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

We obtain the $L^p$ decay of oscillatory integral operators $T_\lambda$ with certain homogeneous polynomial phase of degree $d$ in $(n+n)$-dimensions. In this paper 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. In the case of $p=d/n$ or $d/(d-n)$, we also obtain the almost sharp decay, here "almost" means the decay contains a $\log(\lambda)$ term. For otherwise, the $L^p$ decay of $T_\lambda$ is also obtained but not sharp. A counterexample also arises in this paper to show that $d/(d-n)\leq p\leq d/n$ is not necessary to guarantee the sharp decay.