Smoothing of commutators for a H\"ormander class of bilinear pseudodifferential operators

TL;DR

This paper demonstrates that commutators of certain bilinear pseudodifferential operators with Lipschitz functions are bilinear Calderón-Zygmund operators, establishing a link to compactness in the bilinear context.

Contribution

It introduces a smoothing technique for commutators of bilinear pseudodifferential operators within the Hörmander class, revealing their Calderón-Zygmund structure.

Findings

Commutators are bilinear Calderón-Zygmund operators.

Connection established between commutator compactness and bilinear setting.

Provides new insights into the structure of bilinear pseudodifferential operators.

Abstract

Commutators of bilinear pseudodifferential operators with symbols in the H\"ormander class BS_{1, 0}^1 and multiplication by Lipschitz functions are shown to be bilinear Calder\'on-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.