  In this paper we obtain new estimates for bilinear pseudodifferential
operators with symbol in the class $BS_{1,1}^m$, when both arguments belong to
Triebel-Lizorkin spaces of the type $F_{p,q}^{n/p}(\mathbb{R}^n)$. The
inequalities are obtained as a consequence of a refinement of the classical
Sobolev embedding
$F^{n/p}_{p,q}(\mathbb{R}^n)\hookrightarrow\mathrm{bmo}(\mathbb{R}^n)$, where
we replace $\mathrm{bmo}(\mathbb{R}^n)$ by an appropriate subspace which
contains $L^\infty(\mathbb{R}^n)$. As an application, we study the product of
functions on $F_{p,q}^{n/p}(\mathbb{R}^n)$ when $1<p<\infty$, where those
spaces fail to be multiplicative algebras.
