Bilinear pseudo-differential operators with exotic class symbols of limited smoothness

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 $L^2$.