  Recently, Steinerberger proved a uniform inequality for the Laplacian serving
as a counterpoint to the standard uniform sublevel set inequality which is
known to fail for the Laplacian. In this paper, we observe that many
inequalities of this type follow from a uniform lower bound on the $L^1$ norm,
and give an analogous result for any linear differential operator, which can
fail for non-linear operators. We consider lower bounds on the $L^p$
quasi-norms for $p<1$ as a stronger property that remains weaker than a uniform
sublevel set inequality and prove this for the Laplacian and heat operators. We
conclude with some naturally arising questions.
