  We analyze the Cauchy problem for the Klein-Gordon equation on the type IIB
supergravity backgrounds $AdS^5 \times Y^{p,q}$, where $Y^{p,q}$ is any of the
Sasaki-Einstein 5-manifolds recently discovered by Gaunlett, Martelli, Sparks
and Waldram (Adv. Theor. Math. Phys. 8 (2004) 711--734). Although these spaces
are not globally hyperbolic, we prove that there exists a unique admissible
propagator and derive an integral representation thereof using
spectral-theoretic techniques.
