--- title: Coorbit Spaces on LCA Phase Spaces url: https://www.emergentmind.com/topics/coorbit-spaces-over-locally-compact-abelian-phase-spaces type: topic --- # Coorbit Spaces on LCA Phase Spaces 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 [2401.12387][2308.02078][2511.18175][2101.05232][1007.3528][2210.04844][2203.07959]. ## 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 $\sigma((x,\xi),(y,\eta)) = \langle x,\eta\rangle\,\langle y,\xi\rangle^{-1}$ [2308.02078][2511.18175]. 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 $K_g$ [2308.02078][2511.18175][2101.05232]. 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 $\|f\|_{\Co(Y)} = \|V_g f\|_Y$ [2401.12387][1007.3528]. 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 $M^p_{r,s}$ [2401.12387][2101.05232]. ## 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 [2401.12387][2101.05232]. 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 [2401.12387][2101.05232]. Romero’s phase-space covering approach [1007.3528] 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 [2511.18175][2308.02078]. 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 [2511.18175]. 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 [2308.02078]. ## 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 [2210.04844]. Generalizations, including quasi-Banach settings ($0