Orlicz-Poincare inequalities

Corresponding to known results on Orlicz-Sobolev inequalities which are stronger than the Poincare inequality, this paper studies the weaker Orlicz-Poincare inequality. More precisely, for any Young function Phi whose growth is slower than quadric, the Orlicz-Poincare inequality parallel to f parallel to(2)(Phi) <= C epsilon(f, f), mu(f) := integral f d mu = 0 is studied by using the well-developed weak Poincare inequalities, where epsilon is a conservative Dirichlet form on L(2) (mu) for some probability measure mu. In particular, criteria and concrete sharp examples of this inequality are presented for Phi(r) = r(p) (p is an element of [1, 2)) and Phi(r) = r(2) log(-delta)(e + r(2)) (delta > 0). Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.