Bilinear pseudo-differential operators with exotic symbols

The boundedness from $L^p \times L^q$ to $L^r$, $1<p,q \le \infty$, $0<1/p+1/q=1/r \le 1$, of bilinear pseudo-differential operators with symbols in the bilinear H\"ormander class $BS^m_{\rho,\rho}$, $0 \le \rho <1$, is proved for the critical order $m$. Related results for the cases $p=1$, $q=1$ or $r=\infty$ are also obtained.