Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 751–795 | Cite as

On the \(L^{r}\) Hodge theory in complete non compact Riemannian manifolds

  • Eric AmarEmail author


We study solutions for the Hodge laplace equation \(\Delta u=\omega \) on p forms with \(\displaystyle L^{r}\) estimates for \(\displaystyle r>1.\) Our main hypothesis is that \(\Delta \) has a spectral gap in \(\displaystyle L^{2}.\) We use this to get non classical \(\displaystyle L^{r}\) Hodge decomposition theorems. An interesting feature is that to prove these decompositions we never use the boundedness of the Riesz transforms in \(\displaystyle L^{s}.\) These results are based on a generalisation of the Raising Steps Method to complete non compact Riemannian manifolds.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Université de BordeauxTalenceFrance

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