Sharp $L^{p}$-Boundedness of Oscillatory Integral Operators with Polynomial Phases

TL;DR

This paper establishes sharp $L^{p}$ decay bounds for oscillatory integral operators with polynomial phases, including specific forms like $S(x^{m_1}, y^{m_2})$, advancing understanding of their boundedness properties.

Contribution

It proves sharp $L^{p}$ endpoint decay estimates for oscillatory integral operators with homogeneous polynomial phases, including special cases with monomial structures.

Findings

Established $L^{p}$ endpoint decay estimates for polynomial phase operators

Derived sharp $L^{p}$ decay bounds for phases of the form $S(x^{m_1}, y^{m_2})$

Extended decay estimates to a broad class of polynomial phases

Abstract

In this paper, we shall prove the $L^{p}$ endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases $S$ in $\mathbb{R} \times \mathbb{R}$. As a consequence, sharp $L^{p}$ decay estimates are also obtained when polynomial phases have the form $S(x^{m_{1}},y^{m_{2}})$ with $m_1$ and $m_2$ being positive integers.