Hörmander oscillatory integral operators: a revisit

TL;DR

This work develops a unified bilinear framework to obtain sharp $L^p$ bounds for Hörmander oscillatory integral operators in all dimensions and provides a PS-based approach to the associated decoupling theorem. By reducing a general phase to an asymptotically flat model via parabolic scaling and a scale-dependent induction on $R$, the authors derive both a linear sharp $L^p$ estimate through a broad-narrow analysis and a decoupling inequality by transferring the problem to translation-invariant flat models at small scales. The key innovations include a rigorous reduction to asymptotically flat phases, a parabolic rescaling mechanism linking scales, and a precise combination of bilinear restriction bounds with flat decoupling to control perturbations of the phase. These results illuminate the stability of decoupling and $L^p$ estimates under small phase perturbations and advance the understanding of oscillatory integral operators in the spirit of restriction theory.

Abstract

In this paper, we present new proofs for both the sharp $L^p$ estimate and the decoupling theorem for the Hörmander 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 Hörmander 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 Hörmander 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.

Theorems & Definitions (26)

• Theorem 1.1: BG,stein1
• Theorem 1.2: Lee
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 3.1
• proof
• Lemma 3.2