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

TL;DR

This paper establishes sharp decay estimates in L^p spaces for a class of degenerate and singular oscillatory integral operators with homogeneous polynomial phases, extending known results beyond monomial phases.

Contribution

It provides the range of p for which the sharp decay rate holds for these operators with general homogeneous polynomial phases, excluding monomials.

Findings

Sharp L^p decay estimates for degenerate oscillatory integrals

Decay rate of -(1-μ)/n preserved in L^p spaces for polynomial phases

Extension beyond monomial phases in oscillatory integral estimates

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.