A short proof on the boundedness of triangular Hilbert transform along curves

TL;DR

This paper presents an elementary proof of the boundedness of the triangular Hilbert transform along certain non-flat curves, and offers a criterion for the associated bilinear operator's smoothing properties.

Contribution

It provides a simplified proof technique for the boundedness of the transform and introduces a criterion for the multiplier related to smoothing inequalities.

Findings

Boundedness of the triangular Hilbert transform along non-flat curves established.

A criterion for the multiplier to admit smoothing inequalities is proposed.

Elementary proof approach simplifies previous methods.

Abstract

We give a short and elementary proof of the boundedness of triangular Hilbert transform along non-flat curves definable in a polynomially bounded o-minimal structure. We also provide a criterion on the multiplier to determine whether the associated fiber-wisely defined bilinear operator admits a smoothing inequality.