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

2015/06/27 by Éric Amar, Eric Amar, Amar, Eric
Mathematics · #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Analysis and Transform Methods #math.CV #math.DG

paper · pdf · doi:10.48550/arxiv.1506.08295

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

openalex publication_date 2015/06/27 · arxiv created 2017/08/16 · arxiv updated 2017/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study solutions for the Hodge laplace equation Δu=ω on p forms with L^r estimates for r>1. Our main hypothesis is that Δ has a spectral gap in L². 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.

Related

• The raising steps method. Applications to the L^r Hodge theory in a compact riemannian manifold
• Discretization of Riemannian manifolds applied to the Hodge Laplacian
• L^p-spectral theory for the Laplacian on forms
• Spectral and Hodge theory of `Witt' incomplete cusp edge spaces
• L^p Hodge theory for bounded subanalytic manifolds