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.
Categories: math.CV math.DG