( A^t x) \\&\lesssim |\det A|^{-s^+ -t} \Big (\rho _A( A^t x + y) + \rho _A(-y) \Big ) \\&\lesssim |\det A|^{-s^+ +(s+t)^+ -t} \Big (\rho _A\big ( A^{-(s+t)^+}(A^t x + y)\big ) + 1 \Big ), \end{aligned}$$

where we used in the last step that $\rho _A(y) \lesssim 1$ for $y \in [-N,N]^d$; see Lemma 2.2.

Note that $(s+t)^+ \le s^+ + t^+ \le s^+ + N$ for $t \in [-N,N]$, and hence $-s^+ +(s+t)^+ -t \le 2 N$ for all $s \in \mathbb {R}$ and $t \in [-N,N]$. Therefore, $|\det A|^{-s^+ +(s+t)^+ -t} \lesssim 1$ and consequently

$$\begin{aligned} \eta _{L}\big ( (y,t) (x,s) \big )&= \eta _{L}( A^t x + y, s+t) \asymp \big ( 1 + \rho _A\big ( A^{-(s+t)^+}(A^t x + y) \big ) \big )^{-L} \\&\lesssim \big ( 1+ \rho _A(A^{-s^+} x) \big )^{-L} \asymp \eta _{L}(x,s) \end{aligned}$$

for all $(x,s) \in G_A$ and $(y,t) \in Q$; here, we used that $L \ge 0$.

Step 3. Similar to Step 2, we show that $M_Q^L \eta _{L} \lesssim \eta _{L}$, for $M^L_Q F (g) = \mathop {\mathrm {ess\,sup}}\limits _{v \in Q} |F(gv)|$. Let ${(x,s) \in G_A}$ and $(y,t) \in Q$ be arbitrary. Then Corollary 2.5 again implies

$$\begin{aligned} \rho _A(A^{-s^+} x)&\asymp |\det A|^{-s^+} \rho _A (x) \\&\lesssim |\det A|^{-s^+} \Big (\rho _A( x + A^s y) + \rho _A(-A^s y) \Big )\\&\lesssim |\det A|^{-s^+ +(s+t)^+} \rho _A\big (A^{-(s+t)^+}( x + A^sy) \big ) + |\det A|^{-s^+ + s}\rho _A(y) \\&\lesssim \rho _A\big (A^{-(s+t)^+}( x + A^sy) \big ) + 1, \end{aligned}$$

since we have $|\det A|^{-s^+ +(s+t)^+}, |\det A|^{-s^+ + s} \lesssim 1$ for all $s \in \mathbb {R}$ and $t \in [-N,N]$, and since $\rho _A(y) \lesssim 1$ for $y \in [-N,N]^d$ by Lemma 2.2. Therefore,

$$\begin{aligned} \eta _{L}\big ( (x,s) (y,t) \big )&= \eta _{L}( x + A^sy, s+t) \asymp \big ( 1 + \rho _A\big ( A^{-(s+t)^+}(x + A^sy) \big ) \big )^{-L} \\&\lesssim \big ( 1+ \rho _A(A^{-s^+} x) \big )^{-L} \asymp \eta _{L}(x,s) \end{aligned}$$

for all $(x,s) \in G_A$ and $(y,t) \in Q$.

In combination, the obtained estimates easily imply the claim. $\square$

Appendix C. Proof of Lemma 5.2

Proof

Step 1. In this step, we prove the bound

$$\begin{aligned} v(y,t) \lesssim \textrm{max}\{ 1, |\det A|^{-t} \} \, \big ( 1 + \min \{ \rho _A(y), \rho _A (A^{-t} y) \} \big ), \end{aligned}$$

(C.1)

which will also imply that v is well-defined. To this end, first note by the quasi-triangle inequality for $\rho _A$ that there exists $H \ge 1$ such that

$$\begin{aligned} 1 + \rho _A (x)&\le \! 1 + H \big ( \rho _A (x-y) + \rho _A(y) \big ) \\&\le \! H (1 + \rho _A(x-y) + \rho _A(y)) \\&\le \! H (1 + \rho _A(x-y)) (1 + \rho _A(y)) , \end{aligned}$$

and hence $(1 + \rho _A(x-y))^{-1} \le H \cdot \frac{1 + \rho _A(y)}{1 + \rho _A(x)}$. Next, we note as a consequence of Corollary 2.5 that $1 + \rho _A (A^{-(u - t)} z) > rsim 1 + |\det A|^t \rho _A (A^{-u} z) \ge \min \{ 1, |\det A|^t \} (1 + \rho _A(A^{-u} z)),$ so that

$$\begin{aligned} \bigl (1 + \rho _A (A^{-(u-t)} z)\bigr )^{-1}&\lesssim \big ( \min \bigl \{ 1, |\det A|^t \bigr \}\big )^{-1} \bigl (1 + \rho _A (A^{-u} z)\bigr )^{-1} \\&= \textrm{max}\bigl \{ 1, |\det A|^{-t} \bigr \} \bigl (1 + \rho _A (A^{-u} z)\bigr )^{-1} . \end{aligned}$$

