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On the stability estimate for the sharp second order uncertainty principle

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发表日期：

2025年4月1日

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Anh Tuan Duong,
Van Hoang Nguyen

摘要

Recently, the following sharp second order uncertainty principle has been proved in Cazacu et al (J Funct Anal 283(10): 109659, 2022) $\begin{aligned} \int _{\mathbb {R}^n}|\Delta u|^2 dx \int _{\mathbb {R}^n}|x|^2 |\nabla u|^2 dx \ge \Big (\frac{n+2}{2}\Big )^2 \Big (\int _{\mathbb {R}^n}|\nabla u|^2 dx\Big )^2. \end{aligned}$ $\begin{matrix} \int_{R^{n}} {| Δ u |}^{2} d x \int_{R^{n}} {| x |}^{2} {| \nabla u |}^{2} d x \geq (\frac{n + 2}{2})^{2} (\int_{R^{n}} {| \nabla u |}^{2} d x)^{2} . \end{matrix}$ In this paper, we prove a stability version of this inequality which shows that the difference of both sides of the inequality controls the distance to the set of extremal functions in $L^2$ $L^{2}$ norm of gradient of functions.