Sharp Lp decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

In this paper, we obtain the Lp decay of oscillatory integral operators Tλ with certain homogeneous polynomial phase functions of degree d in (n + n)-dimensions; we require that d > 2n. If d/(d − n) < p < d/n, the decay is sharp and the decay rate is related to the Newton distance. For p = d/n or d/(d−n), we obtain the almost sharp decay, where “almost" means that the decay contains a log(λ) term. For otherwise, the Lp decay of Tλ is also obtained but not sharp. Finally, we provide a counterexample to show that d/(d−n) ⩽ p ⩽ d/n is not necessary to guarantee the sharp decay.