Combining this with the previous estimate, we see

$$\begin{aligned} \bigl (1 + \rho _A (A^{-u} A^t z - y)\bigr )^{-1} \lesssim \frac{1 + \rho _A (y)}{1 + \rho _A (A^{-u} A^t z)} \lesssim \textrm{max}\{ 1, |\det A|^{-t} \} \frac{1 + \rho _A (y)}{1 + \rho _A (A^{-u} z)}, \end{aligned}$$

which shows that $v(y,t) \lesssim \textrm{max}\{ 1, |\det A|^{-t} \} (1 + \rho _A (y))$.

On the other hand, using Corollary 2.5, we see

$$\begin{aligned} \bigl (1 \!+\! \rho _A (A^{-(u - t)} z - y)\bigr )^{-1}&= \bigl (1 \!+\! \rho _A (A^t [A^{-u} z - A^{-t} y]) \bigr )^{-1} \\&\asymp \bigl (1 \!+\! |\det A|^t \rho _A (A^{-u} z - A^{-t} y) \bigr )^{-1} \\&\le \frac{\textrm{max}\{ 1, |\det A|^{- t} \}}{1 +\rho _A (A^{-u} z - A^{-t} y)} \\&\lesssim \textrm{max}\bigl \{ 1, |\det A|^{- t} \bigr \} \frac{1 + \rho _A(A^{-t} y)}{1 + \rho _A(A^{-u} z)} , \end{aligned}$$

which implies $v(y,t) \lesssim \textrm{max}\{ 1, |\det A|^{-t} \} (1 + \rho _A (A^{-t} y))$. This shows Eq. C.1.

Step 2. As a consequence of (A.1), we see for arbitrary $\theta > 0$ and $\lambda \in \mathbb {R}$ that

$$\begin{aligned} \Big \{ x \in \mathbb {R}^d :\tfrac{\theta }{1 + \rho _A(x)} > \lambda \Big \} = \Big \{ x \in \mathbb {R}^d :\rho _A(x) < \tfrac{\theta }{\lambda } - 1 \Big \} \end{aligned}$$

is open, meaning that $\frac{\theta }{(1+\rho _A)^\beta }$ is lower semi-continuous. Based on this, it is not hard to see that v is lower semi-continuous as a supremum of continuous functions; see [27, Proposition 7.11]. This means that $\{ (y,t) \in G_A :v(y,t) > \lambda \}$ is open for all $\lambda \in \mathbb {R}$, so that v is measurable.

Step 3. Define $\gamma (x,s):= 1 + \rho _A (A^{-s} x)$ for brevity. Note that

$$\begin{aligned} \gamma \big ( (z,u) (y,t)^{-1} \big )&= \gamma \bigl ( z - A^{u} A^{-t} y, \,\, u - t \bigr ) \\&= 1 + \rho _A \bigl (A^{-(u - t)} (z - A^{u} A^{-t} y)\bigr ) \\&= 1 + \rho _A \bigl ( A^{-u} A^t z - y \bigr ) . \end{aligned}$$

Thus, $v(y,t) = \sup _{(z,u) \in G_A} \gamma (z,u) / \gamma \bigl ( (z,u) (y,t)^{-1} \bigr )$ and hence $v(g) = \sup _{\kappa \in G_A}$ $\gamma (\kappa ) / \gamma (\kappa g^{-1})$ for $g \in G_A$. This easily implies that v is submultiplicative; indeed, for $g,h \in G_A$, we see for ${\widetilde{\kappa }}:= \kappa \, h^{-1}$ that

$$\begin{aligned} v(g h)&= \sup _{\kappa \in G_A} \frac{\gamma (\kappa )}{\gamma (\kappa h^{-1} g^{-1})} \\ {}&= \sup _{\kappa \in G_A} \frac{\gamma (\kappa )}{\gamma (\kappa h^{-1})} \frac{\gamma (\kappa h^{-1})}{\gamma (\kappa h^{-1} g^{-1})} \\&\le v(h) \sup _{{\widetilde{\kappa }} \in G_A} \frac{\gamma ({\widetilde{\kappa }})}{\gamma ({\widetilde{\kappa }} g^{-1})} \le v(h) v(g) , \end{aligned}$$

as claimed.

