Indeterminacy estimates, eigenfunctions and lower bounds on Wasserstein distances

Abstract

In the paper we prove two inequalities in the setting of $\mathsf{RCD}(K,\infty )$ 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.

arXiv commentMajor update. Structure of the proof simplified. Added new results on indeterminacy estimates for transport distances. The inequality between $p$-Wasserstein and $p$-Hellinger distances for $p>2$ is now not needed: it has been removed and will be part of a future project