Sharp uncertainty principles on metric measure spaces

TL;DR

This paper investigates fundamental uncertainty principles and interpolation inequalities within metric measure spaces that satisfy measure contraction, revealing that these inequalities hold exclusively on volume cones through localization techniques.

Contribution

It demonstrates that classical inequalities are valid only on volume cones in measure contraction spaces, extending the understanding of uncertainty principles in non-Euclidean settings.

Findings

Inequalities hold only on volume cones in measure contraction spaces

Localization techniques are effective in analyzing inequalities in metric measure spaces

Extends classical uncertainty principles to non-Euclidean metric measure spaces

Abstract

We study the Heisenberg-Pauli-Weyl uncertainty principle and the Caffarelli-Kohn-Nirenberg interpolation inequalities, on metric measure spaces satisfying measure contraction property. Using localization techniques, we show that these inequalities are valid only on volume cones.