$L^p$ estimates for the biest II. The Fourier case

TL;DR

This paper establishes L^p estimates for the biest operator in the Fourier setting, addressing challenges in localization and extending previous Walsh model results to a more general and complex Fourier context.

Contribution

It provides the first proof of L^p bounds for the biest in the Fourier case, overcoming localization difficulties not present in the Walsh model.

Findings

L^p estimates are proven for the Fourier biest.

The work extends previous Walsh model results to the Fourier setting.

Addresses localization challenges in harmonic analysis.

Abstract

We prove L^p estimates for the "biest", a trilinear multiplier with singular symbol which arises naturally in the expansion of eigenfunctions of a Schrodinger operator, and which is also related to the bilinear Hilbert transform. In a previous paper these estimates were obtained for a simpler Walsh model for this operator, but in the Fourier case additional complications arise due to the inability to perfectly localize in both space and frequency.