Motivated from the study on logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examined.
Mu Fa CHEN.
Variational Formulas of Poincare-Type Inequalities in Banach Spaces of Functions on the Line. Acta Mathematica Sinica, Chinese Series, 2005, 48(2): 209-220 https://doi.org/10.12386/A2005sxxb0025