A Fourier approach to the sharp stability of Heisenberg Uncertainty Principles of higher and fractional orders: a new perspective

Abstract

We show that, on the Fourier side, the sharp second order Heisenberg Uncertainty Principle (HUP) and its sharp stability for functions are nothing but the first order $L^{2}$-Caffarelli--Kohn--Nirenberg inequality and its sharp stability, applied to their Fourier transforms. This gives a proof in a few lines of the sharp stability estimate that we established recently by a much longer argument. The same observation then produces two one-parameter families of sharp fractional Heisenberg Uncertainty Principles, in which the gradient is replaced by a fractional power of the Laplacian: for each of them we compute the sharp constants, characterize all the optimizers, and establish the sharp stability estimates. The stability constant of the first family is equal to $1$ for every nonnegative order and every dimension. Negative orders, where the fractional Laplacian becomes a Riesz potential, are treated as well. Finally, we show that the deficit carries much more information than the distance to the optimizers alone: it controls an explicit chain of successive remainder terms whose constants are the successive spectral gaps of an explicit operator. For the classical Heisenberg Uncertainty Principle this gives three remainder terms with optimal explicit constants, and for the second order HUP a chain of four remainder terms measured against explicit confluent hypergeometric profiles and coupled through one single set of parameters. As applications, we also establish the chain of stability of the HUP for curl-free vector fields.