Quasi pieces of the bilinear Hilbert transform incorporated into a paraproduct

TL;DR

This paper establishes the boundedness of a new class of tri-linear operators that incorporate quasi pieces of the bilinear Hilbert transform, extending understanding of their behavior in harmonic analysis.

Contribution

It introduces and proves boundedness for a novel class of tri-linear operators involving quasi pieces of the bilinear Hilbert transform, linked to curved versions of the transform.

Findings

Proves boundedness of the new tri-linear operators

Extends analysis to operators with scale dominance

Connects to curved versions of the Hilbert transform

Abstract

We prove the boundedness of a class of tri-linear operators consisting of a quasi piece of bilinear Hilbert transform whose scale equals to or dominates the scale of its linear counter part. Such type of operators is motivated by the tri-linear Hilbert transform and its curved versions.