The chain rule and a compactness theorem for BV functions in the Heisenberg group Hn ✩

At first in the setting of the Heisenberg group we show the chain rule for a function u ∈ BVH(Ω) when composed with a Lipschitz function f :R→R and prove that v = f ◦ u belongs to BVH(Ω) and |DH v| |DH u|. More precisely the following result is shown: DHv = f ( ˜u)∇H uL2n+1 + 2ω2n−1 ω2n+1 f (u+)− f (u−) νuS Q−1 d Ju + f ( ˜u)Dc Hu. Secondly using the chain rule above we prove a compactness theorem for SBVH functions.  2003 Elsevier Inc. All rights reserved.