Sub-elliptic global high order Poincare inequalities in stratified Lie groups and applications

摘要

Sharp Poincare inequalities on balls or chain type bounded domains have been extensively studied both in classical Euclidean space and Camot-Caratheodory spaces associated with sub-elliptic vector fields (e.g., vector fields satisfying Hormander's condition). In this paper, we investigate the validity of sharp global Poincare inequalities of both first order and higher order on the entire nilpotent stratified Lie groups or on unbounded extension domains in such groups. We will show that simultaneous sharp global Poincare inequalities also hold and weighted versions of such results remain to be true. More precisely, let G be a nilpotent stratified Lie group and f be in the localized non-isotropic Sobolev space W-loc(m,p) (G), where 1 <= p < Q/m and Q is the homogeneous dimension of the Lie group G. Suppose that the mth sub-elliptic derivatives of is globally L-p integrable; i.e., f(G) vertical bar X-m f (x)vertical bar(p) dx is finite (but assume that lower order sub-elliptic derivatives are only locally L-p integrable). We denote the space of such functions as B-m.p(G). We prove a high order Poincare inequality for f minus a polynomial of order m - 1 over the entire space G or unbounded extension domains. As applications, we will prove a density theorem stating that smooth functions with compact support are dense in B-m.p(G) modulus a finite-dimensional subspace. (C) 2007 Elsevier Inc. All rights reserved.