Uniform geometric estimates for sublevel sets

TL;DR

This paper provides a geometric approach to improve uniform decay estimates of sublevel sets for certain differential operators, generalizing previous results and achieving better decay rates in many cases.

Contribution

It introduces a geometric perspective and a class of homogeneous nonlinear differential operators that extend mixed derivatives, leading to improved uniform decay estimates.

Findings

Enhanced decay rates for sublevel set estimates in various scenarios

Introduction of a geometric framework for analyzing differential operators

Generalization of mixed derivatives in the context of uniform estimates

Abstract

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright (1999) and Phong, Stein, and Sturm (2001) from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous, nonlinear differential operators which generalize mixed derivatives in $\R^d$. As a consequence, it is shown that, in the case of both of these previous works, improved uniform decay rates are possible in many situations.