Target Manifold Is The Heisenberg Group Of Sobolev Mappings

In this paper, we study the properties of the Sobolev maps W^1,α(Ω,H^m) where thetarget space is Heisenberg group H^m, 2n/n+1≤α≤2 andΩ(?)Rⁿ is a bounded domain. Since t belongs to the space L^α/2(Ω) (2n/n+1≤α≤2) which is different fromL^p(Ω)(0＜p＜1). Firstly, we compare and analyze the Sobolev spaceW^1,α(Ω,R)(0＜α＜1) and W^1,α(Ω,R)(α＞1).Secondly, we prove the equivalencebetween the Sobolev maps W^1,α(Ω,H^m) and the corresponding energy as 2n/n+1≤α≤2,which is the improvement of Capogna and Fanghua Lin's work.