Bilinear pseudodifferential operators with symbol in BSm1,1 on Triebel–Lizorkin spaces with critical Sobolev index

Abstract

In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class BSm1,1 , when both arguments belong to Triebel-Lizorkin spaces of the type Fn/pp,q(Rn). The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding Fn/pp,q(Rn)↪bmo(Rn), where we replace bmo(Rn) by an appropriate subspace which contains L∞(Rn). As an application, we study the product of functions on Fn/pp,q(Rn) when 1<p<∞, where those spaces fail to be multiplicative algebras.