Kernel theorems in coorbit theory

Abstract

We prove general kernel theorems for operators acting between coorbit spaces. These are Banach spaces associated to an integrable representation of a locally compact group and contain most of the usual function spaces (Besov spaces, modulation spaces, etc.). A kernel theorem describes the form of every bounded operator between a coorbit space of test functions and distributions by means of a kernel in a coorbit space associated to the tensor product representation. As special cases we recover Feichtinger’s kernel theorem for modulation spaces and the recent generalizations by Cordero and Nicola. We also obtain a kernel theorem for operators between the Besov spaces $\dot{B}^0_{1,1}$ and $\dot{B}^{0}_{\infty , \infty }$.

1. Introduction

Kernel theorems assert that every “reasonable” operator can be written as a “generalized” integral operator. For instance, the Schwartz kernel theorem states that a continuous linear operator $A:\mathcal{S}(\mathbb{R}^d)\rightarrow \mathcal{S}^\prime (\mathbb{R}^d)$ possesses a unique distributional kernel $K\in \mathcal{S}^\prime (\mathbb{R}^{2d})$ such that

$$\begin{equation} \langle Af,g\rangle = \langle K,g\otimes f \rangle ,\qquad f,g\in \mathcal{S}(\mathbb{R}^d) \, . {\tag{1}} \end{equation}$$

If $K$ is a locally integrable function, then

$$\begin{equation*} \langle Af, g \rangle =\int _{\mathbb{R}^d}K(x,y)f(y) \overline{g(x)} dy dx , \qquad f, g \in \mathcal{S}(\mathbb{R}^d), \end{equation*}$$

and thus $A$ has indeed the form of an integral operator. Similar kernel theorems hold for continuous operators from $\mathcal{D}(\mathbb{R}^d) \to \mathcal{D}^\prime (\mathbb{R}^d)$ 24, Theorem 5.2 and for Gelfand-Shilov spaces and their distribution spaces 21. The importance of these kernel theorems stems from the fact that they offer a general formalism for the description of linear operators.

In the context of time-frequency analysis, Feichtinger’s kernel theorem 12 (see also 18 and 23, Theorem 14.4.1) states that every bounded linear operator from the modulation space $M^1(\mathbb{R}^d)$ to the modulation space $M^\infty (\mathbb{R}^d)$ can be represented in the form 1 with a kernel in $M^\infty (\mathbb{R}^{2d})$. The advantage of this kernel theorem is that both the space of test functions $M^1(\mathbb{R}^d)$ and the distribution space $M^\infty (\mathbb{R}^d) = M^1(\mathbb{R}^d)^*$ are Banach spaces and thus technically easier than the locally convex spaces $\mathcal{S}(\mathbb{R}^d)$ and $\mathcal{S}^\prime (\mathbb{R}^d)$.

Recently, Cordero and Nicola 8 revisited Feichtinger’s kernel theorem and proved several new kernel theorems that “do not have a counterpart in distribution theory”. They argue that “this reveals the superiority, in some respects, of the modulation space formalism upon distribution theory.” While we agree full-heartedly with this claim, we would like to add a more abstract point of view and argue that the deeper reason for this superiority lies in the theory of coorbit spaces and in the convenience of Schur’s test for integral operators. Indeed, we will prove kernel theorems similar to Feichtinger’s kernel theorem for many coorbit spaces.

The main idea is to investigate operators in a transform domain after taking a short-time Fourier transform, a wavelet transform, or an abstract wavelet transform, i.e., a continuous transform with respect to a unitary group representation. In this new representation every operator between a suitable space of test functions and distributions is an integral operator. The standard boundedness conditions of Schur’s test then yield strong kernel theorems.

The technical framework for this idea is coorbit theory, which was introduced and studied in 15161722 for the construction and analysis of function spaces by means of a generalized wavelet transform. The main idea is that functions in the standard function spaces, such as Besov spaces and modulation spaces, can be characterized by the decay or integrability properties of an associated transform (the wavelet transform or the short-time Fourier transform). In the abstract setting, $G$ is a locally compact group and $\pi :G\rightarrow \mathcal{U}(\mathcal{H})$ is an irreducible, unitary, integrable representation of $G$. Leaving technical details aside, the coorbit space ${\mathcal{C}o_\pi }\,L^p_w(G)$ consists of all distributions $f$ in a suitable distribution space such that the representation coefficient $g\mapsto \langle f,\pi (g)\psi \rangle$ is in the weighted space $L^p_w(G)$.

Next, let $G_1$ and $G_2$ be two locally compact groups, and let $(\pi _1, \mathcal{H}_1)$ and $(\pi _2, \mathcal{H}_2)$ be irreducible, unitary, integrable representations of $G_1$ and $G_2$, respectively.

Let $A$ be a bounded linear operator between ${\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ and ${\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$. Our main insight is that such an operator can be described by a kernel in a coorbit space that is related to the tensor product representation $\pi =\pi _2\otimes \pi _1$ of $G=G_1\times G_2$ on the tensor product space $\mathcal{H}_2\otimes \mathcal{H}_1$. The following non-technical formulation offers a flavor of our main result in Theorem 3:

A linear operator $A$ is bounded from ${\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ to ${\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ if and only if there exists a kernel $K\in {\mathcal{C}o_\pi }\,L^\infty _{w_1^{-1} \otimes w_2^{-1}}(G_1\times G_2)$ such that

$$\begin{equation} \langle A\upsilon ,\varphi \rangle =\langle K,\varphi \otimes \upsilon \rangle \tag{2} \end{equation}$$

for all $\upsilon \in {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,,\ \varphi \in {\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,$.

This statement is not just a mere abstraction and generalization of the classical kernel theorem. With the choice of a specific group and representation one obtains explicit kernel theorems. For instance, using the Schrödinger representation of the Heisenberg group, one recovers Feichtinger’s original kernel theorem. The added value is our insight that the conditions on the kernel of 8 in terms of mixed modulation spaces 4 amount to coorbit spaces with respect to the tensor product representation. Choosing the $ax+b$-group and the continuous wavelet representation, one obtains a kernel theorem for all bounded operators between the Besov spaces $\dot{B}^0_{1,1}$ and $\dot{B}^0 _{\infty ,\infty }$ with a kernel in a space of dominating mixed smoothness. This class of function spaces has been studied extensively 3132 and is by no means artificial.

By using suitable versions of Schur’s test, it is then possible to derive characterizations for the boundedness of operators between other coorbit spaces. For example, in Theorem 7 we will prove the following, with $\frac{1}{p} + \frac{1}{q} = 1$:

(i)

$\text{$A:{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{p}_{w_2}(G_2)}\,$ bounded} \Leftrightarrow K\in {\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }_{ 1/w_1\otimes w_2}(G_1\times G_2)$,

(ii)

$\text{$A:{\mathcal{C}o_{\pi _1}L^{p}_{w_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{w_2}(G_2)}\,$ bounded} \Leftrightarrow K\in {\mathcal{C}o_\pi }\,L^{q,\infty }_{1/w_1\otimes w_2}(G_1 \times G_2)$,

where the mixed-norm Lebesgue spaces $\mathcal{L}^{p,q}$ and $L^{p,q}$ on $G_1 \times G_2$ are defined in 23 and 24, respectively.

The paper is organized as follows. In Section 2 we present the basics of tensor products and coorbit space theory. The theory of coorbit spaces of kernels with respect to products of integrable representations is developed in Section 3. Our main results, the kernel theorems, are proved in Section 4 and applied to particular examples of group representations in Section 5.

We note that our proofs require a meaningful formulation of coorbit theory. One can therefore prove kernel theorems also in the context of other coorbit space theories 69, e.g., for certain reducible representations.

2. Preliminaries on tensor products and coorbit spaces

2.1. Tensor products and Hilbert-Schmidt operators

The theory of tensor products is at the heart of kernel theorems for operators. Algebraically, a simple tensor of two vectors (in two possibly different Hilbert spaces) is a formal product of two vectors $f_1 \otimes f_2$, and the tensor product $\mathcal{H}_1 \otimes \mathcal{H}_2$ is obtained by taking the completion of all linear combinations of simple tensors with respect to the inner product

$$\begin{equation*} \langle f_1 \otimes f_2, g_1 \otimes g_2 \rangle \coloneq \langle f_1, g_1 \rangle \, \langle g_2,f_2 \rangle \, . \end{equation*}$$

This tensor product is homogeneous in the following sense: $\alpha \cdot (f_1\otimes f_2)=(\alpha f_1)\otimes f_2=f_1\otimes (\overline{\alpha }f_2)$. Note explicitly that the product $f_1 \otimes f_2$ is anti-linear in the second factor. In some books this is done by introducing the dual Hilbert space $\mathcal{H}_{2}^\prime$ 25.

If each Hilbert space is an $L^2$-space $\mathcal{H}_1 = L^2(X,\mu ),\ \mathcal{H}_2 = L^2(Y,\nu )$, then the simple tensor $f\otimes g$ is just the product $(x,y) \mapsto$ $f(x)\cdot \overline{g(y)}$, and the tensor product becomes the product space $\mathcal{H}_1 \otimes \mathcal{H}_2 = L^2(X,\mu ) \otimes L^2(Y,\nu ) = L^2(X\times Y, \mu \times \nu )$.

