Lipschitzian Norm Estimate of One-Dimensional Poisson Equations
Résumé
By direct calculus we identify explicitly the Lipschitzian norm of the solution of the Poisson equation $-\LL G=g$ in terms of various norms of $g$, where $\LL$ is a Sturm-Liouville operator or generator of a non-singular diffusion in an interval. This allows us to obtain the best constant in the $L^1$-Poincaré inequality (a little stronger than the Cheeger isoperimetric inequality) and some sharp transportation-information inequalities and concentration inequalities for empirical means. We conclude with several illustrative examples.