On a new type of Inequality related to the Uniform Sublevel Set Problem

TL;DR

This paper presents an elementary proof of a uniform inequality related to the Laplacian, extending it to the heat operator and highlighting adaptable techniques for similar problems.

Contribution

Provides a simple proof of a uniform inequality for the Laplacian and extends it to the heat operator, with potential for broader applications.

Findings

The inequality holds for the Laplacian.

The inequality also applies to the heat operator.

The proof technique is adaptable to other operators.

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 note, we give an elementary proof of this result which highlights a step allowing for adaptations to other situations, for instance, we show that the inequality also holds for the heat operator. We formulate some naturally arising questions.