  In this short note, we consider bilinear pseudo-differential operators with
symbols belonging to the Sj\"ostrand class. We show that those operators are
bounded from the product of the $L^2$-based Sobolev spaces $H^{s_1} \times
H^{s_2}$ to $L^r$ for $s_1,s_2 > 0$, $s_1+s_2=n/2$, and $1 \leq r \leq 2$.
