Dyadic triangular Hilbert transform of two general and one not too general function

TL;DR

This paper studies a dyadic model of the triangular Hilbert transform, establishing bounds that imply dyadic analogues of key harmonic analysis operators like the Carleson maximal operator and bilinear Hilbert transform.

Contribution

It introduces a dyadic model for the triangular Hilbert transform with one function essentially one-dimensional, providing new bounds and implications for related operators.

Findings

Established $L^p$ bounds for the dyadic triangular Hilbert transform model.

Derived dyadic analogues of boundedness for the Carleson maximal operator.

Obtained uniform estimates for the one-dimensional bilinear Hilbert transform.

Abstract

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well studied objects of harmonic analysis. We investigate $L^p$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one-dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.