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

TL;DR

This paper extends $L^{r}$ Hodge theory to complete non-compact Riemannian manifolds by establishing new decomposition theorems without relying on Riesz transform boundedness, using a generalized Raising Steps Method.

Contribution

It introduces a novel approach to $L^{r}$ Hodge decomposition on non-compact manifolds without using Riesz transform boundedness, based on a generalized Raising Steps Method.

Findings

Established $L^{r}$ Hodge decomposition theorems without Riesz transform boundedness.

Proved spectral gap conditions imply $L^{r}$ estimates for solutions of the Hodge Laplace equation.

Generalized the Raising Steps Method to non-compact Riemannian manifolds.

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.