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 a natural bound for the n-dimensional oscillatory integral in terms of an analogous associated non-oscillatory integral. 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.