Coorbit Spaces on LCA Phase Spaces

• Coorbit spaces over locally compact Abelian phase spaces are a unified framework defined via square-integrable group representations and the voice transform, leading to Banach and quasi-Banach function spaces.
• The framework employs robust atomic decompositions, Banach frame constructions, and localization techniques to achieve precise norm equivalence and discrete characterizations in both time-frequency and time-scale analyses.
• It enables comprehensive operator analysis by addressing spectral invariance, compactness criteria, and quasi-Banach generalizations with significant implications in signal processing and harmonic analysis.

Coorbit spaces over locally compact Abelian (LCA) phase spaces provide a unified framework for constructing Banach and quasi-Banach function spaces intrinsically linked to group representations, encompassing classical function spaces such as modulation spaces and Besov spaces. The theory leverages square-integrable (possibly projective) unitary representations of an LCA group or its phase space, employing the associated voice (or wavelet) transform to define distributions whose transform coefficients reside in prescribed Banach function spaces. This approach supports robust atomic decompositions, Banach frame constructions, operator characterizations, and localization techniques that respect the geometry and analysis of the underlying phase-space group (Zimmermann, 22 Jan 2024, Fulsche et al., 2023, Fulsche et al., 22 Nov 2025, Berge, 2021, Romero, 2010, Dörfler et al., 2022, Velthoven et al., 2022).

1. Algebraic and Analytical Setup: Phase Space, Representations, and Transforms

The foundational object is a locally compact Abelian group $G$ and its Pontryagin dual $\widehat G$, producing the phase space $\Xi = G \times \widehat G$ with Haar measure. The standard Heisenberg multiplier $m$ and associated 2-cocycle are defined as $m((x,\xi),(y,\eta)) = \overline{\xi(y)}$, inducing the alternating bicharacter $$\sigma((x,\xi),(y,\eta)) = \langle x,\eta\rangle\,\langle y,\xi\rangle^{-1}$$ (Fulsche et al., 2023, Fulsche et al., 22 Nov 2025).

Given a strongly continuous, square-integrable projective unitary representation $\pi:\Xi \to \mathcal U(H)$, the voice (analysis) transform for $f,g\in H$ is

$$V_g f(z) = \langle f, \pi(z) g \rangle_H, \quad z \in \Xi.$$

For admissible $g$ (i.e., $\int_\Xi |\langle g, \pi(z) g \rangle|\,dz<\infty$, often $g \in S_0(G)$ or $\mathcal S(G)$), the transform $V_g$ is an isometry into $L^2(\Xi)$ and admits a reproducing formula

$f = \int_\Xi V_g f(z)\, \pi(z) g\, dz$

(weakly), establishing a closed reproducing kernel Hilbert subspace $K_g$ (Fulsche et al., 2023, Fulsche et al., 22 Nov 2025, Berge, 2021).

Weighted Lebesgue spaces $L^p_m(\Xi)$ are formulated with submultiplicative, moderate weights $m:\Xi \to (0,\infty)$; coorbit spaces are defined in terms of when $V_g f$ belongs to a solid function space $Y \subset L^1_{\text{loc}}(\Xi)$.

2. Construction and Fundamental Properties of Coorbit Spaces

A coorbit space over an LCA phase space is given by

$\Co(Y) = \{f \in R_w: V_g f \in Y\},$

where $R_w$ is the anti-dual of the Banach test-vector space $H^1_w = \{f \in H: V_g f \in L^1_w(\Xi)\}$, and the norm is $\|f\|_{\Co(Y)} = \|V_g f\|_Y$ (Zimmermann, 22 Jan 2024, Romero, 2010). For $Y = L^p_m(\Xi)$, this specializes to weighted Lebesgue coorbit spaces.

Key analytical features are:

• Banach space structure: $\Co(Y)$ is complete for admissible $g$ and suitable $Y$ (Banach or quasi-Banach).
• π-invariance: $\|\pi(z)f\|_{\Co(Y)} \leq w(z)\|f\|_{\Co(Y)}$ for control weights $w$.
• Correspondence principle: $V_g: \Co(Y) \to \{F \in Y: F = F * V_g g\}$ is an isometric isomorphism.
• Duality: $\Co(L^p_m)$ is anti-dual to $\Co(L^q_{1/m})$ under $\langle f,h \rangle = \int V_g f \cdot V_g h$ ($1/p + 1/q = 1$).

For modulation spaces ($G = \mathbb R^d$, $Y = L^p_{v_{r,s}}$ on $\Xi = \mathbb R^d \times \mathbb R^d$), the classical coorbit setup fully recovers $M^p_{r,s}$ (Zimmermann, 22 Jan 2024, Berge, 2021).

3. Discretization, Frames, and Atomic Decompositions

Central to the utility of coorbit spaces is the existence of Banach frames and atomic decompositions compatible with the group structure. Given a kernel $G \in W(L^1_w) \cap L^q_{1/m}$ with self-convolution $G = G * G$, the function space $M^p_m(G) = \{F \in L^p_m(\Xi): F = F*G\}$ supports discrete characterizations via sampling over $U$-well-spread families $X = \{x_i\}$:

• Frame conditions: $$a\|F\|_{L^p_m} \leq (\sum |F(x_i)|^p m(x_i)^p)^{1/p} \leq b\|F\|_{L^p_m}$$.
• For $\{L_{x_i}G\}$ on $M^p_m(\Xi)$, equivalently $\{\pi(x_i)g\}$ on $\Co(L^p_m)$, this yields Banach frames (Zimmermann, 22 Jan 2024, Berge, 2021).

