H\"ormander oscillatory integral operators: a revisit

TL;DR

This paper offers new, unified proofs for the sharp $L^p$ estimates and decoupling theorem of H"ormander oscillatory integral operators, enhancing understanding through bilinear and approximation methods.

Contribution

It introduces a unified bilinear proof for the $L^p$ estimates and reestablishes the decoupling theorem using the Pramanik-Seeger approximation approach.

Findings

Unified proof for sharp $L^p$ estimates in all dimensions.

Reproved decoupling theorem via Pramanik-Seeger method.

Applicable to perturbation terms in phase functions.

Abstract

In this paper, we present new proofs for both the sharp $L^p$ estimate and the decoupling theorem for the H\"ormander oscillatory integral operator. The sharp $L^p$ estimate was previously obtained by Stein\;\cite{stein1} and Bourgain-Guth \cite{BG} via the $TT^\ast$ and multilinear methods, respectively. We provide a unified proof based on the bilinear method for both odd and even dimensions. The strategy is inspired by Barron's work \cite{Bar} on the restriction problem. The decoupling theorem for the H\"ormander oscillatory integral operator can be obtained by the approach in \cite{BHS}, where the key observation can be roughly formulated as follows: in a physical space of sufficiently small scale, the variable setting can be essentially viewed as translation-invariant. In contrast, we reprove the decoupling theorem for the H\"ormander oscillatory integral operator through the Pramanik-Seeger approximation approach \cite{PS}. Both proofs rely on a scale-dependent induction argument, which can be used to deal with perturbation terms in the phase function.