Uniform geometric estimates for sublevel sets

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 ${\mathbb{R}}^d$. 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.