Lipschitzian norm estimate of one-dimensional Poisson equations and applications

Djellout, Hacene; Wu, Liming

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 450-465.

Par un calcul direct, on identifie explicitement la norme Lipschitzienne de la solution de l'équation de Poisson en terme de différentes normes de g, où est l'opérateur de Sturm-Liouville ou le générateur d'une diffusion non singulière sur un intervalle. Ainsi, nous pouvons obtenir, d'une part la meilleure constante dans l'inégalité de Poincaré L1 (une inégalité un peu plus forte que l'inégalité isopérimétrique de Cheeger) et d'autre part certaines inégalités de transport-information et de concentration fines pour la moyenne empirique. On conclut avec des exemples illustratifs.

By direct calculus we identify explicitly the lipschitzian norm of the solution of the Poisson equation in terms of various norms of g, where 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 L1-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.

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