Paley-Wiener Theorems For Hyperbolic Spaces

. We prove a topological Paley-Wiener theorem for the Fourier transform defined on the real hyperbolic spaces SO o (p; q)=SO o (p \Gamma 1; q), for p; q 2 2N, without restriction to K-types. We also obtain Paley-Wiener type theorems for L oe -Schwartz functions (0 ! oe 2) for fixed K-types. 1. Introduction Consider the real hyperbolic spaces X p;q := SO o (p; q)=SO o (p \Gamma 1; q) for p 1 and q 2. The space X p;q is a semisimple Riemannian symmetric space of the non-compact type when p = 1 and a semisimple symmetric space of the non-Riemannian type when p ? 1. There is a natural embedding of X p;q in R p+q as the hypersurface (with x 1 ? 0 if p = 1): X p;q = fx 2 R p+q jx 2 1 + \Delta \Delta \Delta + x 2 p \Gamma x 2 p+1 \Gamma \Delta \Delta \Delta \Gamma x 2 p+q = 1g: Let Y p;q := S p\Gamma1 \Theta S q\Gamma1 and let ae = 1 2 (p+q \Gamma 2). The Fourier transform of any function f 2 C 1 c (X p;q ) is defined as: Ff(ffl; ; y) := Z X p;q jhx; ..