Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry

Abstract

In this article, we establish a Gaffney type inequality, in Wℓ,p{W}^{\ell ,p}-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p∈]1,∞[p\in ]1,\infty {[} and ℓ=1,2\ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

Keywords

• heisenberg groups
• differential forms
• gaffney inequality
• contact manifolds
• 58a10
• 35r03
• 26d15
• 43a80
• 53d10