Bilinear pseudodifferential operators with symbol in $$BS_{1,1}^m$$ on Triebel–Lizorkin spaces with critical Sobolev index

Abstract

AbstractIn 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.