The connection between functions and operators arises in the analytic approach to tensor products. We interpret a function of two variables as an integral kernel for an operator. Thus a simple tensor $f_1\otimes f_2$ of two functions becomes the rank one operator $f \mapsto \langle f, f_2 \rangle f_1$ with integral kernel $f_1(x)\overline{f_2(y)}$, and a general $k\in L^2(X\times Y,\mu \times \nu )$ becomes a Hilbert-Schmidt operator from $L^2(Y,\nu )$ to $L^2(X,\mu )$. The systematic, analytic treatment of general tensor products of two Hilbert spaces often defines the tensor product as a space of Hilbert-Schmidt operators between $\mathcal{H}_2$ and $\mathcal{H}_1$. We note that his definition is already based on the characterization of Hilbert-Schmidt operators and thus represents a non-trivial kernel theorem 7. Whereas the working mathematician habitually identifies an operator with its distributional kernel, we will make the conceptual distinction between tensor products and operators for our study of kernel theorems.

In the sequel we will denote the (distributional) kernel of an integral operator by $k$ and the abstract kernel in a tensor product by $K$.

2.2. Coorbit space theory

Let $G$ be a locally compact group with left Haar measure $\int _G \dots dg$, let $\mathcal{H}$ be a separable Hilbert space, and let $\mathcal{U}(\mathcal{H})$ be the group of unitary operators acting on $\mathcal{H}$. A continuous unitary group representation $\pi :G\rightarrow \mathcal{U}(\mathcal{H})$ is called square integrable 111 if it is irreducible and there exist $\psi \in \mathcal{H}$ such that

$$\begin{equation} \int _G|\langle \psi ,\pi (g)\psi \rangle |^2dg<\infty \, . {\tag{3}} \end{equation}$$

A non-zero vector $\psi$ satisfying 3 is called admissible. For every square integrable representation there exists a densely defined operator $T$ such that $\forall f_1,f_2\in \mathcal{H},\ \psi _1,\psi _2\in {\mathsf{Dom}}\,(T)$, one has

$$\begin{equation} \int _G\langle f_1,\pi (g)\psi _1\rangle \langle \pi (g)\psi _2,f_2\rangle dg=\langle T\psi _2,T\psi _1\rangle \langle f_1,f_2\rangle . {\tag{4}} \end{equation}$$

For fixed $\psi _1 = \psi _2 = \psi$ the representation coefficient $f \mapsto V_\psi f(g)\coloneq \langle f,\pi (g)\psi \rangle$ is interpreted as a generalized wavelet transform. The orthogonality relation 4 then implies that $V_\psi$ is a multiple of an isometry from $\mathcal{H}$ to $L^2(G)$. By using a weak interpretation of vector-valued integrals, 4 can also be recast as the inversion formula

$$\begin{equation} f=\frac{1}{\|T\psi \|^2}\int _G\langle f,\pi (g)\psi \rangle \pi (g)\psi dg \, . {\tag{5}} \end{equation}$$

For the rest of this paper we assume without loss of generality that the chosen admissible vectors $\psi$ are normalized, i.e., $\|T\psi \|=1$.

The adjoint operator $V_\psi ^\ast : L^2(G) \to \mathcal{H}$ is formally defined by

$$\begin{equation*} V_\psi ^\ast F\coloneq \int _G F(g)\pi (g)\psi dg. \end{equation*}$$

Other domains and convergence properties will be discussed later.

With this notation 5 says that $V_\psi ^\ast V_\psi =I_\mathcal{H}$ for all admissible and normalized vectors $\psi$, which in the language of recent frame theory means that $\{\pi (g)\psi \}_{g\in G}$ is a continuous Parseval frame. By 5, Proposition 2.1 one can always assume that $G$ is $\sigma$-finite since we assume $\mathcal{H}$ to be separable.

In coorbit theory one needs much stronger hypotheses on $\pi$. The representation $\pi$ is called integrable with respect to a weight $w$ if there exists an admissible vector $\psi \in \mathcal{H}$ such that

$$\begin{equation} \int _G|\langle \psi ,\pi (g)\psi \rangle | w(g) \, dg<\infty . {\tag{6}} \end{equation}$$

Let $g_1$, $g_2$, $g_3\in G$. We call a weight $w:G\rightarrow \mathbb{R}^+$ submultiplicative if $w(g_1g_2)\leqslant w(g_1)w(g_2)$ and a function $m:G\rightarrow \mathbb{R}^+$ w-moderate if it satisfies $m(g_1 g_2 g_3)\leqslant w(g_1)m(g_2)w(g_3)$. If $m$ is $w$-moderate, the weighted Lebesgue space $L^p_m(G)$ is then invariant under left translation $L_xf(y)=f(x^{-1}y)$ and under the right translation $R_xf(y)=f(yx)$. Throughout this paper, we will assume that the weight $w$ satisfies

$$\begin{equation} w(x)\geqslant C\max \left\{\alpha (x),\alpha (x^{-1}),\beta (x),\Delta (x^{-1})\beta (x^{-1})\right\}, {\tag{7}} \end{equation}$$

where $\alpha (x)\coloneq \|L_x\|_{L^p_m(G)\rightarrow L^p_m(G)}$, $\beta (x)\coloneq \|R_x\|_{L^p_m(G)\rightarrow L^p_m(G)}$, and $\Delta$ denotes the modular function of $G$.

Our standing assumption is that the representation $\pi$ of $G$ possesses an admissible vector $\psi$ such that $V_\psi \psi \in L^1_w(G)$. We denote the corresponding set by

$$\begin{equation*} \mathcal{A}_w(G)\coloneq \left\{\psi \in \mathcal{H}, \psi \neq 0: \ V_\psi \psi \in L^1_w(G)\right\}. \end{equation*}$$

For fixed $\psi \in \mathcal{A}_w(G)$ the linear version of $\mathcal{A}_w(G)$,

$$\begin{equation} \mathcal{H}^1_w\coloneq \left\{f\in \mathcal{H}:\ V_\psi f\in L^1_w(G)\right\}, \tag{8} \end{equation}$$

is dense in $\mathcal{H}$. Let $(\mathcal{H}^1_w)^\sim$ denote the anti-dual of $\mathcal{H}^1_w$, i.e., the space of anti-linear continuous functionals on $\mathcal{H}^1_w$. As $\mathcal{H}^1_w$ is dense in $\mathcal{H}$, it follows that the inner product on $\mathcal{H}\times \mathcal{H}$ extends to $(\mathcal{H}^1_w)^\sim \times \mathcal{H}^1_w$ and so does the generalized wavelet transform.

The coorbit space with respect to $L^p_m(G)$ is then defined by

$$\begin{equation*} {\mathcal{C}o_\pi }\,L^p_m(G)\coloneq \left\{f\in (\mathcal{H}^1_w)^\sim :\ V_\psi f\in L^p_m(G)\right\} \, \end{equation*}$$

and is equipped with the natural norm

$$\begin{equation*} \|f\|_{{\mathcal{C}o_\pi }\,L^p_m(G)}\coloneq \|V_\psi f\|_{L^p_m(G)} \, . \end{equation*}$$

With our assumptions on $\pi , \psi , m$, the coorbit space ${\mathcal{C}o_\pi }\,L^p_m(G)$ is a Banach space 16. Alternatively, ${\mathcal{C}o_\pi }\,L^p_m(G)$ for $p<\infty$ can be defined as the completion of $H_w^1$ with respect to this norm. Moreover,

$$\begin{equation} {\mathcal{C}o_\pi }\,L^2(G)=\mathcal{H},\ \ {\mathcal{C}o_\pi }\,L^1_w(G)=\mathcal{H}^1_w,\quad \text{and}\quad {\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)=(\mathcal{H}^1_w)^\sim =\mathcal{H}^\infty _{1/w} , {\tag{9}} \end{equation}$$

and

$$\begin{equation} \mathcal{H}^1_w \subseteq {\mathcal{C}o_\pi }\,L^p_m(G) \subseteq \mathcal{H}^\infty _{1/w} {\tag{10}} \end{equation}$$

for $1\leqslant p\leqslant \infty$ and $w$-moderate weight $m$. In the context of coorbit space theory the space $\mathcal{H}_w^1$ serves as a space of test functions, and $\mathcal{H}^\infty _{1/w}$ is the corresponding distribution space.

We quickly recall some of the fundamental properties of coorbit spaces; see for example 16, Theorems 4.1 and 4.2 and Proposition 4.3.

Proposition 1.

Let $\psi ,\phi \in \mathcal{A}_w(G)$, $f\in {\mathcal{C}o_\pi }\,L^p_m(G)$, $g\in {\mathcal{C}o_\pi }\,L^q_{1/m}(G)$, and $F\in L^p_m(G)$. Then the following properties hold:

(i)

$V_\psi :{\mathcal{C}o_\pi }\,L^p_m(G)\rightarrow L^p_m(G)$ is an isometry.

(ii)

$\mathcal{H}_m^p$ is invariant with respect to $\pi$ and$$\begin{equation*} \|\pi (g) f\| _{{\mathcal{C}o_\pi }\,L^p_m(G)} \leqslant w (g) \| f\| _{{\mathcal{C}o_\pi }\,L^p_m(G)} \text{ for all }g\in G, f\in {\mathcal{C}o_\pi }\,L^p_{m}(G) . \end{equation*}$$

(iii)

$V_\psi ^\ast :L^p_m(G)\rightarrow {\mathcal{C}o_\pi }\,L^p_m(G)$ is continuous.

(iv)

$V_\psi ^\ast V_\psi =I_{{\mathcal{C}o_\pi }\,L^p_m(G)}$.

(v)

Correspondence principle: Let $F\in L^p_m(G)$. There exists $f\in {\mathcal{C}o_\pi }\,L^p_m(G)$ such that $F=V_\psi f$ if and only if $F=F\ast V_\psi \psi$, where $\ast$ denotes convolution on $G$.

