Uniform geometric estimates for sublevel sets

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright [5] and Phong, Stein, and Sturm [17] from a geometric perspective. This perspective leads one to consider a natural collection of homogeneous, nonlinear differential operators which generalize mixed derivatives in R. 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.