On the $L^r$ Hodge theory in complete non compact riemannian manifolds

Abstract

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

I correct a mistake in the proof of lemma 5.2 and lemma 5.3. The statements have not changed so all the results remain