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

Abstract

Let $X$ be a complete metric space and $\Omega$ a domain in $X.$ The Raising Steps Method allows to get from local results on solutions $u$ of a linear equation $Du=\omega$ global ones in $\Omega .$\ 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 $L^r$ 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.