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

TL;DR

This paper proves a Gaffney type inequality for differential forms on sub-Riemannian contact manifolds with bounded geometry, extending Sobolev space estimates to non-compact settings using Rumin's complex.

Contribution

It establishes Sobolev-Gaffney inequalities for differential forms on non-compact sub-Riemannian contact manifolds with bounded geometry, utilizing Rumin's complex and geometric properties.

Findings

Proved Gaffney type inequality in $W^{ ext{ell},p}$ spaces for contact manifolds.

Extended inequalities to non-compact manifolds with bounded geometry.

Built on Rumin's complex and previous results in Heisenberg groups.

Abstract

In this paper we establish a Gaffney type inequality, in $W^{\ell,p}$-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind non-compact manifolds). Here $p\in]1,\infty[$ and $\ell=1,2$ 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.