Sharp L^p decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

Abstract

In this paper, we obtain the L^p decay of oscillatory integral operators T_λ 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 L^p decay of T_λ 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.