On a trilinear form related to the Carleson theorem

TL;DR

This paper adapts the multilinear Bellman function technique to time-frequency analysis, proving boundedness of a specific trilinear form related to the Carleson theorem at boundary exponents.

Contribution

It introduces a novel application of the Bellman function method to a trilinear form associated with the Carleson theorem, extending known boundedness results.

Findings

Proves boundedness of the trilinear form at boundary exponents

Adapts Bellman function technique to time-frequency analysis

Links the form to the Carleson operator and Walsh model

Abstract

The main purpose of this short note is to present an adaptation of the multilinear Bellman function technique from [4] to the time-frequency analysis. Demeter and Thiele introduced the two-dimensional bilinear Hilbert transform in [3] and showed that the Carleson operator can be identified in particular instances of the corresponding trilinear form. Demeter considered the Walsh model of one such form in [2], in relation to the discussion of the Walsh-Carleson theorem. We prove boundedness of this trilinear form for a single triple of exponents at the boundary of the previously established range.