Local smoothing estimates for bilinear Fourier integral operators

TL;DR

This paper proposes and analyzes a bilinear local smoothing conjecture for Fourier integral operators (FIOs), deriving the bilinear smoothing from the linear smoothing conjecture of Sogge. The authors show that the linear LLSP implies the bilinear version, and leverage this to obtain local smoothing for FIOs in dimension $d=2$ (full range) and partial progress for $d\ge 3$, including bilinear smoothing in all odd dimensions. The approach decomposes bilinear FIOs into low-frequency parts that behave like compositions of linear FIOs and high-frequency parts that are handled as paraproducts via a Coifman–Meyer type framework, aided by Bourgain’s maximal-function estimates. This work provides a transfer principle from linear to bilinear smoothing, introduces a Bourgain-based square-maximal technique in this setting, and has implications for related dispersive and restriction problems through the smoothing conjecture.

Abstract

We formulate a local smoothing conjecture for bilinear Fourier integral operators in every dimension $d \ge 2,$ derived from the celebrated linear case due to Sogge, which we refer to as the \emph{bilinear smoothing conjecture}. We show that the linear local smoothing conjecture implies this bilinear version. As a consequence of our approach and due to the recent progress on the subject, we establish local smoothing estimates for Fourier integral operators in dimension $d=2,$ that is, on $\mathbb{R}^2_x \times \mathbb{R}_t$. Also, a partial progress is presented for the high-dimensional case $d\geq 3.$ In particular, our method allows us to deduce that the bilinear local smoothing conjecture holds for all odd dimensions $d$.

Theorems & Definitions (27)

• Theorem 1.1
• Definition 1.2: Cinematic curvature condition
• Conjecture 1.3: Linear smoothing conjecture for FIOs
• Remark 1.4
• Conjecture 1.5: Bilinear smoothing conjecture for FIOs
• Remark 1.6
• Theorem 1.7
• Corollary 1.8
• Definition 2.1
• Remark 2.2