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.