Indeterminacy estimates, eigenfunctions and lower bounds on Wasserstein distances

Abstract

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