Cancellation for the simplex Hilbert transform

TL;DR

This paper demonstrates that the truncated simplex Hilbert transform exhibits sublinear growth in its norm with respect to the number of scales, extending Tao's cancellation results for multilinear Hilbert transforms using Gowers' Hilbert space regularity lemma.

Contribution

It introduces a new cancellation property for the simplex Hilbert transform and provides a concise proof leveraging Gowers' regularity lemma, extending previous multilinear results.

Findings

The norm of the truncated simplex Hilbert transform grows sublinearly with scales.

The proof is significantly shortened using Gowers' Hilbert space regularity lemma.

Extends Tao's cancellation results to the simplex Hilbert transform.

Abstract

We show that the truncated simplex Hilbert transform enjoys some cancellation in the sense that its norm grows sublinearly in the number of scales retained in the truncation. This extends the recent result by Tao on cancellation for the multilinear Hilbert transform. Our main tool is the Hilbert space regularity lemma due to Gowers, which enables a very short proof.