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

TL;DR

This paper establishes new estimates for bilinear pseudodifferential operators with symbols in $BS_{1,1}^m$ acting on Triebel-Lizorkin spaces at the critical Sobolev index, refining classical embeddings and analyzing function products.

Contribution

It introduces refined inequalities for bilinear operators on Triebel-Lizorkin spaces at critical indices, extending classical Sobolev embeddings with new subspace considerations.

Findings

Derived new bounds for bilinear pseudodifferential operators in Triebel-Lizorkin spaces.

Refined Sobolev embedding replacing bmo with a larger subspace.

Analyzed function products in spaces where they are not algebraically closed.

Abstract

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.