球面上的 Hardy-Rellich 不等式与不确定性原理
经典分析与常微分方程
2014-11-12 v3
摘要
设 Δ 0 \Delta_0 Δ 0 R d \mathbb{R}^d R d S d − 1 \mathbb{S}^{d-1} S d − 1 ∫ S d − 1 ∣ f ( x ) ∣ 2 d σ ( x ) ≤ c d min e ∈ S d − 1 ∫ S d − 1 ( 1 − ⟨ x , e ⟩ ) ∣ ( − Δ 0 ) 1 2 f ( x ) ∣ 2 d σ ( x ) \int_{\mathbb{S}^{d-1}} \left | f (x)\right|^2 d\sigma(x) \leq c_d \min_{e\in \mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}} (1- \langle x, e \rangle) \left |(-\Delta_0)^{\frac{1}{2}}f(x) \right |^2 d\sigma(x) ∫ S d − 1 ∣ f ( x ) ∣ 2 d σ ( x ) ≤ c d e ∈ S d − 1 min ∫ S d − 1 ( 1 − ⟨ x , e ⟩) ( − Δ 0 ) 2 1 f ( x ) 2 d σ ( x ) d = 2 d =2 d = 2 d ≥ 4 d \ge 4 d ≥ 4 d = 3 d=3 d = 3 d = 2 , 4 , 5 d =2, 4, 5 d = 2 , 4 , 5 d ≥ 6 d\ge 6 d ≥ 6 c d = 8 / ( d − 3 ) 2 c_d = 8/(d-3)^2 c d = 8/ ( d − 3 ) 2 min e ∈ S d − 1 ∫ S d − 1 ( 1 − ⟨ x , e ⟩ ) ∣ f ( x ) ∣ 2 d σ ( x ) ∫ S d − 1 ∣ ∇ 0 f ( x ) ∣ 2 d σ ( x ) ≥ c d ′ \min_{e\in\mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}} (1- \langle x, e \rangle) |f(x)|^2 d\sigma(x) \int_{\mathbb{S}^{d-1}}\left |\nabla_0 f(x)\right |^2 d\sigma(x) \ge c'_d e ∈ S d − 1 min ∫ S d − 1 ( 1 − ⟨ x , e ⟩) ∣ f ( x ) ∣ 2 d σ ( x ) ∫ S d − 1 ∣ ∇ 0 f ( x ) ∣ 2 d σ ( x ) ≥ c d ′
引用
@article{arxiv.1212.3887,
title = {The Hardy-Rellich inequality and uncertainty principle on the sphere},
author = {Feng Dai and Yuan Xu},
journal= {arXiv preprint arXiv:1212.3887},
year = {2014}
}