Calculus of Variations and Geometric Measure Theory

F. Cavalletti - S. Farinelli

Indeterminacy estimates and the size of nodal sets in singular spaces

created by farinelli on 07 Nov 2020
modified by cavallett on 06 Jul 2021


Accepted Paper

Inserted: 7 nov 2020
Last Updated: 6 jul 2021

Journal: Advances Math
Year: 2020


We obtain the sharp version of the uncertainty principle recently introduced by Sagiv and Steinerberger, and improved by Carrol, Massaneda and Ortega-Cerdà, 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.