Local smoothing estimates for bilinear Fourier integral operators

Duván Cardona

Vol. 344 (2026), No. 2, 241–272

DOI: 10.2140/pjm.2026.344.241

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 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 bilinear Fourier integral operators in dimension $d=2$, that is, on ${ℝ}_x^2\times {ℝ}_t$. Also, partial progress is presented for the high-dimensional case $d\ge 3$. In particular, our method allows us to deduce that the bilinear local smoothing conjecture holds for all odd dimensions $d$.

Keywords

local smoothing conjecture, bilinear Fourier integral operators, bilinear smoothing conjecture, cinematic curvature condition, wave equation

Mathematical Subject Classification

Primary: 35S30, 42B20

Secondary: 42B35, 42B37

Milestones

Received: 2 March 2026

Revised: 3 June 2026

Accepted: 14 June 2026

Published: 8 September 2026