$L^p$-Poincaré不等式在Lebesgue和高斯概率测度中的稳定性及谱间隙应用
Stability of the $L^{p}$-Poincaré inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps
AI总结：
本文研究了Lebesgue和高斯测度下$L^p$-Poincaré不等式的稳定性，通过显式几何依赖性恢复了关于拉普拉斯算子基本间隙猜想的结果，并扩展到$p$-拉普拉斯算子的谱间隙估计。
摘要：
本文获得了Lebesgue和高斯概率测度的$L^p$-Poincaré不等式的稳定性结果（定理3.3和定理3.13），这些结果明确依赖于域的几何特性。作为副产品，显式常数使我们能够恢复Yu和Zhong [YZ86]以及Smits [Smi96]的重要结果，这些结果与拉普拉斯算子的基本间隙猜想（由Andrews和Clutterbuck [AC11]解决）相关，从而提供了另一种证明。此外，我们将这一谱间隙结果扩展到$p$-拉普拉斯算子（定理3.6）。此类Dirichlet $p$-拉普拉斯算子的间隙估计似乎尚未见文献，如[DSW18]中所观察到的。我们的方法依赖于(Gaussian) $p$-拉普拉斯算子的第一个特征函数的性质以及凸域上log-concave测度的加权Poincaré不等式。
英文摘要：
In this paper, we obtain stability results for the $L^{p}$-Poincaré inequality for both Lebesgue measure and Gaussian probability measure (Theorem 3.3 and Theorem 3.13) that involve explicit dependence on the geometry of the domain. As a byproduct, the explicit constant allows us to recover important results of Yu, Zhong [YZ86] and Smits [Smi96] (Corollary 3.9), related to the fundamental gap conjecture of the Laplacian (resolved by Andrews and Clutterbuck [AC11]), thereby providing an alternative proof. Moreover, we extend this spectral gap result to the $p$-Laplacian (Corollary 3.6). Such gap estimates for the Dirichlet $p$-Laplacian appear to be unavailable, as also observed in [DSW18]. Our approach relies on properties of the first eigenfunction of the (Gaussian) $p$-Laplacian operator and weighted Poincaré inequalities for log-concave measures on convex domains.
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