Step 4. Starting from the definition (5.2) of v, the substitutions $a = A^{-u} z$ and $b = A^t a - y$ show by Corollary 2.5 that

$$\begin{aligned} v(y,t){} & {} = \sup _{a \in \mathbb {R}^d} \frac{1 + \rho _A(a)}{1 + \rho _A (A^t a - y)} = \sup _{b \in \mathbb {R}^d} \frac{1 + \rho _A(A^{-t} (b + y))}{1 + \rho _A (b)} \\{} & {} \asymp \sup _{b \in \mathbb {R}^d} \frac{1 + |\det A|^{-t} \rho _A(b + y)}{1 + \rho _A (b)}. \end{aligned}$$

By using the second-to-last expression and setting $b = 0$, we see $v(y,t) \ge 1 + \rho _A (A^{-t} y)$.

Next, note that ${ \rho _A (b) \le H \big ( \rho _A (b + y) + \rho _A(-y) \big ) }$ and hence ${\rho _A(b + y) \ge H^{-1}}$ $\rho _A(b) \rho _A(y)$, by the symmetry of $\rho _A$. Furthermore, Lemma 2.2 shows $\rho _A (b) \rightarrow \infty$ as $\Vert b \Vert \rightarrow \infty$. Therefore, as $\Vert b \Vert \rightarrow \infty$,

$$\begin{aligned} v(y,t)&\! > rsim \! \sup _{b \in \mathbb {R}^d} \frac{1 \!+\! |\det A|^{-t} \rho _A(b \!+\! y)}{1 \!+\! \rho _A (b)} \\&\ge \frac{1 \!+\! |\det A|^{-t} \!\cdot \! (H^{-1} \rho _A(b) \!-\! \rho _A(y))}{1 \!+\! \rho _A (b)} \\&\rightarrow H^{-1} |\det A|^{-t} , \end{aligned}$$

so that we also get $v(y,t) > rsim |\det A|^{-t}$. Overall, we see $v(y,t) > rsim 1 + |\det A|^{-t} + \rho _A(A^{-t} y)$.

There are now two cases: If $t \ge 0$, then Corollary 2.5 shows that

$$\begin{aligned}&\textrm{max}\bigl \{ 1, |\det A|^{-t} \bigr \} \bigl (1 + \min \{ \rho _A (y), \rho _A (A^{-t} y) \}\bigr ) \\&\quad = 1 + \min \{ \rho _A (y), \rho _A (A^{-t} y) \} \\&\quad \asymp 1 + \min \{ \rho _A (y), |\det A|^{-t} \rho _A (y) \} \\&\quad \asymp 1 + \rho _A(A^{-t} y) \\&\quad \asymp 1 + |\det A|^{-t} + \rho _A(A^{-t} y) \\&\quad \lesssim v(y,t) . \end{aligned}$$

Otherwise, in case of $t < 0$, we see

$$\begin{aligned}&\textrm{max}\bigl \{ 1, |\det A|^{-t} \bigr \} \bigl (1 \!+\! \min \{ \rho _A (y), \rho _A (A^{-t} y) \}\bigr ) \\&\quad \asymp |\det A|^{-t} \big ( 1 \!+\! \min \{ \rho _A(y), |\det A|^{-t} \rho _A(y) \} \big ) \\&\quad = |\det A|^{-t} \big ( 1 + \rho _A(y) \big ) \\&\quad \asymp |\det A|^{-t} + \rho _A(A^{-t} y) \\&\quad \asymp 1 + |\det A|^{-t} + \rho _A(A^{-t} y) \\&\quad \lesssim v(y,t). \end{aligned}$$

In combination with Eq. C.1, this proves Eq. 5.3. $\square$

Appendix D. Independence of coorbit reservoir

Let $\psi \in L^2 ({\mathbb {R}}^d)$ be an admissible vector satisfying $W_{\psi } \psi \in \mathcal {W}(L^r_w)$ for the standard control weight $w = w^{\alpha , \beta }_{p,q}: G_A \rightarrow [1, \infty )$ provided by Lemma 5.7. Define the space

$$\begin{aligned} {\mathcal {H}}^1_w (\psi ) = \big \{ f \in L^2 ({\mathbb {R}}^d): W_{\psi } f \in L^1_w (G_A) \big \} \end{aligned}$$

and equip it with the norm $\Vert f \Vert _{{\mathcal {H}}^1_w (\psi )}:= \Vert W_{\psi } f \Vert _{L^1_w}$. Let ${\mathcal {R}}_w (\psi ):= ( {\mathcal {H}}^1_w (\psi ) )^*$ be the anti-dual space of ${\mathcal {H}}^1_w (\psi )$ and write $V_{\phi } f:= \langle f, \pi (\cdot ) \phi \rangle _{{\mathcal {R}}_w, {\mathcal {H}}^1_w}$ for $f \in {\mathcal {R}}_w (\psi )$ and $\phi \in {\mathcal {H}}_w^1 (\psi )$.

