The raising steps method. Applications to the $\displaystyle L^{r}$ Hodge theory in a compact riemannian manifold

TL;DR

This paper introduces the Raising Steps Method, a technique to extend local solutions of linear equations to global solutions in metric spaces, and applies it to establish an $L^r$ Hodge decomposition in compact Riemannian manifolds.

Contribution

The paper presents a new, simpler approach to prove the $L^r$ Hodge decomposition theorem using the Raising Steps Method, previously used for $ar ext{d}$ equations in Stein manifolds.

Findings

Proves a strong $L^r$ Hodge decomposition theorem for $p$-forms in compact Riemannian manifolds.

Demonstrates the effectiveness of the Raising Steps Method in globalizing local solutions.

Provides an alternative, simpler proof of a classical result in Riemannian geometry.

Abstract

Let $X$ be a complete metric space and $\displaystyle \Omega $ a domain in $\displaystyle X.$ The Raising Steps Method allows to get from local results on solutions $u$ of a linear equation $\displaystyle Du=\omega $ global ones in $\displaystyle \Omega .$\ It was introduced in \cite{AmarSt13} to get good estimates on solutions of $\bar \partial $ equation in domains in a Stein manifold. As a simple application we shall get a strong $\displaystyle L^{r}$ Hodge decomposition theorem for $p-$forms in a compact riemannian manifold without boundary, and then we retrieve this known result by an entirely different and simpler method.