# Averages of simplex Hilbert transforms

## TL;DR

This paper investigates a multilinear singular integral derived from averages of simplex Hilbert transforms, establishing $L^p$ bounds in low dimensions and providing a conditional general result.

## Contribution

It introduces new $L^p$ bounds for a multilinear form related to simplex Hilbert transforms, Calderón commutators, and twisted paraproducts, with results in dimensions two and three.

## Key findings

- Proved $L^p$ bounds in dimension two.
- Proved $L^p$ bounds in dimension three.
- Provided a conditional result valid in all dimensions.

## Abstract

We study a multilinear singular integral obtained by taking averages of simplex Hilbert transforms. This multilinear form is also closely related to Calder\'on commutators and the twisted paraproduct. We prove $L^p$ bounds in dimensions two and three and give a conditional result valid in all dimensions.