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Recently, Steinerberger ( Potential Analysis , 2020) 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.

LOWER BOUNDS ON L P QUASI‐NORMS AND THE UNIFORM SUBLEVEL SET PROBLEM