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

Eric Amar

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arXiv · 1506.00418

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

Eric Amar

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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.

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Eric Amar. 2015-06-01. The raising steps method. Applications to the $\displaystyle L^{r}$ Hodge theory in a compact riemannian manifold. https://arxiv.org/abs/1506.00418

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