Sharp $L^p$ decay estimates for degenerate and singular oscillatory integral operators

TL;DR

This paper establishes sharp decay estimates in L^p spaces for a class of degenerate and singular oscillatory integral operators with polynomial phases, extending previous L^2 results under certain kernel conditions.

Contribution

It generalizes L^2 decay estimates to L^p spaces for degenerate and singular oscillatory integrals with polynomial phases, under specific kernel conditions.

Findings

Sharp decay estimates on L^p spaces for the operators.

Extension of previous L^2 results to a broader L^p range.

Discussion of cases without the additional kernel condition.

Abstract

We consider the following model of degenerate and singular oscillatory integral operators: \begin{equation*} Tf(x)=\int_{\mathbb{R}} e^{i\lambda S(x,y)}K(x,y)\psi(x,y)f(y)dy, \end{equation*} where the phase functions are homogeneous polynomials of degree $n$ and the singular kernel $K(x,y)$ satisfies suitable conditions related to a real parameter $\mu$. We show that the sharp decay estimates on $L^2$ spaces, obtained in \cite{liu1999model}, can be preserved on more general $L^p$ spaces with an additional condition imposed on the singular kernel. In fact, we obtain that \begin{equation*} \|Tf\|_{L^p}\leq C_{E,S,\psi,\mu,n,p}\lambda^{-\frac{1-\mu}{n}}\|f\|_{L^p},\ \ \frac{n-2\mu}{n-1-\mu}\leq p \leq\frac{n-2\mu}{1-\mu}. \end{equation*} The case without the additional condition is also discussed.