(vi)

Duality: For $1\leqslant p<\infty$, $\frac{1}{p}+\frac{1}{q}=1$, we have $({\mathcal{C}o_\pi }\,L^p_m(G))^* = {\mathcal{C}o_\pi }\,L^{q}_{1/m}(G)$, where the duality is given by$$\begin{equation*} \langle f, g \rangle _{{\mathcal{C}o_\pi }\,L^p_m(G), {\mathcal{C}o_\pi }\,L^{q}_{m}(G)} = \langle V_\psi f , V_\psi g \rangle _{L^p_m(G), L^{q}_{1/m}(G)} \, . \end{equation*}$$

(vii)

The definition of ${\mathcal{C}o_\pi }\,L^p_m(G)$ is independent of the particular choice of the window function from $\mathcal{A}_w(G)$. In particular, $\|V_\psi f\|_{L^p_m(G)}\asymp \|V_\phi f\|_{L^p_m(G)}$ for arbitrary non-zero $\phi ,\psi \in \mathcal{A}_w(G)$.

We furthermore need a result on the existence of atomic decompositions for the space ${\mathcal{C}o_\pi }\,L_{w}^1(G)$; see 15, Theorem 4.7.

Theorem 2.

Let $\psi \in \mathcal{A}_w(G)$. There exists a discrete subset $\{g_i\}_{i\in \mathcal{I}}\subset G$ and a collection of linear functionals $\lambda _i:{\mathcal{C}o_\pi }\,L^1_w(G)\rightarrow { \mathbb{C}},\ i\in \mathcal{I}$, such that

$$\begin{equation} f=\sum _{i\in \mathcal{I}}\lambda _i(f)\pi (g_i)\psi ,\quad \text{with }\sum _{i\in \mathcal{I}}|\lambda _i(f)|w(g_i)\asymp \|f\|_{{\mathcal{C}o_\pi }\,L^1_w(G)}, {\tag{11}} \end{equation}$$

and the sum converges absolutely in ${\mathcal{C}o_\pi }\,L^1_w(G)$.

3. Frames and coorbit spaces via tensor products

Let $G_1,G_2$ be two locally compact groups with unitary square integrable representations $\pi _1:G_1\rightarrow \mathcal{U}(\mathcal{H}_1)$ and $\pi _2:G_2\rightarrow \mathcal{U}(\mathcal{H}_2)$. For $g\coloneq (g_1,g_2)\in G\coloneq G_1\times G_2$ the tensor representation $\pi :G\rightarrow \mathcal{U}(\mathcal{H}_2\otimes \mathcal{H}_1)$,

$$\begin{equation*} \pi (g)\coloneq \pi _2(g_2)\otimes \pi _1(g_1), \end{equation*}$$

acts on a simple tensor $\Psi \coloneq \psi _2\otimes \psi _1\in \mathcal{H}_2\otimes \mathcal{H}_1$ by

$$\begin{equation} \pi (g)(\psi _2\otimes \psi _1)=\pi _2(g_2)\psi _2\otimes \pi _1(g_1)\psi _1. {\tag{12}} \end{equation}$$

It follows immediately that $\pi$ is a unitary representation of $G$ on $\mathcal{H}_2\otimes \mathcal{H}_1$. Moreover, $\pi$ is irreducible (e.g., by 34, Section 4.4, Theorem 6). Note that the order of indices is in agreement with the formulation of the kernel theorem in Theorem 3.

If we interpret the simple tensor $\Psi = \psi _2 \otimes \psi _1$ as the rank-one operator $f\mapsto \psi _1(f) \psi _2$ with $\psi _1\in \mathcal{H}^\prime _1$, then we can write 12 as

$$\begin{equation*} \pi (g)(\Psi )(f)=(\pi _1^\prime (g_1)\psi _1)(f)\cdot \pi _2(g_2)\psi _2=\big (\pi _2(g_2)\psi _2\otimes \pi _1^\prime (g_1)\psi _1\big )(f), \end{equation*}$$

where the contragredient representation $\pi ^\prime _1:G_1\rightarrow GL(\mathcal{H}^\prime _1)$ of $\pi _1$ is defined as $(\pi _1^\prime (g_1)\psi _1)(f)=\psi _1(\pi _1(g^{-1}_1)f)$; see 34, Section 3.1.

In case we treat the tensor product as a space of Hilbert-Schmidt operators, $\pi$ acts on $A\in \mathcal{HS}(\mathcal{H}_1,\mathcal{H}_2)$ as

$$\begin{equation*} \pi (g)A=\pi _2(g_2)A\pi _1(g_1)^\ast . \end{equation*}$$

The generalized wavelet transform of a simple tensor $f_2\otimes f_1$ with respect to a “wavelet” $\Psi =\psi _2\otimes \psi _1$ is given by

$$\begin{align} V_{\Psi } (f_2\otimes f_1)(g) & =\langle f_2\otimes f_1,(\pi _2(g_2)\otimes \pi _1(g_1))(\psi _2\otimes \psi _1)\rangle \\ & = \langle f_2,\pi _2(g_2)\psi _2\rangle \overline{\langle f_1,\pi _1(g_1)\psi _2\rangle } {\tag{13}}\\ & =V_{\psi _2}f_2(g_2)\overline{V_{ \psi _1}f_1(g_1)}. \end{align}$$

Thus, the wavelet transform of the tensor product representation factors into the product of wavelet transforms on $G_1$ and $G_2$. Strictly speaking, we would have to write $V_{\psi _i}^{\pi _i}f_i$ to indicate the underlying representation, but we omit the reference to the group to keep notation simple.

Throughout this paper we consider only separable weights $w:G\rightarrow \mathbb{R}_+$ with $w(g)= (w_1\otimes w_2)(g) = w_1(g_1) w_2(g_2)$ and $m(g)= (m_1\otimes m_2)(g) = m_1(g_1) m_2(g_2)$, where $w_i$ is submultiplicative and $m_i$ is $w_i$-moderate. Moreover we write $(1/w)(g)=(w_1\otimes w_2)(g)^{-1}$. It follows from 13 that the tensor representation $\pi _2 \otimes \pi _1$ of two square-integrable representations is again square-integrable and that the tensor $\Psi = \psi _2 \otimes \psi _1$ of two admissible vectors $\psi _2$ and $\psi _1$ is admissible for $\pi$. Likewise, if $w=w_1 \otimes w_2$ and $\psi _1\in \mathcal{A}_{w_1}(G_1)$, $\psi _2\in \mathcal{A}_{w_2}(G_2)$, then $\psi _2 \otimes \psi _1 \in \mathcal{A}_{w}(G_1\times G_2)$ (where we assume that $w_i$, $i=1,2$, satisfies 7). Therefore all definitions and results of Section 2.2 hold for the representation $\pi = \pi _2 \otimes \pi _1$ and $\Psi = \psi _2 \otimes \psi _1$ . In particular, the orthogonality relation 4, the inversion formula 5, Proposition 1, and Theorem 2 hold for suitable admissible vectors $\Psi =\psi _2\otimes \psi _1$.

4. Kernel theorems

In this section we derive the general kernel theorems for operators between coorbit spaces. The basic idea comes from linear algebra, where a linear operator is identified with its matrix with respect to a basis. In coorbit theory the basic structure consists of the vectors $\pi (g)\psi$. Thus in analogy to linear algebra we try to describe an operator $A: \mathcal{H}_1 \to \mathcal{H}_2$ by the kernel (= continuous matrix)

$$\begin{equation} k_A(g_1,g_2) = \langle A\pi _1(g_1)\psi _1,\pi _2(g_2)\psi _2\rangle . {\tag{14}} \end{equation}$$

This can be seen as a continuous Galerkin-like representation of the operator $A$ 23. The idea goes back to coherent state theory 30, Ch. 1.6. One of its goals is to associate to every operator $A$ a function or symbol $k_A$, and 14 is one of the many possibilities to do so.

Assume that $A:{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ and $f\in {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$, i.e., $A$ maps “test functions” to “distributions”. By using the inversion formula 5 for $f$ and applying $A$ to it, it follows formally that

$$\begin{equation*} Af=\int _{G_1}\langle f,\pi _1(g_1)\psi _1\rangle A\pi _1(g_1)\psi _1 dg_1, \end{equation*}$$

and furthermore that

$$\begin{align} V_{\psi _2} (Af)(g_2) & =\langle Af,\pi _2(g_2)\psi _2\rangle =\int _{G_1}\langle f,\pi _1(g_1)\psi _1\rangle \langle A\pi _1(g_1)\psi _1,\pi _2(g_2)\psi _2\rangle dg_1 \\ & =\int _G\langle f,\pi _1(g_1)\psi _1\rangle k_A(g_1,g_2)dg_1 \, . {\tag{15}} \end{align}$$

Let

$$\begin{equation} \mathfrak{ A}F(g_2)=\int _{G_1} F(g_1)k_A(g_1,g_2) dg_1 {\tag{16}} \end{equation}$$

be the integral operator with the kernel $k_A$. Then 15 can be written as

$$\begin{equation} V_{\psi _2} Af = \mathfrak{A} V_{\psi _1}f \, {\tag{17}} \end{equation}$$

or, equivalently,

$$\begin{equation} A = V_{\psi _2}^* \mathfrak{A} V_{\psi _1} \, . {\tag{18}} \end{equation}$$

Using this factorization, the computation in 15 can be given a precise meaning on coorbit spaces. Identity 18 is the heart of the kernel theorems. The combination of the properties of the generalized wavelet transform (Proposition 1) and boundedness properties of integral operators yields powerful and very general kernel theorems.

We will first show the existence of a generalized kernel for operators mapping the space of test functions ${\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ into the distribution space ${\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$. Subsequently, we will characterize continuous operators in certain subclasses.

Theorem 3.

Let $G_1$ and $G_2$ be two locally compact groups, and let $(\pi _j,\mathcal{H}_j)$ be integrable, unitary, irreducible representations of $G_j$, such that $\mathcal{A}_{w_j}(G_j) \neq \emptyset$ for $j=1,2$.

