On the Lr Hodge theory in complete non compact Riemannian manifolds

We study solutions for the Hodge laplace equation on p forms with estimates for Our main hypothesis is that has a spectral gap in We use this to get non classical Hodge decomposition theorems. An interesting feature is that to prove these decompositions we never use the boundedness of the Riesz transforms in These results are based on a generalisation of the Raising Steps Method to complete non compact Riemannian manifolds.