Extremizers and Stability for Fractional $L^p$ Uncertainty Principles
S. Hashemi Sababe, Amir Baghban
TL;DR
The paper develops a fractional $L^p$ uncertainty principle tied to the operator $(-\Delta)^{\gamma/2}$ and a variational problem with a best constant $\kappa_{\alpha,\beta,p}$. It proves the existence of extremizers, conjectures their fractional Gaussian structure, and establishes scaling and translation invariances that underpin sharp constants and stability. A quantitative stability result shows near-extremizers are close in $L^p$ to the extremal manifold $\mathcal{M}$, via a second-variation analysis. Applications to the fractional Schrödinger equation and fractional Sobolev embeddings are discussed, highlighting the impact on dispersive estimates and functional inequalities and outlining directions for further research in nonlocal harmonic analysis.
Abstract
We extend the classical Heisenberg uncertainty principle to a fractional $L^p$ setting by investigating a novel class of uncertainty inequalities derived from the fractional Schrödinger equation. In this work, we establish the existence of extremal functions for these inequalities, characterize their structure as fractional analogues of Gaussian functions, and determine the sharp constants involved. Moreover, we prove a quantitative stability result showing that functions nearly attaining the equality in the uncertainty inequality must be close -- in an appropriate norm -- to the set of extremizers. Our results provide new insights into the fractional analytic framework and have potential applications in the analysis of fractional partial differential equations.