(i)

Every kernel $K \in {\mathcal{C}o_\pi }\,L^\infty _{1/w} (G_1 \times G_2)$ defines a unique linear operator $A: {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ by means of$$\begin{equation} \langle A\upsilon ,\varphi \rangle =\langle K, \varphi \otimes \upsilon \rangle {\tag{19}} \end{equation}$$

for all $\upsilon \in {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ and $\varphi \in {\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,$. The operator norm satisfies$$\begin{equation} \|A\|_{Op} \asymp \|K\|_{{\mathcal{C}o_\pi }\,L^{\infty }_{1/w}(G)} {\tag{20}} \end{equation}$$

and$$\begin{equation} k_A=V_\Psi K. {\tag{21}} \end{equation}$$

(ii)

Kernel theorem: Conversely, if $A:{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ is bounded, then there exists a unique kernel $K\in {\mathcal{C}o_\pi }\,L^\infty _{1/w} (G_1 \times G_2)$ such that 19 holds.

Proof.

(i) Fix $K\in {\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)$ with $G=G_1\times G_2$, and let $\upsilon \in {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$, $\varphi \in {\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,$ be arbitrary. By 13 it follows that $\varphi \otimes \upsilon \in {\mathcal{C}o_\pi }\,L^1_{w}(G)$. Therefore, the duality in 19 is well-defined and

$$\begin{align} |\langle K,\varphi \otimes \upsilon \rangle | & \leqslant \|K\|_{{\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)}\|\varphi \otimes \upsilon \|_{{\mathcal{C}o_\pi }\,L^1_{w}(G)} \\ & =\|K\|_{{\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)}\|\varphi \|_{{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,}\|\upsilon \|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,}. {\tag{22}} \end{align}$$

Therefore, if we fix $\upsilon$, the mapping $\varphi \mapsto \langle K,\varphi \otimes \upsilon \rangle$ is a bounded, anti-linear functional on ${\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,$, which we call $A\upsilon \in {\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$. The map $\upsilon \mapsto A\upsilon$ is clearly linear, and 19 defines a linear operator $A:{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$. The estimate 22 implies that

$$\begin{equation*} \|Av\|_{{\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,} \leqslant \|K\|_{{\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)} \|v\|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,} \, , \end{equation*}$$

and thus

$$\begin{equation*} \|A\|_{\mathrm{Op}} \leqslant \|K\|_{{\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)} \, . \end{equation*}$$

(ii) To prove the converse, we need to show that the mapping $K\mapsto A$ is one-to-one and onto.

Uniqueness: Let us assume that the kernel $\mathcal{K}\in {\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)$ also satisfies

$$\begin{equation*} \langle A\upsilon ,\varphi \rangle =\langle K,\varphi \otimes \upsilon \rangle =\langle \mathcal{K},\varphi \otimes \upsilon \rangle {} \end{equation*}$$

for every $\upsilon \in {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$, $\varphi \in {\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,$. By Theorem 2, there exists a discrete set $\{\gamma _i\}_{i\in \mathcal{I}}\subset G$ such that every $F\in {\mathcal{C}o_\pi }\,L_w^1(G)$ can be written as

$$\begin{equation*} F=\sum _{i\in \mathcal{I}}\lambda _i(F)\pi ( \gamma _i)\big (\psi _2\otimes \psi _1\big ), \end{equation*}$$

with unconditional convergence in ${\mathcal{C}o_\pi }\,L_w^1(G)$ and

$$\begin{equation*} \sum _i |\lambda _i(F)| w(\gamma _i) \leqslant C \|F\|_{{\mathcal{C}o_\pi }\,L^1_w(G_1\times G_2)}. \end{equation*}$$

Since $\pi ( \gamma _i)\big (\psi _2\otimes \psi _1\big ) = \pi (\gamma _{i,2}) \psi _2 \otimes \pi ( \gamma _{i,1}) \psi _1$, we conclude that

$$\begin{align*} \langle K,F\rangle & =\sum _{i\in \mathcal{I}}\overline{\lambda _i(F)}\langle K,\pi (\gamma _{i,2}) \psi _2 \otimes \pi ( \gamma _{i,1}) \psi _1 \rangle \\ & =\sum _{i\in \mathcal{I}}\overline{\lambda _i(F)}\langle \mathcal{K},\pi (\gamma _{i,2}) \psi _2 \otimes \pi ( \gamma _{i,1}) \psi _1 \rangle \\ & =\langle \mathcal{K},F\rangle . \end{align*}$$

As this equality holds for every $F\in {\mathcal{C}o_\pi }\,L^1_w(G)$, it follows that $K=\mathcal{K}$.

Surjectivity: Let us assume that $A:{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ is bounded. Then the kernel $k_A$ defined in 14 is an element of $L^\infty _{1/w}(G_1 \times G_2)$, because

$$\begin{align*} |k_A(g)| & =\left|\langle A \pi _1(g_1)\phi ,\pi _2(g_2)\psi \rangle \right| \\ & \leqslant \|A\|_{Op} \, \, \|\pi _1(g_1)\phi \|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,}\,\, \|\pi _2(g_2)\psi \|_{{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,} \\ & \leqslant \|A\|_{Op} \,\, w_1(g_1)\,\, \|\phi \|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,}\,\, w_2(g_2)\,\, \|\psi \|_{{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,}. \end{align*}$$

We claim that $k_A$ is a generalized wavelet transform. Precisely, there exists $K\in {\mathcal{C}o_\pi }\,L^\infty _{1/w}(G_1 \times G_2)$ such that $k_A=V_{\Psi } K$. To prove this claim, we use Proposition 1v, which asserts that $k_A = V_\psi K$ for some $K\in {\mathcal{C}o_\pi }\,L^\infty _{1/w}(G)$ if and only if $k_A=k_A\ast V_\Psi \Psi$.

As $k_A\cdot V_\Psi (\pi (g)\Psi )\in L^1(G_1\times G_2)$, we may choose the most convenient order of integration and apply the reproducing formula of Proposition 1v consecutively to the representations $\pi _1$ and $\pi _2$. Using 13 we obtain

$$\begin{align*} (k_A\ast V_\Psi \Psi )(g) & =\int _{G}k_A(h) V_\Psi \Psi (h^{-1}g) \, dh \\ & =\int _{G_1}\int _{G_2} V_{\psi _2}\big (A\pi _1(h_1)\psi _1\big )(h_2) V_{\psi _2}\psi _2(h_2^{-1}g_2) dh_2\ \overline{V_{\psi _1}\psi _1(h_1^{-1}g_1)} dh_1 \\ & =\int _{G_1}\Big (V_{\psi _2}\big (A\pi _1(h_1)\psi _1\big )\ast V_{\psi _2}\psi _2\Big )(g_2) \ \overline{V_{\psi _1}\psi _1(h_1^{-1}g_1)} dh_1 \\ & =\int _{G_1}\langle A \pi _1(h_1)\psi _1,\pi _2(g_2)\psi _2\rangle \ \overline{V_{\psi _1}\psi _1(h_1^{-1}g_1)} dh_1 = (\ast ) \, . \end{align*}$$

At this point we note that by the assumption on $A$ there exists a unique operator $A^\prime :{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,\rightarrow {\mathcal{C}o_{\pi _1}L^{\infty }_{1/w_1}(G_1)}\,$ that satisfies

$$\begin{equation*} \langle A\upsilon ,\varphi \rangle =\langle \upsilon ,A^\prime \ \varphi \rangle \end{equation*}$$

for every $\upsilon \in {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ and $\varphi \in {\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,$. By its definition, $A^\prime$ is weak$^\ast$-continuous. We continue with the integration over $G_1$ and obtain

$$\begin{align*} (\ast ) & =\overline{\int _{G_1}\langle A^\prime \pi _2(g_2)\psi _2,\pi _1(h_1)\psi _1\rangle \ V_{\psi _1}\psi _1(h_1^{-1}g_1) dh_1} \\ & =\overline{\Big ( V_{\psi _1}\big (A^\prime \pi _2(g_2)\psi _2\big )\ast V_{\psi _1}\psi _1\Big )(g_1)}=\overline{\langle A^\prime \pi _2(g_2)\psi _2,\pi _1(g_1)\psi _1\rangle } \\ & =\langle A \pi _1(g_1)\psi _1,\pi _2(g_2)\psi _2\rangle =k_A(g). \end{align*}$$

By Proposition 1v there exists a kernel $K\in {\mathcal{C}o_\pi }\,L^\infty _{1/w}(G_1 \times G_2)$ such that $k_A(g_1,g_2)$ $= V_\Psi K(g_1,g_2)$. By the first part of the proof $K$ defines an operator $B: {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ $\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ by means of $\langle Bv, \phi \rangle =\langle K, \phi \otimes v \rangle$. In particular,

$$\begin{align*} \langle B \pi _1(g_1)\psi _1 ,\pi _2(g_2)\psi _2\rangle & = \langle K, \pi _2(g_2)\psi _2 \otimes \pi _1(g_1) \psi _1\rangle = V_\Psi K(g_1,g_2) \\ & = k_A (g_1,g_2) = \langle A \pi _1(g_1)\psi _1 ,\pi _2(g_2)\psi _2\rangle \, . \end{align*}$$

Consequently, $B\pi _1(g_1)\psi _1 = A\pi _1(g_1)\psi _1$ for all $g_1\in G_1$. This identity extends to all finite linear combinations of vectors $\pi _1(g_1)\psi _1$ and by Theorem 2 to $\mathcal{C}o _{\pi _1} L^1_{w_1}(G_1)$. Thus $B=A$, and we have shown that the map from kernels to operators is onto.

