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

TL;DR

This paper proves an $L^p$-Hodge decomposition theorem on compact contact manifolds, utilizing Sobolev classes and recent advances in Rumin complex analysis, extending classical results to sub-Riemannian geometry.

Contribution

It establishes a new $L^p$-Hodge decomposition in sub-Riemannian contact manifolds with Sobolev regularity, building on recent developments in Rumin complex theory.

Findings

Decomposition with higher regularity primitives in Sobolev classes

Extension of Hodge theory to sub-Riemannian contact manifolds

Utilization of recent results in Rumin complex analysis

Abstract

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].