Local smoothing estimates for bilinear Fourier integral operators

TL;DR

This paper introduces a bilinear local smoothing conjecture for Fourier integral operators, shows its relation to the linear case, and proves it in dimension two with partial results in higher dimensions, especially odd dimensions.

Contribution

The paper formulates a bilinear local smoothing conjecture, proves it in dimension two, and establishes partial results for higher dimensions, extending the linear case insights.

Findings

Proves local smoothing estimates for Fourier integral operators in 2D.

Establishes the bilinear local smoothing conjecture in all odd dimensions.

Shows the linear local smoothing conjecture implies the bilinear version.

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 $d$.