Uniform geometric estimates of sublevel sets

This paper reconsiders the uniform sublevel set estimates of Carbery, Christ, and Wright [7], Phong, Stein, and Sturm [23], and Carbery and Wright [8] 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 comparison to these previous works, improved uniform estimates are possible in all but certain explicitly “flat” situations.