Improved Poincaré inequalities
نویسندگان
چکیده
Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy-Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities.
منابع مشابه
On weighted isoperimetric and Poincaré-type inequalities
Weighted isoperimetric and Poincaré-type inequalities are studied for κ-concave probability measures (in the hierarchy of convex measures).
متن کاملSome Discrete Poincaré-type Inequalities
Some discrete analogue of Poincaré-type integral inequalities involving many functions of many independent variables are established. These in turn can serve as generators of further interesting discrete inequalities. 2000 Mathematics Subject Classification. Primary 39A10, 39A12, 39B72.
متن کاملPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L-norm control of such smoothing procedures in terms of the gradient of the function involved. When available, pseudo-Poincaré inequalities are an efficient way to prove Sobolev type inequalities. We review this technique and its applications in various geometric setups.
متن کاملOn Poincaré-wirtinger Inequalities in Spaces of Functions of Bounded Variation
The goal of this paper is to extend Poincaré-Wirtinger inequalities from Sobolev spaces to spaces of functions of bounded variation of second order.
متن کامل