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

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In this paper we establish some endpoint estimates for bilinear pseudodifferential operators with symbol in the class BS m 1,1 , involving the space of functions with local bounded mean oscillation bmo(R n ). As a consequence we also obtain an endpoint estimate of Kato-Ponce type.