Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces

Abstract

Recently, in the article [LW], the authors use the notion of polynomials in metric spaces \((\mathcal S, \rho, \mu)\) of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincare inequalities and representation formulas involving fractional integrals of high order, assuming only that \(\mu\) is a doubling measure and that geodesics exist. Motivated by this and by recent work in [H], [FHK], [KS] and [FLW] about first order Sobolev spaces in metric spaces, we define Sobolev spaces of high order in such metric spaces \((\mathcal S, \rho, \mu)\). We prove that several definitions are equivalent if functions of polynomial type exist. In the case of stratified groups, where polynomials do exist, we show that our spaces are equivalent to the Sobolev spaces defined by Folland and Stein in [FS]. Our results also give some alternate definitions of Sobolev spaces in the classical Euclidean case.