Uniform estimates for cubic oscillatory integrals

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. The purpose of this paper is to establish decay estimates for the scalar oscillatory integral I(λ, ξ) := ∫