Characterizations of Sobolev spaces in Euclidean spaces and Heisenberg groups

Recently, many new features of Sobolev spaces W (k,p) (a"e (N) ) 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 different characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].