  This paper establishes the optimal decay rate for scalar oscillatory
integrals in $n$ variables which satisfy a nondegeneracy condition on the third
derivatives. The estimates proved are stable under small linear perturbations,
as encountered when computing the Fourier transform of surface-carried
measures. The main idea of the proof is to construct a nonisotropic family of
balls which locally capture the scales and directions in which cancellation
occurs.
