$L^p$-Hodge Decomposition with Sobolev classes in Sub-Riemannian Contact Manifolds

Let $1<p<\infty$. In this article we establish an $L^p$-Hodge decomposition theorem on sub-Riemannian compact contact manifolds without boundary, related to the Rumin complex of differential forms. Given an $L^p$- Rumin's form, we adopt an approach in the spirit of Morrey's book to obtain a decomposition with higher regular ``primitives'' i.e. that belong to suitable Sobolev classes. Our proof relies on recent results obtained in [4] and [6].