  In this paper we extend classical Titchmarsh theorems on the Fourier
transform of H\"older-Lipschitz functions to the setting of compact homogeneous
manifolds. As an application, we derive a Fourier multiplier theorem for
$L^2$-H\"older-Lipschitz spaces on compact Lie groups. We also derive
conditions and a characterisation for Dini-Lipschitz classes on compact
homogeneous manifolds in terms of the behaviour of their Fourier coefficients.
