The Hardy Uncertainty Principle Revisited

TL;DR

This paper provides a new real-variable proof of the Hardy uncertainty principle using energy estimates, convexity, and Fourier transform invertibility, offering a fresh perspective on this fundamental result.

Contribution

It introduces a novel real-variable proof of the Hardy uncertainty principle, expanding the methods beyond traditional approaches.

Findings

New proof technique based on energy estimates and convexity

Demonstrates invertibility of Fourier transform in relevant function spaces

Provides insights into Gaussian decay properties at two different times

Abstract

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic $L^2$-estimates and the invertibility of the Fourier transform on $L^2(\Rn)$ and $\mathcal S'(\Rn)$.