Paley–Wiener Theorems for Hyperbolic Spaces

We prove a topological Paley–Wiener theorem for the Fourier transform defined on the real hyperbolic spaces SOo(p, q)/SOo(p−1, q), for p, q∈2N, without restriction to K-types. We also obtain Paley–Wiener type theorems for Lσ-Schwartz functions (0<σ⩽2) for fixed K-types.