A method for bounding oscillatory integrals in terms of non-oscillatory integrals

We describe an elementary method for bounding a one-dimensional oscillatory integral in terms of an associated non-oscillatory integral. The bounds obtained are efficient in an appropriate sense and behave well under perturbations of the phase. As a consequence, for an $n$-dimensional oscillatory integral with a critical point at the origin, we may apply the one-dimensional estimates in the radial direction and then integrate the result, thereby obtaining natural bounds for the $n$-dimensional oscillatory integral in terms of the measures of the sublevel sets associated with the phase. To illustrate, we provide several classes of examples, including situations where the phase function has a critical point at which it vanishes to infinite order.