The map $K \mapsto A$ is bounded and invertible. By the inverse mapping theorem we obtain that $\|K\| _{ {\mathcal{C}o_\pi }\,L^\infty _{1/w} (G) } \leqslant C \|A\|_{\mathrm{Op}}$, which proves 20.

■

Remark 4.

It is crucial to interpret the brackets in 19 correctly. For utmost precision, we would have to write

$$\begin{equation*} \langle A\upsilon ,\varphi \rangle _{{\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,\!,\;{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,} = \langle K, \varphi \otimes \upsilon \rangle _{{\mathcal{C}o_\pi }\,L^\infty _{1/w }(G),\;{\mathcal{C}o_\pi }\,L^1_{w}(G)} \, , \end{equation*}$$

but we feel that this notation would distract from the analogy to distribution theory.

The injectivity of the mapping $K\mapsto A$ from kernels to operators is closely related to an important property of the coorbit spaces ${\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$. This so-called tensor product property has gained considerable importance in certain special cases 13, Theorem 7D and 26. We therefore state and prove a general version. Recall that the projective tensor product of two Banach spaces $B_1$ and $B_2$ is defined to be

$$\begin{equation*} B_1\widehat{\otimes }B_2 = \{f=\sum _{i\in \mathcal{I}} \phi _i \otimes \psi _i: \phi _i\in B_1, \psi _i \in B_2 \quad \text{and}\quad \sum _{i\in \mathcal{I}}\|\phi _i\|_{B_1}\|\psi _i\|_{B_2} <\infty \} \, . \end{equation*}$$

The norm is given as $\|f\|_{\widehat{\otimes }}=\inf \sum _{i\in \mathcal{I}}\|\phi _i\|_{B_1}\|\psi _i\|_{B_2}$ over all representations of $f=\sum _{i\in \mathcal{I}}\phi _i\otimes \psi _i$.

The following identification of the projective tensor product of ${\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ and ${\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,$ with the coorbit space ${\mathcal{C}o_\pi }\,L^1_w(G_1\times G_2)$ is a generalization of Feichtinger’s original result for modulation spaces 13, Theorem 7D.

Theorem 5.

Under the general assumptions on the groups $G_i$ and the representations $(\pi _i,\mathcal{H}_i)$ we have

$$\begin{equation*} {\mathcal{C}o_\pi }\,L^1_w(G_1 \times G_2) ={\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,\widehat{\otimes }\ {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,. {} \end{equation*}$$

Proof.

Let $F\in {\mathcal{C}o_\pi }\,L^1_w(G)$. Then by Theorem 2 applied to $\pi = \pi _2 \otimes \pi _1$, $F$ possesses the representation $F=\sum _{i\in \mathcal{I}} \lambda _i(F)\pi (\gamma _i)\Psi \in {\mathcal{C}o_\pi }\,L^1_w(G)$ with $\gamma _i = (\gamma _{i,1} , \gamma _{i,2}) \in G_1\times G_2$ and $\sum _{i\in \mathcal{I}} |\lambda _i(F)| w(\gamma _i) \leqslant C \|F\|_{{\mathcal{C}o_\pi }\,L^1_w(G)}$. Using Proposition 1(ii) we obtain that

$$\begin{align*} \sum _{i\in \mathcal{I}} \| & \lambda _i(F) \pi _1(\gamma _{i,1})\psi _1 \|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,}\|\pi _2(\gamma _{i,2})\psi _2\|_{{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,} \\ & \leqslant \sum _{i\in \mathcal{I}} |\lambda _i(F)|w_1(\gamma _{i,1}) w_2(\gamma _{i,2}) \|\psi _1 \|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,}\|\psi _2\|_{{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,} \\ & \leqslant C\|F\|_{{\mathcal{C}o_\pi }\,L^1_w(G)}. \end{align*}$$

Thus $F\in {\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,\widehat{\otimes }\ {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$, and ${\mathcal{C}o_\pi }\,L^1_w(G)$ is continuously embedded into ${\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,\widehat{\otimes }\ {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$.

Conversely, let $F \in {\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,\widehat{\otimes }\ {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$. Choose a representation $F = \sum \limits _{i\in \mathcal{I}} f_{i,2} \otimes f_{i,1}$ with $\sum \limits _{i\in \mathcal{I}} \| f_{i,1} \|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,} \|f_{i,2}\|_{{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,} < \infty$. Using Fubini’s theorem and Proposition 1(ii) yields

$$\begin{align*} \|F\|_{{\mathcal{C}o_\pi }\,L^1_w(G)} & = \int _G \left| V_{\Psi } F (g) \right| w(g) dg \\ & \leqslant \sum \limits _{i\in \mathcal{I}} \left( \int _{G_1} \left| V_{\psi _1} f_{i,1} (g_1) \right| w_1(g_1) dg_1 \right) \cdot \left( \int _{G_2} \left| V_{\psi _2} f_{i,2} (g_2) \right| w_2(g_2) dg_2 \right) \\ & = \sum \limits _{i\in \mathcal{I}} \|f_{i,1}\|_{{\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,} \| f_{i,2} \|_{{\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,} < \infty . \end{align*}$$

Thus, ${\mathcal{C}o_{\pi _2}L^{1}_{w_2}(G_2)}\,\widehat{\otimes }\ {\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,\subseteq {\mathcal{C}o_\pi }\,L^1_w(G)$. The equivalence of the norms follows from the inverse mapping theorem.

■

Once the kernel theorem provides a general description of operators between test functions and distributions, we may try to characterize certain classes of operators by properties of their kernel. Since on the level of the generalized wavelet transform such operators correspond to integral operators (see diagram in Figure 1), we may translate the various versions of Schur’s test to kernel theorems for operators between coorbit spaces. Following the procedure in 8, Theorem 3.3, we first formulate a general version of Schur’s test and then derive the abstract kernel theorem.

We introduce two classes of mixed norm spaces. For two $\sigma$-finite measure spaces $(X,\mu )$ and $(Y,\nu )$, $1\leqslant p\leqslant \infty$, and $m:X\times Y\rightarrow \mathbb{R}_+,$ we define the spaces $L^{p,\infty }_m(X\times Y)$ and $\mathcal{L}^{p,\infty }_m(X\times Y)$ by the norms

$$\begin{equation} \|F\|_{L^{p,\infty }_m(X\times Y)}\coloneq \operatorname *{ess\,\sup }_{y\in Y}\left(\int _X |F(x,y)|^pm(x,y)^p d\mu (x)\right)^{1/p} {\tag{23}} \end{equation}$$

and

$$\begin{equation} \|F\|_{\mathcal{L}^{p,\infty }_m(X\times Y)}\coloneq \operatorname *{ess\,\sup }_{x\in X}\left(\int _Y |F(x,y)|^pm(x,y)^p d\nu (y)\right)^{1/p}. {\tag{24}} \end{equation}$$

The following version of Schur’s test is folklore and can be found in 33, Propositions 5.2 and 5.4 or 27.

Proposition 6.

Let $(X,\mu )$ and $(Y,\nu )$ be $\sigma$-finite measure spaces, let $1\leqslant p\leqslant \infty$, let $\frac{1}{p}+\frac{1}{q}=1$, and let $T$ be the integral operator $Tf(y)=\int _X f(x)k_T(x,y)d\mu (x)$ with kernel $k_T:X\times Y\rightarrow \mathbb{C}$.

(i)

The operator $T$ is bounded from $L^1_{m_1}(X)$ to $L^p_{m_2}(Y)$ if and only if $k_T \in \mathcal{L}^{p,\infty }_{m_1^{-1}\otimes m_2}(X\times Y)$. In that case$$\begin{equation} \|T\|_{L^1_{m_1}(X)\rightarrow L^p_{m_2}(Y)} =\|k_T\|_{\mathcal{L}^{p,\infty }_{m_1^{-1}\otimes m_2}(X\times Y)} \, . \tag{25} \end{equation}$$

(ii)

The operator $T$ is bounded from $L^p_{m_1}(X)$ to $L^\infty _{m_2}(Y)$ if and only if $k_T \in L^{q,\infty }_{m_1^{-1}\otimes m_2}(X\times Y)$. In this case$$\begin{equation} \|T\|_{L^p_{m_1}(X)\rightarrow L^\infty _{m_2}(Y)} =\|k_T\|_{L^{q,\infty }_{m_1^{-1}\otimes m_2}(X\times Y)}. \tag{26} \end{equation}$$

We now characterize the boundedness of operators between certain coorbit spaces.

Theorem 7.

Let $1\leqslant p,q \leqslant \infty$ with $\frac{1}{p}+\frac{1}{q}=1$, and let $m_j$ be $w_j$-moderate weights on $G_j$. If $A$ is a bounded operator from ${\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ to ${\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ with kernel $K$, then the following hold:

(i)

$A$ is bounded from ${\mathcal{C}o_{\pi _1}L^{1}_{m_1}(G_1)}\,$ to ${\mathcal{C}o_{\pi _2}L^{p}_{m_2}(G_2)}\,$ if and only if its kernel $K$ is in ${\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }_{m_1 ^{-1}\otimes m_2}(G_1\times G_2)$. Its operator norm satisfies$$\begin{equation*} \|A\|_{Op} \asymp \|K\|_{{\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }_{m_1^{-1}\otimes m_2}(G)} \, . \end{equation*}$$

(ii)

$A$ is bounded from ${\mathcal{C}o_{\pi _1}L^{p}_{m_1}(G_1)}\,$ to ${\mathcal{C}o_{\pi _2}L^{\infty }_{m_2}(G_2)}\,$ if and only if its kernel $K$ is in ${\mathcal{C}o_\pi }\,L^{q,\infty }_{m_1^{-1}\times m_2}(G_1\times G_2)$. Its operator norm satisfies$$\begin{equation*} \|A\|_{Op}\asymp \|K\|_{{\mathcal{C}o_\pi }\,L^{q,\infty }_{m_1^{-1}\otimes m_2}(G)}\, . \end{equation*}$$

Proof.

