Sharp $L^p$ decay estimates for degenerate and singular oscillatory integral operators: Homogeneous polynomial phases

Abstract

In this paper, we consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calder\'{o}n-Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $\mu$. For any given homogeneous polynomial phases, except monomial phases, of degreee $n$, we give the range of $p$ for which the sharp decay rate $-\frac{1-\mu}{n}$ on $L^2$ spaces can be preserved on $L^p$ spaces.