Hardy Uncertainty Principle, Convexity and Parabolic Evolutions

TL;DR

This paper presents a novel proof of Hardy's uncertainty principle using convexity properties and heat equation dynamics, extending results to variable coefficient heat equations with Gaussian decay bounds.

Contribution

It introduces a new proof method based on log-convexity and extends Hardy's uncertainty principle to heat equations with variable coefficients.

Findings

New proof of Hardy's uncertainty principle using convexity

Optimal Gaussian decay bounds for heat equation solutions

Extension to heat equations with variable coefficients

Abstract

We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time. We extend the result to heat equations with lower order variable coefficient.