On the family of singular Brascamp-Lieb inequalities with dimension datum $(1, 2, 2, 1)$

TL;DR

This paper classifies a family of singular Brascamp-Lieb inequalities with a specific dimension datum, determines their valid Lebesgue exponents, and explores related counterexamples and estimates in harmonic analysis.

Contribution

It provides a complete classification and exponent range for a particular family of singular Brascamp-Lieb inequalities, including new proofs and counterexamples.

Findings

Exact range of Lebesgue exponents for the inequalities

A simple proof of a variant of the dyadic triangular Hilbert transform estimate

Counterexamples to boundedness with certain exponents

Abstract

We classify a certain family of singular Brascamp-Lieb forms which we associate with the dimension datum $(1, 2, 2, 1)$. We determine the exact range of Lebesgue exponents, for which one has singular Brascamp Lieb inequalities within this family. One key observation is a simple proof of a variant of an estimate in dyadic triangular Hilbert transform of two general and one not too general function. The remaining observations concern counter examples to boundedness. We compare with a counter example showing that the triangular Hilbert form does not satisfy singular Brascamp Lieb bounds with exponents $(\infty,p,p')$.