Higher order transversality in harmonic analysis

TL;DR

This paper generalizes the concept of transversality in harmonic analysis, showing the equivalence of three different notions and connecting recent advances in Brascamp--Lieb inequalities.

Contribution

It introduces a broader notion of transversality for multiple submanifolds and proves the equivalence of three concepts relevant to harmonic analysis.

Findings

Three concepts of transversality are shown to be equivalent.

The work unifies recent developments related to Brascamp--Lieb inequalities.

Provides a survey of current research in the area.

Abstract

In differential topology two smooth submanifolds $S_1$ and $S_2$ of euclidean space are said to be transverse if the tangent spaces at each common point together form a spanning set. The purpose of this article is to explore a much more general notion of transversality pertaining to a collection of submanifolds of euclidean space. In particular, we show that three seemingly different concepts of transversality arising naturally in harmonic analysis, are in fact equivalent. This result is an amalgamation of several recent works on variants of the Brascamp--Lieb inequality, and we take the opportunity here to briefly survey this growing area. This is not intended to be an exhaustive account, and the choices made reflect the particular perspectives of the authors.