Indeterminacy estimates and the size of nodal sets in singular spaces

TL;DR

This paper establishes a sharp uncertainty principle in metric measure spaces with synthetic Ricci curvature bounds, leading to new bounds on the size of nodal sets of Laplacian eigenfunctions, including in non-smooth spaces.

Contribution

It extends the uncertainty principle and nodal set estimates to non-smooth metric measure spaces satisfying MCP(K,N) or CD(K,N) conditions, a novel advancement.

Findings

Derived sharp uncertainty principle for metric measure spaces.

Established lower bounds on nodal set sizes in non-smooth spaces.

Extended results to non-linear Laplacians and eigenfunction combinations.

Abstract

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.