L^p estimates for a singular integral operator motivated by Calder\'on's second commutator

TL;DR

This paper establishes a broad spectrum of L^p estimates for a novel trilinear singular integral operator inspired by Calderón's second commutator, utilizing adapted time-frequency analysis techniques.

Contribution

It introduces a new approach to analyze a non-standard symbol in a trilinear operator, avoiding the complexities of the trilinear Hilbert transform.

Findings

Proved L^p bounds for the operator across a wide range.

Developed adapted time-frequency analysis methods for non-standard symbols.

Enhanced understanding of singular integrals related to Calderón's commutator.

Abstract

We prove a wide range of L^p estimates for a trilinear singular integral operator motivated by dropping one average in Calder\'{o}n's second commutator. For comparison by dropping two averages in Calder\'{o}n's second commutator one faces the trilinear Hilbert transform. The novelty in this paper is that in order to avoid difficulty of the level of the trilinear Hilbert transform, we choose to view the symbol of the operator as a non-standard symbol. The methods used come from time-frequency analysis but must be adapted to the fact that our symbol is non-standard.