Since $\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)\,\subseteq \mathcal{C}o_{\pi _1}L^{1}_{m_1}(G_1)\,$ and $\mathcal{C}o_{\pi _2}L^{p}_{m_2}(G_2)\,\subseteq \mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)\,$ by 10, the kernel theorem is applicable to the operator $A$, and there exists a kernel $K\in {\mathcal{C}o_\pi }\,L^\infty _{1/w} (G_1\otimes G_2)$ such that

$$\begin{equation*} V_\Psi K (g_1,g_2) = k_A(g_1,g_2) = \langle A \pi _1(g_1)\psi _1, \pi _2(g_2)\psi _2 \rangle . \end{equation*}$$

Assume first that $K \!\in \! {\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }_{m_1^{-1}\otimes m_2}(G)$, which means that $V_\Psi K \!\in \! \mathcal{L}^{p,\infty }_{1/m_1\otimes m_2}(G)$. By Proposition 6, the integral operator $\mathfrak{A}$ defined by the integral kernel $k_A$ is bounded from $L^1_{m_1}(G_1)$ to $L^p_{m_2}(G_2)$. According to 18, $A$ factors as $A = V_{\psi _2}^* \mathfrak{A} V_{\psi _1}$, where $V_{\psi _1}$ is an isometry from $\mathcal{C}o _{\pi _1}L^1_{m_1}(G_1)$ to $L^1_{m_1}(G_1)$, and $V_{\psi _2}^*$ is bounded from $L^p_{m_2}(G_2)$ to $\mathcal{C}o _{\pi _2 } L^p_{m_2}(G_2)$ by Proposition 1. Consequently $A$ is bounded from ${\mathcal{C}o_{\pi _1}L^{1}_{m_1}(G_1)}\,$ to ${\mathcal{C}o_{\pi _2}L^{p}_{m_2}(G_2)}\,$. The boundedness estimate follows from

$$\begin{align*} \|A\|_{Op} & \leqslant \|V_{\psi _2}^*\|_{Op} \, \| \mathfrak{A}\|_{L^1_{m_1}(G_1)\rightarrow L_{m_2}^p(G_2)} \, \|V_{\psi _1}\|_{Op} \\ & \leqslant C \|k_A\|_{\mathcal{L}^{p,\infty }_{m_1^{-1}\otimes m_2}(G)} = C \|K\|_{{\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }_{m_1^{-1}\otimes m_2}(G)}. \end{align*}$$

Conversely, let $A$ be bounded from ${\mathcal{C}o_{\pi _1}L^{1}_{m_1}(G_1)}\,$ to ${\mathcal{C}o_{\pi _2}L^{p}_{m_2}(G_2)}\,$. Then $A \pi _1(g_1)\psi _1$ $\in \mathcal{C}o _{\pi _2} L^p_{m_2}(G_2)$ and the following estimates make sense:

$$\begin{align*} \|K\|_{{\mathcal{C}o_\pi }\,\mathcal{L}_{m_1^{-1}\otimes m_2}^{p,\infty }(G)} & = \|V_\Psi K\| _{\mathcal{L}_{m_1^{-1}\otimes m_2}^{p,\infty }(G)} = \|k_A\|_{\mathcal{L}_{m_1^{-1}\otimes m_2}^{p,\infty }(G)} \\ & =\sup _{g_1\in G_1}\!\left(\int _{G_2}|\langle A\pi _1(g_1)\psi _1,\pi _2(g_2)\psi _2\rangle m_2(g_2)|^p dg_2\right)^{1/p}\! m_1(g_1)^{-1} \\ & =\sup _{g_1\in G_1}\|A\pi _1(g_1)\psi _1\|_{{\mathcal{C}o_{\pi _2}L^{p}_{m_2}(G_2)}\,} m_1(g_1)^{-1} \\ & \leqslant \|A\|_{Op}\sup _{g_1\in G_1}\|\pi _1(g_1)\psi _1\|_{{\mathcal{C}o_{\pi _1}L^{1}_{m_1}(G_1)}\,} m_1(g_1)^{-1}. \end{align*}$$

Since $V_{\psi _1} \psi _1 \in L^1_{w_1}(G_1)$ and $m_1$ is $w_1$-moderate and thus satisfies $m_1(g_1h) m_1(g_1)^{-1} \leqslant w_1(h)$, the last expression is bounded by

$$\begin{align*} \sup _{g_1\in G_1} \|\pi _1(g_1) & \psi _1\|_{{\mathcal{C}o_{\pi _1}L^{1}_{m_1}(G_1)}\,}\cdot m_1(g_1)^{-1} =\sup _{g_1\in G_1}\int _{G_1}|\langle \psi _1,\pi _1(g_1^{-1}h)\psi _1 \rangle | \frac{m_1(h)}{ m_1(g_1)}dh \\ & = \sup _{g_1\in G_1}\int _{G_1}|\langle \psi _1,\pi _1(h)\psi _1 \rangle \frac{m_1(g_1h)}{ m_1(g_1)}dh\leqslant \|V_{\psi _1}\psi _1\|_{L^1_{w_1}(G_1)}. \end{align*}$$

Thus $K\in {\mathcal{C}o_\pi }\,\mathcal{L}_{m_1^{-1}\otimes m_2}^{p,\infty }(G)$.

Part (ii) follows by using Proposition 6(ii) instead of (i) and is proved similarly.

■

The following diagram (Figure 1) shows the connection between the different operators and spaces.

Using interpolation between $L^p$-spaces, Schur’s test can also be formulated as saying that an integral operator is bounded on all $L^p$ simultaneously if and only if its kernel belongs to $L^{1,\infty } \cap \mathcal{L}^{1,\infty }$. The corresponding version for coorbit spaces is a consequence of Theorem 7 and an interpolation argument.

Corollary 8.

The following conditions are equivalent:

(i)

$A:{\mathcal{C}o_{\pi _1}L^{p}_{m_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{p}_{m_2}(G_2)}\,$ is bounded for every $1\leqslant p\leqslant \infty$.

(ii)

Both $A:{\mathcal{C}o_{\pi _1}L^{1}_{m_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{1}_{m_2}(G_2)}\,$ and $A:{\mathcal{C}o_{\pi _1}L^{\infty }_{m_1}(G_1)}\,\rightarrow {\mathcal{C}o_{\pi _2}L^{\infty }_{m_2}(G_2)}\,$ are bounded.

(iii)

$K \in {\mathcal{C}o_\pi }\,\mathcal{L}^{1,\infty }_{ m_1^{-1}\otimes m_2}(G) \bigcap {\mathcal{C}o_\pi }\,L^{1,\infty }_{m_1^{-1}\otimes m_2}(G)$.

Clearly one can now translate every boundedness result for an integral operator into a kernel theorem for coorbit spaces. As a simple but important example we offer a sufficient condition for regularizing operators, i.e., operators that map distributions to test functions.

Theorem 9.

Under the assumptions of Theorem 3, if the unique kernel of the operator $A$ satisfies $K\in \mathcal{C}o _\pi L^1_{ w} (G)$, then $A$ is bounded from $\mathcal{C}o _{\pi _1} L^\infty _{1/w_1} (G_1)$ to $\mathcal{C}o _{\pi _2} L^1_{w_2} (G_2)$.

Proof.

Consider the integral operator $\mathfrak{A}$ as in the proof of Theorem 7 and observe that $V_\Psi K=k_A\in L^1_{w}(G)$ is a sufficient condition for $\mathfrak{A}:L^\infty _{1/w_1}(G_1)\rightarrow L^1_{w_2}(G_2)$ to be bounded by Schur’s test.

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4.1. Discretization

Coorbit theory guarantees the discretization of the coorbit spaces via atomic decompositions and Banach frames. For our purposes, it is sufficient to state a shortened and simplified version of 22, Theorem 5.3. Let $Y$ be one of the function spaces $L^p_m(G),L^{p,\infty }_m(G),$ or $\mathcal{L}^{p,\infty }_m(G)$, and let $Y_d$ be the natural sequence space associated to $Y$.

Proposition 10.

If $\psi$ satisfies

$$\begin{equation} \int _G \sup _{h\in gQ}|V_\psi \psi (h)|w(g)dg<\infty , {\tag{27}} \end{equation}$$

for a compact neighborhood $Q$ of $e$, then there exist a discrete subset $\Lambda \subset G$ and constants $C_1,C_2>0$ such that

$$\begin{equation} C_1\|f\|_{{\mathcal{C}o_\pi }\,Y}\leqslant \|V_\psi f\|_{Y_d}\leqslant C_2\|f\|_{{\mathcal{C}o_\pi }\,Y} \qquad \text{ for every } f\in {\mathcal{C}o_\pi }\,Y \, . {\tag{28}} \end{equation}$$

Corollary 11.

