  We show that the Fourier transform of Patterson-Sullivan measures associated
to convex cocompact groups of isometries of real hyperbolic space decays
polynomially quickly at infinity. The proof is based on the $L^2$-flattening
theorem obtained in prior work of the author, combined with a method based on
dynamical self-similarity for ruling out the sparse set of potential
frequencies where the Fourier transform can be large.
