Local smoothing estimates for bilinear Fourier integral operators

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 ${\mathbb{R}}_x^2\times {\mathbb{R}}_t$. Also, a 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 $d$.