Moment estimates derived from Poincaré and logarithmic Sobolev inequalities (Q1897758)

Assume that the space on which the probability measure \(\mu\) is defined possesses a natural gradient operation \(\nabla\). Then a test for mixing properties of the measure \(\mu\) is to examine what (if any) a priori estimates \(\mu\) satisfy relative to \(\nabla\). For example, such estimates are expressed by the Poincaré inequality \[ |f-\mathbb{E}^\mu [f] |^2_{L^2(\mu)}\leq P|\nabla f|^2_{L^2(\mu)} \] or by the logarithmic Sobolev inequality \[ \int f^2\log{f^2\over |f|^2_{L^2(\mu)}} d\mu\leq S|\nabla f|^2_{L^2(\mu)}. \] The aim of the paper is to show that each of these a priori estimates allows to obtain bounds of \(f\) in terms of bounds on \(|\nabla f|\).