In the Euclidean phase-space case, Gabor frames on $M^p_{r,s}(\mathbb R^d)$ are constructed for lattices $$\Lambda = \alpha \mathbb Z^d \times \beta \mathbb Z^d$$: $$\{M_{\beta l} T_{\alpha k} g\}_{k,l \in \mathbb Z^d}$$ with frame inequalities and unconditional reconstruction (Zimmermann, 22 Jan 2024, Berge, 2021).

Romero’s phase-space covering approach (Romero, 2010) further generalizes discretization by partitions of unity $\{\Phi_i\}$ subordinate to arbitrary covers, with norm equivalence: $\|f\|_{\Co(L^p_w)} \approx \left( \sum_i \|M_i f\|_B^p \right)^{1/p}$ for local multipliers $M_i f$ defined via the voice transform and envelope control from Wiener amalgam spaces. This subsumes both time-frequency and time-scale settings, establishing norm equivalence for irregular, non-lattice decompositions.

4. Operator Theory: Compactness, Fredholmness, and Quantum Harmonic Analysis

Operator-theoretic aspects on coorbit spaces over LCA phase spaces are established using band-dominated operators and quantum harmonic analytic convolution (Fulsche et al., 22 Nov 2025, Fulsche et al., 2023). Limit operator techniques define the compactness and Fredholm property via boundary operators $\alpha_x(A)$ (translations in the maximal ideal space $\sigma \Xi$):

• An operator $A$ is compact iff $\alpha_x(A) = 0$ for all $x$ on the boundary.
• $A$ is Fredholm iff each limit operator $\alpha_x(A)$ is invertible on $\Co_p(U)$, with invertibility implying uniform boundedness of inverses (Fulsche et al., 22 Nov 2025).

The Wiener-type theorem for coorbit spaces asserts spectral invariance for operators induced by symbols in the Wiener algebra $W(L^\infty, \ell^1)(\Xi)$: If $a \in W(L^\infty, \ell^1)(\Xi)$, $\inf_{z \in \Xi} |a(z)| > 0$, then the corresponding pseudodifferential operator is invertible on $\Co(Y)$, and its inverse again lies in the Wiener algebra (Fulsche et al., 2023).

5. Flexibility: Window Independence, Quasi-Banach Generalizations, and Phase-Space Covers

Coorbit spaces exhibit robustness with respect to the choice of analyzing window $g$, provided admissibility conditions hold (decay and integrability, typically $g \in M^1_v(G)$ or $S_0(G)$). Classification theorems in the operator-valued setting guarantee norm equivalence across admissible windows via twisted convolution identities and Young’s inequality (Dörfler et al., 2022).

Generalizations, including quasi-Banach settings ($0$Y$, permitting construction of molecular frames and Riesz sequences under weaker group and integrability requirements (Velthoven et al., 2022). The atomic decomposition remains valid, and the existence of dual molecule families yields unconditional expansions and interpolation solutions.

Phase-space covers—arbitrary partitions of unity on $\Xi$ with localized support—facilitate discrete norm characterizations and localization operator representations for both time-frequency and time-scale analyses, extending to highly irregular grids (Romero, 2010). This enables flexible adaptation to domain-specific requirements such as non-uniform sampling or randomized coverings.

6. Special Cases and Applications

Coorbit spaces over LCA phase spaces encapsulate a wide variety of classical function spaces:

• Modulation spaces $M^{p,q}_m(G)$: Realized as coorbits for $L^{p,q}_m(G \times \widehat G)$ under the Schrödinger representation (Fulsche et al., 2023, Berge, 2021).
• Besov and Triebel–Lizorkin spaces: Obtained via the affine group and wavelet transforms (Zimmermann, 22 Jan 2024, Romero, 2010).
• Sequence spaces and digital analogues: For discrete $G$, coorbit spaces become $\ell^p(G)$, and band-dominated operator theory recovers classical Fredholm criteria on $\ell^p$ (Fulsche et al., 22 Nov 2025).
• Operator coorbit spaces: Spaces of operators over Hilbert–Schmidt class $HS(L^2(G))$ are characterized analogously, with vector-valued reproducing kernel structures and atomic decompositions (Dörfler et al., 2022).
• Toeplitz and Fock/Bergman spaces: Embedding of Toeplitz algebras into the coorbit operator framework using phase spaces like $\mathbb C^n$, unifying Fredholm criteria (Fulsche et al., 22 Nov 2025).

7. Outlook and Research Directions

Coorbit theory over Abelian phase spaces integrates group-theoretic, functional-analytic, and operator-theoretic methods into a flexible, geometrically compatible framework supporting discretization, localization, operator analysis, and robust invariance to window choice and covering structure. This unification has extended spectral invariance, frame theory, and functional space construction far beyond classical metric and countability constraints. Open directions include further generalization to quasi-Banach spaces, adaptation to non-unimodular and non-separable groups, and exploration of irregular phase-space covers for applications in signal analysis, time-frequency localization, and non-commutative harmonic analysis (Velthoven et al., 2022, Romero, 2010, Fulsche et al., 22 Nov 2025).