Inequalities For Vector Fields Research Articles

On the Poincaré inequality for vector fields

We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

Sharp Geometric Poincaré Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimate the Lq(d μ) norm of a function on a metric ball when μ is an arbitrary Borel measure. The estimate is in terms of the L1(d ν) norm on the ball of a vector field gradient of the function, where d ν dx is a power of a fractional maximal function of μ. We show that the estimates are sharp in several senses, and we derive isoperimetric inequalities as corollaries. 1991 Mathematics Subject Classification: 46E35, 42B25.

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.

Subelliptic Poincaré inequalities: The case $p < 1$

We obtain (weighted) Poincare type inequalities for vector fields satisfying the Hormander condition for $p < 1$ under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying Carnot-Caratheodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for $p\ge 1$.

Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications

In this paper we mainly prove weighted Poincaré inequalities for vector fields satisfying Hörmander's condition. A crucial part here is that we are able to get a pointwise estimate for any function over any metric ball controlled by a fractional integral of certain maximal function. The Sobolev type inequalities are also derived. As applications of these weighted inequalities, we will show the local regularity of weak solutions for certain classes of strongly degenerate differential operators formed by vector fields.

The Poincaré inequality for vector fields satisfying Hörmander’s condition

The Poincaré inequality for vector fields satisfying Hörmander’s condition