  We extend the known result that the bilinear pseudo-differential operators
with symbols in the bilinear H\"ormander class $BS^{-n/2}_{0,0}(\mathbb{R}^n)$
are bounded from $L^2 \times L^2$ to $h^1$. We show that those operators are
also bounded from $L^2 \times L^2$ to $L^r $ for every $1< r \le 2$. Moreover
we give similar results for symbol classes wider than
$BS^{-n/2}_{0,0}(\mathbb{R}^n)$. We also give results for symbols of limited
smoothness.
