  We consider bilinear pseudo-differential operators with symbols in the
bilinear H\"ormander class, $BS_{\rho, \rho}^m$, $m \in \mathbb{R}$, $0 \leq
\rho < 1$. The aim of this paper is to discuss low regularity conditions for
symbols to assure the boundedness from $L^2 \times L^2$ to $h^1$ and from $L^2
\times bmo$ to $L^2$.
