On a trilinear singular integral form with determinantal kernel

TL;DR

This paper investigates a highly symmetric trilinear singular integral form with a determinantal kernel, establishing precise L^p bounds and connecting it to classical operators like the bilinear Hilbert transform and Calderón commutator.

Contribution

It introduces and analyzes a new trilinear form with large symmetry groups, bridging the gap between bilinear Hilbert transforms and Calderón commutators, and determines its L^p boundedness range.

Findings

Established the exact L^p estimates for the form.

Connected the form to known singular integral operators.

Identified the symmetry properties and invariance groups.

Abstract

We study a trilinear singular integral form acting on two-dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calder\'on commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the ${L}^p$ estimates hold for the considered form.