On the H\"ormander classes of bilinear pseudodifferential operators

TL;DR

This paper extends the theory of bilinear pseudodifferential operators by characterizing their H"ormander classes, analyzing transposition properties, and establishing boundedness results, thereby broadening the scope of symbolic calculus in harmonic analysis.

Contribution

It introduces a comprehensive analysis of bilinear H"ormander classes, including transposition closure and asymptotic expansion characterizations, expanding prior limited results.

Findings

Classes are closed under transposition in certain cases.

Symbolic calculus can recover boundedness on Lebesgue space products.

Boundedness properties for classes with Leibniz-type estimates are established.

Abstract

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the H\"ormander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calder\'on-Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz' rule are presented as well.