Lipschitzian norm estimate of one-dimensional Poisson equations and applications
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, p. 450-465

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.

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.

DOI : https://doi.org/10.1214/10-AIHP360
Classification:  47B38,  60E15,  60J60,  34L15,  35P15
Keywords: Poisson equations, transportation-information inequalities, concentration and isoperimetric inequalities
@article{AIHPB_2011__47_2_450_0,
     author = {Djellout, Hacene and Wu, Liming},
     title = {Lipschitzian norm estimate of one-dimensional Poisson equations and applications},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     pages = {450-465},
     doi = {10.1214/10-AIHP360},
     zbl = {1233.47029},
     mrnumber = {2814418},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_2_450_0}
}
Djellout, Hacene; Wu, Liming. Lipschitzian norm estimate of one-dimensional Poisson equations and applications. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, pp. 450-465. doi : 10.1214/10-AIHP360. http://www.numdam.org/item/AIHPB_2011__47_2_450_0/

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