A note on one-dimensional Poincaré inequalities by Stein-type integration (Q2692556)

Given a density \(p\) on the real line, and a positive weight function \(w\), then \(p\) is said to satisfy a Poincaré inequality with weight \(w\) if there exists \(C>0\) satisfying \[ \ \mathrm{Var}_p[h]\leq C\mathbb{E}_p[|h^\prime|^2w]\,, \] for all \(h\) in an appropriate Sobolev space. The smallest such \(C\) for which this inequality holds is the Poincaré constant of \(p\) with respect to the weight \(w\), denoted by \(C(p,w)\). The authors develop connections with Stein's method to derive a new version of a variational formula for \(C(p,w)\) due to Chen and Wang [\textit{M.-F. Chen} and \textit{F.-Y. Wang}, Trans. Am. Math. Soc. 349, No. 3, 1239--1267 (1997; Zbl 0872.35072)], which leads to upper and lower bounds for \(C(p,w)\) in terms of the so-called Stein kernel associated with \(p\). This variational formula is also used to derive sequences of nested intervals which contain \(C(p,w)\) and a sequence which converges to \(C(p,w)\), among other results. Several examples are used to illustrate these main results.