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

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A Correction to this article was published on 12 February 2019

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Abstract

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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Change history

  • 12 February 2019

    Our aim is to correct the proofs of Lemma 5.2 and Lemma 5.3

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  1. Université de Bordeaux, Talence, France

    Eric Amar

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  1. Eric Amar
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Amar, E. On the \(L^{r}\) Hodge theory in complete non compact Riemannian manifolds. Math. Z. 287, 751–795 (2017). https://doi.org/10.1007/s00209-017-1844-9

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