Oscillatory integral operators and variable Schr\"odinger propagators: beyond the universal estimates

TL;DR

This paper extends the understanding of oscillatory integral operators and variable Schrödinger propagators by establishing new $L^p$ bounds under weaker phase conditions, using advanced geometric and analytic techniques.

Contribution

It introduces novel sublevel set estimates for real analytic functions and demonstrates their application to improve bounds for oscillatory operators beyond classical limits.

Findings

Established $L^p$ bounds beyond Stein's universal range.

Developed new sublevel set estimates for real analytic functions.

Highlighted differences between H"ormander-type and variable coefficient Schrödinger operators.

Abstract

We consider a class of H\"ormander-type oscillatory integral operators in $\mathbb{R}^n$ for $n \geq 3$ odd with real analytic phase. We derive weak conditions on the phase which ensure $L^p$ bounds beyond the universal $p \geq 2 \cdot \frac{n+1}{n-1}$ range guaranteed by Stein's oscillatory integral theorem. This expands and elucidates pioneering work of Bourgain from the early 1990s. We also consider a closely related class of variable coefficient Schr\"odinger propagator-type operators, and show that the corresponding theory differs significantly from that of the H\"ormander-type operators. The main ingredient in the proof is a curved Kakeya/Nikodym maximal function estimate. This is established by combining the polynomial method with certain uniform sublevel set estimates for real analytic functions. The sublevel set estimates are the main novelty in the argument and can be interpreted as a form of quantification of linear independence in the $C^{\omega}$ category.