Polynomials, Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

Abstract

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincare-type inequalities for Hormander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inqualities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on ?. Some very usefull properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (ɛ,δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.