  We prove that the geodesic flow on a geometrically finite locally symmetric
space of negative curvature is exponentially mixing with respect to the
Bowen-Margulis-Sullivan measure. The approach is based on constructing a
suitable anisotropic Banach space on which the infinitesimal generator of the
flow admits an essential spectral gap. A key step in the proof involves
estimating certain oscillatory integrals against the Patterson-Sullivan
measure. For this purpose, we prove a general result of independent interest
asserting that measures on $\mathbb{R}^d$ that do not concentrate near proper
affine subspaces enjoy polynomial Fourier decay outside of a sparse set of
frequencies. As an intermediate step, we show that the $L^q$-dimension
($1<q\leq \infty$) of iterated self-convolutions of such measures tend towards
that of the ambient space. Our analysis also yields that the Laplace transform
of the correlation function of smooth observables extends meromorphically to
the entire complex plane in the convex cocompact case and to a strip of
explicit size beyond the imaginary axis in the case the manifold admits cusps.
