Oscillatory integral operators and variable Schrödinger propagators: beyond the universal estimates

TL;DR

The paper advances the understanding of oscillatory integral operators with real-analytic phases in odd dimensions by identifying weak curvature-type hypotheses—Kakeya non-compression and Nikodym non-compression—for translation-invariant phases. Under these assumptions, the authors prove bounds beyond the universal Stein–Tomas range for Hörmander-type operators and related maximal functions, and compare with variable-coefficient Schrödinger propagators, which exhibit distinct behavior. The core methodology blends geometric Kakeya/Nikodym maximal estimates with polynomial partitioning and novel uniform sublevel-set estimates for real-analytic functions, yielding non-concentration results and improved $L^p\to L^q$ bounds. The work connects to classical Bourgain-type constructions, Wisewell’s observations, and recent analyses of Bourgain-type conditions, and includes applications to Pierce–Yung operators and Carleson-type Radon problems. Overall, the paper provides a robust framework to surpass universal estimates in a controlled analytic setting, with implications for Fourier restriction, local smoothing, and related harmonic analysis topics.

Abstract

We consider a class of Hörmander-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 Schrödinger propagator-type operators, and show that the corresponding theory differs significantly from that of the Hörmander-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 $C^ω$ category.

Theorems & Definitions (96)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Universal oscillatory estimate Stein1986
• Proposition 1.4: Universal geometric estimate
• Theorem 1.5: Bourgain Bourgain1991
• Theorem 1.6: Wisewell Wisewell2005
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4