Weighted Poincare Inequalities Research Articles

Local and global interpolation inequalities on the Folland-Stein Sobolev spaces and polynomials on stratified groups

We derive both local and global Sobolev interpolation inequalities of anyhigher orders for the Folland-Stein Sobolev on stratified nilpotent Lie groups G and on domains satisfying a certain chain condition. Weighted versions of such inequalities are also included for doubling weights satisfying a weighted Poincare inequality. This paper appears to be the first one to deal with gen- eral Sobolev interpolation inequalities for vector fields on Lie groups; Despite the extensive research for Poincare type inequalities for vector fields over the years, interpolation inequalities given here even in the nonweighted case appear to be new. Such interpolation inequalities have important applications to subelliptic or parabolic pde's involving vector fields. The main tools to prove such inequalities are approximating the functions bypolynomials on G. Some veryuseful proper- ties for projections of polynomials associated with the functions are given here and theyappear to have independent interests in their own rights. Main ideas of the proofs of the theorems are explained here. Detailed proofs of the results pre- sented here are given in the paper Polynomials on stratified groups and Sobolev interpolation inequalities on nonisotropic Folland-Stein spaces bythe author.

Weighted Poincar� inequalities for H�rmander vector fields and local regularity for a class of degenerate elliptic equations

In this note we state weighted Poincare inequalities associated with a family of vector fields satisfying Hormander rank condition. Then, applications are given to relative isoperimetric inequalities and to local regularity (Harnack's inequality) for a class of degenerate elliptic equations with measurable coefficients.

Poincaré-type inequalities via stochastic integrals

Martingales and stochastic integrals are applied to prove Poincare-type inequalities involving probability distributions on the Euclidean space. These inequalities generalize and improve several results in the literature and are shown to yield weighted Poincare inequalities on some special compact manifolds. This leads to a new method of calculating all the eigenvalues and eigenfunctions of the Laplacian on then-sphere. As a by-product the eigenvalues are shown to be related to the moments of a probability distribution.