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We obtain the sharp version of the uncertainty principle recently introducedin [47], and improved by [13], relating the size of the zero set of acontinuous function having zero mean and the optimal transport cost between themass of the positive part and the negative one. The result is actually validfor the wide family of metric measure spaces verifying a synthetic lower boundon the Ricci curvature, namely the MCP(K,N) or CD(K,N) condition, thus alsoextending the scope beyond the smooth setting of Riemannian manifolds. Applyingthe uncertainty principle to eigenfunctions of the Laplacian in possiblynon-smooth spaces, we obtain new lower bounds on the size of their nodal setsin terms of the eigenvalues. Those cases where the Laplacian is possiblynon-linear are also covered and applications to linear combinations ofeigenfunctions of the Laplacian are derived. To the best of our knowledge, noprevious results were known for non-smooth spaces.

Indeterminacy estimates and the size of nodal sets in singular spaces