We obtain the sharp version of the uncertainty principle recently introduced in [47], and improved by [13], relating the size of the zero set of a continuous function having zero mean and the optimal transport cost between the mass of the positive part and the negative one. The result is actually valid for the wide family of metric measure spaces verifying a synthetic lower bound on the Ricci curvature, namely the MCP(K, N ) or CD(K, N ) condition, thus also extending the scope beyond the smooth setting of Riemannian manifolds.Applying the uncertainty principle to eigenfunctions of the Laplacian in possibly non-smooth spaces, we obtain new lower bounds on the size of their nodal sets in terms of the eigenvalues. Those cases where the Laplacian is possibly non-linear are also covered and applications to linear combinations of eigenfunctions of the Laplacian are derived. To the best of our knowledge, no previous results were known for non-smooth spaces.
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.
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