Let $\Lambda =\Lambda _1\times \Lambda _2\subset G$ be a discrete set such that $\{\pi (\lambda )\Psi \}_{\lambda \in \Lambda }$ satisfies 28 for ${\mathcal{C}o_\pi }\,L_{m_1^{-1}\otimes m_2}^{p,\infty }(G)$ and ${\mathcal{C}o_\pi }\,\mathcal{L}_{m_1^{-1}\otimes m_2}^{p,\infty }(G)$. If $A$ is a bounded operator from ${\mathcal{C}o_{\pi _1}L^{1}_{w_1}(G_1)}\,$ to ${\mathcal{C}o_{\pi _2}L^{\infty }_{1/w_2}(G_2)}\,$ with kernel $K$, then the following hold:

(i)

$A:\mathcal{C}o _{\pi _1} L^1_{m_1}(G_1) \rightarrow \mathcal{C}o _{\pi _2} L^p_{m_2}(G_2)$ is bounded if and only if$$\begin{equation} \sup _{\lambda _1\in \Lambda _1}\left(\sum _{\lambda _2\in \Lambda _2}|V_{\Psi }K(\lambda )(m_1^{-1}\otimes m_2)(\lambda )|^p\right)^{1/p}<\infty . {\tag{29}} \end{equation}$$

(ii)

Likewise $A:\mathcal{C}o _{\pi _1} L^p_{m_1}(G_1) \rightarrow \mathcal{C}o _{\pi _2} L^\infty _{m_2}(G_2)$ is bounded if and only if$$\begin{equation} \sup _{\lambda _2\in \Lambda _2}\left(\sum _{\lambda _1\in \Lambda _1}|V_{\Psi }K(\lambda )(m_1^{-1}\otimes m_2)(\lambda )|^q\right)^{1/q}<\infty . {\tag{30}} \end{equation}$$

Proof.

(i) By Theorem 7 $A$ has a kernel in ${\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }(G)$, and

$$\begin{equation*} \|A\|_{Op}\asymp \| K\|_{{\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }(G)}. \end{equation*}$$

By 28, the expression in 29 is an equivalent norm for $\| K\|_{{\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }(G)}$. The proof of (ii) works in exactly the same way.

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5. Examples

5.1. Modulation spaces

The Weyl-Heisenberg group $G_{WH}=\mathbb{R}^d\times \mathbb{R}^d\times \mathbb{T}$ is defined by the group law

$$\begin{equation*} (x,\omega ,e^{2\pi i\tau })\cdot (x^\prime ,\omega ^\prime ,e^{2\pi i\tau ^\prime })=(x+x^\prime ,\omega +\omega ^\prime ,e^{2\pi i(\tau +\tau ^\prime -x\cdot \omega ^\prime )}). \end{equation*}$$

Let $T_xf(t)\coloneq f(t-x)$ denote the translation, and let $M_\omega f(t)\coloneq e^{2\pi i\omega t}f(t)$ be the modulation operator. The operator $\pi _{WH}(x,\omega ,\tau )=e^{2\pi i \tau }M_\omega T_x$ for $(x,\omega ,\tau ) \in G_{WH}$ defines a unitary square-integrable representation of $G_{WH}$ acting on $L^2(\mathbb{R}^d)$, for which every non-zero vector in $L^2(\mathbb{R}^d)$ is admissible. Since the phase factor $e^{2\pi i \tau }$ is irrelevant for the definition of coorbit spaces, it is convenient to drop the trivial third component and consider the time-frequency shift $\pi (x,\omega )=\pi _{WH}(x,\omega ,1)= M_\omega T_x$. Formally, we treat the projective representation $\pi$ of $\mathbb{R}^{2d}$ instead of the unitary representation $\pi _{WH}$ of $G_{WH}$. The transform corresponding to $\pi$ is nothing else but the short-time Fourier transform

$$\begin{equation*} V_\psi f(x,\omega )=\langle f,M_\omega T_x\psi \rangle =\int _{\mathbb{R}^d}f(t)\overline{\psi (t-x)}e^{-2\pi i\omega \cdot t}dt,\qquad f,\psi \in L^2(\mathbb{R}^d). \end{equation*}$$

The coorbit spaces associated to $\pi _{WH}$ coincide therefore with the coorbit spaces associated to $\pi$. These are the modulation spaces $M_m^p(\mathbb{R}^d)$ which were first introduced by Feichtinger in 14 as certain decomposition spaces and subsequently were identified with the coorbit spaces of the Heisenberg group $\mathcal{C}o _{\pi _{WH}} L^p_m (G_{WH}) = \mathcal{C}o _\pi L^p_m(\mathbb{R}^{2d})=M^p_m(\mathbb{R}^d)$ 18. We refer to the standard textbooks 2023 for more information on time-frequency analysis.

Theorem 3 asserts that every bounded operator from $M^1_w(\mathbb{R}^d) \!=\! \mathcal{C}o _{\pi _{WH}} L^1_w(G_{WH}) = {\mathcal{C}o_\pi }\,L^1_w(\mathbb{R}^{2d})$ to $M^\infty _{1/w}(\mathbb{R}^d) = \mathcal{C}o _{\pi _{WH}} L^\infty _{1/w}(G_{WH}) = {\mathcal{C}o_\pi }\,L^\infty _{1/w}(\mathbb{R}^{2d})$ possesses a kernel $K\in \mathcal{C}o _{\pi _{WH} \otimes \pi _{WH} } L^\infty _{w^{-1} \otimes w^{-1} } (G_{WH} \times G_{WH})$ such that $\langle Af,g \rangle = \langle K, g \otimes f \rangle$ for $f,g\in M^1_w(\mathbb{R}^d)$. Let us elaborate in detail what the kernel theorem asserts in this case: for $g_i = (x_i,\omega _i, \tau _i) \in G_{WH}, i=1,2$, the tensor representation $\pi _{WH} \otimes \pi _{WH}$ acts on the simple tensor $(\psi _2 \otimes \psi _1)(t_2,t_1) = \psi _2(t_2)\overline{\psi _1(t_1)} \in L^2(\mathbb{R}^d) \otimes L^2(\mathbb{R}^d) \cong L^2(\mathbb{R}^{2d})$ as

$$\begin{multline*} \pi _{WH}\otimes \pi _{WH} (g_2,g_1) (\psi _2\otimes \psi _1)(t_2,t_1) = e^{2\pi i (\tau _1 - \tau _2)} M_{\omega _2}T_{x_2}\psi _2(t_2) \overline{M_{\omega _1}T_{x_1}\psi _1(t_1)} \\ = e^{2\pi i (\tau _1 - \tau _2)} M_{(\omega _2,-\omega _1)} T_{(x_2,x_1)}(\psi _2 \otimes \psi _1)(t_2,t_1) \, . \end{multline*}$$

Thus except for the phase factor $e^{2\pi i (\tau _1 - \tau _2)}$ the tensor representation $\pi _{WH}\otimes \pi _{WH}$ is just the time-frequency shift $M_{(\omega _2,-\omega _1)} T_{(x_2,x_1)}$ acting on $L^2(\mathbb{R}^{2d})$. Consequently, the coorbit spaces with respect to the product group $G_{WH} \otimes G_{WH}$ are again modulation spaces, this time on $\mathbb{R}^{2d}$. For the coorbit of $L^\infty$ we compare the norms

$$\begin{equation*} \|K\|_{M^\infty (\mathbb{R}^{2d})}=\sup _{(x_1,x_2,\omega _1,\omega _2)\in \mathbb{R}^{4d}}\big |\big \langle K,M_{(\omega _1, \omega _2)} T_{(x_1,x_2)}(\psi _2\otimes \psi _1)\big \rangle \big | \end{equation*}$$

and

$$\begin{equation*} \|K\|_{\mathcal{C}o_{\pi \otimes \pi } L^\infty (\mathbb{R}^{4d})}=\sup _{\big ((x_1,\omega _1),(x_2,\omega _2)\big )\in \mathbb{R}^{4d}}\left|\big \langle K,\big (\pi (x_1,\omega _1)\otimes \pi (x_2,\omega _2)\big )(\psi _2\otimes \psi _1)\big \rangle \right|, \end{equation*}$$

which are obviously equal. In this case Theorem 3 is therefore just Feichtinger’s kernel theorem: For $A: M^1(\mathbb{R}^d) \to M^\infty (\mathbb{R}^d)$ there exists a unique kernel $K\in M^\infty (\mathbb{R}^{2d})$ such that $\langle Af,g\rangle = \langle K, g\otimes f \rangle$.

The recent extension of Feichtinger’s kernel theorem by Cordero and Nicola 8 can be seen in the same light. Let us explain the difference in the formulations. Our approach considers the generalized wavelet transform

$$\begin{equation*} V_{\Psi }K(x_1,\omega _1,x_2,\omega _2)=\langle K,\pi _{WH}(x_2,\omega _2,1)\otimes \pi _{WH}(x_1,\omega _1,1)(\psi _2 \otimes \psi _1)\rangle \end{equation*}$$

of the kernel. The conditions of Theorem 7 are formulated by mixed norms acting simultaneously on the variables $(x_2,\omega _2)$ and $(x_1,\omega _1)$. The treatment in 8 uses the short-time Fourier transform on $\mathbb{R}^{2d}$

$$\begin{equation*} V_{\Psi }K(x_1,x_2,\omega _1,\omega _2)=\langle K,M_{(\omega _1,\omega _2)} T_{(x_1,x_2)} \Psi \rangle \, , \end{equation*}$$

which is the same transform, except for the order of the variables. In 8 it was therefore necessary to reshuffle the order of integration of time-frequency shifts and to use the notion of mixed modulation spaces, which were studied in 429. The new insight of our formulation is that the mixed modulation spaces are simply the coorbit spaces with respect to the tensor product representation.

