Lower bounds on $L^p$ quasi-norms and the Uniform Sublevel Set Problem

TL;DR

This paper explores lower bounds on $L^p$ quasi-norms for $p<1$ and their relation to uniform sublevel set inequalities, extending results to linear differential operators like the Laplacian and heat operators.

Contribution

It introduces lower bounds on $L^p$ quasi-norms for linear differential operators, providing a new perspective on inequalities related to the Laplacian and heat operators.

Findings

Established lower bounds on $L^p$ quasi-norms for $p<1$ for Laplacian and heat operators.

Connected these bounds to uniform sublevel set inequalities and their failures for non-linear operators.

Proposed questions for further research in the area.

Abstract

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.