Indeterminacy estimates, eigenfunctions and lower bounds on Wasserstein distances

TL;DR

This paper establishes new inequalities in ${\sf RCD}(K,\infty)$ spaces, including an indeterminacy estimate related to Wasserstein distances and a lower bound conjecture for eigenfunctions, advancing understanding of metric measure space analysis.

Contribution

The paper introduces two novel inequalities in ${\sf RCD}(K,\infty)$ spaces, including a conjectured lower bound on Wasserstein distances between eigenfunction parts.

Findings

Proved an indeterminacy estimate involving Wasserstein distance and measure of the interface.

Established a conjectured lower bound on Wasserstein distance for Laplace eigenfunctions.

Abstract

In the paper we prove two inequalities in the setting of ${\sf RCD}(K,\infty)$ spaces using similar techniques. The first one is an indeterminacy estimate involving the $p$-Wasserstein distance between the positive part and the negative part of an $L^{\infty}$ function and the measure of the interface between the positive part and the negative part. The second one is a conjectured lower bound on the $p$-Wasserstein distance between the positive and negative parts of a Laplace eigenfunction.