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

Abstract

We study solutions for the Hodge laplace equation $Δu=ω$ on $p$ forms with $\displaystyle L^{r}$ estimates for $\displaystyle r>1.$ Our main hypothesis is that $Δ$ 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.