The following lemma is a special case of [64, Corollary 4.9].

Lemma D.1

Let $\alpha \in \mathbb {R}$, $\beta > 0$, and $p \in (0,\infty ),q \in (0,\infty ]$, with $r:= \min \{1, p,q\}$. Let $w = w^{\alpha , \beta }_{p,q}: G_A \rightarrow [1, \infty )$ be a standard control weight for as provided by Lemma 5.7. Suppose $\varphi \in {\mathcal {S}}_0 ({\mathbb {R}}^d)$ is admissible and $\psi \in L^2 ({\mathbb {R}}^d)$ is admissible satisfying $W_{\psi } \psi \in \mathcal {W}(L^r_w)$ and $W_{\varphi } \psi \in \mathcal {W}(L^r_w)$. Then

in the sense that the restriction $f \mapsto f|_{{\mathcal {S}}_0}$ is a well-defined bijection. Furthermore, given the unique extension of , then

$$\begin{aligned} \langle {\widetilde{f}}, \phi \rangle _{{\mathcal {R}}_w, {\mathcal {H}}^1_w} = \langle f, \phi \rangle _{\varphi } \quad \phi \in {\mathcal {H}}^1_w (\varphi ), \end{aligned}$$

where $\langle \cdot , \cdot \rangle _{\varphi }$ denotes the extended pairing of Definition 6.5.

Proof

We first verify that the Peetre-type spaces satisfy the standing assumptions of [64]. As shown in Lemma 5.3, the Peetre space is a solid, translation-invariant quasi-Banach space, and Lemma 5.4 shows that is an r-norm. Moreover, the standard control weight $w:= w_{p,q}^{\alpha , \beta }: G_A \rightarrow [1,\infty )$ defined in Lemma 5.7 is continuous, submultiplicative and satisfies $w(g) = w(g^{-1}) \Delta ^{1/r}(g^{-1})$. Furthermore, Lemma 5.7 shows that and for all $h \in G_A$. Together, this shows that w is a strong control weight for in the terminology of [64, Definition 3.1]. By [64, Corollary 3.9], this implies that the pair is $L_w^r$-compatible in the sense of [64, Definition 3.5].

Since $\psi , \varphi \in L^2 ({\mathbb {R}}^d)$ are admissible and satisfy $W_\psi \psi , W_{\varphi } \varphi , W_{\varphi } \psi \in \mathcal {W}(L^r_w)$, it follows from [64, Lemma 4.3 and Proposition 4.8] that ${\mathcal {H}}_w^1(\psi ) = {\mathcal {H}}_w^1(\varphi )$ and hence also ${\mathcal {R}}_w(\psi ) = {\mathcal {R}}_w (\varphi )$. Therefore,

(D.1)

Lemma 5.8 easily shows that ${\mathcal {S}}_0(\mathbb {R}^d) \hookrightarrow {\mathcal {H}}_w^1(\varphi )$, and Lemma 4.7 shows that [64, Equation (4.14)] is satisfied for ${\mathcal {S}}_0(\mathbb {R}^d)$. Therefore, invoking [64, Corollary 4.9], it follows that the restriction map $f \mapsto f|_{{\mathcal {S}}_0(\mathbb {R}^d)}$ is a bijection from onto . Combining this with Eq. D.1 yields that

is a well-defined bijection.

By the above, any uniquely extends to an element of denoted by ${\widetilde{f}}$. Note that $V_{\varphi } {\widetilde{f}} = W_{\varphi } f$. Then, a combination of [64, Lemma 4.6(iii)] and Definition 6.5 shows for any $\phi \in L^2(\mathbb {R}^d)$ with $W_{\varphi } \phi \in L_w^1 (\mathbb {R}^d)$ (i.e., $\phi \in {\mathcal {H}}_w^1 (\varphi )$) that

$$\begin{aligned} \langle {\widetilde{f}}, \phi \rangle _{{\mathcal {R}}_w, {\mathcal {H}}_w^1} = \langle W_{\varphi } f, W_{\varphi } \phi \rangle _{L_{1/w}^\infty , L_w^1} = \langle f, \phi \rangle _{\varphi }. \end{aligned}$$

(D.2)

This completes the proof. $\square$