Stability of the $L^{p}$-Poincar\'e inequality for the Lebesgue and Gaussian probability measures with explicit geometric dependence and applications to spectral gaps

Abstract

In this paper, we obtain stability results for the $L^{p}$-Poincar\'e inequality for both Lebesgue and Gaussian probability measures (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 Poincar\'e inequalities for log-concave measures on convex domains.