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 any higher orders for the Folland-Stein Sobolev spaces on stratified nilpotent Lie groups $\Bbb G$ and on domains satisfying a certain chain condition. Weighted versions of such inequalities are also included for doubling weights satisfying a weighted Poincaré inequality. This paper appears to be the first one to deal with general Sobolev interpolation inequalities for vector fields on Lie groups; Despite the extensive research for Poincaré 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 by polynomials on $\Bbb G$. Some very useful properties for projections of polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Main ideas of the proofs of the theorems are explained here. Detailed proofs of the results presented here are given in the paper “Polynomials on stratified groups and Sobolev interpolation inequalities on nonisotropic Folland-Stein spaces” by the author.

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Published 1 January 1997