Variational bounds for a dyadic model of the bilinear Hilbert transform

TL;DR

This paper establishes variation-norm estimates for a Walsh model of the truncated bilinear Hilbert transform, advancing the understanding of its boundedness properties using novel analytical tools.

Contribution

It introduces a variational extension of Bourgain's lemma and a variation-norm Rademacher-Menshov theorem, providing new techniques for analyzing bilinear Hilbert transforms.

Findings

Proved variation-norm estimates for the Walsh model of the bilinear Hilbert transform.

Extended analytical tools with two new ingredients: a variational lemma extension and a variation-norm Rademacher-Menshov theorem.

Enhanced understanding of the boundedness and behavior of bilinear Hilbert transforms in the Walsh model.

Abstract

We prove variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele, and Demeter. The proof uses analysis on the Walsh phase plane and two new ingredients: (i) a variational extension of a lemma of Bourgain by Nazarov-Oberlin-Thiele, and (ii) a variation-norm Rademacher-Menshov theorem of Lewko-Lewko.