Bilinear pseudodifferential operators with symbol in $B{S}_{1,1}^m$ on Triebel-Lizorkin spaces with critical Sobolev index

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