Characterizations of Sobolev spaces in Euclidean spaces and Heisenberg groups  

作　　者：CUI Xiao-yue LAM Nguyen LU Guo-zhen 

机构地区：[1]Department of Mathematics, Wayne State University,Detroit, MI 48202, USA

出　　处：《Applied Mathematics(A Journal of Chinese Universities)》2013年第4期531-547,共17页高校应用数学学报（英文版）（B辑）

基　　金：supported by a US NSF grant DMS-1301595

摘　　要：Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].

关 键 词：characterization of Sobelev spaces Folland-Stein space Poincar′e inequalities Heisenberg group second order Sobolev space 

分 类 号：O152[理学—数学] O177[理学—基础数学]