Uniform estimates for cubic oscillatory integrals Philip Gressman 42B99oscillatory integralssymmetric spacesstationary phase 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. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3403 10.1512/iumj.2008.57.3403 en Indiana Univ. Math. J. 57 (2008) 3419 - 3442 state-of-the-art mathematics http://iumj.org/access/