$L^p$ estimates for the biest I. The Walsh case

TL;DR

This paper establishes L^p estimates for the Walsh model of the biest, a complex trilinear operator linked to eigenfunction expansions and the bilinear Hilbert transform, with Fourier model results to follow.

Contribution

It provides the first L^p estimates for the Walsh biest, advancing understanding of multilinear singular integrals in this setting.

Findings

Proved L^p bounds for the Walsh biest.

Connected the biest to eigenfunction expansions and the bilinear Hilbert transform.

Set the stage for Fourier model estimates in future work.

Abstract

We prove L^p estimates for the Walsh model of 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. The corresponding estimates for the Fourier model will be obtained in the sequel of this paper.