  Oscillatory integrals arise in many situations where it is important to
obtain decay estimates which are stable under certain perturbations of the
phase. Examining the structural problems underpinning these estimates leads one
to consider sublevel set estimates, which behave nicely under certain algebraic
operations such as composition with a polynomial. This motivates us to ask how
oscillatory integral estimates behave under such transformations of the phase,
and under some natural higher order convexity assumptions we obtain stable
estimates under composition with polynomial phases in one dimension, and in
higher dimensions in the setting of the higher dimensional van der Corput's
lemma of Carbery-Christ-Wright.
