Scalar Oscillatory Integrals are an important basic object in Harmonic Analysis & its applications. A common goal is to give estimates on the rate at which these integrals decay to zero as the frequency of the oscillation is increased. This is understood to be related to estimates for the size of non-oscillatory "sublevel sets", on both an intuitive level and in the sense that oscillatory integral estimates imply sublevel set estimates. We explore some situations in which the reverse may be established, necessarily utilizing some additional geometric structure. Our results also feature stability within a class of phase functions, which is important in applications such as Fourier transform estimates where uniformity of estimates under certain perturbations of the phase is required.