Indeterminacy estimates, eigenfunctions and lower bounds on Wasserstein distances. (English) Zbl 1505.53058
Summary: In the paper we prove two inequalities in the setting of \(\mathsf{RCD}(K, \infty)\) 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 \(L^\infty\) 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.
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35P05 | General topics in linear spectral theory for PDEs |
| 28A75 | Length, area, volume, other geometric measure theory |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 51K05 | General theory of distance geometry |
Keywords:
Wasserstein distance between eigenfunctions; Laplace eigenfunction; Hellinger-Kantorovich distanceReferences:
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