The special case of Theorem 7 for the Weyl-Heisenberg group and the weights $m_s(x,\omega ,\tau )=(1+|x|+|\omega |)^s$ for $s\in \mathbb{R}$ states the following: Fix $\sigma >0$ and let $A$ be an operator from $M^1_{m_\sigma }(\mathbb{R}^d)$ to $M^\infty _{m_{-\sigma }}(\mathbb{R}^d)$. Then for $|r|, |s| \leqslant \sigma$, $1\leqslant p,q\leqslant \infty$, and $1/p+1/q=1$ we have

(i)

$\text{$A:M_{m_s}^{1}(\mathbb{R}^d)\rightarrow M_{m_r}^{p}(\mathbb{R}^d)$ bounded} \Leftrightarrow K\in {\mathcal{C}o_\pi }\,\mathcal{L}^{p,\infty }_{m_{-s}\otimes w_{r}}(\mathbb{R}^{4d})$,

(ii)

$\text{$A:M_{m_s}^{p}(\mathbb{R}^d)\rightarrow M^{\infty }_{m_r}(\mathbb{R}^d)$ bounded} \Leftrightarrow K\in {\mathcal{C}o_\pi }\,L^{q,\infty }_{ m_{-s}\otimes m_{r}}(\mathbb{R}^{4d})$.

Regularizing operators from $M^\infty$ to $M^1$ were recently studied by Feichtinger and Jakobsen 19: they characterized a subclass of this space of operators by an integral kernel in $M^1(\mathbb{R}^{2d})$. The sufficiency of this result in a coorbit version is contained in Theorem 9.

5.2. Wavelet coorbit spaces and Besov spaces

The affine group $G_{\mathrm{aff}}=\mathbb{R}\times \mathbb{R}^*$ is given by the group law $(x,a)\cdot (y,b)=(x+ay,ab)$, where $x,y\in \mathbb{R}$ and $a,b \in \mathbb{R}\setminus \{0\}$. Its left Haar measure is given by $\frac{dxda}{a^{2}}$. Let $D_af(t)=|a|^{-1/2}f(t/a)$ denote the dilation operator. Then $(x,a) \to \pi _{\mathrm{aff}}(x,a)= T_x D_a$ defines a unitary, square-integrable representation of $G_{\mathrm{aff}}$ on $L^2(\mathbb{R})$.

Now let $f,\psi \in L^2 (\mathbb{R})$. The continuous wavelet transform is defined as

$$\begin{equation*} W_\psi f(x,a)\coloneq \langle f,\pi _{\mathrm{aff}}(x,a)\psi \rangle =|a|^{-1/2}\int _\mathbb{R}f(t)\overline{\psi (a^{-1}(t-x))}dt, \end{equation*}$$

and the admissibility condition 3 reads as

$$\begin{equation*} \int _{\mathbb{R}^*}|\widehat{\psi }(\omega )|^2\frac{d\omega }{|\omega |}<\infty . \end{equation*}$$

It is well-known that the coorbit spaces associated to the representation $\pi _{\mathrm{aff}}$ are the homogeneous Besov spaces. See the textbooks 1028 for details and further expositions of wavelet theory. For brevity, we consider only the coorbit spaces with respect to the weighted $L^p(G_{\mathrm{aff}})$-spaces with the weight function $\nu _s(x,a)=\nu _s(a)=|a|^{-s}$ for $s\in \mathbb{R}$. Note that $\nu _{-s}=1/\nu _s$. Then $\mathcal{C}o _{\pi _{\mathrm{aff}}} L^p_{\nu _s}(G_{\mathrm{aff}})=\dot{B}_{p,p}^{s-1/2+1/p}(\mathbb{R})$ by 15, Section 7.2. In particular $\mathcal{C}o _{\pi _{\mathrm{aff}}} L^1_{\nu _s}(G_{a})=\dot{B}_{1,1}^{s+1/2}(\mathbb{R})$ and $\mathcal{C}o_{\pi _{\mathrm{aff}}} L^\infty _{\nu _s}(G_{\mathrm{aff}})=\dot{B}_{\infty ,\infty }^{s-1/2}(\mathbb{R})$. In this example Theorem 3 states that an operator $A:\dot{B}_{1,1}^s(\mathbb{R})\rightarrow \dot{B}_{\infty ,\infty }^{-r}(\mathbb{R})$ is bounded if and only if its associated kernel $K$ is in $\mathcal{C}o _{\pi _{\mathrm{aff}}\otimes \pi _{\mathrm{aff}}} \, L^{\infty }_{\nu _{-s-1/2}\otimes \nu _{-r-1/2}}(G_{\mathrm{aff}}^2)$. At first glance not much seems to have been gained by this formulation, but it turns out that the coorbit spaces of the tensor product $\pi _{\mathrm{aff}}\otimes \pi _{\mathrm{aff}}$ of $G_{\mathrm{aff}}^2$ are well understood in the theory of function spaces under the name of Besov spaces of dominating mixed smoothness. In particular, $\mathcal{C}o _{\pi _{\mathrm{aff}}\otimes \pi _{\mathrm{aff}}} \, L^{\infty }_{\nu _{-s-1/2}\otimes \nu _{-r-1/2}}(G_{\mathrm{aff}}^2)$ can be identified with the Besov space of dominating mixed smoothness $S^{-s,-r}_{\infty ,\infty }B(\mathbb{R}^2)$. See 32, Definition A.4 and 31. Moreover, Theorem 7 yields a characterization of continuous operators between certain Besov spaces:

(i)

$A:\dot{B}_{1,1}^{s}(\mathbb{R})\rightarrow \dot{B}_{p,p}^{r}(\mathbb{R})$ bounded $\Leftrightarrow$ $K\in \mathcal{C}o_{\pi _{\mathrm{aff}}\otimes \pi _{\mathrm{aff}}}\mathcal{L}^{p,\infty }_{\nu _{-s+1/2}\otimes \nu _{r+1/2-1/p}}(G_{\mathrm{aff}}^2)$,

(ii)

$A:\dot{B}_{p,p}^{s}(\mathbb{R}) \rightarrow \dot{B}_{\infty ,\infty }^{r}(\mathbb{R})$ bounded $\Leftrightarrow$ $K\in \mathcal{C}o_{\pi _{\mathrm{aff}}\otimes \pi _{\mathrm{aff}}} L^{q,\infty }_{\nu _{-s-1/2+1/p}\otimes \nu _{r+1/2}}(G_{\mathrm{aff}}^2)$.

The case (i) for $p=1$ was already formulated in a discrete version by Meyer 28, Section 6.9, Proposition 6.

Theorem 12.

Let $\{\psi _{k,j}\}_{(k,j)\in \mathbb{Z}^2}$ be a wavelet basis with $\psi _{k,j}(t)=2^{j/2}\psi (2^{j}t-k)$, and assume that $\psi$ has compact support and satisfies sufficiently many moment conditions so that the assumption of Proposition 10 is satisfied. An operator $A:\dot{B}_{1,1}^0(\mathbb{R})\rightarrow \dot{B}^0_{1,1}(\mathbb{R})$ is bounded if and only if

$$\begin{equation*} \sup _{(k\prime ,j\prime )\in \mathbb{Z}^2}\sum _{(k,j)\in \mathbb{Z}^2}\big |\langle A\psi _{k\prime ,j\prime },\psi _{k,j}\rangle \big |2^{-j/2+j^\prime /2}\leqslant C. \end{equation*}$$

Proof.

Set $p=1$, $s=-1/2$, recall that $k_A=V_\Psi K$, and apply Corollary 11.

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5.3. The case of two distinct representations

For most applications it suffices to consider a single group $G$ and its product group $G\times G$. Our formulation with two different groups allows us to study operators acting between coorbit spaces associated with different group representations. Using the representations of the Weyl-Heisenberg group and the affine group of Sections 5.1 and 5.2, one can characterize the boundedness of operators between certain modulation spaces and Besov spaces by properties of their associated kernels. Theorem 7 now reads as follows:

(i)

$A:M^1_{m_s}(\mathbb{R}^d) \rightarrow \dot{B}^r_{p,p}(\mathbb{R}) \ \text{bdd.} \Leftrightarrow K \in \mathcal{C}o_{\pi _{\mathrm{aff}}\otimes \pi _{WH}}\mathcal{L}^{p,\infty }_{ \widetilde{m}_{-s}\otimes \nu _{r+1/2-1/p}}(G_{WH}\times G_{\mathrm{aff}})$,

(ii)

$A:M^p_{m_s}(\mathbb{R}^d)\rightarrow \dot{B}^r_{\infty ,\infty }(\mathbb{R}) \ \text{bdd.} \Leftrightarrow K \in \mathcal{C}o_{\pi _{\mathrm{aff}}\otimes \pi _{WH}}L^{q,\infty }_{m_{-s}\otimes \nu _{r+1/2}}(G_{WH}\times G_{\mathrm{aff}})$,

(iii)

$A:\dot{B}^r_{1,1}(\mathbb{R})\rightarrow M_{m_s}^p(\mathbb{R}^d) \ \text{bdd.}\Leftrightarrow K \in \mathcal{C}o_{\pi _{WH}\otimes \pi _{\mathrm{aff}}}\mathcal{L}^{p,\infty }_{m_{-r+1/2}\otimes \nu _s}(G_{a}\times G_{WH})$,

(iv)

$A:\dot{B}^r_{p,p}(\mathbb{R}) \rightarrow M_{m_s}^\infty (\mathbb{R}^d) \ \text{bdd.} \Leftrightarrow K \in \mathcal{C}o_{\pi _{\mathrm{WH}}\otimes \pi _{\mathrm{aff}}}L^{q,\infty }_{\nu _{-r-1/2+1/p}\otimes m_s}(G_{a}\times G_{WH})$.

As a special case one obtains a characterization of the bounded operators $A$ from $\dot{B}^r_{1,1}(\mathbb{R})$ to $L^2(\mathbb{R}^d)$. Since $M^2(\mathbb{R}^d)=L^2(\mathbb{R}^d)$, they are completely characterized by the membership of their kernel in $\mathcal{C}o_{\pi _{\mathrm{aff}}\otimes \pi _{WH}}\mathcal{L}^{2,\infty }_{1\otimes m_{-r+1/2}}( G_{\mathrm{aff}}\times G_{WH})$.

Figure 1.