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Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces
Authors:Yongping Liu  Guozhen Lu  Richard L Wheeden
Institution:(1) Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China (e-mail: ypliu@bnu.edu.cn) , CN;(2) Department of Mathematics, Wayne State University, Detroit, MI 48202, USA (e-mail: gzlu@math.wayne.edu) , US;(3) Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA (e-mail: wheeden@math.rutgers.edu) , US
Abstract:Recently, in the article LW], the authors use the notion of polynomials in metric spaces of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincaré inequalities and representation formulas involving fractional integrals of high order, assuming only that 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 . 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. Received: 10 February 1999 / Published online: 1 February 2002
Keywords:Mathematics Subject Classification (1991): 46E35  41A10  22E25
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