Sharp estimates for oscillatory integral operators via polynomial partitioning

TL;DR

This paper establishes precise $L^p$ bounds for H"ormander-type oscillatory integral operators across all dimensions, using polynomial partitioning techniques inspired by Fourier extension operator analysis.

Contribution

It generalizes polynomial partitioning methods to obtain sharp $L^p$ estimates for oscillatory integral operators with positive-definite phases.

Findings

Established sharp $L^p$ estimates in all dimensions

Extended polynomial partitioning techniques to oscillatory integrals

Provided a unified approach for H"ormander-type operators

Abstract

The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.