A bilinear pseudodifferential calculus

TL;DR

This paper develops a new bilinear pseudodifferential calculus with symbolic classes extending Coifman-Meyer, enabling analysis of operators related to the bilinear Hilbert transform and their behavior on Sobolev spaces.

Contribution

It introduces new symbolic classes for bilinear pseudodifferential operators that improve understanding of their properties and interactions, especially regarding the bilinear Hilbert transform.

Findings

Defined new symbolic classes extending Coifman-Meyer

Analyzed operators' actions on Sobolev spaces

Established properties under transposition and composition

Abstract

In this paper, we are interested in the construction of a bilinear pseudodifferential calculus. We define some symbolic classes which contains those of Coifman-Meyer. These new classes allow us to consider operators closely related to the bilinear Hilbert transform. We give a description of the action of our bilinear operators on Sobolev spaces. These classes also have a ``nice'' behavior through the transposition and the composition operations that we will present.