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

Eric Amar (1)

1 IMB - Institut de Mathématiques de Bordeaux

Let $X$ be a smooth manifold and $\displaystyle \Omega $ a domain in $\displaystyle X.$ The Raising Steps Method allows to get from local results on solutions $u$ of 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. It is extended here to linear partial differential operator of any finite order. As a simple application we shall get a $\displaystyle L^{r}$ Hodge decomposition theorem for $p$ forms in a compact riemannian manifold without boundary, and then we retrieve known results of C. Scott by an entirely different method.

Mots clés: Hodge Laplacian; $L^r$ Hodge decomposition; compact Riemannian manifold

Domaines: Variables complexes [math.CV]; Equations aux dérivées partielles [math.AP]

Eric Amar. The raising steps method. Applications to the $\displaystyle L^{r}$ Hodge theory in a compact riemannian manifold.. 2015. ⟨hal-01158323v1⟩